"TMS320C55x DSP Library DSPLIB Programmer's Reference"

TMS320C55x DSP LibraryProgrammer’s ReferenceSPRU422COctober 2001Printed on Recycled Paper

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PrefaceRead This FirstAbout This ManualThe Texas Instruments TMS320C55x DSPLIB is an optimized DSP FunctionLibrary for C programmers on TMS320C55x devices. It includes over 50C-callable assembly-optimized general-purpose signal processing routines.These routines are typically used in computationally intensive real-timeapplications where optimal execution speed is critical. By using these routinesyou can achieve execution speeds considerable faster than equivalent codewritten in standard ANSI C language. In addition, by providing ready-to-useDSP functions, TI DSPLIB can shorten significantly your DSP applicationdevelopment time.Related DocumentationThe MathWorks, Inc. Matlab Signal Processing Toolbox User’s Guide. Natick,MA: The MathWorks, Inc., 1996. .Lehmer, D.H. “Mathematical Methods in large-scale computing units.”Proc. 2nd Sympos. on Large-Scale Digital Calculating Machinery, Cambridge,MA, 1949. Cambridge, MA: Harvard University Press, 1951.Oppenheim, Alan V. and Ronald W Schafer. Discrete-Time Signal Processing.Englewood Cliffs, NJ: Prentice Hall, 1989.Digital Signal Processing with the TMS320 Family (SPR012)TMS320C55x DSP CPU Reference Guide (SPRU371)TMS320C55x Optimizing C Compiler User’s Guide (SPRU281)TrademarksTMS320, TMS320C55x, and C55x are trademarks of Texas Instruments.Matlab is a trademark of Mathworks, Inc.Contentsiii

ContentsContents1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1Introduction to the TMS320C55x DSP Library.1.1 DSP Routines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-21.2 Features and Benefits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-21.3 DSPLIB: Quality Freeware That You Can Build On and Contribute To . . . . . . . . . . . . . . 1-22 Installing DSPLIB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1Describes how to install the DSPLIB.2.1 DSPLIB Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-22.2 How to Install DSPLIB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-22.2.1 Read README.1ST File for Specific Details of Release . . . . . . . . . . . . . . . . . . 2-22.3 How to Rebuild DSPLIB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-32.3.1 For Full Rebuild of 55xdsp.lib . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-32.3.2 For Partial Rebuild of 55xdsp.lib(modification of a specific DSPLIB function, for example fir.asm) . . . . . . . . . . . 2-33 Using DSPLIB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1Describes how to use the DSPLIB.3.1 DSPLIB Arguments and Data Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-23.1.1 DSPLIB Arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-23.1.2 DSPLIB Data Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-23.2 Calling a DSPLIB Function from C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-33.3 Calling a DSPLIB Function from Assembly Language Source Code . . . . . . . . . . . . . . . 3-53.4 Where to Find Sample Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-53.5 How DSPLIB is Tested – Allowable Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-63.6 How DSPLIB Deals with Overflow and Scaling Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-63.7 Where DSPLIB Goes from Here . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-84 Function Descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-1Provides descriptions for the TMS320C55x DSPLIB Functions.4.1 Arguments and Conventions Used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-24.2 DSPLIB Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-3acorr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-7add . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-9atan2_16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-10v

Contentsatan16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-11bexp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-13cbrev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-14cfft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-15cfir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-17cifft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-21convol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-23convol1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-25convol2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-27corr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-30dlms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-31expn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-33fir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-34fir2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-37firdec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-40firinterp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-42firlat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-44firs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-46fltoq15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-49hilb16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-50iir32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-53iircas4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-56iircas5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-58iircas51 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-59iirlat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-61ldiv16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-63log_10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-64log_2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-66logn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-67maxidx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-69maxidx34 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-70maxval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-70maxvec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-71minidx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-72minval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-73minvec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-73mmul . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-74mtrans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-76mul32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-76neg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-77neg32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-78power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-79q15tofl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-80rand16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-81vi

Chapter 1IntroductionThe Texas Instruments TMS320C55x DSP Library (DSPLIB) is an optimizedDSP Function Library for C programmers on TMS320C55x devices. It includesover 50 C-callable assembly-optimized general-purpose signal processingroutines. These routines are typically used in computationally intensive realtimeapplications where optimal execution speed is critical. By using these routinesyou can achieve execution speeds considerable faster than equivalentcode written in standard ANSI C language. In addition, by providing ready-touseDSP functions, TI DSPLIB can shorten significantly your DSP applicationdevelopment time.TopicPage1.1 DSP Routines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-21.2 Features and Benefits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-21.3 DSPLIB: Quality Freeware That You Can Build Onand Contribute To . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-21-1

DSP Routines1.1 DSP RoutinesThe TI DSPLIB includes commonly used DSP routines. Source code isprovided to allow you to modify the functions to match your specific needs.The routines included within the library are organized into eight differentfunctional categories:Fast-Fourier Transforms (FFT)Filtering and convolutionAdaptive filteringCorrelationMathTrigonometricMiscellaneousMatrix1.2 Features and BenefitsHand-coded assembly optimized routinesC-callable routines fully compatible with the TI C55x compilerFractional Q15-format operand supportedComplete set of examples on usage providedBenchmarks (time and code) providedTested against Matlab scripts1.3 DSPLIB: Quality Freeware That You Can Build On and Contribute ToDSPLIB is a free-of-charge product. You can use, modify, and distribute TIC55x DSPLIB for usage on TI C55x DSPs with no royalty payments. SeeLicensing and Warranty, section 6.1, and Where DSPLIB Goes from Here,section 3.7, for details.1-2

Chapter 2Installing DSPLIBThis chapter describes how to install the DSPLIB.TopicPage2.1 DSPLIB Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-22.2 How to Install DSPLIB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-22.3 How to Rebuild DSPLIB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-32-1

DSPLIB Content / How to Install DSPLIB2.1 DSPLIB ContentThe TI DSPLIB software consists of 4 parts:1) a header file for C programmers:dsplib.h2) One object library:55xdsp.lib3) One source library to allow function customization by the end user55xdsp.src4) Example programs and linker command files used under the “55x_test”sub-directory.2.2 How to Install DSPLIB2.2.1 Read README.1ST File for Specific Details of ReleaseStep 1:De-archive DSPLIBDSPLIB is distributed in the form of an executable self-extracting ZIP file(55xdsplib.exe). The zip file automatically restores the DSPLIB individualcomponents in the same directory you execute the self extracting file. Followingis an example on how to install DSPLIB, just type:55xdsplib.exe –dThe DSPLIB directory structure and content you will find is:55xdsplib (dir)55xdsp.libblt55x.batexamples(dir)include(dir)dsplib.htms320.hmisc.hdoc(dir)55x_src (dir): use for standards short-call mode: re-generate 55xdsp.lib based on 55xdsp.src: contains one subdirectory for each routine included inthe library where you can find complete test cases: include file with data types and function prototypes: include file with type definitions to increase TMS320portability: include file with useful miscellaneous definitions: contains assembly source files for functions2-2

How to Install DSPLIB / How to Rebuild DSPLIBStep 2:Relocate library fileCopy the C55x DSPLIB object library file, 55xdsp.lib, to your C5500 runtimesupport library folder.For example, if your TI C5500 tools are located in c:\ti\c5500\cgtools\bin andc runtime support libraries (rts55.lib etc.) in c:\ti\c5500\cgtools\lib, copy55xdsplib.lib to this folder. This allows the C55x compiler/linker to find55xdsp.lib.2.3 How to Rebuild DSPLIB2.3.1 For Full Rebuild of 55xdsp.libTo rebuild 55xdsp.lib, execute the blt55x.bat. This will overwrite any existing55xdsp.lib.2.3.2 For Partial Rebuild of 55xdsp.lib(modification of a specific DSPLIB function, for example fir.asm)1) Extract the source for the selected function from the source archive:ar55 x 55xdsp.src fir.asm2) Re-assemble your new fir.asm assembly source file:asm55 –g fir.asm3) Replace the object , fir.obj, in the dsplib.lib object library with the newlyformed object:ar55 r 55xdsp.lib fir.objInstalling DSPLIB2-3

Chapter 3Using DSPLIBThis chapter describes how to use the DSPLIB.TopicPage3.1 DSP Arguments and Data Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-23.2 Calling a DSPLIB Function from C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-33.3 Calling a DSPLIB Function from Assembly LanguageSource Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-53.4 Where to Find Sample Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-53.5 How DSPLIB is Tested — Allowable Error . . . . . . . . . . . . . . . . . . . . . . . 3-63.6 How DSPLIB Deals with Overflow and Scaling Issues . . . . . . . . . . . . 3-63.7 Where DSPLIB Goes from Here . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-83-1

DSPLIB Arguments and Data Types3.1 DSPLIB Arguments and Data Types3.1.1 DSPLIB ArgumentsDSPLIB functions typically operate over vector operands for greater efficiency.Though these routines can be used to process short arrays or scalars (unlessa minimum size requirement is noted) , the execution times will be longer inthose cases.Vector stride is always equal 1: vector operands are composed of vectorelements held in consecutive memory locations (vector stride equal to 1).Complex elements are assumed to be stored in a Re-Im format.In-place computation is allowed (unless specifically noted): Sourceoperand can be equal to destination operand to conserve memory.3.1.2 DSPLIB Data TypesDSPLIB handles the following fractional data types:Q.15 (DATA) : A Q.15 operand is represented by a short data type (16 bit)that is predefined as DATA, in the dsplib.h header file.Q.31 (LDATA) : A Q.31 operand is represented by a long data type (32 bit)that is predefined as LDATA, in the dsplib.h header file.Q.3.12 : Contains 3 integer bits and 12 fractional bits.Unless specifically noted, DSPLIB operates on Q15-fractional data typeelements. Appendix A presents an overview of Fractional Q formats3-2

Calling a DSPLIB Function from C3.2 Calling a DSPLIB Function from CIn addition to installing the DSPLIB software, to include a DSPLIB function inyour code you have to:Include the dsplib.h include fileLink your code with the DSPLIB object code library, 55xdsp.lib.Use a correct linker command file describing the memory configurationavailable in your C55x board.For example, the following code contains a call to the recip16 and q15toflroutines in DSPLIB:#include ”dsplib.h”DATA x[3] = { 12398 , 23167, 564};DATA r[NX];DATA rexp[NX];float rf1[NX];float rf2[NX];void main(){short i;}for (i=0;i

Calling a DSPLIB Function from CIn this example, the q15tofl DSPLIB function is used to convert Q15 fractionalvalues to floating point fractional values. However, in many applications, yourdata is always maintained in Q15 format so that the conversion between floatand Q15 is not required.The above code, ug.c, is available to you in the /doc/code subdirectory. Tocompile and link this code with 55xdsp.lib, issue the following command:cl55 –pk –g –o3 –i. ug.c –z –v0 ld3.cmd 55xdsp.lib –m ug.map –oug.outThe examples presented in this document have been tested using the TexasInstruments C55x Simulator. Customization may be required to use it with adifferent simulator or development board.Refer to the TMS320C55x Optimizing C Compiler User’s Guide (SPRU281).3-4

Calling a DSPLIB Function from Assembly Language Source Code3.3 Calling a DSPLIB Function from Assembly Language Source CodeThe TMS320C55x DSPLIB functions were written to be used from C. Callingthe functions from assembly language source code is possible as long as thecalling-function conforms with the Texas Instruments C55x C compiler callingconventions. Refer to the TMS320C55x Optimizing C Compiler User’s Guide,if a more in-depth explanation is required.Realize that the TI DSPLIB is not an optimal solution for assembly-onlyprogrammers. Even though DSPLIB functions can be invoked from anassembly program, the result may not be optimal due to unnecessary C-callingoverhead.3.4 Where to Find Sample CodeYou can find examples on how to use every single function in DSPLIB, in theexamples subdirectory. This subdirectory contains one subdirectory for eachfunction. For example, the examples/araw subdirectory contains the followingfiles:araw_t.c: main driver for testing the DSPLIB acorr (raw) function.test.h: contains input data(a) and expected output data(yraw) for the acorr(raw) function as. This test.h file is generated by using Matlab scripts.test.c: contains function used to compare the output of araw function withthe expected output data.ftest.c: contains function used to compare two arrays of float data types.ltest.c: contains function used to compare two arrays of long data types.ld3.cmd: an example of a linker command you can use for this function.Using DSPLIB3-5

How DSPLIB is Tested – Allowable Error3.5 How DSPLIB is Tested – Allowable ErrorVersion 1.0 of DSPLIB is tested against Matlab scripts. Expected data outputhas been generated from Matlab that uses double-precision (64-bit) floatingpointoperations (default precision in Matlab). Test utilities have been addedto our test main drivers to automate this checking process. Note that a maximumabsolute error value (MAXERROR) is passed to the test function, to setthe trigger point to flag a functional error.We consider this testing methodology a good first pass approximation. Furthercharacterization of the quantization error ranges for each function (under randominput) as well as testing against a set of fixed-point C models is plannedfor future releases. We welcome any suggestions you may have on thisrespect.3.6 How DSPLIB Deals with Overflow and Scaling IssuesOne of the inherent difficulties of programming for fixed-point processors isdetermining how to deal with overflow issues. Overflow occurs as a result ofaddition and subtraction operations when the dynamic range of the resultingdata is larger than what the intermediate and final data types can contain.The methodology used to deal with overflow should depend on the specificsof your signal, the type of operation in your functions, and the DSP architectureused. In general, overflow handling methodologies can be classified in fivecategories: saturation, input scaling, fixed scaling, dynamic scaling, andsystem design considerations.It’s important to note that a TMS320C55x architectural feature that makesoverflow easier to deal with is the presence of guard bits in all four accumulators.The 40-bit accumulators provide eight guard bits that allow up to 256 consecutivemultiply-and-accumulate (MAC) operations before an accumulatoroverrun – a very useful feature when implementing, for example, FIR filters.There are 4 specific ways DSPLIB deals with overflow, as reflected in eachfunction description:Scaling implemented for overflow prevention: In this type of function,DSPLIB scales the intermediate results to prevent overflow. Overflowshould not occur as a result. Precision is affected but not significantly. Thisis the case of the FFT functions, in which scaling is used after each FFTstage.3-6

How DSPLIB Deals with Overflow and Scaling IssuesNo scaling implemented for overflow prevention: In this type of function,DSPLIB does not scale to prevent overflow due to the potentiallystrong effect in data output precision or in the number of cycles required.This is the case, for example, of the MAC-based operations like filtering,correlation, or convolutions. The best solution on those cases is to designyour system , for example your filter coefficients with a gain less than 1 toprevent overflow. In this case, overflow could happen unless you inputscale or you design for no overflow.Saturation implemented for overflow handling: In this type of function,DSPLIB has enabled the TMS320C55x 32-bit saturation mode (SATDbit = 1). This is the case of certain basic math functions that require thesaturation mode to be enabled.Not applicable: In this type of function, due to the nature of the functionoperations, there is no overflow.DSPLIB reporting of overflow conditions (overflow flag): Due to thesometimes unpredictible overflow risk, most DSPLIB functions have beenwritten to return an overflow flag (oflag) as an indication of a potentiallydangerous 32-bit overflow. However, because of the guard-bits, the C55xis capable of handling intermediate 32-bit overflows and still produce thecorrect final result. Therefore, the oflag parameter should be taken in thecontext of a warning but not a definitive error.As a final note, DSPLIB is provided also in source format to allow customizationof DSPLIB functions to your specific system needs.Using DSPLIB3-7

Where DSPLIB Goes from Here3.7 Where DSPLIB Goes from HereWe anticipate DSPLIB to improve in future releases in the following areas:Increased number of functions: We anticipate the number of functionsin DSPLIB will increase. We welcome user-contributed code. If during theprocess of developing your application you develop a DSP routine thatseems like a good fit to DSPLIB, let us know. We will review and test yourroutine and possibly include it in the next DSPLIB software release. Yourcontribution will be acknowledged and recognized by TI in the Acknowledgmentssection. Use this opportunity to make your name known by yourDSP industry peers. Simply email your contribution To Whom It May Concern:dsph@ti.com and we will contact you.Increased Code portability: DSPLIB looks to enhance code portabilityacross different TMS320-based platforms. It is our goal to provide similarDSP libraries for other TMS320 devices, working in conjunction withC55x compiler intrinsics to make C-developing easier for fixed-pointdevices. However, it’s anticipated that a 100% portable library acrossTMS320 devices may not be possible due to normal device architecturaldifferences. TI will continue monitoring DSP industry standardization activitiesin terms of DSP function libraries.3-8

Chapter 4Function DescriptionsThis chapter provides descriptions for the TMS330C55x DSPLIB functions.TopicPage4.1 Arguments and Conventions Used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-24.2 DSPLIB Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-34-1

Arguments and Conventions Used4.1 Arguments and Conventions UsedTable 4–1. Function DescriptionsThe following convention has been followed when describing the argumentsfor each individual function:Argumentx,yrnx,ny,nrDescriptionargument reflecting input data vectorargument reflecting output data vectorarguments reflecting the size of vectors x,y, and r respectively. Infunctions where nx = nr = nr, only nx has been used.hnhDATAArgument reflecting filter coefficient vector (filter routines only)Argument reflecting the size of vector hdata type definition equating a short, a 16-bit value representing aQ15 number. Usage of DATA instead of short is recommended toincrease future portability across devices.LDATAushortdata type definition equating a long, a 32-bit value representing aQ31 number. Usage of LDATA instead of long is recommended toincrease future portability across devices.Unsigned short (16 bit). You can use this data type directly,because it has been defined in dsplib.h4-2

DSPLIB Functions4.2 DSPLIB FunctionsThe routines included within the library are organized into 8 different functionalcategories:FFTFiltering and convolutionAdaptive filteringCorrelationMathTrigonometricMiscellaneousMatrixTable 4–2 lists the functions by these 8 functional catagories.Table 4–2. Summary Table(a) FFTFunctionsvoid cfft (DATA *x, ushort nx, type)void cifft (DATA *x, ushort nx, type)void cbrev (DATA *x, DATA *r, ushort n)void rfft (DATA *x, ushort nx, type)void rifft (DATA *x, ushort nx, type)DescriptionRadix-2 complex forward FFT – MACRORadix-2 complex inverse FFT – MACROComplex bit-reverse functionRadix-2 real forward FFT – MACRORadix-2 real inverse FFT – MACRO(b) Filtering and ConvolutionFunctionsushort fir (DATA *x, DATA *h, DATA *r, DATA *dbuffer,ushort nx, ushort nh)ushort fir2 (DATA *x, DATA *h, DATA *r, DATA *dbuffer,ushort nx, ushort nh)ushort firs (DATA *x, DATA *h, DATA *r, DATA *dbuffer,ushort nx, ushort nh2)ushort cfir (DATA *x, DATA *h, DATA *r, DATA *dbuffer,ushort nx, ushort nh)ushort convol (DATA *x, DATA *h, DATA *r, ushort nr,ushort nh)ushort convol1 (DATA *x, DATA *h, DATA *r, ushort nr,ushort nh)DescriptionFIR direct formFIR direct form (Optimized to use DUAL–MAC)Symmetric FIR direct form (generic routine)Complex FIR direct formConvolutionConvolution (Optimized to use DUAL–MAC)Function Descriptions4-3

DSPLIB FunctionsTable 4–2. Summary Table (Continued)(b) Filtering and Convolution (Continued)Functionsushort convol2 (DATA *x, DATA *h, DATA *r, ushort nr,ushort nh)ushort iircas4 (DATA *x, DATA *h, DATA *r, DATA *dbuffer,ushort nbiq, ushort nx)ushort iircas5 (DATA *x, DATA *h, DATA *r, DATA *dbuffer,ushort nbiq, ushort nx)ushort iircas51 (DATA *x, DATA *h, DATA *r, DATA *dbuffer,ushort nbiq, ushort nx)ushort iirlat (DATA *x, DATA *h, DATA *r, DATA *pbuffer,int nx, int nh)ushort firlat (DATA *x, DATA *h, DATA *r, DATA *pbuffer,int nx, int nh)ushort firdec (DATA *x, DATA *h, DATA *r, DATA *dbuffer,ushort nh, ushort nx, ushort D)ushort firinterp (DATA *x, DATA *h, DATA *r, DATA *dbuffer,ushort nh, ushort nx, ushort I)ushort hilb16 (DATA *x, DATA *h, DATA *r, DATA *dbuffer,ushort nx, ushort nh)ushort iir32 (DATA *x, LDATA *h, DATA *r, LDATA *dbuffer,ushort nbiq, ushort nr)DescriptionConvolution (Optimized to use DUAL–MAC)IIR cascade direct form II. 4 coefficients perbiquad.IIR cascade direct form II. 5 coefficients perbiquadIIR cascade direct form I. 5 coefficients perbiquadLattice inverse IIR filterLattice forward FIR filterDecimating FIR filterInterpolating FIR filterFIR Hilbert TransformerDouble-precision IIR filter(c) Adaptive filteringFunctionsushort dlms (DATA *x, DATA *h, DATA *r, DATA *des,DATA *dbuffer, DATA step, ushort nh, ushort nx)DescriptionLMS FIR (delayed version)(d) CorrelationFunctionsushort acorr (DATA *x, DATA *r, ushort nx, ushort nr, type)ushort corr (DATA *x, DATA *y, DATA *r, ushort nx, ushortny, type)DescriptionAutocorrelation (positive side only) – MACROCorrelation (full-length)4-4

DSPLIB FunctionsTable 4–2. Summary Table (Continued)(e) TrigonometricFunctionsushort sine (DATA *x, DATA *r, ushort nx)ushort atan2_16 (DATA *i, DATA *q, DATA *r, ushort nx)ushort atan16 (DATA *x, DATA *r, ushort nx)Descriptionsine of a vectorFour quadrant inverse tangent of a vectorArctan of a vector(f) MathFunctionsushort add (DATA *x, DATA *y, DATA *r, ushort nx,ushort scale)ushort expn (DATA *x, DATA *r, ushort nx)short bexp (DATA *x, ushort nx)ushort logn (DATA *x, LDATA *r, ushort nx)ushort log_2 (DATA *x, LDATA *r, ushort nx)ushort log_10 (DATA *x, LDATA *r, ushort nx)short maxidx (DATA *x, ushort ng, ushort ng_size)DescriptionOptimized vector additionExponent of a vectorExponent of all values in a vectorNatural log of a vectorLog base 2 of a vectorLog base 10 of a vectorIndex for maximum magnitude in a vectorshort maxidx34 (DATA *x, ushort nx) Index of the maximum element of a vector ≤ 34short maxval (DATA *x, ushort nx)void maxvec (DATA *x, ushort nx, DATA *r_val,DATA *r_idx)short minidx (DATA *x, ushort nx)short minval (DATA *x, ushort nx)void minvec (DATA *x, ushort nx, DATA *r_val,DATA *r_idx)ushort mul32 (LDATA *x, LDATA *y, LDATA *r, ushort nx)short neg (DATA *x, DATA *r, ushort nx)short neg32 (LDATA *x, LDATA *r, ushort nx)short power (DATA *x, LDATA *r, ushort nx)void recip16 (DATA *x, DATA *r, DATA *rexp, ushort nx)void ldiv16 (LDATA *x, DATA *y, DATA *r, DATA *rexp,ushort nx)Maximum magnitude in a vectorIndex and value of the maximum element of avectorIndex for minimum magnitude in a vectorMinimum element in a vectorIndex and value of the minimum element of avector32-bit vector multiply16-bit vector negate32-bit vector negatesum of squares of a vector (power)Vector reciprocal32-bit by 16-bit long divisionFunction Descriptions4-5

DSPLIB FunctionsTable 4–2. Summary Table (Continued)(f) Math (Continued)Functionsushort sqrt_16 (DATA *x, DATA *r, short nx)short sub (DATA *x, DATA *y, DATA *r, ushort nx,ushort scale)DescriptionSquare root of a vectorVector subtraction(g) MatrixFunctionsushort mmul (DATA *x1, short row1, short col1,DATA *x2, short row2, short col2, DATA *r)ushort mtrans (DATA *x, short row, short col, DATA *r)Descriptionmatrix multiplymatrix transponse(h) MiscellaneousFunctionsushort fltoq15 (float *x, DATA *r, ushort nx)ushort q15tofl (DATA *x, float *r, ushort nx)ushort rand16 (DATA *r, ushort nr)void rand16init(void)DescriptionFloating-point to Q15 conversionQ15 to floating-point conversionRandom number generationRandom number generation initialization4-6

acorracorrFunctionArgumentsAutocorrelationushort oflag = acorr (DATA *x, DATA *r, ushort nx, ushort nr, type)(defined in araw.asm, abias.asm , aubias.asm)x [nx]Pointer to real input vector of nx real elements. nx ≥ nrr [nr]nxnrtypeoflagPointer to real output vector containing the first nr elementsof the positive side of the autocorrelation function of vectora. r must be different than a (in-place computation is notallowed).Number of real elements in vector xNumber of real elements in vector rAutocorrelation type selector. Types supported: If type = raw, r contains the raw autocorrelation of x If type = bias, r contains the biased autocorrelation of x If type = unbias, r contains the unbiased autocorrelation ofxOverflow flag. If oflag = 1, a 32-bit overflow has occurred If oflag = 0, a 32-bit overflow has not occurredDescriptionAlgorithmComputes the first nr points of the positive side of the autocorrelation of thereal vector x and stores the results in real output vector r. The full-length autocorrelationof vector x will have 2*nx–1 points with even symmetry around thelag 0 point (r[0]). This routine provides only the positive half of this for memoryand computational savings.Raw Autocorrelationr[j] nxj1k0x[j k] x[k]0 j nrBiased Autocorrelationr[j] nx 1 nxj1 x[j k] x[k]k00 j nrUnbiased Autocorrelationr[j] 1(nx abs(j))nxj1k0x[j k] x[k]0 j nrFunction Descriptions4-7

acorrOverflow Handling MethodologyNo scaling implemented for overflow preventionSpecial Requirements noneImplementation NotesSpecial debugging consideration: This function is implemented as a macrothat invokes different autocorrelation routines according to the typeselected. As a consequence the acorr symbol is not defined. Instead theacorr_raw, acorr_bias, acorr_unbias symbols are defined.Autocorrelation is implemented using time-domain techniquesExampleBenchmarksSee examples/abias, examples/aubias, examples/araw subdirectories(preliminary)Cycles †Abias:Core:nr even: [(4 * nx – nr * (nr + 2) + 20) / 8] * nrnr odd: [(4 * nx – (nr – 1) * (nr + 1) + 20) / 8] * (nr – 1) + 10nr = 1: (nx + 2)Overhead:nr even: 90nr odd: 83nr = 1: 59Araw:Core:nr even: [(4 * nx – nr * (nr + 2) + 28) / 8] * nrnr odd: [(4 * nx – (nr – 1) * (nr + 1) + 28) / 8] * (nr – 1) + 13nr = 1: (nx + 1)Overhead:nr even: 34nr odd: 35nr = 1: 30† Assumes all data is in on-chip dual-access RAM and that there is no bus conflict due to twiddletable reads and instruction fetches (provided linker command file reflects those conditions).4-8

addCycles †Code size(in bytes)Aubias:Core:nreven: [(8 * nx – 3 * nr * (nr + 2) + 68) / 8] * nrnr odd: [(8 * nx – 3 * (nr–1) * (nr+1) + 68)/8] * (nr – 1) + 33nr = 1: nx + 26Overhead:nr even: 64nr odd: 55nr = 1: 47Abias: 226Araw: 178Aubias: 308† Assumes all data is in on-chip dual-access RAM and that there is no bus conflict due to twiddletable reads and instruction fetches (provided linker command file reflects those conditions).addFunctionVector Addushort oflag = add (DATA *x, DATA *y, DATA *r, ushort nx, ushort scale)(defined in add.asm)Argumentsx[nx]y[nx]r[nx]nxscaleoflagPointer to input data vector 1 of size nx. In-place processingallowed (r can be = x = y)Pointer to input data vector 2 of size nxPointer to output data vector of size nx containing (x+y) if scale = 0 (x+y) /2 if scale = 1Number of elements of input and output vectors.nx ≥ 4Scale selection If scale = 1, divide the result by 2 to prevent overflow If scale = 0, do not divide by 2Overflow flag. If oflag = 1, a 32-bit overflow has occurred If oflag = 0, a 32-bit overflow has not occurredDescriptionThis function adds two vectors, element by element.Function Descriptions4-9

atan2_16Algorithm for (i 0; i nx; i ) z(i) x(i) y(i)Overflow Handling MethodologyScaling implemented for overflow prevention (User selectable)Special Requirements noneImplementation Notes noneExampleBenchmarksSee examples/add subdirectory(preliminary)Cycles † Core: 3 * nxOverhead: 23Code size(in bytes)60† Assumes all data is in on-chip dual-access RAM and that there is no bus conflict due to twiddletable reads and instruction fetches (provided linker command file reflects those conditions).atan2_16FunctionArctangent 2 Implementationushort oflag = atan2_16 (DATA *i, DATA *q, DATA *r, ushort nx)(defined in arct2.asm)Argumentsq[nx]i[nx]r[nx]nxoflagPointer to quadrature input vector of size nx.Pointer to in-phase input vector of size nxPointer to output data vector (in Q15 format) numberrepresentation of size nx containing. In-place processingallowed (r can be equal to x ) on output, r contains thearctangent of (q/I) /PINumber of elements of input and output vectors.Overflow flag. If oflag = 1, a 32-bit overflow has occurred If oflag = 0, a 32-bit overflow has not occurred4-10

atan16Description This function calculates the arctangent of the ratio q/I, where –1

atan16nxoflagNumber of elements of input and output vectors.Overflow flag. If oflag = 1, a 32-bit overflow has occurred If oflag = 0, a 32-bit overflow has not occurredDescriptionThis function calculates the arc tangent of each of the elements of vector x. Theresult is placed in the resultant vector r and is in the range [–pi/2 to pi/2]radians. For example,if x = [0x7fff, 0x3505, 0x1976, 0x0] (equivalent to tan(PI/4), tan(PI/8),tan(PI/16), 0 in float):atan16(x,r,4) should giver = [0x6478, 0x3243, 0x1921, 0x0] equivalent to [PI/4, PI/8, PI/16, 0]Algorithm for (i 0; i nx; i ) r(i) atan(x(i))Overflow Handling MethodologyNot applicableSpecial Requirements Linker command file: you must allocate .data section (for polynomialcoefficients)Implementation Notesatan(x), with 0

expbexpFunctionBlock Exponent Implementationshort r = bexp (DATA *x, ushort nx)(defined in bexp.asm)Argumentsx [nx]rnxPointer to input vector of size nxReturn value. Maximum exponent that may be used inscaling.Length of input data vectorDescriptionAlgorithmComputes the exponents (number of extra sign bits) of all values in the inputvector and returns the minimum exponent. This will be useful in determiningthe maximum shift value that may be used in scaling a block of data.Not applicableOverflow Handling MethodologyNot applicableSpecial Requirements noneImplementation Notes noneExampleBenchmarksSee examples/bexp subdirectory(preliminary)Cycles Core: 3 * nxOverhead: 4Code size(in bytes)19Function Descriptions4-13

cbrevcbrevComplex Bit ReverseFunction void cbrev (DATA *a, DATA *r, ushort n)(defined in cbrev.asm)Argumentsx[2*nx]Pointer to complex input vector xr[2*nx] Pointer to complex output vector r.nx Number of complex elements of vectors x and r. To bit-reverse the input of a complex FFT, nx should be thecomplex FFT size.To bit-reverse the input of a real FFT, nx should be half thereal FFT size.DescriptionAlgorithmThis function bit-reverses the position of elements in complex vector x into outputvector r. In-place bit-reversing is allowed. Use this function in conjunctionwith FFT routines to provide the correct format for the FFT input or output data.If you bit-reverse a linear-order array, you obtain a bit-reversed order array. Ifyou bit-reverse a bit-reversed order array, you obtain a linear-order array.Not applicableOverflow Handling MethodologyNot applicableSpecial RequirementsImplementation NotesAlignment of input database address: (n+1) LSBs must be zeros, wheren = log 2 (nx).Ensure that the entire array fits within a 64K boundary (the largest possiblearray addressable by the 16-bit auxiliary register).x is read with bit-reversed addressing and r is written in normal linear addressing.Off-place bit-reversing (x = r) requires half as many cycles as in-place bitreversing(x r). However, this is at the expense of doubling the datamemory requirements.ExampleSee examples/cfft and examples/rfft subdirectories4-14

cfftBenchmarks(preliminary)Cycles †Code size(in bytes)Core:2 * nx (off-place)4 * nx + 6 (in-place)Overhead: 1781 (includes support for both in-place and off-placebit-reverse)† Assumes all data is in on-chip dual-access RAM and that there is no bus conflict due to twiddletable reads and instruction fetches (provided linker command file reflects those conditions).cfftFunctionForward Complex FFTvoid cfft (DATA *x, ushort nx, type);(defined in cfft.asm)Argumentsx [2*nx]nxtypePointer to input vector containing nx complex elements(2*nx real elements) in bit-reversed order. On output, vectora contains the nx complex elements of the FFT(x). Complexnumbers are stored in interleaved Re-Im format.Number of complex elements in vector x. Must be between8 and 1024.FFT type selector. Types supported: If type = SCALE, scaled version selected If type = NOSCALE, non-scaled version selectedDescriptionAlgorithmComputes a complex nx-point FFT on vector x, which is in bit-reversed order.The original content of vector x is destroyed in the process. The nx complexelements of the result are stored in vector x in normal-order.(DFT)y[k] 1nx1(scale factor) * x[i]*cos 2* nx* i * k j sin 2* nx* i * ki0Overflow Handling MethodologyIf scale==1, scaling before each stage is implemented for overflowpreventionSpecial RequirementsThis function requires the inclusion of file “twiddle.inc” which contains thetwiddle table (automatically included).Function Descriptions4-15

cfftImplementation NotesData memory alignment (reference cfft.cmd in examples/cfft directory):Alignment of input database address: (n+1) LSBs must be zeros,where n = log 2 (nx).Ensure that the entire array fits within a 64K boundary (the largest possiblearray addressable by the 16-bit auxiliary register).For best performance the data buffer has to be in a DARAM block.If the twiddle table and the data buffer are in the same block then theradix-2 kernal is 7 cycles and the radix-4 kernel is not affected.Radix-2 DIF version of the FFT algorithm is implemented. Theimplementation is optimized for MIPS, not for code size. The first stage isunrolled and the last two stages are implemented in radix-4, for MIPSoptimization.This routine prevents overflow by scaling by 2 before each FFT stage,assuming that the parameter scale is set to 1.ExampleBenchmarksSee examples/cfft subdirectory(preliminary)5 cycles (radix-2 butterfly – used in both scaled and non-scaled versions)10 cycles (radix-4 butterfly – used in non-scaled version)CFFT – SCALEFFT Size Cycles † Code Size (in bytes)8 149 10916 325 46232 591 46264 1177 462128 2483 462256 5389 462512 11815 4621024 25921 462† Assumes all data is in on-chip dual-access RAM and that there is no bus conflict due to twiddletable reads and instruction fetches (provided linker command file reflects those conditions).4-16

cfirCFFT – NOSCALEFFT Size Cycles † Code Size (in bytes)8 139 32516 246 32532 477 32564 996 325128 2171 325256 4818 325512 10729 3251024 23808 325† Assumes all data is in on-chip dual-access RAM and that there is no bus conflict due to twiddletable reads and instruction fetches (provided linker command file reflects those conditions).cfirFunctionComplex FIR Filterushort oflag = cfir (DATA *x, DATA *h, DATA *r, DATA *dbuffer, ushort nx,ushort nh)Argumentsx[2*nx]Pointer to input vector of nx complex elements.h[2*nh] Pointer to complex coefficient vector of size nh innormal order. For example, if nh=6, then h[nh] ={h0r, h0i, h1r, h1i h2r, h2i, h3r, h3i, h4r, h4i, h5r, h5i}where h0 resides at the lowest memory address inthe array.This array must be located in internal memory sinceit is accessed by the C55x coefficient bus.r[2*nx]Pointer to output vector of nx complex elements.In-place computation (r = x) is allowed.Function Descriptions4-17

cfirdbuffer[2*nh + 2] Pointer to delay buffer of length nh =2 * nh + 2In the case of multiple-buffering schemes, thisarray should be initialized to 0 for the first filter blockonly. Between consecutive blocks, the delay bufferpreserves the previous r output elements needed.The first element in this array is present for alignmentpurposes, the second element is special inthat it contains the array index–1 of the oldest inputentry in the delay buffer. This is needed for multiplebufferingschemes, and should be initialized to 0(like all the other array entries) for the first blockonly.nxnhoflagNumber of complex input samplesThe number of complex coefficients of the filter. Forexample, if the filter coefficients are {h0, h1, h2, h3,h4, h5}, then nh = 6. Must be a minimum value of 3.For smaller filters, zero pad the coefficients to meetthe minimum value.Overflow error flag (returned value) If oflag = 1, a 32-bit data overflow has occurred inan intermediate or final result.If oflag = 0, a 32-bit overflow has not occurred.DescriptionComputes a complex FIR filter (direct-form) using the coefficients stored invector h. The complex input data is stored in vector x. The filter output resultis stored in vector r. This function maintains the array dbuffer containing theprevious delayed input values to allow consecutive processing of input datablocks. This function can be used for both block-by-block (nx ≥ 2) and sampleby-samplefiltering (nx = 1). In-place computation (r = x) is allowed.Algorithmnh1r[j] h[k] x[j k]k00 j nxOverflow Handling MethodologyNo scaling implemented for overflow prevention.Special Requirements nh must be a minimum value of 3. For smaller filters, zero pad the h[] array.4-18

cfirImplementation Notes The first element in the dbuffer array is present only for alignment purposes.The second element in this array (index=0) is the entry index for the inputhistory. It is treated as an unsigned 16-bit value by the function even thoughit has been declared as signed in C. The value of the entry index is equal tothe index – 1 of the oldest input entry in the array. The remaining elementsmake up the input history. Figure 4–1 shows the array in memory with an entryindex of 2. The newest entry in the dbuffer is denoted by x(j–0), which in thiscase would occupy index = 3 in the array. The next newest entry is x(j–1), andso on. It is assumed that all x() entries were placed into the array by theprevious invocation of the function in a multiple-buffering scheme.Figure 4–1, Figure 4–2, and Figure 4–3 show the dbuffer, x, and r arrays asthey appear in memory.Figure 4–1. dbuffer Array in Memory at Time joldest x( ) entrynewest x( ) entrydummy valueentry index = 2xr(j–nh–2)xi(j–nh–2)xr(j–nh–1)xi(j–nh–1)xr(j–nh)xi(j–nh)xr(j–0)xi(j–0)xr(j–1)xi(j–1)xr(j–2)xi(j–2)•lowest memory addressxr(j–nh–3)xi(j–nh–3)xr(j–nh–4)xi(j–nh–4)xr(j–nh–3)xi(j–nh–3)highest memory addressFunction Descriptions4-19

cfirFigure 4–2. x Array in Memoryoldest x( ) entryxr(0)xi(0)xr(1)xi(1)lowest memory address•newest x( ) entryxr(nx–2)xi(nx–2)xr(nx–1)xi(nx–1)highest memory addressFigure 4–3. r Array in Memoryoldest x( ) entrynewest x( ) entryrr(0)ri(0)rr(1)ri(1)•rr(nx–2)ri(nx–2)rr(nx–1)ri(nx–1)lowest memory addresshighest memory addressExampleBenchmarksSee examples/cfir subdirectory(preliminary)Cycles † Core: nx * [8 + 2(nh–2)]Overhead: 51Code size(in bytes)136† Assumes all data is in on-chip dual-access RAM and that there is no bus conflict due to twiddletable reads and instruction fetches (provided linker command file reflects those conditions).4-20

cifftcifftFunctionInverse Complex FFTvoid cifft (DATA *x, ushort nx, type);(defined in cifft.asm)Argumentsx [2*nx]nxtypePointer to input vector containing nx complex elements(2*nx real elements) in bit-reversed order. On output, vectora contains the nx complex elements of the IFFT(x).Complex numbers are stored in interleaved Re-Im format.Number of complex elements in vector x. Must be between8 and 1024.FFT type selector. Types supported: If type = SCALE, scaled version selected If type = NOSCALE, non-scaled version selectedDescriptionAlgorithmComputes a complex nx-point IFFT on vector x, which is in bit-reversed order.The original content of vector x is destroyed in the process. The nx complexelements of the result are stored in vector x in normal-order.(IDFT)y[k] 1nx1(scale factor) * x[w]*cos 2* nx* i * k j sin 2* nx* i * ki0Overflow Handling MethodologyIf scale==1, scaling before each stage is implemented for overflowpreventionSpecial RequirementsThis function requires the inclusion of file “twiddle.inc” which contains thetwiddle table (automatically included).Data memory alignment (reference cfft.cmd in examples/cifft directory):Alignment of input database address: (n+1) LSBs must be zeros,where n = log 2 (nx).Ensure that the entire array fits within a 64K boundary (the largest possiblearray addressable by the 16-bit auxiliary register).For best performance the data buffer has to be in a DARAM block.If the twiddle table and the data buffer are in the same block then theradix-2 kernal is 7 cycles and the radix-4 kernel is not affected.Function Descriptions4-21

cifftImplementation NotesRadix-2 DIF version of the IFFT algorithm is implemented. Theimplementation is optimized for MIPS, not for code size. The first stage isunrolled and the last two stages are implemented in radix-4, for MIPSoptimization.This routine prevents overflow by scaling by 2 before each IFFT stage,assuming that the parameter scale is set to 1.IFFT implementation is directly derived from FFT implementation bychanging signs (or replacing adds with subtracts or vice versa) where appropriate.ExampleBenchmarksSee examples/cifft subdirectory(preliminary)5 cycles (radix-2 butterfly – used in both scaled and non-scaled versions)10 cycles (radix-4 butterfly – used in non-scaled version)CIFFT – SCALEFFT Size Cycles † Code Size (in bytes)8 149 46216 325 46232 591 46264 1177 462128 2483 462256 5389 462512 11815 4621024 25921 462† Assumes all data is in on-chip dual-access RAM and that there is no bus conflict due to twiddletable reads and instruction fetches (provided linker command file reflects those conditions).4-22

convolCIFFT – NOSCALEFFT Size Cycles † Code Size (in bytes)8 139 32516 246 32532 477 32564 996 325128 2171 325256 4818 325512 10729 3251024 23808 325† Assumes all data is in on-chip dual-access RAM and that there is no bus conflict due to twiddletable reads and instruction fetches (provided linker command file reflects those conditions).convolFunctionConvolutionushort oflag = convol (DATA *x, DATA *h, DATA *r, ushort nr, ushort nh)Argumentsx[nr+nh–1]h[nh]r[nr]Pointer to input vector of nr + nh – 1 real elements.Pointer to input vector of nh real elements.Pointer to output vector of nr real elements.nrNumber of elements in vector r. In-place computation(r = x) is allowed (see Description section for comment).nh Number of elements in vector h.oflagOverflow error flag (returned value) If oflag = 1, a 32-bit data overflow occurred in an intermediateor final result.If oflag = 0, a 32-bit overflow has not occurred.Function Descriptions4-23

convolDescriptionComputes the real convolution of two real vectors x and h, and places theresults in vector r. Typically used for block FIR filter computation when thereis no need to retain an input delay buffer. This function can also be used toimplement single-sample FIR filters (nr = 1) provided the input delay historyfor the filter is maintained external to this function. In-place computation (r = x)is allowed, but be aware that the r output vector is shorter in length than thex input vector; therefore, r will only overwrite the first nr elements of the x.Algorithmnh1r[j] h[k] x[j k]k00 j nrOverflow Handling MethodologyNo scaling implemented for overflow prevention.Special Requirements noneImplementation Notes Figure 4–4, Figure 4–5, and Figure 4–6 show the x, r, and h arrays as theyappear in memory.Figure 4–4. x Array in Memoryx(0)x(1)•x(nr+nh–2)x(nr+nh–1)lowest memory addresshighest memory addressFigure 4–5. r Array in Memoryr(0)r(1)•r(nr–2)r(nr–1)lowest memory addresshighest memory address4-24

convol1Figure 4–6. h Array in Memoryh(0)h(1)•h(nh–2)h(nh–1)lowest memory addresshighest memory addressExampleBenchmarksSee examples/convol subdirectory(preliminary)Cycles † Core: nr * (1 + nh)Overhead: 44Code size(in bytes)88† Assumes all data is in on-chip dual-access RAM and that there is no bus conflict due to twiddletable reads and instruction fetches (provided linker command file reflects those conditions).convol1FunctionConvolution (fast)ushort oflag = convol1 (DATA *x, DATA *h, DATA *r, ushort nr, ushort nh)Argumentsx[nr+nh–1]h[nh]r[nr]nrPointer to input vector of nr+nh–1 real elements.Pointer to input vector of nh real elements.Pointer to output vector of nr real elements. In-placecomputation (r = x) is allowed (see Description section forcomment).Number of elements in vector r. Must be an even number.nh Number of elements in vector h.oflagOverflow error flag (returned value) If oflag = 1, a 32-bit data overflow occurred in an intermediateor final result.If oflag = 0, a 32-bit overflow has not occurred.Function Descriptions4-25

convol1DescriptionComputes the real convolution of two real vectors x and h, and places theresults in vector r. This function utilizes the dual-MAC capability of the C55xto process in parallel two output samples for each iteration of the inner functionloop. It is, therefore, roughly twice as fast as CONVOL, which implements onlya single-MAC approach. However, the number of output samples (nr) must beeven. Typically used for block FIR filter computation when there is no need toretain an input delay buffer. This function can also be used to implement singlesampleFIR filters (nr = 1) provided the input delay history for the filter is maintainedexternal to this function. In-place computation (r = x) is allowed, but beaware that the r output vector is shorter in length than the x input vector; therefore,r will only overwrite the first nr elements of the x.Algorithmnh1r[j] h[k] x[j k]k00 j nrOverflow Handling MethodologyNo scaling implemented for overflow prevention.Special Requirementsnr must be an even value.The vector h[nh] must be located in internal memory since it is accessedusing the C55x coefficient bus, and that bus does not have access to externalmemory.Implementation Notes Figure 4–7, Figure 4–8, and Figure 4–9 show the x, r, and h arrays as theyappear in memory.Figure 4–7. x Array in Memoryx(0)x(1)•x(nr+nh–2)x(nr+nh–1)lowest memory addresshighest memory address4-26

convol2Figure 4–8. r Array in Memoryr(0)r(1)•r(nr–2)r(nr–1)lowest memory addresshighest memory addressFigure 4–9. h Array in Memoryh(0)h(1)•h(nh–2)h(nh–1)lowest memory addresshighest memory addressExampleBenchmarksSee examples/convol1 subdirectory(preliminary)Cycles † Core: nr/2 * [3+(nh–2)]Overhead: 58Code size(in bytes)101† Assumes all data is in on-chip dual-access RAM and that there is no bus conflict due to twiddletable reads and instruction fetches (provided linker command file reflects those conditions).convol2FunctionArgumentsConvolution (fastest)ushort oflag = convol2 (DATA *x, DATA *h, DATA *r, ushort nr, ushort nh)x[nr+nh–1] Pointer to input vector of nr + nh – 1 real elements.h[nh]r[nr]Pointer to input vector of nh real elements.Pointer to output vector of nr real elements. In-placecomputation (r = x) is allowed (see Description section forcomment). This array must be aligned on a 32-bitboundary in memory.Function Descriptions4-27

convol2nrNumber of elements in vector r. Must be an even number.nh Number of elements in vector h.oflagOverflow error flag (returned value) If oflag = 1, a 32-bit data overflow has occurred in anintermediate or final result.If oflag = 0, a 32-bit overflow has not occurred.DescriptionComputes the real convolution of two real vectors x and h, and places theresults in vector r. This function utilizes the dual-MAC capability of the C55xto process in parallel two output samples for each iteration of the inner functionloop. It is, therefore, roughly twice as fast as CONVOL, which implements onlya single-MAC approach. However, the number of output samples (nr) must beeven. In addition, this function offers a small performance improvement overCONVOL1 at the expense of requiring the r array to be 32-bit aligned in memory.Typically used for block FIR filter computation when there is no need toretain an input delay buffer. This function can also be used to implement singlesampleFIR filters (nr = 1) provided the input delay history for the filter is maintainedexternal to this function. In-place computation (r = x) is allowed, but beaware that the r output vector is shorter in length than the x input vector; therefore,r will only overwrite the first nr elements of the x.Algorithmnh1r[j] h[k] x[j k]k00 j nrOverflow Handling MethodologyNo scaling implemented for overflow prevention.Special Requirementsnr must be an even value.The vector h[nh] must be located in internal memory since it is accessedusing the C55x coefficient bus, and that bus does not have access to externalmemory.The vector r[nr] must be 32-bit aligned in memory.Implementation Notes Figure 4–10, Figure 4–11, and Figure 4–12 show the x, r, and h arrays as theyappear in memory.4-28

convol2Figure 4–10. x Array in Memoryx(0)x(1)•x(nr+nh–2)x(nr+nh–1)lowest memory addresshighest memory addressFigure 4–11.r Array in Memoryr(0)r(1)•r(nr–2)r(nr–1)lowest memory addresshighest memory addressFigure 4–12. h Array in Memoryh(0)h(1)•h(nh–2)h(nh–1)lowest memory addresshighest memory addressExampleBenchmarksSee examples/convol2 subdirectory(preliminary)Cycles † Core: nr/2 * (1 + nh)Overhead: 24Code size(in bytes)100† Assumes all data is in on-chip dual-access RAM and that there is no bus conflict due to twiddletable reads and instruction fetches (provided linker command file reflects those conditions).Function Descriptions4-29

corrcorrFunctionCorrelation, full-lengthushort oflag = corr (DATA *x, DATA *y, DATA *r, ushort nx, ushort ny, type)Argumentsx [nx]x [ny]r[nx+ny–1]nxnytypeoflagPointer to real input vector of nx real elements.Pointer to real input vector of ny real elements.Pointer to real output vector containing the full-lengthcorrelation (nx + ny – 1 elements) of vector x with y. rmust be different than both x and y (in-placecomputation is not allowed).Number of real elements in vector xNumber of real elements in vector yCorrelation type selector. Types supported: If type = raw, r contains the raw correlation If type = bias, r contains the biased-correlation If type = unbias, r contains the unbiased-correlationOverflow flag If oflag = 1, a 32-bit overflow has occurred If oflag = 0, a 32-bit overflow has not occurredDescriptionAlgorithmComputes the full-length correlation of vectors x and y and stores the resultin vector r. using time-domain techniques.Raw correlationnrj1r[j] x[j k]*y[k] 0 j nr nx ny 1koBiased correlationnrj1r[j] 1nr x[j k]*y[k] 0 j nr nx ny 1koUnbiased correlationnrj1r[j] 1(nx abs(j)) x[j k]*y[k] 0 j nr nx ny 1koOverflow Handling MethodologyNo scaling implemented for overflow preventionSpecial Requirements none4-30

dlmsImplementation NotesSpecial debugging consideration: This function is implemented as amacro that invokes different correlation routines according to the typeselected. As a consequence the corr symbol is not defined. Instead thecorr_raw, corr_bias, corr_unbias symbols are defined.Correlation is implemented using time-domain techniquesBenchmarks(preliminary)Cycles Raw: 2 times faster than C54xUnbias: 2.14 times faster than C54xBias: 2.1 times faster than C54xCode size(in bytes)Raw: 318Unbias: 417Bias: 356dlmsFunctionAdaptive Delayed LMS Filterushort oflag = dlms (DATA *x, DATA *h, DATA *r, DATA *des, DATA *dbuffer,DATA step, ushort nh, ushort nx)(defined in dlms.asm)Argumentsx[nx]h[nh]r[nx]des[nx]dbuffer[nh+2]Pointer to input vector of size nxPointer to filter coefficient vector of size nh. h is stored in reversed order : h(n–1), ... h(0) whereh[n] is at the lowest memory address.Memory alignment: h is a circular buffer and must startin a k-bit boundary(that is, the k LSBs of the startingaddress must be zeros) where k = log2(nh)Pointer to output data vector of size nx. r can be equalto x.Pointer to expected output arrayPointer to the delay buffer structure.The delay buffer is a structure comprised of an indexregister and a circular buffer of length nh + 1. The indexregister is the index into the circular buffer of the oldestdata sample.Function Descriptions4-31

dlmsnh Number of filter coefficients. Filter order = nh – 1.nh ≥ 3nxoflagLength of input and output data vectorsOverflow flag. If oflag = 1, a 32-bit overflow has occurred If oflag = 0, a 32-bit overflow has not occurredDescriptionAdaptive delayed least-mean-square (LMS) FIR filter using coefficients storedin vector h. Coefficients are updated after each sample based on the LMSalgorithm and using a constant step = 2*mu. The real data input is stored invector dbuffer. The filter output result is stored in vector r .LMS algorithm uses the previous error and the previous sample (delayed) totake advantage of the C55x LMS instruction.The delay buffer used is the same delay buffer used for other functions in theC55x DSP Library. There is one more data location in the circular delay bufferthan there are coefficients. Other C55x DSP Library functions use this delaybuffer to accommodate use of the dual-MAC architecture. In the DLMS function,we make use of the additional delay slot to allow coefficient updating aswell as FIR calculation without a need to update the circular buffer in the interimoperations.The FIR output calculation is based on x(i) through x(i–nh+1). The coefficientupdate for a delayed LMS is based on x(i–1) through x(i–nh). Therefore, byhaving a delay buffer of nh+1, we can perform all calculations with the givendelay buffer containing delay values of x(i) through x(i–nh). If the delay bufferwas of length nh, the oldest data sample, x(i–nh), would need to be updatedwith the newest data sample, x(i), sometime after the calculation of the first coefficientupdate term, but before the calculation of the last FIR term.AlgorithmFIR portionnh1r[j] h[k]* x[i k] 0 i nx 1k0Adaptation using the previous error and the previous sample:e(i) des(i 1) r (i 1)h k (i 1) h k (i) 2*mu * e(i 1) * x(i k 1)Overflow Handling MethodologyNo scaling implemented for overflow prevention.Special Requirements Minimum of 2 input and desired data samples. Minimum of 2 coefficients4-32

expnImplementation NotesDelayed version implemented to take advantage of the C55x LMS instruction.Effect of using delayed error signal on convergence minimum:For reference, following is the algorithm for the regular LMS (nondelayed):FIR portionnh1r[j] h[k]* x[i k] 0 i nx 1k0Adaptation using the current error and the current sample:e(i) des(i) r (i)h k (i 1) h k (i) 2*mu * e(i)*x(i k)ExampleBenchmarksSee examples/dlms subdirectory(preliminary)Cycles † Core: nx * (7 + 2*(nh – 1)) = nx * (5 + 2 * nh)Overhead: 26Code size(in bytes)122† Assumes all data is in on-chip dual-access RAM and that there is no bus conflict due to twiddletable reads and instruction fetches (provided linker command file reflects those conditions).expnFunctionExponential Base eushort oflag = expn (DATA *x, DATA *r, ushort nx)(defined in expn.asm)Argumentsx[nx]r[nx]nxoflagPointer to input vector of size nx. x contains the numbersnormalized between (–1,1) in q15 formatPointer to output data vector (Q3.12 format) of size nx. r canbe equal to x.Length of input and output data vectorsOverflow flag. If oflag = 1, a 32-bit overflow has occurred If oflag = 0, a 32-bit overflow has not occurredFunction Descriptions4-33

firDescriptionComputes the exponent of elements of vector x using Taylor series.Algorithm for (i 0; i nx; i ) y(i) e x(i) where –1 x(i) 1Overflow Handling MethodologyNot applicableSpecial Requirements Linker command file: you must allocate .data section (for polynomial coefficients)Implementation Notes Computes the exponent of elements of vector x. It uses the following Taylorseries:exp(x) c0 (c1*x) (c2*x 2 ) (c3*x 3 ) (c4*x 4 ) (c5*x 5 )wherec0 = 1.0000c1 = 1.0001c2 = 0.4990c3 = 0.1705c4 = 0.0348c5 = 0.0139ExampleBenchmarksSee examples/expn subdirectory(preliminary)Cycles † Core: 11 * nxOverhead: 18Code size(in bytes)57† Assumes all data is in on-chip dual-access RAM and that there is no bus conflict due to twiddletable reads and instruction fetches (provided linker command file reflects those conditions).firFunctionArgumentsFIR Filterushort oflag = fir (DATA *x, DATA *h, DATA *r, DATA *dbuffer, ushort nx,ushort nh)x[nx]Pointer to input vector of nx real elements.h[nh] Pointer to coefficient vector of size nh in normal order.For example, if nh=6, then h[nh] = {h0, h1, h2, h3, h4,h5} where h0 resides at the lowest memory addressin the array.This array must be located in internal memory sinceit is accessed by the C55x coefficient bus.4-34

firr[nx]Pointer to output vector of nx real elements. In-placecomputation (r = x) is allowed.dbuffer[nh+2] Pointer to delay buffer of length nh = nh + 2 In the case of multiple-buffering schemes, this arrayshould be initialized to 0 for the first filter block only.Between consecutive blocks, the delay buffer preservesthe previous r output elements needed.The first element in this array is special in that it containsthe array index–1 of the oldest input entry in thedelay buffer. This is needed for multiple-bufferingschemes, and should be initialized to 0 (like all the otherarray entries) for the first block only.nxnhoflagNumber of input samplesThe number of coefficients of the filter. For example, ifthe filter coefficients are {h0, h1, h2, h3, h4, h5}, then nh= 6. Must be a minimum value of 3. For smaller filters,zero pad the coefficients to meet the minimum value.Overflow error flag (returned value) If oflag = 1, a 32-bit data overflow occurred in an intermediateor final result.If oflag = 0, a 32-bit overflow has not occurred.DescriptionComputes a real FIR filter (direct-form) using the coefficients stored in vectorh. The real input data is stored in vector x. The filter output result is stored invector r. This function maintains the array dbuffer containing the previousdelayed input values to allow consecutive processing of input data blocks. Thisfunction can be used for both block-by-block (nx ≥ 2) and sample-by-samplefiltering (nx = 1). In place computation (r = x) is allowed.Algorithmnh1r[j] h[k] x[j k]k00 j nxOverflow Handling MethodologyNo scaling implemented for overflow prevention.Special Requirements nh must be a minimum value of 3. For smaller filters, zero pad the h[] array.Function Descriptions4-35

firImplementation Notes The first element in the dbuffer array (index = 0) is the entry index for the inputhistory. It is treated as an unsigned 16-bit value by the function even thoughit has been declared as signed in C. The value of the entry index is equal tothe index – 1 of the oldest input entry in the array. The remaining elementsmake up the input history. Figure 4–13 shows the array in memory with anentry index of 2. The newest entry in the dbuffer is denoted by x(j–0), whichin this case would occupy index = 3 in the array. The next newest entry isx(j–1), and so on. It is assumed that all x() entries were placed into the arrayby the previous invocation of the function in a multiple-buffering scheme.The dbuffer array actually contains one more history value than is needed toimplement this filter. The value x(j–nh) does not enter into the calculations forfor the output r(j). However, this value is required in other DSPLIB filter functionsthat utilize the dual-MAC units on the C55x, such as FIR2. Including thisextra location ensures compatibility across all filter functions in the C55xDSPLIB.Figure 4–13, Figure 4–14, and Figure 4–15 show the dbuffer, x, and r arraysas they appear in memory.Figure 4–13. dbuffer Array in Memory at Time joldest x( ) entrynewest x( ) entryentry index = 2x(j–nh–2)x(j–nh–1)x(j–nh–1)x(j–nh)x(j–0)x(j–1)x(j–2)•x(j–nh–5)x(j–nh–4)x(j–nh–3)lowest memory addresshighest memory address4-36

fir2Figure 4–14. x Array in Memoryoldest x( ) entrynewest x( ) entryx(0)x(1)•x(nx–2)x(nx–1)lowest memory addresshighest memory addressFigure 4–15. r Array in Memoryoldest x( ) entrynewest x( ) entryr(0)r(1)•r(nx–2)r(nx–1)lowest memory addresshighest memory addressExampleBenchmarksSee examples/fir subdirectory(preliminary)Cycles † Core: nx * (2 + nh)Overhead: 25Code size(in bytes)107† Assumes all data is in on-chip dual-access RAM and that there is no bus conflict due to twiddletable reads and instruction fetches (provided linker command file reflects those conditions).fir2FunctionBlock FIR Filter (fast)ushort oflag = fir2 (DATA *x, DATA *h, DATA *r, DATA *dbuffer, ushort nx,ushort nh)Argumentsx[nx]r[nx]Pointer to input vector of nx real elements.Pointer to output vector of nx real elements. In-placecomputation (r = x) is allowed.Function Descriptions4-37

fir2h[nh] Pointer to coefficient vector of size nh in normal order.For example, if nh=6, then h[nh] = {h0, h1, h2, h3, h4,h5} where h0 resides at the lowest memory addressin the array.This array must be located in internal memory sinceit is accessed by the C55x coefficient bus.dbuffer[nh + 2] Pointer to delay buffer of length nh = nh + 2 In the case of multiple-buffering schemes, this arrayshould be initialized to 0 for the first filter block only.Between consecutive blocks, the delay buffer preservesthe previous r output elements needed.The first element in this array is special in that it containsthe array index–1 of the oldest input entry in thedelay buffer. This is needed for multiple-bufferingschemes, and should be initialized to 0 (like all the otherarray entries) for the first block only.nxnhoflagNumber of input samples. Must be an even number.The number of coefficients of the filter. For example, ifthe filter coefficients are {h0, h1, h2, h3, h4, h5}, then nh= 6. Must be a minimum value of 3. For smaller filters,zero pad the coefficients to meet the minimum value.Overflow error flag (returned value) If oflag = 1, a 32-bit data overflow occurred in an intermediateor final result.If oflag = 0, a 32-bit overflow has not occurred.DescriptionComputes a real block FIR filter (direct-form) using the coefficients stored invector h. This function utilizes the dual-MAC capability of the C55x to processin parallel two output samples for each iteration of the inner function loop. Itis, therefore, roughly twice as fast as FIR, which implements only a single-MAC approach. However, the number of input samples (nx) must be even Thereal input data is stored in vector x. The filter output result is stored in vectorr. This function maintains the array dbuffer containing the previous delayed inputvalues to allow consecutive processing of input data blocks. This functioncan be used for block-by-block filtering only (nx ≥ 2). It cannot be used for sample-by-samplefiltering (nx = 1). In-place computation (r = x) is allowed.nh1Algorithmr[j] h[k] x[j k] 0 j nxk0Overflow Handling Methodology No scaling implemented for overflow prevention.4-38

fir2Special Requirementsnh must be a minimum value of 3. For smaller filters, zero pad the h[] array.nx must be an even value.Coefficient array h[nh] must be located in internal memory since it is accessedusing the C55x coefficient bus, and that bus does not have accessto external memory.Implementation Notes The first element in the dbuffer array (index = 0) is the entry index for the inputhistory. It is treated as an unsigned 16-bit value by the function even thoughit has been declared as signed in C. The value of the entry index is equal tothe index – 1 of the oldest input entry in the array. The remaining elementsmake up the input history. Figure 4–16 shows the array in memory with anentry index of 2. The newest entry in the dbuffer is denoted by x(j–0), whichin this case would occupy index = 3 in the array. The next newest entry isx(j–1), and so on. It is assumed that all x() entries were placed into the arrayby the previous invocation of the function in a multiple-buffering scheme.Figure 4–16, Figure 4–17, and Figure 4–18 show the dbuffer, x, and r arraysas they appear in memory.Figure 4–16. dbuffer Array in Memory at Time joldest x( ) entrynewest x( ) entryentry index = 2x(j–2*nh2–2) x(j–2*nh2–1)x(j–2*nh2) x(j–0)x(j–1)x(j–2)•x(j–2*nh2–5x(j–2*nh2–4) )x(j–2*nh2–3)lowest memory addresshighest memory addressFunction Descriptions4-39

firdecFigure 4–17. x Array in Memoryoldest x( ) entrynewest x( ) entryx(0)x(1)•x(nx–2)x(nx–1)lowest memory addresshighest memory addressFigure 4–18. r Array in Memoryoldest x( ) entrynewest x( ) entryr(0)r(1)•r(nx–2)r(nx–1)lowest memory addresshighest memory addressExampleBenchmarksSee examples/fir2 subdirectory(preliminary)Cycles † Core: (nx/2) * [5+(nh–3)]Overhead: 32Code size(in bytes)134† Assumes all data is in on-chip dual-access RAM and that there is no bus conflict due to twiddletable reads and instruction fetches (provided linker command file reflects those conditions).firdecFunctionDecimating FIR Filterushort oflag = firdec (DATA *x, DATA *h, DATA *r, DATA *dbuffer , ushort nh,ushort nx, ushort D)(defined in decimate.asm)Argumentsx [nx]h[nh]Pointer to real input vector of nx real elements.Pointer to coefficient vector of size nh in normal order:H = b0 b1 b2 b3 …4-40

firdecr[nx/D]dbuffer[nh+1]nxnhDoflagPointer to real input vector of nx/D real elements.In-place computation (r = x) is allowedDelay buffer In the case of multiple-buffering schemes, this arrayshould be initialized to 0 for the first block only. Betweenconsecutive blocks, the delay buffer preservesprevious delayed input samples. It also preserves aptr to the next new entry into the dbuffer. This ptr ispreserved across function calls in dbuffer[0].Memory alignment: this is a circular buffer and muststart in a k-bit boundary(that is, the k LSBs of the startingaddress must be zeros) where k = log2 (nh).Number of real elements in vector xNumber of coefficientsDecimation factor. For example a D = 2 means you dropevery other sample. Ideally, nx should be a multiple ofD. If not, the trailing samples will be lost in the process.Overflow error flag If oflag = 1, a 32-bit data overflow occurred in an intermediateor final result.If oflag = 0, a 32-bit overflow has not occurred.DescriptionComputes a decimating real FIR filter (direct-form) using coefficient stored invector h. The real data input is stored in vector x. The filter output result isstored in vector r. This function retains the address of the delay filter memoryd containing the previous delayed values to allow consecutive processing ofblocks. This function can be used for both block-by-block and sample-bysamplefiltering (nx = 1).Algorithmr[j] nhk0h[k]x[j * D k]0 j nxOverflow Handling MethodologyNo scaling implemented for overflow prevention.Special Requirements noneImplementation Notes noneExampleSee examples/decim subdirectoryFunction Descriptions4-41

firinterpBenchmarks(preliminary)Cycles Core: (nx/D)*(10+nh+(D–1))Overhead 67Code size(in bytes)144firinterpInterpolating FIR FilterFunctionushort oflag = firinterp (DATA *x, DATA *h, DATA *r, DATA *dbuffer , ushort nh,ushort nx, ushort I)(defined in interp.asm)Argumentsx [nx]h[nh]r[nx*I]dbuffer[(nh/I)+1]nxPointer to real input vector of nx real elements.Pointer to coefficient vector of size nh in normalorder:H = b0 b1 b2 b3 …Pointer to real output vector of nx real elements.In-place computation (r = x) is allowedDelay buffer of (nh/I)+1 elements In the case of multiple-buffering schemes, thisarray should be initialized to 0 for the first blockonly. Between consecutive blocks, the delay bufferpreserves delayed input samples in dbuffer[1…(nh/I)+1].It also preserves a ptr to the nextnew entry into the dbuffer. This ptr is preservedacross function calls in dbuffer[0].The delay buffer is only nh/I elements and holdsonly delayed x inputs. No zero-samples are insertedinto dbuffer (since only non-zero productscontribute to the filter output)Number of real elements in vector x and rnh Number of coefficients, with (nh/I) 34-42

firinterpIoflagInterpolation factor. I is effectively the number ofoutput samples for every input sample. This routinecan be used with I=1.Overflow error flag If oflag = 1, a 32-bit data overflow occurred in anintermediate or final result.If oflag = 0, a 32-bit overflow has not occurred.DescriptionComputes an interpolating real FIR filter (direct-form) using coefficient storedin vector h. The real data input is stored in vector x. The filter output result isstored in vector r. This function retains the address of the delay filter memoryd containing the previous delayed values to allow consecutive processing ofblocks. This function can be used for both block-by-block and sample-bysamplefiltering (nx = 1).Algorithmr[t] nhk0h[k]x[tI k]0 j nrOverflow Handling MethodologyNo scaling implemented for overflow prevention.Special Requirements noneImplementation Notes noneExampleBenchmarksSee examples/decimate subdirectory(preliminary)CyclesCore:If I > 1nx*(2+I*(1+(nh/I)))If I=1 :nx*(2+nh)Overhead 72Code size(in bytes)164Function Descriptions4-43

firlatfirlatFunctionLattice Forward (FIR) Filterushort oflag = firlat (DATA *x, DATA *h, DATA *r, DATA *pbuffer, int nx, int nh)Argumentsx [nx]h[nh]r[nx]Pointer to real input vector of nx real elements in normalorder:x[0]x[1]..x[nx–2]x[nx–1]Pointer to lattice coefficient vector of size nh in normalorder:h[0]h[1]..h[nh–2]h[nh–1]Pointer to output vector of nx real elements. In-placecomputation (r = x) is allowed.r[0]r[1]..r[nx–2]r[nx–1]4-44

firlatpbuffer [nh]nxnhoflagDelay bufferIn the case of multiple-buffering schemes, this arrayshould be initialized to 0 for the first block only. Betweenconsecutive blocks, the delay buffer preserves the previousr output elements needed.pbuffer: procession buffer of nh length in order:e′0[n–1]e′1[n–1]..e′nh–2[n–1]e′nh–1[n–1]Number of real elements in vector x (input samples)Number of coefficientsOverflow error flag If oflag = 1, a 32-bit data overflow has occurred in anintermediate or final resultIf oflag = 0, a 32-bit overflow has not occurred.DescriptionComputes a real lattice FIR filter implementation using coefficient stored invector h. The real data input is stored in vector x. The filter output result isstored in vector r. This function retains the address of the delay filter memoryd containing the previous delayed values to allow consecutive processing ofblocks. This function can be used for both block-by-block and sample-bysamplefiltering (nx=1)Algorithme 0 [n] e 0 [n] x[n],e i [n] e i1 [n] hie i1 [n 1],e i [n] h i e i1 [n] e i1 [n 1],y[n] e N [n]i 1, 2, , Ni 1, 2, , NOverflow Handling MethodologyNo scaling implemented for overflow prevention.Special Requirements noneImplementation Notes noneExampleSee examples/firlat subdirectoryFunction Descriptions4-45

firsBenchmarks(preliminary)Cycles † Core: nx{4 + 4(nh–1)]Overhead: 23Code size(in bytes)53† Assumes all data is in on-chip dual-access RAM and that there is no bus conflict due to twiddletable reads and instruction fetches (provided linker command file reflects those conditions).firsFunctionSymmetric FIR Filterushort oflag = firs (DATA *x, DATA *h, DATA *r, DATA *dbuffer, ushort nx,ushort nh2)Argumentsx[nx]Pointer to input vector of nx real elements.r[nx]Pointer to output vector of nx real elements.In-place computation (r = x) is allowed.h[nh2] Pointer to coefficient vector containing the firsthalf of the symmetric filter coefficients. For example,if the filter coefficients are {h0, h1, h2, h2, h1,h0}, then h[nh2] = {h0, h1, h2} where h0 residesat the lowest memory address in the array.This array must be located in internal memorysince it is accessed by the C55x coefficient bus.dbuffer[2*nh2 + 2] Pointer to delay buffer of length nh = 2*nh2 + 2 In the case of multiple-buffering schemes, thisarray should be initialized to 0 for the first filterblock only. Between consecutive blocks, the delaybuffer preserves the previous r output elementsneeded.nxThe first element in this array is special in that itcontains the array index of the oldest input entryin the delay buffer. This is needed for multiplebufferingschemes, and should be initialized to 0(like all the other array entries) for the first blockonly.Number of input samples4-46

firsnh2oflagHalf the number of coefficients of the filter (due tosymmetry there is no need to provide the otherhalf). For example, if the filter coefficients are {h0,h1, h2, h2, h1, h0}, then nh2 = 3. Must be aminimum value of 3. For smaller filters, zero pad thecoefficients to meet the minimum value.Overflow error flag (returned value) If oflag = 1, a 32-bit data overflow occurred in anintermediate or final result.If oflag = 0, a 32-bit overflow has not occurred.DescriptionComputes a real FIR filter (direct-form) with nh2 symmetric coefficients usingthe FIRS instruction approach. The filter is assumed to have a symmetric impulseresponse, with the first half of the filter coefficients stored in the array h.The real input data is stored in vector x. The filter output result is stored in vectorr. This function maintains the array dbuffer containing the previous delayedinput values to allow consecutive processing of input data blocks. This functioncan be used for both block-by-block (nx ≥ 2) and sample-by-sample filtering(nx = 1). In-place computation (r = x) is allowed.Algorithmnh21r[j] h,...,[k]* (x[j k] x[j k 2*nh2 1] )k00 j nxOverflow Handling MethodologyNo scaling implemented for overflow prevention.Special Requirementsnh must be a minimum value of 3. For smaller filters, zero pad the h[] array.Coefficient array h[nh2] must be located in internal memory since it is accessedusing the C55x coefficient bus, and that bus does not have accessto external memory.Implementation Notes The first element in the dbuffer array (index = 0) is the entry index for the inputhistory. It is treated as an unsigned 16-bit value by the function even thoughit has been declared as signed in C. The value of the entry index is equal tothe index – 1 of the oldest input entry in the array. The remaining elementsmake up the input history. Figure 4–19 shows the array in memory with anentry index of 2. The newest entry in the dbuffer is denoted by x(j–0), whichin this case would occupy index = 3 in the array. The next newest entry isx(j–1), and so on. It is assumed that all x() entries were placed into the arrayby the previous invocation of the function in a multiple-buffering scheme.The dbuffer array actually contains one more history value than is needed toimplement this filter. The value x(j–2*nh2) does not enter into the calculationsFunction Descriptions4-47

firsfor for the output r(j). However, this value is required in other DSPLIB filter functionsthat utilize the dual-MAC units on the C55x, such as FIR2. Including thisextra location ensures compatibility across all filter functions in the C55xDSPLIB.Figure 4–19, Figure 4–20, and Figure 4–21 show the dbuffer, x, and r arraysas they appear in memory.Figure 4–19. dbuffer Array in Memory at Time joldest x( ) entrynewest x( ) entryentry index = 2x(j–2*nh2–2) x(j–2*nh2–1)x(j–2*nh2) x(j–0)x(j–1)x(j–2)•x(j–2*nh2–5x(j–2*nh2–4) )x(j–2*nh2–3)lowest memory addresshighest memory addressFigure 4–20. x Array in Memoryoldest x( ) entrynewest x( ) entryx(0)x(1)•x(nx–2)x(nx–1)lowest memory addresshighest memory address4-48

fltoq15Figure 4–21. r Array in Memoryoldest x( ) entrynewest x( ) entryr(0)r(1)•r(nx–2)r(nx–1)lowest memory addresshighest memory addressExampleBenchmarksSee examples/firs subdirectory(preliminary)Cycles † Core: nx[5 + (nh–2)]Overhead: 72Code size(in bytes)133† Assumes all data is in on-chip dual-access RAM and that there is no bus conflict due to twiddletable reads and instruction fetches (provided linker command file reflects those conditions).fltoq15FunctionFloating-point to Q15 Conversionushort errorcode = fltoq15 (float *x, DATA *r, ushort nx)(defined in fltoq15.asm)Argumentsx[nx]r[nx]nxerrorcodePointer to floating-point input vector of size nx. x shouldcontain the numbers normalized between (–1,1). Theerrorcode returned value will reflect if that condition is notmet.Pointer to output data vector of size nx containing the q15equivalent of vector x.Length of input and output data vectorsThe function returns the following error codes:1 – if any element is too large to represent in Q15 format2 – if any element is too small to represent in Q15 format3 – both conditions 1 and 2 were encounteredDescriptionConvert the IEEE floating-point numbers stored in vector x into Q15 numbersstored in vector r. The function returns the error codes if any element x[i] is notrepresentable in Q15 format.Function Descriptions4-49

hilb16All values that exceed the size limit will be saturated to a Q15 1 or –1 dependingon sign (0x7fff if value is positive, 0x8000 if value is negative). All valuestoo small to be correctly represented will be truncated to 0.AlgorithmNot applicableOverflow Handling MethodologySaturation implemented for overflow handlingSpecial Requirements noneImplementation Notes noneExampleBenchmarksSee examples/expn subdirectory(preliminary)Cycles † Core: 17 * nx (if x[n] ==0)23 * nx (if x[n] is too small for Q15representation)32 * nx (if x[n] is too large for Q15representation)38 * nx (otherwise)Overhead: 23Code size(in bytes)157† Assumes all data is in on-chip dual-access RAM and that there is no bus conflict due to twiddletable reads and instruction fetches (provided linker command file reflects those conditions).hilb16FunctionFIR Hilbert Transformerushort oflag = hilb16 (DATA *x, DATA *h, DATA *r, DATA *dbuffer, ushort nx,ushort nh)Argumentsx[nx]Pointer to input vector of nx real elements.h[nh] Pointer to coefficient vector of size nh in normalorder. H= {h0, h1, h2, h3, h4, …} Every odd valuedfilter coefficient has to 0, i.e. h1 = h3 = … = 0. AndH = {h0, 0, h2, 0, h4, 0, …} where h0 resides at thelowest memory address in the array.This array must be located in internal memorysince it is accessed by the C55x coefficient bus.r[nx]Pointer to output vector of nx real elements.In-place computation (r = x) is allowed.4-50

hilb16dbuffer[nh + 2] Pointer to delay buffer of length nh = nh + 2In the case of multiple-buffering schemes, thisarray should be initialized to 0 for the first filterblock only. Between consecutive blocks, the delaybuffer preserves the previous r output elementsneeded.The first element in this array is special in that itcontains the array index-1 of the oldest input entryin the delay buffer. This is needed for multiplebufferingschemes, and should be initialized tozero (like all the other array entries) for the firstblock only.nxnhoflagNumber of real elements in vector x (input samples)The number of coefficients of the filter. For exampleif the filter coefficients are {h0, h1, h2, h3, h4, h5},then nh = 6. Must be a minimum value of 6. Forsmaller filters, zero pad the coefficients to meet theminimum value.Overflow error flag (returned value) If oflag = 1, a 32-bit data overflow occurred in anintermediate or final result.If oflag = 0, a 32-bit overflow has not occurred.DescriptionComputes a real FIR filter (direct-form) using the coefficients stored in vectorh. The real input data is stored in vector x. The filter output result is stored invector r. This function maintains the array dbuffer containing the previousdelayed input values to allow consecutive processing of input data blocks. Thisfunction can be used for both block-by-block (nx >= 2) and sample-by-samplefiltering (nx = 1). In place computation (r = x) is allowed.Algorithmnh1r[j] h[k]x[j k]k00 j nxOverflow Handling MethodologyNo scaling implemented for overflow prevention.Special RequirementsEvery odd valued filter coefficient has to be 0. This is a requirement for theHilbert transformer. For example, a 6 tap filter may look like this: H = [0.8670 –0.324 0 –0.002 0]Always pad 0 to make nh as a even number. For example, a 5 tap filter witha zero pad may look like this: H = [0.867 0 –0.324 0 –0.002 0]Function Descriptions4-51

hilb16nh must be a minimum value of 6. For smaller filters, zero pad the H[] array.Implementation Notes The first element in the dbuffer array (index = 0) is the entry index for the inputhistory. It is treated as an unsigned 16-bit value by the function even thoughit has been declared as signed in C. The value of the entry index is equal tothe index – 1 of the oldest input entry in the array. The remaining elementsmake up the input history. Figure 4–22 shows the array in memory with anentry index of 2. The newest entry in the dbuffer is denoted by x(j–0), whichin this case would occupy index = 3 in the array. The next newest entry isx(j–1), and so on. It is assumed that all x() entries were placed into the arrayby the previous invocation of the function in a multiple-buffering scheme.The dbuffer array actually contains one more history value than is needed toimplement this filter. The value x(j–nh) does not enter into the calculations forfor the output r(j). However, this value is required in other DSPLIB filter functionsthat utilize the dual-MAC units on the C55x, such as FIR2. Including thisextra location ensures compatibility across all filter functions in the C55xDSPLIB.Figure 4–22, Figure 4–23, and Figure 4–24 show the dbuffer, x, and r arraysas they appear in memory.Figure 4–22. dbuffer Array in Memory at Time joldest x( ) entrynewest x( ) entryentry index = 2x(j–nh–2)x(j–nh–1)x(j–nh)x(j–0)x(j–1)x(j–2)•x(j–nh–5)x(j–nh–4)x(j–nh–3)lowest memory addresshighest memory address4-52

iir32Figure 4–23. x Array in Memoryoldest x( ) entrynewest x( ) entryx(0)x(1)•x(nx–2)x(nx–1)lowest memory addresshighest memory addressFigure 4–24. r Array in Memoryoldest x( ) entrynewest x( ) entryr(0)r(1)•r(nx–2)r(nx–1)lowest memory addresshighest memory addressExampleBenchmarksSee examples/hilb16 subdirectory(preliminary)Cycles Core: nx*(2+nh/2)Overhead: 28Code size(in bytes)108iir32FunctionDouble-precision IIR Filterushort oflag = iir32 (DATA *x, LDATA *h, DATA *r, LDATA *dbuffer, ushort nbiq,ushort nr)(defined in iir32.asm)Argumentsx [nr]h[5*nbiq]Pointer to input data vector of size nrPointer to the 32-bit filter coefficient vector with thefollowing format. For example for nbiq= 2, h is equalto:Function Descriptions4-53

iir32b21 – highb21 – lowb11 – highb11 – lowb01 – highb01 – lowa21 – higha21 – lowa11 – higha11 – lowb22 – highb22 – lowb12 – highb12 – lowb02 – highb02 – lowa22 – higha22 – lowa12 – higha12 – lowbeginning of biquad 1beginning of biquad 2 coefsr[nr]dbuffer[2*nbiq+2]Pointer to output data vector of size nr. r can beequal or less than x.Pointer to address of 32-bit delay line dbuffer. Eachbiquad has 3 consecutive delay line elements. Forexample for nbiq = 2:d1(n–2) – lowd1(n–2) – highd1(n–1) – lowd1(n–1) – highd2(n–2) – lowd2(n–2) – highd2(n–1) – lowd2(n–1) – highbeginning of biquad 1beginning of biquad 24-54

iir32In the case of multiple-buffering schemes, thisarray should be initialized to 0 for the first blockonly. Between consecutive blocks, the delay bufferpreserves the previous r output elements needed.nbiqnroflagMemory alignment: none required for C5510. Thisis a group of circular buffers. Each biquad’s delaybuffer is treated separately. The Buffer Start Address(BSAxx) updated to a new location for eachbiquad.Number of biquadsNumber of elements of input and output vectorsOverflow flag. If oflag = 1, a 32-bit overflow has occurred If oflag = 0, a 32-bit overflow has not occurredDescriptionComputes a cascaded IIR filter of nbiquad biquad sections using 32-bit coefficientsand 32-bit delay buffers. The input data is assumed to be single-precision(16 bits).Each biquad section is implemented using Direct-form II. All biquad coefficients(5 per biquad) are stored in vector h. The real data input is stored in vectorx. The filter output result is stored in vector r .This function retains the address of the delay filter memory d containing theprevious delayed values to allow consecutive processing of blocks. This functionis more efficient for block-by-block filter implementation due to the C-callingoverhead. However, it can be used for sample-by-sample filtering (nx = 1).Algorithm(for biquad)d(n) x(n) a1*d(n 1) a2*d(n 2)y(n) b0*d(n) b1*d(n 1) b2*d(n 2)Overflow Handling MethodologyNo scaling implemented for overflow prevention.Special Requirements noneImplementation Notes See program iircas32.asmExampleSee examples/iir32 subdirectoryFunction Descriptions4-55

iircas4Benchmarks(preliminary)Cycles Core: nx*(7+ 31*nbiq)Overhead: 77Code size(in bytes)203iircas4FunctionCascaded IIR Direct Form II Using 4 Coefficients per Biquadushort oflag = iircas4 (DATA *x, DATA *h, DATA *r, DATA *dbuffer, ushort nbiq,ushort nx)(defined in iir4cas4.asm)Argumentsx [nx]Pointer to input data vector of size nxh[4*nbiq]r[nx]Pointer to filter coefficient vector with the followingformat:h = a11 a21 b21 b11 ....a1i a2i b2i b1iwhere i is the biquad index (a21 is the a2 coefficient ofbiquad 1). Pole (recursive) coefficients = a. Zero(non-recursive) coefficients = bPointer to output data vector of size nx. r can be equalthan x.dbuffer[2*nbiq] Pointer to address of delay line d.Each biquad has 2 delay line elements separated bynbiq locations in the following format:d1(n–1), d2(n–1),..di(n–1) d1(n–2), d2(n–2)...di(n–2)where i is the biquad index (d2(n–1) is the (n–1)thdelay element for biquad 2).nbiqIn the case of multiple-buffering schemes, this arrayshould be initialized to 0 for the first block only. Betweenconsecutive blocks, the delay buffer preservesthe previous r output elements needed.Memory alignment: this is a circular buffer and muststart in a k-bit boundary(that is, the k LSBs of thestarting address must be zeros) where k = log2(2*nbiq).Number of biquads4-56

iircas4nxoflagNumber of elements of input and output vectorsOverflow flag. If oflag = 1, a 32-bit overflow has occurred If oflag = 0, a 32-bit overflow has not occurredDescriptionComputes a cascade IIR filter of nbiq biquad sections. Each biquad section isimplemented using Direct-form II. All biquad coefficients (4 per biquad) arestored in vector h. The real data input is stored in vector x. The filter outputresult is stored in vector r.This function retains the address of the delay filter memory d containing theprevious delayed values to allow consecutive processing of blocks. This functionis more efficient for block-by-block filter implementation due to the C-callingoverhead. However, it can be used for sample-by-sample filtering (nx = 1).Algorithm(for biquad)d(n) x(n) a1*d(n 1) a2*d(n 2)y(n) d(n) b1*d(n 1) b2*d(n 2)Overflow Handling MethodologyNo scaling implemented for overflow prevention.Special Requirements noneImplementation Notes noneExampleBenchmarksSee examples/iircas4 subdirectory(preliminary)Cycles † Core: nx * (2 + 3 * nbiq)Overhead: 44Code size(in bytes)122† Assumes all data is in on-chip dual-access RAM and that there is no bus conflict due to twiddletable reads and instruction fetches (provided linker command file reflects those conditions).Function Descriptions4-57

iircas5iircas5FunctionArgumentsCascaded IIR Direct Form II (5 Coefficients per Biquad)ushort oflag = iircas5 (DATA *x, DATA *h, DATA *r, DATA *dbuffer, ushort nbiq,ushort nx)(defined in iircas5.asm)x [nx]Pointer to input data vector of size nxh[5*nbiq]r[nx]Pointer to filter coefficient vector with the followingformat:h = a11 a21 b21 b01 b11 ... a1i a2i b2i b0i b1iwhere i is the biquad index a21 is the a2 coefficient ofbiquad 1). Pole (recursive) coefficients = a. Zero(non-recursive) coefficients = bPointer to output data vector of size nx. r can be equalthan x.dbuffer[2*nbiq] Pointer to address of delay line d. Each biquad has 2delay line elements separated by nbiq locations in thefollowing format:d1(n–1), d2(n–1),..di(n–1) d1(n–2), d2(n–2)...di(n–2)where i is the biquad index(d2(n–1) is the (n–1)thdelay element for biquad 2).nbiqIn the case of multiple-buffering schemes, this arrayshould be initialized to 0 for the first block only. Betweenconsecutive blocks, the delay buffer preservesthe previous r output elements needed.Memory alignment: this is a circular buffer and muststart in a k-bit boundary(that is, the k LSBs of thestarting address must be zeros) where k = log2(2*nbiq).Number of biquadsnxoflagNumber of elements of input and output vectorsOverflow flag. If oflag = 1, a 32-bit overflow has occurred If oflag = 0, a 32-bit overflow has not occurredDescriptionComputes a cascade IIR filter of nbiq biquad sections. Each biquad section isimplemented using Direct-form II. All biquad coefficients (5 per biquad) arestored in vector h. The real data input is stored in vector x. The filter outputresult is stored in vector r.4-58

iircas51This function retains the address of the delay filter memory d containing theprevious delayed values to allow consecutive processing of blocks. This functionis more efficient for block-by-block filter implementation due to the C-callingoverhead. However, it can be used for sample-by-sample filtering (nx = 1).The usage of 5 coefficients instead of 4 facilitates the design of filters with aunit gain of less than 1 (for overflow avoidance), typically achieved by filtercoefficient scaling.Algorithm(for biquad)d(n) x(n) a1*d(n 1) a2*d(n 2)y(n) b0*d(n) b1*d(n 1) b2*d(n 2)Overflow Handling MethodologyNo scaling implemented for overflow prevention.Special Requirements noneImplementation Notes noneExampleBenchmarksSee examples/iircas5 subdirectory(preliminary)Cycles † Core: nx * (5 + 5 * nbiq)Overhead: 60Code size(in bytes)126† Assumes all data is in on-chip dual-access RAM and that there is no bus conflict due to twiddletable reads and instruction fetches (provided linker command file reflects those conditions).iircas51FunctionCascaded IIR Direct Form I (5 Coefficients per Biquad)ushort oflag = iircas51 (DATA *x, DATA *h, DATA *r, DATA *dbuffer, ushort nbiq,ushort nx)(defined in iircas51.asm)Argumentsx [nx]h[5*nbiq]Pointer to input data vector of size nxPointer to filter coefficient vector with the followingformat:h = b01 b11 b21 a11 a21 ....b0i b1i b2i a1i a2Iwhere i is the biquad index (a21 is the a2 coefficient ofbiquad 1). Pole (recursive) coefficients = a. Zero(non-recursive) coefficients = bFunction Descriptions4-59

iircas51r[nx]dbuffer[4*nbiq]nbiqnxoflagPointer to output data vector of size nx. r can be equalthan x.Pointer to address of delay line dbuffer. Each biquadhas 4 delay line elements stored consecutively inmemory in the following format:x1(n–1), x1(n–2), y1(n–1), y1(n–2) ... xi(n–2), xi(n–2),yi(n–1),yi(n–2)where i is the biquad index(x1(n–1) is the (n–1)thdelay element for biquad 1).In the case of multiple-buffering schemes, this arrayshould be initialized to 0 for the first block only. Betweenconsecutive blocks, the delay buffer preservesthe previous r output elements needed.Memory alignment: No need for memory alignment.Number of biquadsNumber of elements of input and output vectorsOverflow flag. If oflag = 1, a 32-bit overflow has occurred. If oflag = 0, a 32-bit overflow has not occurred.DescriptionComputes a cascade IIR filter of nbiq biquad sections. Each biquad section isimplemented using Direct-form I. All biquad coefficients (5 per biquad) arestored in vector h. The real data input is stored in vector x. The filter outputresult is stored in vector r.This function retains the address of the delay filter memory d containing theprevious delayed values to allow consecutive processing of blocks. This functionis more efficient for block-by-block filter implementation due to the C-callingoverhead. However, it can be used for sample-by-sample filtering (nx = 1).The usage of 5 coefficients instead of 4 facilitates the design of filters with aunit gain of less than 1 (for overflow avoidance), typically achieved by filtercoefficient scaling.Algorithm(for biquad)y(n) b0*x(n) b1*x(n 1) b2*x(n 2) a1*y(n 1) a2*y(n 2)Overflow Handling MethodologySpecial Requirements noneImplementation Notes noneExampleNo scaling implemented for overflow prevention.See examples/iircas51 subdirectory4-60

iirlatBenchmarks(preliminary)Cycles † Core: nx * (5 + 8 * nbiq)Overhead: 68Code size(in bytes)154† Assumes all data is in on-chip dual-access RAM and that there is no bus conflict due to twiddletable reads and instruction fetches (provided linker command file reflects those conditions).iirlatFunctionLattice Inverse (IIR) Filterushort oflag = iirlat (DATA *x, DATA *h, DATA *r, DATA *pbuffer, int nx, int nh)Argumentsx [nx]h[nh]r[nx]Pointer to real input vector of nx real elements in normalorder:x[0]x[1]..x[nx–2]x[nx–1]Pointer to lattice coefficient vector of size nh in normalorder with the first element zero-padded:0h[0]h[1]..h[nh–2]h[nh–1]Pointer to output vector of nx real elements. In-placecomputation (r = x) is allowed.r[0]r[1]..r[nx–2]r[nx–1]Function Descriptions4-61

iirlatpbuffer[nh]nxnhoflagDelay bufferIn the case of multiple-buffering schemes, this array shouldbe initialized to 0 for the first block only. Betweenconsecutive blocks, the delay buffer preserves theprevious r output elements needed.Number of real elements in vector x (input samples)Number of coefficientsOverflow error flag If oflag = 1, a 32-bit data overflow has occurred in anintermediate or final result.If oflag = 0, a 32-bit overflow has not occurred.DescriptionAlgorithmComputes a real lattice IIR filter implementation using coefficient stored in vectorh. The real data input is stored in vector x. The filter output result is storedin vector r . This function retains the address of the delay filter memory d containingthe previous delayed values to allow consecutive processing of blocks.This function can be used for both block-by-block and sample-by-sample filtering(nx = 1)e N [n] x[n],e i1 [n] e i [n] hie i1 [n 1], i N,(N1), ,1e i [n] –k i e i1 e i1 [n 1], i N,(N1), ,1y[n] e 0 [n] e 0 [n]Overflow Handling MethodologyNo scaling implemented for overflow prevention.Special Requirements noneImplementation Notes noneExampleBenchmarksSee examples/iirlat subdirectory(preliminary)Cycles † Core: 4 * (nh – 1) * nxOverhead: 24Code size(in bytes)54† Assumes all data is in on-chip dual-access RAM and that there is no bus conflict due to twiddletable reads and instruction fetches (provided linker command file reflects those conditions).4-62

ldiv16ldiv16Function32-bit by 16-bit Long Division Functionvoid ldiv16 (LDATA *x, DATA *y, DATA *r, DATA *rexp, ushort nx)Argumentsx [nx]r[nx]rexp[nx]nxPointer to input data vector 1 of size nxx[0]x[1]..x[nx–2]x[nx–1]Pointer to output data bufferr[0]r[1]..r[nx–2]r[nx–1]Pointer to exponent buffer for output values. Theseexponent values are in integer format.rexp[0]rexp[1]..rexp[nx–2]rexp[nx–1]Number of elements of input and output vectorsDescriptionThis routine implements a long division function of a Q31 value divided by aQ15 value. The reciprocal of the Q15 value, y, is calculated then multiplied bythe Q31 value, x. The result is returned as an exponent such that:r[i] * rexp[i] = true reciprocal in floating-pointAlgorithmThe reciprocal of the Q15 number is calculated using the following equation:Ym 2*Ym Ym 2 * XnormIf we start with an initial estimate of Ym, the equation converges to a solutionvery rapidly (typically 3 iterations for 16-bit resolution).Function Descriptions4-63

log_10The initial estimate can be obtained from a look-up table, from choosing a midpoint,or simply from linear interpolation. The method chosen for this problemis linear interpolation and is accomplished by taking the complement of theleast significant bits of the Xnorm value.The reciprocal is multiplied by the Q31 number to generate the output.Overflow Handling MethodologynoneSpecial Requirements noneImplementation Notes noneExampleBenchmarksSee examples/ldiv16 subdirectory(preliminary)Cycles † Core: 4 * nxOverhead: 14Code size(in bytes)91† Assumes all data is in on-chip dual-access RAM and that there is no bus conflict due to twiddletable reads and instruction fetches (provided linker command file reflects those conditions).log_10FunctionBase 10 Logarithmushort oflag = log_10 (DATA *x, LDATA *r, ushort nx)(defined in log_10.asm)Argumentsx[nx]r[nx]nxoflagPointer to input vector of size nx.Pointer to output data vector (Q31 format) of size nx.Length of input and output data vectorsOverflow flag. If oflag = 1, a 32-bit overflow has occurred. If oflag = 0, a 32-bit overflow has not occurred.DescriptionComputes the log base 10 of elements of vector x using Taylor series.Algorithm for (i 0; i nx; i ) y(i) log 10x(i) where –1 x(i) 1Overflow Handling MethodologyNo scaling implemented for overflow prevention4-64

log_10Special Requirements noneImplementation Notes y = 0.4343 * ln(x) with x = M(x)*2^P(x) = M*2^Py = 0.4343 * (ln(M) + ln(2)*P)y = 0.4343 * (ln(2*M) + (P–1)*ln(2))y = 0.4343 * (ln((2*M–1)+1) + (P–1)*ln(2))y = 0.4343 * (f(2*M–1) + (P–1)*ln(2))with f(u) = ln(1+u).We use a polynomial approximation for f(u) :f(u) = (((((C6*u+C5)*u+C4)*u+C3)*u+C2)*u+C1)*u+C0for 0

log_2log_2Base 2 LogarithmFunctionushort oflag = log_2 (DATA *x, LDATA *r, ushort nx)(defined in log_2.asm)Argumentsx[nx]r[nx]nxoflagPointer to input vector of size nx.Pointer to output data vector (Q31 format) of size nx.Length of input and output data vectorsOverflow flag. If oflag = 1, a 32-bit overflow has occurred. If oflag = 0, a 32-bit overflow has not occurred.DescriptionComputes the log base 2 of elements of vector x using Taylor series.Algorithm for (i 0; i nx; i ) y(i) log 12x(i) where 0 x(i) 1Overflow Handling MethodologyNo scaling implemented for overflow preventionSpecial Requirements noneImplementation Notes y = 1.4427 * ln(x) with x = M(x)*2^P(x) = M*2^Py = 1.4427 * (ln(M) + ln(2)*P)y = 1.4427 * (ln(2*M) + (P–1)*ln(2))y = 1.4427 * (ln((2*M–1)+1) + (P–1)*ln(2))y = 1.4427 * (f(2*M–1) + (P–1)*ln(2))with f(u) = ln(1+u).We use a polynomial approximation for f(u) :f(u) = (((((C6*u+C5)*u+C4)*u+C3)*u+C2)*u+C1)*u+C0for 0

lognThe coefficients Bi used in the calculation are derived from the Ci as follows:B0 Q30 1581d 0062DhB1 Q14 16381d 03FFDhB2 Q15 –16298d 0C056hB3 Q16 20693d 050D5hB4 Q17 –24950d 09E8AhB5 Q18 21677d 054ADhB6 Q19 –9130d 0DC56hExampleBenchmarksSee examples/log_2 subdirectory(preliminary)Cycles † Core: 36 * nxOverhead: 37Code size(in bytes)166† Assumes all data is in on-chip dual-access RAM and that there is no bus conflict due to twiddletable reads and instruction fetches (provided linker command file reflects those conditions).lognFunctionBase e Logarithm (natural logarithm)ushort oflag = logn (DATA *x, LDATA *r, ushort nx)(defined in logn.asm)Argumentsx[nx]r[nx]nxoflagPointer to input vector of size nx.Pointer to output data vector (Q31 format) of size nx.Length of input and output data vectorsOverflow flag. If oflag = 1, a 32-bit overflow has occurred. If oflag = 0, a 32-bit overflow has not occurred.DescriptionComputes the log base e of elements of vector x using Taylor series.Algorithm for (i 0; i nx; i ) y(i) log nx(i) where –1 x(i) 1Overflow Handling MethodologyNo scaling implemented for overflow preventionSpecial Requirements Special RequirementsnoneFunction Descriptions4-67

lognImplementation Notes y = 0.4343 * ln(x) with x = M(x)*2^P(x) = M*2^Py = 0.4343 * (ln(M) + ln(2)*P)y = 0.4343 * (ln(2*M) + (P–1)*ln(2))y = 0.4343 * (ln((2*M–1)+1) + (P–1)*ln(2))y = 0.4343 * (f(2*M–1) + (P–1)*ln(2))with f(u) = ln(1+u).We use a polynomial approximation for f(u):f(u) = (((((C6*u+C5)*u+C4)*u+C3)*u+C2)*u+C1)*u+C0for 0

maxidxmaxidxFunctionIndex of the Maximum Element of a Vectorshort r = maxidx (DATA *x, ushort ng, ushort ng_size);(defined in maxidx.asm)Argumentsx[nx]rngng_sizePointer to input vector of size nx.Index for vector element with maximum value.Number of groups.Size of group.DescriptionThe vector x is divided in ng groups of size ng_size.Size of x = ng x ng_size. ng_size must be an even number between 2 and 34.The larger ng_size, the better the performance.Returns the index of the maximumelement of a vector x. In case of multiple maximum elements, r containsthe index of the first maximum element found.Example 1: size of x is 64.Choose ng_size = 32, ng = 2Example 2: size of x is 100.Choose ng_size = 20, ng = 5Example 3: size of x is 90.Choose ng_size = 30, ng = 3AlgorithmNot applicableOverflow Handling MethodologyNot applicableSpecial Requirements ng_size is an even number between 2 and 34Implementation Notes noneExampleBenchmarksSee examples/maxidx subdirectory(preliminary)Cycles † Core: nx/2 + ng16Overhead: 40Code size(in bytes)143† Assumes all data is in on-chip dual-access RAM and that there is no bus conflict due to twiddletable reads and instruction fetches (provided linker command file reflects those conditions).Function Descriptions4-69

maxidx34maxidx34FunctionIndex of the Maximum Element of a Vector ≤ 34short r = maxidx34 (DATA *x, ushort nx)(defined in maxidx34.asm)Argumentsx[nx]rPointer to input vector of size nx.Index for vector element with maximum value.nx Lenght of input data vector (nx ≤ 34).DescriptionAlgorithmReturns the index of the maximum element of a vector x. In case of multiplemaximum elements, r contains the index of the first maximum element found.Not applicableOverflow Handling MethodologyNot applicableSpecial Requirements Size of the vector, nx ≤ 34Implementation Notes noneExampleBenchmarksSee examples/maxidx34 subdirectory(preliminary)Cycles † Core: nx/2Overhead: 42Code size(in bytes)26† Assumes all data is in on-chip dual-access RAM and that there is no bus conflict due to twiddletable reads and instruction fetches (provided linker command file reflects those conditions).maxvalFunctionMaximum Value of a Vectorshort r = maxval (DATA *x, ushort nx)(defined in maxval.asm)Argumentsx[nx]rnxPointer to input vector of size nx.Maximum value of a vectorLength of input data vector4-70

maxvecDescription Returns the maximum element of a vector x.AlgorithmNot applicableOverflow Handling MethodologyNot applicableSpecial Requirements noneImplementation Notes noneExampleBenchmarksSee examples/maxval subdirectory(preliminary)Cycles † Core: nxOverhead: 3Code size(in bytes)20† Assumes all data is in on-chip dual-access RAM and that there is no bus conflict due to twiddletable reads and instruction fetches (provided linker command file reflects those conditions).maxvecFunctionIndex and Value of the Maximum Element of a Vectorvoid maxvec (DATA *x, ushort nx, DATA *r_val, DATA *r_idx)(defined in maxvec.asm)Argumentsx[nx]r_valr_idxPointer to input vector of size nx.maximum valueIndex for vector element with maximum valuenx Lenght of input data vector (nx 6)DescriptionAlgorithmThis function finds the index for vector element with maximum value. In caseof multiple maximum elements, r_idx contains the index of the first maximumelement found. r_val contains the maximum value.Not applicableOverflow Handling MethodologyNot applicableSpecial Requirements noneImplementation Notes noneExampleSee examples/maxvec subdirectoryFunction Descriptions4-71

minidxBenchmarks(preliminary)Cycles Core: nx*3Overhead: 8Code size(in bytes)26minidxIndex of the Minimum Element of a VectorFunctionshort r = minidx (DATA *x, ushort nx)(defined in minidx.asm)Argumentsx[nx]rnxPointer to input vector of size nx.Index for vector element with minimum valueLength of input data vectorDescriptionAlgorithmReturns the index of the minimum element of a vector x. In case of multipleminimum elements, r contains the index of the first minimum element found.Not applicableOverflow Handling MethodologyNot applicableSpecial Requirements noneImplementation Notes noneExampleBenchmarksSee examples/minidx subdirectory(preliminary)Cycles † Core: nx * 3Overhead: 7Code size(in bytes)26† Assumes all data is in on-chip dual-access RAM and that there is no bus conflict due to twiddletable reads and instruction fetches (provided linker command file reflects those conditions).4-72

minvecminvalFunctionMinimum Value of a Vectorshort r = minval (DATA *x, ushort nx)(defined in minval.asm)Argumentsx[nx]rnxPointer to input vector of size nx.Minimum value of a vectorLength of input data vectorDescription Returns the minimum element of a vector x.AlgorithmNot applicableOverflow Handling MethodologyNot applicableSpecial Requirements noneImplementation Notes noneExampleBenchmarksSee examples/minval subdirectory(preliminary)Cycles † Core: nx/2Overhead: 7Code size(in bytes)20† Assumes all data is in on-chip dual-access RAM and that there is no bus conflict due to twiddletable reads and instruction fetches (provided linker command file reflects those conditions).minvecFunctionIndex and Value of the Minimum Element of a Vectorvoid minvec (DATA *x, ushort nx, DATA *r_val, DATA *r_idx)(defined in minvec.asm)Argumentsx[nx]r_valr_idxPointer to input vector of size nx.Minimum valueIndex for vector element with minimum valuenx Length of input data vector (nx 6)Function Descriptions4-73

mmulDescriptionAlgorithmThis function finds the index for vector element with minimum value. In caseof multiple minimum elements, r_idx contains the index of the first minimumelement found. r_val contains the minimum value.Not applicableOverflow Handling MethodologyNot applicableSpecial Requirements noneImplementation Notes noneExampleBenchmarksSee examples/minvec subdirectory(preliminary)CyclesCode size(bytes)Core: nx*3Overhead: 826mmulFunctionArgumentsMatrix Multiplicationushort oflag = mmul (DATA *x1,short row1,short col1,DATA *x2,shortrow2,short col2,DATA *r)(defined in mmul.asm)x1[row1*col1]:Pointer to input vector of size nxPointer to input matrix of size row1*col1; row1 :; :; :; r[row1*col2] : Pointer to output data vector of sizerow1*col2row1 number of rows in matrix 1col1 number of columns in matrix 1x2[row2*col2]:Pointer to input matrix of size row2*col2row2 number of rows in matrix 2col2 number of columns in matrix 2r[row1*col2]Pointer to output matrix of size row1*col24-74

mmulDescriptionAlgorithmThis function multiplies two matricesMultiply input matrix A (M by N) by input matrix B (N by P) using 2 nested loops:for i = 1 to Mfor k = 1 to P{temp = 0for j = 1 to Ntemp = temp + A(i,j) * B(j,k)C(i,k) = temp}Overflow Handling MethodologyNot applicableSpecial Requirements Verify that the dimensions of input matrices are legal, i.e. col1 == row2Implementation Notes In order to take advantage of the dual MAC architecture of the C55x, this implementationchecks the size of the matrix x1. For small matrices x1 (row1 < 4 orcol1 < 2), single MAC loops are used. For larger matrices x1 (row1 ≥ 4 andcol1 ≥ 2), Dual MAC loops are more efficient and quickly make up for the additionalinitialization overhead.ExampleBenchmarksSee examples/mmul subdirectory(preliminary)Cycles †Code size(in bytes)Core: if(row1 < 4 || col1 < 2), use single MAC((col1 + 2)*row1 + 4)*col2if((row1==even)&&(row1 ≥ 4)&&(col1 ≥ 2)), use dual MAC((col1 + 4)*0.5*row1 + 10)col2 if((row1==odd)&&(row1 ≥ 4)&&(col1 ≥ 2), use dual MAC((col1 + 4)*0.5*(row1 – 1) + col1 + 12)col2Overhead: 30215† Assumes all data is in on-chip dual-access RAM and that there is no bus conflict due to twiddletable reads and instruction fetches (provided linker command file reflects those conditions).Function Descriptions4-75

mtransmtransFunctionMatrix Transposeushort oflag = mtrans (DATA *x, short row, short col, DATA *r)(defined in mtrans.asm)Argumentsx[row*col]rowcolr[row*col]Pointer to input matrix. In-place processing is not allowed.number of rows in matrixnumber of columns in matrixPointer to output data vectorDescriptionAlgorithmThis function transposes matrix xfor i = 1 to Mfor j = 1 to NC(j,i) = A(i,j)Overflow Handling MethodologyNot applicableSpecial Requirements noneImplementation Notes noneExampleBenchmarksSee examples/mtrans subdirectory(preliminary)Cycles † Core: (1 + col) * rowOverhead: 23Code size(in bytes)65† Assumes all data is in on-chip dual-access RAM and that there is no bus conflict due to twiddletable reads and instruction fetches (provided linker command file reflects those conditions).mul32Function32-bit Vector Multiplicationushort oflag = mul32 (LDATA *x, LDATA *y, LDATA *r, ushort nx)(defined in mul32.asm)Argumentsx[nx]y[nx]Pointer to input data vector 1 of size nx. In-place processingallowed (r can be = x = y)Pointer to input data vector 2 of size nx4-76

negr[nx]Pointer to output data vector of size nx containingnx Number of elements of input and output vectors. Nx 4oflagOverflow flagIf oflag = 1, a 32-bit overflow has occurredIf oflag = 0, a 32-bit overflow has not occurredDescriptionAlgorithmThis function multiplies two 32-bit Q31 vectors, element by element, andproduces a 32-bit Q31 vector.for (i=0; i < nx; i++)z (i) = x (i) * y (i)Overflow Handling MethodologyScaling implemented for overflow prevention (user selectable)Special Requirements noneImplementation Notes noneExampleSee examples/add subdirectoryBenchmarksCycles Core: 4*nx + 4Overhead 21Code size(in bytes)73negFunctionVector Negateushort oflag = neg (DATA *x, DATA *r, ushort nx)(defined in neg.asm)Argumentsx[nx]r[nx]Pointer to input data vector 1 of size nx. In-place processingallowed (r can be = x = y)Pointer to output data vector of size nx. In-place processingallowedSpecial cases: if x[I] = –1 = 32768 , then r = 1 = 321767 with oflag = 1 if x= 1 = 32767 , then r = –1 = 321768 with oflag = 1Function Descriptions4-77

neg32nxoflagNumber of elements of input and output vectors.nx ≥ 4Overflow flag. If oflag = 1, a 32-bit overflow has occurred. If oflag = 0, a 32-bit overflow has not occurred.Caution: overflow in negation of a Q15 number can happennaturally when negating (–1).DescriptionThis function negates each of the elements of a vector (fractional values).Algorithm for (i 0; i nx; i ) x(i) –x(i)Overflow Handling MethodologySpecial Requirements noneImplementation Notes noneExampleBenchmarksSaturation implemented for overflow handlingSee examples/neg subdirectory(preliminary)Cycles † Core: 4 * nxOverhead: 13Code size(in bytes)61† Assumes all data is in on-chip dual-access RAM and that there is no bus conflict due to twiddletable reads and instruction fetches (provided linker command file reflects those conditions).neg32FunctionArgumentsVector Negate (double precision)ushort oflag = neg32 (LDATA *x, LDATA *r, ushort nx)(defined in neg.asm)x[nx]r[nx]Pointer to input data vector 1 of size nx. In-place processingallowed (r can be = x = y)Pointer to output data vector of size nx. In-place processingallowedSpecial cases: if x = –1 = 32768 * 2 16 , then r = 1 = 321767 * 2 16 with oflag= 1if x= 1 = 32767 * 2 16 , then r = –1 = 321768 * 2 16 with oflag= 14-78

powernxoflagNumber of elements of input and output vectors.nx ≥ 4Overflow flag. If oflag = 1, a 32-bit overflow has occurred. If oflag = 0, a 32-bit overflow has not occurred.Caution: overflow in negation of a Q31 number can happennaturally when negating (–1).DescriptionThis function negates each of the elements of a vector (fractional values).Algorithm for (i 0; i nx; i ) x(i) –x(i)Overflow Handling MethodologySpecial Requirements noneImplementation Notes noneExampleBenchmarksSaturation implemented for overflow handlingSee examples/neg32 subdirectory(preliminary)Cycles † Core: 4 * nxOverhead: 13Code size(in bytes)61† Assumes all data is in on-chip dual-access RAM and that there is no bus conflict due to twiddletable reads and instruction fetches (provided linker command file reflects those conditions).powerFunctionArgumentsVector Powerushort oflag = power (DATA *x, LDATA *r, ushort nx)(defined in power.asm)x[nx]r[1]Pointer to input data vector 1 of size nx. In-place processingallowed (r can be = x = y)Pointer to output data vector element in Q31 formatSpecial cases: if x= –1 = 32768*2 16 , then r = 1 = 321767*2 16with oflag = 1 if x= 1 = 32767*2 16 , then r = –1 = 321768*2 16with oflag = 1Function Descriptions4-79

q15toflnxoflagNumber of elements of input vectors.nx ≥ 4Overflow flag. If oflag = 1, a 32-bit overflow has occurred. If oflag = 0, a 32-bit overflow has not occurred.DescriptionThis function calculates the power (sum of products) of a vector.Algorithm Power = 0 for (i 0; i nx; i ) power x(i)*x(I)Overflow Handling MethodologyNo scaling implemented for overflow handlingSpecial Requirements noneImplementation Notes noneExampleBenchmarksSee examples/power subdirectory(preliminary)Cycles † Core: nx – 1Overhead: 12Code size(in bytes)54† Assumes all data is in on-chip dual-access RAM and that there is no bus conflict due to twiddletable reads and instruction fetches (provided linker command file reflects those conditions).q15toflFunctionQ15 to Floating-point Conversionushort q15tofl (DATA *x, float *r, ushort nx)(defined in q152fl.asm)Argumentsx[nx]r[nx]nxPointer to Q15 input vector of size nx.Pointer to floating-point output data vector of size nxcontaining the floating-point equivalent of vector x.Length of input and output data vectorsDescriptionConverts the Q15 stored in vector x to IEEE floating-point numbers stored invector r.4-80

and16AlgorithmNot applicableOverflow Handling MethodologySaturation implemented for overflow handlingSpecial Requirements noneImplementation Notes noneExampleBenchmarksSee examples/ug subdirectory(preliminary)Cycles † Core: 7 * nx (if x[n] ==0)32 * nxOverhead: 18Code size(in bytes)124† Assumes all data is in on-chip dual-access RAM and that there is no bus conflict due to twiddletable reads and instruction fetches (provided linker command file reflects those conditions).rand16FunctionArgumentsRandom Number Generation Algorithmushort oflag= rand16 (DATA *r, ushort nr)*r Pointer to the array where the 16-bit random numbers arestorednrNumber of random numbers that are generatedoflagOverflow error flag (returned value) If oflag = 1, a 32-bit data overflow occurred in anintermediate or final result.If oflag = 0, a 32-bit overflow has not occurred.DescriptionAlgorithmThis algorithm computes an array of random numbers based on the linear congruentialmethod introduced by D. Lehmer in 1951. This is one of the fastestand simplest techniques of generating random numbers. The code shownhere generates 16-bit integers, however, if a 32-bit value is desired the codecan be modified to perform 32-bit multiplies using the defined constantsRNDMULT and RNDINC. The disadvantage of this technique is that it is verysensitive to the choice of RANDMULT and RNDINC.r [n] = [(r[n–1]*RNDMULT) + RNDINC ] % Mwhere 0 n nr and 0 M 65536Function Descriptions4-81

and16Overflow Handling MethodologyNo scaling implemented for overflow prevention.Special Requirements No special requirements.Implementation Notes Rand16() is written so that it can be called from a C program. Prior to callingrand16(), rand16i() can be called to initialize the random number generatorseed value. The C routine passes two parameters to rand16(): A pointer to therandom number array *r and the count of random numbers (nr) desired. Therandom numbers are declared as short or 16 bit values. Two constantsRNDMULT and RNDINC are defined in the function. The algorithm is sensitiveto the choice of RNDMULT and RNDINC so exercise caution when changingthese.MThis value is based on the system that the routine runs. Thisroutine returns a random number from 0 to 65536 (64K) andis NOT internally bounded. If you need a min/max limit, thismust be coded externally to this routine.RNDSEEDRNDMULTAn arbitrary constant that can be any value between 0 and64K. If 0 (zero) is chosen, then RNDINC should be somevalue greater than 1. Otherwise, the first two values will be 0and 1. To change the set of random numbers generated bythis routine, change the RNDSEED value. In this routine,RNDSEED is initialized to 21845, which is 65536/3.Should be chosen such that the last three digits fall in thepattern even_digit–2–1 such as xx821, xx421 etc.RNDMULT = 31821 is used in this routine.RNDINCIn general, this constant can be any prime number related toM. Research shows that RNDINC (the increment value)should be chosen by the following formula:RNDINC = ((1/2 – (1/6 * SQRT(3))) * M). Using M=65536,RNDINC was picked as 13849.The random seed initialized in rand16i() is used to generate the first randomnumber. Each random number generated is used to generate the next numberin the series. The random number is generated in the accumulator (32 bits) byusing the multiply-accumulate (MAC) unit to do the computation. In the courseof the algorithm if there is intermediate overflow, the overflow flag bit in statusregister is set. At the end of the algorithm, the overflow flag is tested for anyintermediate overflow conditions.ExampleSee examples/rand16 subdirectory4-82

and16initBenchmarksCycles Core: 13 + nr*2Overhead: 10Code size(in bytes)49C54x Benchmark for ComparisonCycles Core: 10 + nr*4Overhead: 16Code size(in bytes)56rand16initFunctionArgumentsDescriptionAlgorithmRandom Number Generation Initializationvoid rand16init(void)noneInitializes seed for 16-bit random number generator.Not applicableOverflow Handling MethodologyNo scaling implemented for overflow prevention.Special Requirements Allocation of .bss section is required in linker command file.Implementation Notes This function initializes a global variable rndseed in global memory to be usedfor the 16 bit random number generation routine (rand16)ExampleSee examples/rand16i subdirectoryBenchmarksCycles 6Code size(in bytes)9C54x Benchmark for ComparisonCycles 7Code size(in bytes)10Function Descriptions4-83

ecip16recip1616-bit Reciprocal FunctionFunctionvoid recip16 (DATA *x, DATA *r, DATA *rexp, ushort nx)Argumentsx[nx]r[nx]rexp[nx]nxPointer to input data vector 1 of size nx.x[0]x[1]..x[nx–2]x[nx–1]Pointer to output data bufferr[0]r[1]..r[nx–2]r[nx–1]Pointer to exponent buffer for output values. Theseexponent values are in integer format.rexp[0]rexp[1]..rexp[nx–2]rexp[nx–1]Number of elements of input and output vectorsDescriptionThis routine returns the fractional and exponential portion of the reciprocal ofa Q15 number. Since the reciprocal is always greater than 1, it returns an exponentsuch that:r[i] * rexp[i] = true reciprocal in floating-pointAlgorithmYm 2*Ym Ym 2 * XnormIf we start with an initial estimate of Ym, the equation converges to a solutionvery rapidly (typically 3 iterations for 16-bit resolution).4-84

fftThe initial estimate can be obtained from a look-up table, from choosing a midpoint,or simply from linear interpolation. The method chosen for this problemis linear interpolation and is accomplished by taking the complement of theleast significant bits of the Xnorm value.Overflow Handling MethodologynoneSpecial Requirements noneImplementation Notes noneExampleBenchmarksSee examples/recip16 subdirectory(preliminary)Cycles † Core: 33 * nxOverhead: 12Code size(in bytes)69† Assumes all data is in on-chip dual-access RAM and that there is no bus conflict due to twiddletable reads and instruction fetches (provided linker command file reflects those conditions).rfftFunctionForward Real FFT (in-place)void rfft (DATA *x, ushort nx, type);(reference cfft.asm and unpack.asm)Argumentsx [nx]nxPointer to input vector containing nx real elements. On output,vector x contains the first half (nx/2 complex elements) of theFFT output in the following order. Real FFT is a symmetricfunction around the Nyquist point, and for this reason only halfof the FFT(x) elements are required.On output x will contain the FFT(x) = y in the following format:y(0)Re y(nx/2)im → DC and Nyquisty(1)Re y(1)Imy(2)Re y(2)Im....y(nx/2)Re y(nx/2)ImComplex numbers are stored in Re-Im formatNumber of real elements in vector x. can take the followingvalues.nx = 16, 32, 64, 128, 256, 512, 1024Function Descriptions4-85

ffttypeRFFT type selector. Types supported: If type = SCALE, scaled version selected If type = NOSCALE, non-scaled version selectedDescriptionAlgorithmSpecial RequirementsComputes a Radix-2 real DIT FFT of the nx real elements stored in vector xin bit-reversed order. The original content of vector x is destroyed in theprocess. The first nx/2 complex elements of the FFT(x) are stored in vector xin normal-order.(DFT)See CFFTThis function should work with unpack.asm and/or rfft.asm for properresult. See example in examples/rifft directory.Memory alignment: input data (x) must be aligned at even word boundary.Data memory alignment (referenced rfft.cmd in examples/rfft directory): Alignment of input database address: (n+1) LSBs must be zeros,where n = log 2 (nx). Ensure that the entire array fits within a 64K boundary (the largest possiblearray addressable by the 16-bit auxiliary register). For best performance, the data buffer has to be in a DARAM block. If the twiddle table and the data buffer are in the same block, then theradix-2 kernel is 7 cycles and the radix-4 kernel is not affected. The twiddle table should be aligned to the nx word boundary; i.e., thelowest log 2 (nx) least significant bits of the twiddle table starting addressshould be zero.Implementation Notes Implemented as a complex FFT of size nx/2 followed by an unpack stage tounpack the real FFT results. Therefore, Implementation Notes for the cfft functionapply to this case.ExampleNotice that normally an FFT of a real sequence of size N, produces a complexsequence of size N (or 2*N real numbers) that will not fit in the input sequence.To accommodate all the results without requiring extra memory locations, theoutput reflects only half of the spectrum (complex output). This still providesthe full information because an FFT of a real sequence has even symmetryaround the center or nyquist point(N/2).When scale = 1, this routine prevents overflow by scaling by 2 at each FFTintermediate stages and at the unpacking stage.See examples/rfft Zip file4-86

ifftrifftFunctionInverse Real FFT (in-place)void rifft (DATA *x, ushort nx, type);(reference cifft.asm)Argumentsx [nx]nxtypePointer to input vector x containing nx real elements. Theunpacki routine should be called to unpack the rfft sequencebefore calling the bit reversal routine. (See examples directoryfor calling sequence)On output, the vector x contains nx complex elementscorresponding to RIFFT(x) or the signal itself.Number of real elements in vector x. nx can take the followingvalues.nx =16, 32, 64, 128, 256, 512, 1024RFFT type selector. Types supported: If type = SCALE, scaled version selected If type = NOSCALE, non-scaled version selectedDescriptionAlgorithmComputes a Radix-2 real DIT IFFT of the nx real elements stored in vector xin bit-reversed order. The original content of vector x is destroyed in theprocess. The first nx/2 complex elements of the IFFT(x) are stored in vectorx in normal-order.(IDFT)See CIFFTSpecial RequirementsThis function should work with unpacki.asm and/or cbrev.asm for properresult. See example in examples/rifft directory.Memory alignment: input data (x) must be aligned at even word boundary.Data memory alignment (referenced rfft.cmd in examples/rfft directory):Alignment of input database address: (n+1) LSBs must be zeros,where n = log 2 (nx).Ensure that the entire array fits within a 64K boundary (the largest possiblearray addressable by the 16-bit auxiliary register).For best performance, the data buffer has to be in a DARAM block.If the twiddle table and the data buffer are in the same block, then theradix-2 kernel is 7 cycles and the radix-4 kernel is not affected.Function Descriptions4-87

sineThe twiddle table should be aligned to the nx word boundary; i.e., thelowest log 2 (nx) least significant bits of the twiddle table starting addressshould be zero.Implementation Notes Implemented as a complex IFFT of size nx/2 followed by an unpack stage tounpack the real IFFT results. Therefore, Implementation Notes for the cfftfunction apply to this case.Notice that normally an IFFT of a real sequence of size N, produces a complexsequence of size N (or 2*N real numbers) that will not fit in the input sequence.To accommodate all the results without requiring extra memory locations, theoutput reflects only half of the spectrum (complex output). This still providesthe full information because an IFFT of a real sequence has even symmetryaround the center or nyquist point(N/2).When scale = 1, this routine prevents overflow by scaling by 2 at each IFFTintermediate stages and at the unpacking stage.ExampleSee examples/rifft subdirectorysineFunctionSineushort oflag = sine (DATA *x, DATA *r, ushort nx)(defined in sine.asm)Argumentsx[nx]r[nx]nxoflagPointer to input vector of size nx. x contains the angle inradians between [–pi, pi] normalized between (–1,1) in q15formatx = xrad /piFor example:45 o = pi/4 is equivalent to x = 1/4 = 0.25 = 0x200 in q15format.Pointer to output vector containing the sine of vector x inq15 formatNumber of elements of input and output vectors.nx ≥ 4Overflow flag. If oflag = 1, a 32-bit overflow has occurred. If oflag = 0, a 32-bit overflow has not occurred.4-88

sqrt_16DescriptionComputes the sine of elements of vector x. It uses Taylor series to computethe sine of angle x.Algorithm for (i 0; i nx; i ) y(i) sin(x(i)) where x(i) xradpiOverflow Handling MethodologyNot applicableSpecial Requirements noneImplementation Notes Computes the sine of elements of vector x. It uses the following Taylor seriesto compute the angle x in quadrant 1 (0–pi/2).sin(x) = c1*x + c2*x^2 + c3*x^3 + c4*x^4 + c5*x^5c1 = 3.140625xc2 = 0.02026367c3 = – 5.3251c4 = 0.5446778c5 = 1.800293The angle x in other quadrant is calculated by using symmetries that map theangle x into quadrant 1.ExampleBenchmarksSee examples/sine subdirectory(preliminary)Cycles † Core: 19 * nxOverhead: 17Code size(in bytes)93 program; 3 data† Assumes all data is in on-chip dual-access RAM and that there is no bus conflict due to twiddletable reads and instruction fetches (provided linker command file reflects those conditions).sqrt_16FunctionSquare Root of a 16-bit Numberushort oflag = sqrt_16 (DATA *x, DATA *r, short nx)(defined in sqrtv.asm)Argumentsx[nx]r[nx]Pointer to input vector of size nx.Pointer to output vector of size nx containing the sqrt(x).Function Descriptions4-89

subnxoflagNumber of elements of input and output vectors.Overflow flag. If oflag = 1, a 32-bit overflow has occurred. If oflag = 0, a 32-bit overflow has not occurred.DescriptionCalculates the square root for each element in input vector x, storing resultsin output vector r.Algorithm for (i 0; i nx; i ) r[i] (x(i)0 i nxOverflow Handling MethodologyNot applicableSpecial Requirements noneImplementation Notes The square root of a number(x) can be calculated using Newton’s method. Aninitial approximation is guessed and then the approximation gets recomputedusing the formula,new approximation old approximation (old approximation2 x).2The new approximation then becomes the old approximation and the processis repeated until the desired accuracy is reached.ExampleBenchmarksSee examples/sqrtv subdirectory(preliminary)Cycles † Core: 35 * nxOverhead: 14Code size(in bytes)84 program; 5 data† Assumes all data is in on-chip dual-access RAM and that there is no bus conflict due to twiddletable reads and instruction fetches (provided linker command file reflects those conditions).subFunctionVector Subtractshort oflag = sub (DATA *x, DATA *y, DATA *r, ushort nx, ushort scale)(defined in sub.asm)Argumentsx[nx]y[nx]Pointer to input data vector 1 of size nx. In-place processingallowed (r can be = x = y)Pointer to input data vector 2 of size nx4-90

subr[nx]nxscaleoflagPointer to output data vector of size nx containing (x–y) if scale =0 (x–y)/2 if scale =1Number of elements of input and output vectors.nx ≥ 4Scale selection If scale = 1, divide the result by 2 to prevent overflow. If scale = 0, do not divide by 2.Overflow flag. If oflag = 1, a 32-bit overflow has occurred. If oflag = 0, a 32-bit overflow has not occurred.DescriptionThis function subtracts two vectors, element by element.Algorithm for (i 0; i nx; i ) z(i) x(i) y(i)Overflow Handling MethodologyScaling implemented for overflow prevention (User selectable)Special Requirements noneImplementation Notes noneExampleBenchmarksSee examples/sub subdirectory(preliminary)Cycles † Core: 3 * nxOverhead: 23Code size(in bytes)60† Assumes all data is in on-chip dual-access RAM and that there is no bus conflict due to twiddletable reads and instruction fetches (provided linker command file reflects those conditions).Function Descriptions4-91

Chapter 5DSPLIB Benchmarks and Performance IssuesAll functions in the DSPLIB are provided with execution time and code sizebenchmarks. While developing the included functions, we tried to compromisebetween speed, code size, and ease of use. However, with few exceptions, thehighest priority was given to optimize for speed and ease of use, and last forcode size.Even though DSPLIB can be used as a first estimation of processor performancefor a specific function, you should know that the generic nature ofDSPLIB may add extra cycles not required for customer specific usage.TopicPage5.1 What DSPLIB Benchmarks are Provided . . . . . . . . . . . . . . . . . . . . . . . . 5-25.2 Performance Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-25-1

What DSPLIB Benchmarks are Provided / Performance Considerations5.1 What DSPLIB Benchmarks are ProvidedDSPLIB documentation includes benchmarks for instruction cycles andmemory consumption. The following benchmarks are typically included:Calling and register initialization overheadNumber of cycles in the kernel code: Typically provided in the form of anequation that is a function of the data size parameters. We consider thekernel (or core) code, the instructions contained between the _start and_end labels that you can see in each of the functions.Memory consumption: Typically program size in bytes is reported. Forfunctions requiring significant internal data allocation, data memory consumptionis also provided. When stack usage for local variables is minimum,that data consumption is not reported.For functions in which it is difficult to determine the number of cycles in the kernelcode as a function of the data size parameters, we have included directcycle count for specific data sizes.5.2 Performance ConsiderationsBenchmark cycles presented assume best case conditions, typically assuming:0 wait-state memory external memory for program and datadata allocation to on-chip DARAMno pipeline hitsA linker command file showing the memory allocation used during testing andbenchmarking in the Code Composer C55x Simulator is included under theexample subdirectory.Remember, execution speed in a system is dependent on where the differentsections of program and data are located in memory. Be sure to account forsuch differences, when trying to explain why a routine is taking more time thatthe reported DSPLIB benchmarks.5-2

Chapter 6Software Updates and Customer SupportThis chapter details the software updates and customer support issues for theTMS320C55x DSPLIB.TopicPage6.1 DSPLIB Software Updates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-26.2 DSPLIB Customer Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-26-1

DSPLIB Software Updates6.1 DSPLIB Software UpdatesC54x DSPLIB software updates will be periodically released, incorporatingproduct enhancement and fixes.DSPLIB software updates will be posted as they become available in the samelocation you download this information. Source code for previous releases willbe kept public to prevent any customer problem in case we decide to discontinueor change the functionality of one of the DSPLIB functions. Make sure toread the readme.1 st file available in the root directory of every release.6.2 DSPLIB Customer SupportIf you have question or want to report problems or suggestions regarding theC54x DSPLIB, contact Texas Instruments at dsph@ti.com. We encourage theuse of the software report form (report.txt) contained in the DSPLIB doc directoryto report any problem associated with the C54xDSPLIB.6-2

Appendix AOverview of Fractional Q FormatsUnless specifically noted, DSPLIB functions use Q15 format or to be more exactQ0.15. In a Qm.n format, there are m bits used to represent the two’s complementinteger portion of the number, and n bits used to represent the two’scomplement fractional portion. m+n+1 bits are needed to store a general Qm.nnumber. The extra bit is needed to store the sign of the number in the most-significantbit position. The representable integer range is specified by (–2 m ,2 m )and the finest fractional resolution is 2 n .For example, the most commonly used format is Q.15. Q.15 means that a16-bit word is used to express a signed number between positive and negative1. The most-significant binary digit is interpreted as the sign bit in any Q formatnumber. Thus in Q.15 format, the decimal point is placed immediately to theright of the sign bit. The fractional portion to the right of the sign bit is storedin regular two’s complement format.TopicPageA.1 Q3.12 Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-2A.2 Q.15 Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-2A.3 Q.31 Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-2A-1

Q3.12 Format / Q.15 Format / Q.31 FormatA.1 Q3.12 FormatTable A–1. Q3.12 Bit FieldsQ.3.12 format places the sign bit after the fourth binary digit from the right, andthe next 12 bits contain the two’s complement fractional component. Theapproximate allowable range of numbers in Q.3.12 representation is (–8,8)and the finest fractional resolution is 2 12 2.441 10 4 .Bit 15 14 13 12 11 10 9 … 0Value S I3 I2 I1 Q11 Q10 Q9 … Q0A.2 Q.15 FormatQ.15 format places the sign bit at the leftmost binary digit, and the next 15 leftmostbits contain the two’s complement fractional component. The approximate allowablerange of numbers in Q.15 representation is (–1,1) and the finest fractional resolutionis 2 15 3.05 10 5 .Table A–2. Q.15 Bit FieldsBit 15 14 13 12 11 10 9 … 0Value S Q14 Q13 Q12 Q11 Q10 Q9 … Q0A.3 Q.31 FormatQ.31 format spans two 16-bit memory words. The 16-bit word stored in the lowermemory location contains the 16 least-significant bits, and the higher memorylocation contains the most-significant 15 bits and the sign bit. The approximateallowable range of numbers in Q.31 representation is (–1,1) and the finest fractionalresolution is 2 31 4.66 10 10 .Table A–3. Q.31 Low Memory Location Bit FieldsBit 15 14 13 12 … 3 2 1 0Value Q15 Q14 Q13 Q12 … Q3 Q2 Q1 Q0Table A–4. Q.31 High Memory Location Bit FieldsBit 15 14 13 12 … 3 2 1 0Value S Q30 Q29 Q28 … Q19 Q18 Q17 Q16A-2

Appendix BACalculating the Reciprocal of a Q15 NumberThe most optimal method for calculating the inverse of a fractional number(Y=1/X) is to normalize the number first. This limits the range of the numberas follows:0.5 Xnorm 1–1 Xnorm –0.5 (1)The resulting equation becomesY 1(Xnorm *2 n )orY 2nXnormwhere n = 1, 2, 3, …, 14, 15Letting Ye 2 n :Ye 2 n (3)Substituting (3) into equation (2):Y Ye *1(4)XnormLetting Ym 1Xnorm :Ym 1(5)XnormSubstituting (5) into equation (4):Y Ye * Ym (6)For the given range of Xnorm, the range of Ym is:1 Ym 2–2 Ym –1 (7)To calculate the value of Ym, various options are possible:a) Taylor Series Expansionb) 2nd,3rd,4th,.. Order Polynomial (Line Of Best Fit)c) Successive Approximation(2)B-1

Calculating the Reciprocal of a Q15 NumberThe method chosen in this example is (c). Successive approximation yieldsthe most optimum code versus speed versus accuracy option. The methodoutlined below yields an accuracy of 15 bits.Assume Ym(new) exact value of1Xnorm :Ym(new) 1XnormorYm(new)*X 1(c1)(c2)Assume Ym(old) estimate of value 1 X :Ym(old)*Xnorm 1 DyxorDxy Ym(old)*Xnorm 1where Dyx = error in calculationAssume that Ym(new) and Ym(old) are related as follows:Ym(new) Ym(old) Dywhere Dy = difference in valuesSubstituting (c2) and (c4) into (c3):Ym(old)*Xnorm Ym(new)*Xnorm Dxy(Ym(new) Dy)*Xnorm Ym(new)*Xnorm DxyYm(new)*Xnorm Dy * Xnorm Ym(new)*Xnorm DxyDy * Xnorm DxyDy Dxy *1XnormAssume that 1/Xnorm is approximately equal to Ym(old):Dy Dxy * Ym(old) (approx)Substituting (c6) into (c4):Ym(new) Ym(old) Dxy * Ym(old)Substituting for Dxy from (c3) into (c7):Ym(new) Ym(old) (Ym(old)*Xnorm 1) * Ym(old)Ym(new) Ym(old) Ym(old) 2 * Xnorm Ym(old)Ym(new) 2*Ym(old) Ym(old) 2 * Xnorm(c3)(c4)(c5)(c6)(c7)(c8)B-2

Calculating the Reciprocal of a Q15 NumberIf after each calculation we equate Ym(old) to Ym(new):Ym(old) Ym(new) YmThen equation (c8) evaluates to:Ym 2*Ym Ym 2 * Xnorm(c9)If we start with an initial estimate of Ym, then equation (c9) converges to a solutionvery rapidly (typically 3 iterations for 16-bit resolution).The initial estimate can be obtained from a look-up table, from choosing a midpoint,or simply from linear interpolation. The method chosen for this problemis linear interpolation and accomplished by taking the complement of the leastsignificant bits of the Xnorm value.Calculating the Reciprocal of a Q15 NumberB-3

IndexIndex16-bit reciprocal function 4-8432-bit by 16-bit long division function 4-6332-bit vector multiplication 4-76Aacorr 4-7adaptive delayed LMS filter 4-31add 4-9arctangent 2 implementation 4-10arctangent implementation 4-11atan16 4-11atan2_16 4-10autocorrelation 4-7Bbase 10 logarithm 4-64base 2 logarithm 4-66base e logarithm 4-67bexp 4-13block exponent implementation 4-13block FIR filter (fast) 4-37Ccascaded IIR direct form I 4-59cascaded IIR direct form II 4-56, 4-58cbrev 4-14cfft 4-15cfir 4-17cifft 4-21complex bit reverse 4-14complex FIR filter 4-17conversionfloating-point to Q15 (fltoq15) 4-49Q15 to floating-point (q15tofl) 4-80convol 4-23convol1 4-25convol2 4-27convolution 4-23convolution (fast) 4-25convolution (fastest) 4-27corr 4-30correlationauto (acorr) 4-7full-length (corr) 4-30correlation (full-length) 4-30Ddecimating FIR filter 4-40dlms 4-31double-precision IIR filter 4-53DSPLIBarguments 3-2calling a function from assembly language sourcecode 3-5calling a function from C 3-3content 2-2data types 3-2dealing with overflow and scaling issues 3-6how to install 2-2how to rebuild 2-3Eexpn 4-33exponential base e 4-33Index-1

IndexFFFTforward complex (cfft) 4-15forward real, in-place (rfft) 4-85inverse complex (cifft) 4-21inverse real, in-place (rifft) 4-87fir 4-34FIR filter 4-34block filter, fast (fir2) 4-37complex (cfir) 4-17decimating (firdec) 4-40direct form (fir) 4-34Hilbert Transformer 4-50interpolating (firinterp) 4-42lattice forward (firlat) 4-44symmetric (firs) 4-46FIR Hilbert Transformer 4-50fir2 4-37firdec 4-40firinterp 4-42firlat 4-44firs 4-46floating-point to Q15 conversion 4-49fltoq15 4-49forward complex FFT 4-15forward real FFT (in-place) 4-85Hhilb16 4-50IIIR filtercascaded, direct form I (iircas51) 4-59cascaded, direct form II (iircas4) 4-56cascaded, direct form II (iircas5) 4-58double-precision (iir32) 4-53lattice inverse (iirlat) 4-61iir32 4-53iircas4 4-56iircas5 4-58iircas51 4-59iirlat 4-61index and value of maximum element of avector 4-71index and value of minimum element of avector 4-73index of maximum element of a vector 4-69index of maximum element of a vector less than orequal to 34 4-70index of minimum element of a vector 4-72interpolating FIR filter 4-42inverse complex FFT 4-21inverse real FFT (in-place) 4-87Llattice forward (FIR) filter 4-44lattice inverse (IIR) filter 4-61ldiv16 4-63log_10 4-64log_2 4-66logarithmbase 10 (log_10) 4-64base 2 (log_2) 4-66base e (logn) 4-67logn 4-67Mmatrix multiplication 4-74matrix transpose 4-76maxidx 4-69maxidx34 4-70maximum element of a vectorindex (maxidx) 4-69index and value (maxvec) 4-71maximum element of a vector less than or equal to34, index (maxidx34) 4-70maximum value of a vector 4-70maxval 4-70maxvec 4-71minidx 4-72minimum element of a vectorindex (minidx) 4-72index and value (minvec) 4-73minimum value of a vector 4-73minval 4-73minvec 4-73Index-2

Indexmmul 4-74mtrans 4-76mul32 4-76Nnatural logarithm (logn) 4-67neg 4-77neg32 4-78Ppower 4-79QQ15 to floating-point conversion 4-80q15tofl 4-80Rrand16 4-81rand16init 4-83random number generationalgorithm 4-81initialization 4-83recip16 4-84rfft 4-85rifft 4-87Ssine 4-88sqrt_16 4-89square root of a 16-bit number 4-89sub 4-90symmetric FIR filter 4-46Vvector add 4-9vector negate 4-77vector negate, double-precision 4-78vector power 4-79vector subtract 4-90Index-3