Practical Digital Signal Processing using Microcontrollers - Elektor

Practical Digital Signal Processing using
Microcontrollers
Prof Dr Dogan Ibrahim
Near East University

ACKNOWLEDGEMENTS
The following material is reproduced in this book with the kind permission of the
respective copyright holders and may not be reprinted, or reproduced in any way, without
their prior consent.
Figures A.1 – A.11 are taken from Microchip Technology Inc. Data Sheet
PIC18(L)F2X/4XK22 (DS41412F). Figures 3.1 – 3.3, 3.9, 12.4 are taken from the web site
of Microchip Technology Inc.
Figure 3.4 is taken from the web site of Custom Computer Services Inc.
Figures 3.6 – 3.8, 12.2, 12.4 are taken from the web site of mikroElektronica.
Figure 3.5 is taken from the web site of Olimex.
Figures 3.10 - 3.12, 14.1 – 14.3 are taken from the web site of Analog Devices Inc.
Figure 3.13 is taken from the web site of Texas Instruments Inc.
Figure 3.14 is taken from the web site of Freescale Semiconductors Inc.
Copyright Elektor
PIC®, PICSTART®, and MPLAB® are all trademarks of Microchip Technology Inc.

Contents
CHAPTER 1 -
INTRODUCTION ........................................................................... 17
1.1 BACKGROUND ....................................................................................................... 17
1.2 A TYPICAL DSP APPLICATION ................................................................................... 18
1.3 THE ANTI-ALIASING ANALOG FILTER ........................................................................... 18
1.4 SAMPLE AND HOLD CIRCUIT ..................................................................................... 21
1.5 A/D CONVERTER ................................................................................................... 21
1.6 DSP ................................................................................................................... 24
1.7 D/A CONVERTER ................................................................................................... 24
1.8 EXERCISES ............................................................................................................ 25
CHAPTER 2 -
NUMBER SYSTEMS ...................................................................... 27
2.1 FIXED POINT NUMBERS ........................................................................................... 27
2.1.1 Decimal Number System ............................................................................ 28
2.1.2 Binary Number System .............................................................................. 28
2.1.3 Octal Number System ................................................................................ 29
2.1.4 Hexadecimal Number System .................................................................... 29
2.1.5 Converting Binary Numbers into Decimal .................................................. 30
2.1.6 Converting Decimal Numbers into Binary .................................................. 31
2.1.7 Converting Binary Numbers into Hexadecimal ........................................... 33
2.1.8 Converting Hexadecimal Numbers into Binary ........................................... 34
2.1.9 Converting Hexadecimal Numbers into Decimal ........................................ 35
2.1.10 Converting Decimal Numbers into Hexadecimal ........................................ 36
2.1.11 Converting Octal Numbers into Decimal .................................................... 37
2.1.12 Converting Decimal Numbers into Octal .................................................... 38
2.1.13 Converting Octal Numbers into Binary ....................................................... 39
2.1.14 Converting Binary Numbers Into Octal ....................................................... 40
2.1.15 Negative Numbers ..................................................................................... 40
2.1.16 Adding Binary Numbers ............................................................................. 42
2.1.17 Subtracting Binary Numbers ...................................................................... 43
2.1.18 Multiplication of Binary Numbers .............................................................. 43
2.1.19 Division of Binary Numbers ........................................................................ 45
2.1.20 Signed Magnitude Representation ............................................................. 45
2.1.21 Offset Binary Representation ..................................................................... 46
Copyright Elektor
5

2.1.22 Fractional Fixed Point Numbers ................................................................. 47
2.1.23 Converting Fractional Number into Integer ................................................ 48
2.1.24 Converting from Integer into Fractional Number ....................................... 48
2.2 FLOATING POINT NUMBERS ..................................................................................... 49
2.2.1 Converting a Floating Point Number into Decimal ..................................... 50
2.2.2 Normalizing the Floating Point Numbers ................................................... 51
2.2.3 Converting a Decimal Number into Floating Point ..................................... 52
2.2.4 Multiplication and Division of Floating Point Numbers............................... 53
2.2.5 Addition and Subtraction of Floating Point Numbers ................................. 55
2.3 FIXED POINT OR FLOATING POINT NUMBERS IN DSP OPERATIONS .................................... 56
2.4 BCD NUMBERS ..................................................................................................... 57
2.5 THE ASCII TABLE ................................................................................................... 58
2.6 SUMMARY ........................................................................................................... 59
2.7 EXERCISES ............................................................................................................ 60
CHAPTER 3 -
DIGITAL SIGNAL PROCESSORS ............................................... 65
3.1 CHOOSING A DDSP PROCESSOR ............................................................................... 66
3.2 DSP HARDWARE DEVELOPMENT TOOLS ..................................................................... 68
3.2.1 General Purpose Microcontroller Development Tools ................................ 68
3.2.2 DSP-Enhanced Processor Development Tools ............................................. 75
3.2.3 Dedicated DSP Processor Development Tools ............................................ 80
3.3 DSP SOFTWARE DEVELOPMENT TOOLS ...................................................................... 83
3.3.1 General Purpose Microcontroller Software Development Tools ................. 83
3.3.2 DSP-Enhanced Processor Software Development Tools .............................. 84
3.3.3 Dedicated DSP Processor Software Development Tools.............................. 85
3.4 EXERCISES ............................................................................................................ 86
Copyright Elektor
CHAPTER 4 -
MIKROC PRO PROGRAMMING LANGUAGE FOR
MICROCONTROLLERS .................................................................. 87
4.1 C LANGUAGES FOR MICROCONTROLLERS ..................................................................... 87
4.2 YOUR FIRST MIKROC PRO FOR PIC PROGRAM .............................................................. 89
4.2.1 Comments ................................................................................................. 90
4.2.2 Beginning and Ending a Program .............................................................. 90
4.2.3 White Spaces ............................................................................................. 91
4.2.4 Variable Names ......................................................................................... 92
4.2.5 Reserved Names ........................................................................................ 92
4.2.6 Variable Types ........................................................................................... 93
6

4.2.7 Constants .................................................................................................. 95
4.2.8 Escape Sequences ...................................................................................... 96
4.2.9 Accessing Bits of a Variable ....................................................................... 96
4.2.10 sbit Type .................................................................................................... 97
4.2.11 bit Type ..................................................................................................... 97
4.2.12 Arrays ........................................................................................................ 97
4.2.13 Pointers ................................................................................................... 100
4.2.14 Structures ................................................................................................ 102
4.2.15 Operators in mikroC Pro for PIC ............................................................... 105
4.2.16 The Flow of Control ................................................................................. 109
4.3 FUNCTIONS IN MIKROC PRO FOR PIC ....................................................................... 118
4.3.1 void Functions ......................................................................................... 119
4.3.2 Passing Parameters to Functions ............................................................. 119
4.3.3 Passing Arrays to Functions ..................................................................... 120
4.3.4 Interrupt Processing ................................................................................ 121
4.4 MIKROC PRO FOR PIC BUILT-IN FUNCTIONS ............................................................... 122
4.5 MIKROC PRO FOR PIC LIBRARIES ............................................................................. 123
4.6 USING THE MIKROC PRO FOR PIC COMPILER.............................................................. 124
4.6.1 mikroC Pro for PIC IDE ............................................................................. 125
4.6.2 Creating a New Source File ...................................................................... 127
4.6.3 Compiling the Source File......................................................................... 130
4.7 SUMMARY ......................................................................................................... 131
4.8 EXERCISES .......................................................................................................... 132
CHAPTER 5 -
Copyright Elektor
INTRODUCTION TO MATLAB ................................................. 135
5.1 GETTING STARTED ............................................................................................... 135
5.1.1 disp Command ........................................................................................ 137
5.1.2 format Command .................................................................................... 138
5.1.3 input Command ....................................................................................... 139
5.1.4 Comment lines......................................................................................... 139
5.1.5 Programming in MATLAB ........................................................................ 139
5.1.6 The Working Environment ....................................................................... 149
5.1.7 Vectors in MATLAB .................................................................................. 149
5.1.8 Matrices in MATLAB ................................................................................ 154
5.1.9 Graphics Plotting Using MATLAB ............................................................. 157
5.2 SUMMARY ......................................................................................................... 170
5.3 EXERCISES .......................................................................................................... 170
7

CHAPTER 6 -
DISCRETE TIME SIGNALS ....................................................... 173
6.1 THE SAMPLING PROCESS ....................................................................................... 173
6.2 SOME DIGITAL SIGNAL TYPES.................................................................................. 174
6.2.1 Unit Step Function ................................................................................... 174
6.2.2 Unit Impulse Function .............................................................................. 175
6.2.3 Sinusoidal Signal ...................................................................................... 175
6.2.4 Exponential Signal ................................................................................... 177
6.2.5 Complex Exponential Signal ..................................................................... 177
6.2.6 Exponential Sinusoidal Decay Signal ........................................................ 178
6.3 OPERATIONS ON DISCRETE TIME SIGNALS .................................................................. 179
6.3.1 Delay or Shift ........................................................................................... 179
6.3.2 Vector Addition and Subtraction .............................................................. 181
6.3.3 Scalar Addition and Subtraction .............................................................. 182
6.3.4 Scalar Multiplication and Division ............................................................ 182
6.3.5 Vector Multiplication and Division ........................................................... 183
6.4 BLOCK DIAGRAM REPRESENTATION.......................................................................... 184
6.4.1 Block Diagram Manipulation ................................................................... 184
6.5 DISCRETE SYSTEM CLASSIFICATION ........................................................................... 186
6.5.1 Linear and Non-linear .............................................................................. 186
6.5.2 Time-variant and Time-invariant ............................................................. 186
6.5.3 Causal and Non-causal ............................................................................ 186
6.5.4 Stable and Unstable ................................................................................ 187
6.5.5 Linear Time-invariant Systems ................................................................. 187
6.6 THE CONVOLUTION PROCESS .................................................................................. 187
6.7 EXERCISES .......................................................................................................... 192
Copyright Elektor
CHAPTER 7 -
THE Z-TRANSFORM .................................................................. 195
7.1 UNIT STEP FUNCTION ........................................................................................... 195
7.2 UNIT RAMP FUNCTION.......................................................................................... 196
7.3 EXPONENTIAL FUNCTION ....................................................................................... 197
7.4 SINE FUNCTION ................................................................................................... 198
7.5 DISCRETE IMPULSE FUNCTION ................................................................................. 199
7.6 TABLES OF Z-TRANSFORMS .................................................................................... 199
7.7 THE Z-TRANSFORM OF A FUNCTION EXPRESSED AS A LAPLACE TRANSFORM ....................... 200
7.8 PROPERTIES OF THE Z-TRANSFORM .......................................................................... 201
7.9 INVERSE Z-TRANSFORMS ....................................................................................... 204
7.9.1 Coefficients of Partial Fraction Expansion ................................................ 211
8

7.10 EXERCISES .......................................................................................................... 214
CHAPTER 8 -
THE DISCRETE FOURIER TRANSFORM ............................... 217
8.1 THE SPECTRA OF PERIODIC SIGNALS ......................................................................... 217
8.2 PROPERTIES OF THE DFT ....................................................................................... 231
8.3 FOURIER TRANSFORM OF APERIODIC DIGITAL SIGNALS ................................................. 232
8.4 EXERCISES .......................................................................................................... 235
CHAPTER 9 -
DIGITAL FILTERS ...................................................................... 237
9.1 FIR FILTERS AND IIR FILTERS .................................................................................. 238
9.2 THE DIGITAL FILTER DESIGN PROCESS ....................................................................... 239
CHAPTER 10 -
DESIGN OF FIR DIGITAL FILTERS ........................................ 245
10.1 THE FOURIER TRANSFORM METHOD ........................................................................ 245
10.2 TRUNCATION AND WINDOWING.............................................................................. 249
10.3 SOME COMMONLY USED WINDOWING FUNCTIONS ..................................................... 250
10.3.1 Rectangular Window ............................................................................... 250
10.3.2 Hamming Window................................................................................... 252
10.3.3 Hanning Window ..................................................................................... 256
10.3.4 Blackman Window................................................................................... 256
10.3.5 Remez Exchange Based Design ................................................................ 260
10.4 USING COMPUTER AIDED TECHNIQUES FOR THE DESIGN OF FIR FILTERS .......................... 260
10.4.1 The ScopeFIR Software ............................................................................ 260
10.5 FIR DIGITAL FILTER STRUCTURES ............................................................................. 268
10.5.1 Direct FIR Structures ................................................................................ 268
10.5.2 Cascade FIR Structures ............................................................................ 269
10.6 EXERCISES .......................................................................................................... 269
Copyright Elektor
CHAPTER 11 -
DESIGN OF IIR DIGITAL FILTERS ......................................... 271
11.1 IIR FILTER TRANSFER FUNCTION .............................................................................. 271
11.2 ANALOG FILTER DESIGN REVIEW ............................................................................. 272
11.3 BUTTERWORTH FILTERS ......................................................................................... 273
11.3.1 Bilinear transformation method .............................................................. 275
11.4 CHEBYSHEV FILTERS.............................................................................................. 282
11.5 ELLIPTIC FILTERS .................................................................................................. 286
11.6 THE SCOPEIIR FILTER DESIGN PROGRAM................................................................... 288
11.7 IIR FILTER STRUCTURES ......................................................................................... 289
11.7.1 Direct Structure ....................................................................................... 290
9

11.7.2 Cascade Structure .................................................................................... 290
11.8 EXERCISES .......................................................................................................... 292
CHAPTER 12 -
DESIGN OF FIR DIGITAL FILTERS USING
MICROCONTROLLERS ................................................................ 295
12.1 DESIGN OF A LOW-PASS FILTER ............................................................................... 295
12.1.1 The Project Hardware .............................................................................. 295
12.1.2 Calculating the Filter Coefficients with Scope FIR Filter Design Program .. 301
12.1.3 Implementing the FIR Filter ..................................................................... 304
12.1.4 The Algorithm.......................................................................................... 305
12.1.5 Program Listing ....................................................................................... 306
12.2 DESIGN OF A BAND-PASS FILTER ............................................................................. 310
12.3 MORE EFFICIENT ALGORITHM ................................................................................. 311
12.3.1 Filter Specifications.................................................................................. 315
12.4 DESIGNING A BAND-PASS FILTER WITH MORE EFFICIENT ALGORITHM .............................. 318
12.5 CALCULATING THE TIME IN THE FIR FILTER LOOP......................................................... 320
CHAPTER 13 - DESIGN OF IIR DIGITAL FILTERS USING
MICROCONTROLLERS ................................................................ 323
13.1 DESIGN OF A LOW-PASS FILTER ............................................................................... 323
13.1.1 The Project Hardware .............................................................................. 323
13.1.2 Cascade Implementation ......................................................................... 324
13.1.3 The Filter Coefficients .............................................................................. 326
13.1.4 The Filter Program ................................................................................... 328
13.2 SECOND ORDER HIGH-PASS FILTER .......................................................................... 331
13.3 SIXTH ORDER LOW-PASS FILTER .............................................................................. 332
13.4 PROGRAM FOR ANY EVEN ORDER IIR FILTER .............................................................. 335
13.5 IIR FILTERS OF ANY ORDER .................................................................................... 337
13.6 CALCULATING THE TIME IN THE IIR FILTER LOOP ......................................................... 341
Copyright Elektor
CHAPTER 14 -
GETTING STARTED WITH DEDICATED DSPS ..................... 343
14.1 THE DEDICATED DSP MARKET ................................................................................ 343
14.2 THE ADSP-21469 DSP PROCESSOR ....................................................................... 344
14.3 THE ADSP-21469 EZ-BOARD DEVELOPMENT KIT ...................................................... 347
14.4 AUDIO INPUT-OUTPUT CONNECTIONS ...................................................................... 351
14.5 THE SHARC FAMILY DEVELOPMENT TOOLS ............................................................... 353
14.6 THE VISUALDSP++ .............................................................................................. 356
10

APPENDIX A BUTTERWORTH DIGITAL FILTER COEFFICIENTS .......... 365
APPENDIX B CHEBYSHEV DIGITAL FILTER COEFFICIENTS ................. 371
APPENDIX C
ARCHITECTURE OF THE PIC18F45K22
MICROCONTROLLER .................................................................. 377
C.1 THE PIC18F45K22 MICROCONTROLLER .................................................................. 377
C.2 PIC18F45K22 ARCHITECTURE ............................................................................... 378
C.2.1 The Program Memory ............................................................................. 380
C.2.2 The Data Memory ................................................................................... 381
C.2.3 Oscillator Configurations ......................................................................... 382
C.2.4 The Reset ................................................................................................. 382
C.2.5 Parallel I/O Ports ..................................................................................... 383
C.2.6 Timer Modules ........................................................................................ 384
C.2.7 Analog-to-Digital Converter Module ........................................................ 387
C.2.8 Interrupts ................................................................................................ 391
APPENDIX D PROGRAM LISTING FIR1.C ..................................................... 397
APPENDIX E PROGRAM LISTING FIR1.C (INPUT BUFFER) ...................... 401
APPENDIX F PROGRAM LISTING IIR1.C ...................................................... 405
APPENDIX G 6 TH ORDER IIR DIGITAL FILTER PROGRAM ....................... 409
APPENDIX H PROGRAM LISTING EVEN NUMBER IIR FILTER ............... 413
APPENDIX I PROGRAM LISTING: MODIFIED IIR FILTER DESIGN ....... 417
Copyright Elektor
INDEX .......................................................................................................................... 423
11

PREFACE
Digital Signal Processing (DSP) is the process of capturing, analysing, and manipulation of
an analog signal by a digital processor, e.g. a digital computer.
The theory of DSP is quite complex and requires good understanding of high level of
mathematics. Students new to DSP are usually taught the theory in great detail with very
little or no practical applications. For example, in many cases a student can derive complex
equations for digital filters, but is unable to implement a digital filter in real life. Some
institutions use tools such as MATLAB to derive the coefficients of digital filters and then
to simulate the behaviour of these filters on a PC. Although simulation can be an invaluable
tool in teaching, it is never the same as real-time and real-life implementations.
The aim of this book is to teach the basic principles of DSP and to introduce DSP from a
practical point of view using minimum of mathematics. Only the basic level of theory is
given just enough to implement DSP applications in real-time. The emphasis of the book is
on practical aspects of DSP, such as design issues and real-time implementation issues. The
practical implementation is described using the widely available low-cost general purpose
microcontrollers so that the readers can very easily and quickly design and implement DSP
applications in real-time. The architectures of dedicated DSP processors are complex and
this adds additional difficulty to students who may want to implement DSP applications
using such tools.
In this book the popular and highly powerful PIC18F45K22 microcontroller is used as the
processor to implement DSP applications in real-time. Using this microcontroller the
readers should be able to implement DSP applications with sampling frequencies within the
audio range. Programming is done using the popular and powerful mikroC PRO for PIC,
which is a “C” language and compiler developed specifically for the PIC family of
microcontrollers. The DSP projects given in the book are based on using the EasyPIC 7
microcontroller development kit, although any other development kit could also be used.
Copyright Elektor
The DSP development environment used in the book consists of:
• EasyPIC 7 microcontroller development board with PIC18F45K22 chip
• A/D converter (built-in on the microcontroller chip)
• D/A converter (MCP4921)
• PCSGU250 Frequency generator and Bode plotter
• mikroC PRO for PIC compiler
This book is primarily intended for students and practising engineers who may want to
learn the practical implementation of DSPs, such as digital filters in real-time.
13

The structure of the book is as follows:
Chapter 1 is an introduction to the world of DSPs and gives a brief overview of the basic
parts forming a typical DSP system.
Chapter 2 is about the important topic of number systems which describes the various
number systems used in microcontroller and DSP applications.
Chapter 3 is an introduction to the processor architectures used in DSP applications. The
features of various hardware and software development tools are given in this chapter.
Chapter 4 is about “C” programming and describes the features of the popular mikroC PRO
for PIC language and compiler in detail and with many examples.
Chapter 5 is an introduction to the powerful engineering design and simulation tool
MATLAB. The basic programming and graph plotting details are given with examples.
Chapter 6 provides the basic principles of discrete time systems required for the
understanding of DSP systems in general.
Chapter 7 is about the z-transforms and covers the principles of this important topic,
including the inverse transformations.
Chapter 8 is an introduction to the important topic of Discrete Fourier Transforms and gives
examples of finding the transform of some common functions.
Chapter 9 is about the features of digital filters and provides a comparison of various types
of filters used in practice.
Copyright Elektor
Chapter 10 provides the theory of designing FIR digital filters. The design of low-pass and
high-pass filters is given with examples. In addition, the ScopeFIR digital filter design
software package is described with examples.
Chapter 11 provides the theory of designing IIR digital filters. The design of Butterworth
and Chebyshev type digital filters are given with examples. In addition, the ScopeIIR
digital filter design software package is described with examples.
Chapter 12 is about the implementation of FIR digital filters using general purpose
microcontrollers. Various tested and working example projects are given in this chapter,
including the implementation of several different windowed designs.
Chapter 13 is about the implementation of IIR digital filters using general purpose
microcontrollers. Various tested and working examples are given in this chapter, including
the implementation of Butterworth and Chebyshev type filters.
14

Chapter 14 is an introduction to the dedicated DSP processors where the use of
VisualDSP++ software package is given with a simple example.
Finally, Appendix A and Appendix B give tables for the design of Butterworth and
Chebyshev type digital filters respectively. Appendix C provides the architectural details of
the PIC18F45K22 microcontroller used in the projects in this book. Appendix D to
Appendix I give program listings of designed digital filters.
Prof Dr Dogan Ibrahim
London, 2013
Copyright Elektor
15

6.4 | Block Diagram Representation
6.4 Block Diagram Representation
Operations on discrete time sequences can be represented by using block diagrams
as shown in Figure 6-14. Block diagrams can be useful when discrete time
sequences are used in systems with discrete time inputs and outputs.
Figure 6-14 Block diagram representation of discrete time sequences
6.4.1 Block Diagram Manipulation
Block diagram manipulation is useful for deriving the equations at various parts of
the system. An example is given below.
Copyright Elektor
Example 6.1
The block diagram of a discrete time system is shown in Figure 6-15. Derive an
expression for the output y[n].
184

7.8 | Properties of the z-Transform
Z[ax(n)] = aZ[x(n)] = aX[z]
2. Left shift property
Suppose that the z-transform of x[n] is X[z] and let y[n] = x(n + m). Then,
m−1
Y[z] = z m X[Z] − x(i)z m−i
i=0
If the initial conditions are all zero, i.e. x(i) = 0, i = 0,1,2,….,m-1, then,
Z[x(n + m)] = z m X[z]
3. Right shift property
Suppose that the z-transform of x[n] is X[z] and let y[n] = x(n - m). Then,
m−1
Y[z] = z −m X[Z] − x(i − m)z −i
i=0
If the initial conditions are all zero, i.e. x(i) = 0, i = 0,1,2,….,m-1, then,
Z[x(n − m)] = z − m X[z]
4. Attenuation property
Suppose that the z-transform of x[n] is X[z], then,
Z[e −an x(n)] = X[ze a ]
Copyright Elektor
This result states that if a function is multiplied by the exponential e −an
then in the z-transform of this function z is replaced by ze a .
5. Initial value theorem
Suppose that the z-transform of x[n] is X[z], then the initial value of the
time response is given by:
lim x(n) = lim X[z]
n→0 z→∞
6. Final value theorem
Suppose that the z-transform of x[n] is X[z], then the initial value of the
time response is given by:
lim x(n) = lim X[1 −
n→∞ z→1 z−1 ]X[z]
202

Chapter 10 - | Design of FIR Digital Filters
Figure 10-2 shows a graph of the impulse response of the filter.
Figure 10-2
Impulse response of the filter
In practice, it is impossible to realize a filter with the ideal characteristics of Figure
10-1, as the impulse response is not causal and is of infinite length. In order to
develop stable and realizable filters, the frequency response is relaxed and the sharp
edges are removed and replaced with a transition band, and small ripple is allowed
in the pass-band and stop band.
Copyright Elektor
As seen from Figure 10-2, the impulse response decays to either side of the vertical
axis, but is never 0. This is because we have defined the frequency response with
very sharp edges. To realize the ideal filter we would need infinite number of
impulse response coefficients which is not practical. In practice, the small sample
values at the tails of the response are ignored and then h[n] is set to begin from n =
0, so that we have a causal, realizable, linear phase filter. Typically depending upon
the applications between 10 and 100 coefficients are taken in practical applications.
10.2 Truncation and Windowing
As we have seen in Example 10.1, the impulse response of an ideal filter has
infinite number of coefficients and is not realizable. A finite length filter is obtained
from the ideal response by setting all its coefficients to zero outside the range of
− M ≤ n ≤ M , thus giving a filter with a length of N = 2 M + 1. If we then
249

Chapter 13 - | Design of IIR Digital Filters using Microcontrollers
There are basically two methods that we can implement a second-order section in
cascade and these are described below.
In the first method the filter transfer function is assumed to be as follows:
b + b z + b z
H (13.1)
−1
−2
0 1 2
( z)
=
−1
−2
1+
a1z
+ a2
z
Using equation (13.1), the second order section is implemented as shown in Figure
13-4, requiring 2 delays, 5 multiplications, and 2 summations.
Figure 13-4
Implementing a second order section
In the second method, the transfer function is written as in equation (13.2) below.
G(
b + b z + b z )
H (13.2)
−1
−2
0 1 2
( z)
=
−1
−2
1+
a1z
+ a2
z
Copyright Elektor
Where G is the gain and b 0 = 1 and equation (13.2) reduces to
G(1
+ b z + b z )
H (13.3)
−1
−2
1 2
( z)
=
−1
−2
1+
a1z
+ a2
z
Figure 13-5 shows a second order filter implemented using the second method. The
ScopeIIR digital filter design program gives the filter parameters as shown in Figure
13-5.
325

Index
Index
8
8-Bit Mode, 384
A
Butterworth, 19, 271, 365
Butterworth Filters, 273
Butterworth Polynomials, 275
C
A/D converter, 21, 387
A/D converters, 23
Adding Binary Numbers, 42
ADSP-21469 DSP Processor, 344
ADSP-21469 EZ, 80
ADSP-21469 EZ-Board, 347
Analog Filter, 18
Analog-to-digital converters, 173
Anti-aliasing, 18
Aperiodic Digital Signals, 232
Arithmetic format, 66
Arrays, 97
ASCII characters
Extended, 59
Standard, 59
ASCII Table, 58
Audio Blackfin, 78
B
Band-Pass Filter, 310
band-pass filters, 237
BCD numbers, 57
Bessel, 19, 271
Bilinear transformation, 275
Binary Number System, 28
Bit Fields, 104
bit Type, 97
Blackman Window, 256
Bode plotter, 300
Bode Plotter, 301
Bookmarks, 125
Built-in Functions, 122
C language, 87
Cascade FIR, 269
Cascade Implementation, 324
Cascade Structure, 290
Causal, 186
CCS C compiler, 88
Character Arrays, 99
Character Constants, 95
Chebyshev, 19, 271, 371
Chebyshev Filters, 282
Code Assistant, 125
Code Composer, 86
Code Editor Window, 125
Code Explorer Window, 126
Code Template, 125, 126
Comment lines, 139
Comments, 90
Complex Exponential Signal, 177
Connector locations, 350
const, 95
Constant Strings, 100
Constants, 95
Converting Binary, 30, 33, 40
Converting Decimal, 31, 36, 38
Converting Hexadecimal, 34, 35
Converting Octal, 37, 39
Convolution Process, 187
Cost and system complexity, 68
CrossCore, 85
Copyright Elektor
D
D/A Converter, 24
423

Data Memory, 381
Data width, 67
DDSP processor, 66
Debug Agent, 359
Decimal equivalent, 31
Decimal Number System, 28
Dedicated DSP, 85
Dedicated DSP processors, 66
Dedicated DSPs, 343
Development time, 67
DFT
Differentiation, 232
Linearity, 231
Parseval's Relation, 232
Periodicity, 231
Real Sequences, 231
Time delay, 232
Diagram Manipulation, 184
Digital Filters, 237
frequency spectrum, 237
Digital Signal Processing, 17
Digital Signal Processors, 65
Digital-to-analog converter, 173
Direct FIR Structures, 268
Direct Structure, 290
Discrete Fourier Transform, 217
Discrete Impulse Function, 199
Discrete time signals, 173
disp Command, 137
Distinct real roots, 211
Division of Binary Numbers, 45
do – while Statements, 115
DS1843, 21
DSP, 24
DSP algorithms, 56
DSP Application, 18
DSP applications, 195
DSP Hardware, 68
DSP-Enhanced Processor, 75
DSP-enhanced processors, 66
dsPIC33E Starter Kit, 77
E
Copyright Elektor
EasyPIC 7, 295, 296
EasyPIC Fusion V7 Development Board,
76
EasyPIC V7 Development Board, 74
Elliptic Filters, 286
Escape Sequences, 96
Even Order IIR Filter, 335
EXOR, 53
Exponential Function, 197
Exponential signal, 177
Extended ASCII characters, 59
EZ-Kit Lite, 79
F
FFT implementations, 343
Filter Algorithm, 326
Filter Coefficients, 301, 326
Filter order, 241, 273, 283
Filter Program, 328
Filter response, 239
Filter Specifications, 315
Filter type, 241
FilterLab, 20
Finite Impulse Response filter, 238
FIR Digital Filters, 245
FIR Filter Loop, 320
FIR Filters, 238
Fixed Point, 27
Fixed Point Numbers, 27
float, double, 94
Floating Point, 27, 49
Floating Point Constants, 95
for Statements, 115
format Command, 138
Fourier Series, 217
Fourier Transform, 232
Fourier Transform Method, 245
Fraction Expansion, 211
424

Index
Fractional Fixed Point Numbers, 47
function declaration, 118
G
General purpose processors, 66
goto Statement, 118
Graphics Plotting, 157
H
Hamming Window, 252
Hanning Window, 256
Hexadecimal equivalent, 34
Hexadecimal Number System, 29
High-Pass Filter, 331
high-pass filters, 237
HS2420, 21
I
I/O Ports, 383
IDE, 88
IEEE 754, 27
IIR Digital Filters, 271
IIR Filter Loop, 341
IIR Filter Structures, 289
IIR filters, 371
IIR Filters, 238, 337
Illegal names, 92
Impulse function, 175
Infinite Impulse Response filters, 238
input Command, 139
Integer Constants, 95
Interrupt Processing, 121
Interrupts, 395
Inverse z-Transforms, 204
L
Laplace transform, 195, 200
LED connections, 361
LEDs, 361
LF398, 21
Library Manager Window, 127
Linear, 186
Linear Time-invariant, 187
Low-Pass Filter, 295, 323, 332
M
mantissa, 55
MATLAB, 135
matrix operations, 154
MCP4921, 299
Memory, 67
Message Window, 126
Microcontroller, 173, 377
Microcontrollers, 295, 323
mikroC PRO, 83, 84
MPLAB C18, 84
MSC8156ADS StarCore DSP, 82
MT090, 21
Multi-channel, 23
Multi-dimensional array, 99
Multiple order roots, 212
Multiplication of Binary Numbers, 43
Copyright Elektor
N
Negative Numbers, 40
New project window, 128
Non-causal, 186
Non-linear, 186
non-recursive filters, 238
Normalizing, 51
Number Systems, 27
O
Octal equivalent, 38
Octal Number System, 29
425

Offset Binary, 46
Operators, 105
Arithmetic Operators, 105
Bitwise, 107
Conditional, 107
Logical, 107
Preprocessor, 108
Relational, 107
Oscillator Configurations, 382
P
R
random noise, 23
Rectangular Window, 250
Remez Exchange, 260
Repetition Statements, 113
Reserved names, 92
Reserved Names, 92
Reset, 382
S
Parallel converters, 24
Parameter Assistant, 125
Passing Arrays, 120
Passing Parameters, 119
PCSGU250, 300
Periodic Signals, 217
PIC Compiler, 124
PIC IDE, 125
PIC Libraries, 123
PIC Program, 89
PIC variable types, 93
PIC18 Explorer Board, 69
PIC18F45K22, 295, 297
PIC18F45K22 Development Kit, 72
PIC18F4XK20 Starter Kit, 70
PICDEM 4 Demo Board, 71
PIC-USB-STK Development Board, 73
Pointer Arithmetic, 101
Pointers, 100
Power consumption, 68
Program Memory, 380
Project Hardware, 295
Project Manager Window, 127
Q
Q-Format, 47
quantization error, 23
Sampling frequency, 243
Sampling Process, 173
sbit Type, 97
Scalar addition, 182
Scalar division, 182
Scalar multiplication, 182
Scalar subtraction, 182
ScopeFIR, 260
ScopeIIR Filter, 288
Selection Statements, 109
Shannon sampling theorem, 18
Shannon theorem, 174
SHARC family, 345
signed char, 94
signed long int, 94
Signed Magnitude, 45
Sine Function, 198
Sinusoidal Decay, 178
Sinusoidal function, 175
Special Matrices, 156
Speed, 67
Stable, 187
Standard ASCII characters, 59
String Constants, 96
String Operations, 101
Strings, 99
Structures, 102
Subtracting Binary Numbers, 43
Switch Selection Statements, 110
Copyright Elektor
426

Index
Symphony Studio, 86
Syntax, 89
System Classification, 186
T
Time-invariant, 186
Timer Modules, 384
Time-variant, 186
TMS320C5416 DSP, 81
Transfer Function, 271
Truncation, 249
Types
Derived, 93
Fundamental types, 93
U
Unit Ramp Function, 196
Unit step function, 174
Unit Step Function, 195
unsigned int, 94
unsigned long int, 94
Unstable, 187
V
Variable Names, 92
Variable Types, 93
Vector Addition, 181
Vector arithmetic operators, 153
Vector division, 183
Vector multiplication, 183
Vector Subtraction, 181
Vectors in MATLAB, 149
Velleman PCSGU250, 295
VisualDSP++, 356
void, 90
void Functions, 119
W
while Statement, 113
White spaces, 91
Windowing, 249
Z
z-Transform
Attentuation property, 202
Final value theorem, 202
Initial value theorem, 202
iztrans, 204
Left shift property, 202
Linearity property, 201
Right shift property, 202
Z-transform, 195
Z-transforms tables, 199
Copyright Elektor
427