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DIGITAL FILTERING 517 NOTE: Xl, Y, X2, Y2 X3, Y3 X4, V4 GIVEN FDA 3rd DEGREE POLYNOMIAL INTERPOLATION CUglC FORMULA FOR X SPACING ~ 1.0 Vu"" - 1/6 Yo[(XU~Xl) (XU-X2) (XU-X3)] + 1/2 Y,[(Xo-Xo) (Xc-X,) (Xc-X'I] - 1/2 y,[(Xc-Xo) (Xc-X,) (Xc-X'I] + i/6 Y3[(XU-Xo) (Xu-X,) (XU-X2)] Xu, Yu 0= UNKNOWN POINT Xo,Vo Xl,V, X2 .Y2 X3,YJ = FOUR TABULATED POINTS SURROUNDING THE UNKNOWN POINT GENERAL POLYNOMIAL INTERPOLATION FORMULA N N Yo " L YK [J [X~:: ~'J K ~O I ~ 0 I" K WHERE N "" DEGREE OF POLYNOMIAL = NUMBER OF POINTS TO CONSIDER MINUS 1 XI,VK = TABULATED POINTS iXI SPACING = 1.0) Xc,Yo ~ UNKNOWN POINT Fig. 14-26. Third- and fifth-degree interpolation polynorr,ials cubic is used with four data points, the procedure is called a cubic spline, which is detailed in Fig. 14-26 and compared with a fifth-degree interpolation polynomial.
518 MUSICAL ApPLICATIONS OF MICROPROCESSORS SAMPLE STREAM AT RATE F2 Fig. 14-27. Sample-rate conversion Interpolation Filters In digital synthesis, one most often needs to interpolate between sample points in a single string ofsamples. One excellent application, which will be used throughout the remaining discussion, is sample-rate conversion. This is the case in which one has a string of samples taken at, say, 25 ksls and wishes to convert it to, say, 28.371 ks/s. If it is true that the 25 ks/s samples captured all of the information in the originallow-pass-filtered signal, then the conversion should be possible with no degradation of the signal whatsoever. Downward conversion is also possible provided the signal is lowpass filtered to less than one-half the new sample rate. Ptobably the first indination in solving the rate-conversion problem is to linearly interpolate between the input samples as required to obtain the output samples. Unfortunately, this simple method generates an unacceptably large amount of noise. Cubic and quintic splines are better, but the noise is still excessive when high-frequency signals are being converted. These simple procedures fail primarily because the sample points are sparse compared with the variability of the curve before it was sampled, even though we know that they are dense enough for accurate reconstruction of the curve by a low-pass filter. Figure 14-27 illustrates another samp1e-rate-changing process based on reconstruction of the original waveform and resampling at a different rate. The digital sample stream is converted into analog form at the original sample rate by the DAC and filtered by the low-pass filter. The filtered waveform is then resampled at the converted sample rate to provide the result. The low-pass filter cutoff should be less than one-half of the lower of the two sample rates. By studying the diagram, it is apparent that the filter actually does the interpolation and the digital-analog-digital conversion is just overhead. What is desired is a way to simulate this sample-rateconversion system completely in the digital domain. Earlier it was learned that the output waveform ofa filter is actually the superposition of impulse responses, one for each input sample. Figure 14-28 shows the ideal low-pass impulse response, where the cutoff frequency is exactly one-half of an arbitrary sample frequency. Note that the response is zero at all sample times except the one that corresponds to the input impulse!
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Musical Applications of Microproces
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© 19B5 by Hayden Books A Division
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serve to acquaint the general reade
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SECfrON II. Computer-Controlled Ana
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17. Digital Hardware 589 AnalogModu
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1 Music Synthesis Principles Creati
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MUSIC SYNTHESIS PRINCIPLES 5 own ar
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MUSIC SYNTHESIS PRINCIPLES 19 SIN (
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MUSIC SYNTHESIS PRINCIPLES 21 lILt!
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MUSIC SYNTHESIS PRINCIPLES 23 - ,--
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MUSIC SYNTHESIS PRINOPLES 25 + or--
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MUSIC SYNTHESIS PRINCIPLES 27 Human
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MUSIC SYNTHESIS PRINOPLES 29 fundam
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MUSIC SYNTHESIS PRINCIPLES 31 frequ
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MUSIC SYNTHESIS PRINCIPLES 39 No li
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vvv·(B) 86 MUSICAL ApPLICATIONS OF
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100 MUSICAL ApPLICAnONS OF MICROPRO
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Table 5-1. 6502 Instruction Set Lis
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Coding Table 5-3. 68000 Addressing
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6 BasicAnalogModules Probably the m
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BASIC ANALOG MODULES 179 tage remai
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BASIC ANALOG MODULES 181 Op-amps ma
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BASIC ANALOG MODULES 183 '0: :? u
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BASIC ANALOG MODULES 185 with an id
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BASIC ANALOG MODULES 187 EXPONENTIA
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BASIC ANALOG MODULES 189 (or 0.9766
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BASIC ANALOG MODULES 191 4R Vr'o'M
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BASIC ANALOG MODULES 193 tooth with
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BASIC ANALOG MODULES 195 +IOVrel PU
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BASIC ANALOG MODULES 197 Table 6-1.
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BASIC ANALOG MODULES 199 constants
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BASIC ANALOG MODULES 203 R2 R3 100
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BASIC ANALOG MODULES 205 fore, must
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BASIC ANALOG MODULES 207 For peak p
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BASIC ANALOG MODULES 209 22 pF RI S
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BASIC ANALOG MODULES 211 INPUT~OUTP
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BASIC ANALOG MODULES 213 Voltage-Tu
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BASIC ANALOG MODULES 217 100 ko. BA
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BASIC ANALOG MODULES 219 +15 V0----
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Digital-to-Analog and tlnalog-to-Di
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DIGITAL-TO-ANALOG AND ANALOG-TO-DIG
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11 CORtrolSequeRce Display andEditi
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CONTROL SEQUENCE DISPLAY AND EDITIN
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SECTIONIII DigitalSynthesis antl So
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Table 12·1. Design Data for Chebys
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~ o Table 12·1. (Cont.) Ripple = 1
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13 Digital Tone Generation Teehniqu
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DIGITAL TONE GENERATION TECHNIQUES
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Sample Number Harm. No. o 2 3 4 5 6
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Sample Spectrum Number 0 1 2 3 4 5
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SOURCE-SIGNAL ANALYSIS 567 • ....
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SOURCE-SIGNAL ANALYSIS 569 from an
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SOURCE-SIGNAL ANALYSIS 573 and some
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SOURCE-SIGNAL ANALYSIS 575 185491 A
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SOURCE-SIGNAL ANALYSIS 577 SYSTEM F
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SOURCE-SIGNAL ANALYSIS 579 (AI (BI
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SOURCE-SIGNAL ANALYSIS 581 (Gl II'
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SOURCE-SIGNAL ANALYSIS 583 PULSE PE
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SOURCE-SIGNAL ANALYSIS 585 again, t
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SOURCE-SIGNAL ANALYSIS 587 Quite ob
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17 DigitalHardware At this point, w
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DIGITAL HARDWARE 591 Lsa I N+I DIVI
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DIGITAL HARDWARE 593 (~OCK[ L _ LSB
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DIGITAL HARDWARE 595 FREQUENCY CONT
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DIGITAL HARDWARE 597 Fig. 17-7. Pha
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DIGITAL HARDWARE 599 MOST SIGNIFICA
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DIGITAL HARDWARE 601 waveform chang
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DIGITAL HARDWARE 603 As an example,
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DIGITAL HARDWARE 605 1. Sixteen ind
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DIGITAL HARDWARE 607 discussed, the
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DIGITAL HARDWARE 609 between the 41
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DIGITAL HARDWARE 611 8-BIT PORT I M
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FREOUENCY CONTROL IN I I 20 FREOUEN
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DIGITAL HARDWARE 615 TO MULTIPLEXED
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FREQ. :a.-8 CNTL. WRITE FREQ. I 20
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DIGITAL HARDWARE 619 effective freq
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DIGITAL HARDWARE 621 The signal mem
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DIGITAL HARDWARE 623 into a filter
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DIGITAL HARDWARE 625 made using con
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DIGITAL HARDWARE 627 A Hybrid Voice
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DIGITAL HARDWARE 629 o -5 -10 -15 ~
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DIGITAL HARDWARE 631 o -5 -10 -15 ~
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DIGITAL HARDWARE 633 D3 02 01 10 00
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DIGITAL HARDWARE 635 plementers on
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DIGITAL HARDWARE 637 taken care of
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SECTIONIV ProductApplications andth
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716 MUSICAL ApPLICATIONS OF MJCROPR
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RING MOD VCD A PITCHo--+lcNTL OUT A
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20 Low-CostSYllthesis Techniques Wh
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LOW-COST SYNTHESIS TECHNIQUES 759 g
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LOW-COST SYNTHESIS TECHNIQUES 777 f
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21 Future Frequently, when glvmg ta
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LOW-COST SYNTHESIS TECHNIQUES 785 "
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LOW-COST SYNTHESIS TECHNIQUES 789 a
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Bibliography The following publicat
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BIBLIOGRAPHY 793 DMA DOS FET FFT FI
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(Communication) Minor - Computer Sc
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newsletter, writing numerous articl
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conversion subsystem, especially if
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In the beginning the company was su
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HAL : I really haven't kept up with
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was clear he was serious, I named m
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influences too like interrupts from
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SONIK : What activities do you do o
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as a homework assignment. So much o
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