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Sample Spectrum Number 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Harm. No. .310 .694 .473 .515 .339 .334 .171 .142 -.013 -.060 -.209 -.277 -.419 -.517 -.652 -.830 C 0 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 C 8 1.000 -1.000 1.000 -1.000 1.000 -1.000 1.000 -1.000 1.000 -1.000 1.000 -1.000 1.000 -1.000 1.000 -1.000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 C 1 1.000 .924 .707 .383 .000 -.383 -.707 -.924 -1.000 -.924 -.707 -.383 .000 .383 .707 .924 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 0 .... S 1 .000 .383 .707 .924 1.000 .924 .707 .383 .000 -.383 -.707 -.924 -1.000 -.924 -.707 -.383 .500 (l .... .000 .192 .353 .462 .500 .462 .353 .192 .000 -.192 -.353 -.462 -.500 -.462 -.353 -.192 C 2 1.000 .707 .000 -.707 -1.000 -.707 .000 .707 1.000 .707 .000 -.707 -1.000 -.707 .000 .707 .037 ~ t-' -.037 .026 .000 -.026 -.037 -.026 .000 -.026 -.037 -.026 .000 -.026 -.037 -.026 .000 .026 S 2 .000 .707 1.000 .707 .000 -.707 -1.000 -.707 '.000 .707 1.000 .707 .000 -.707 -1.000 -.707 .247 Ql .000 .175 .247 .175 .000 -.175 -.247 -.175 .000 .175 .247 .175 .000 -.175 -.247 -.175 Z C 3 1.000 .383 -.707 -.924 .000 .924 .707 -.383 -1.000 -.383 .707 .924 .000 -.924 -.707 -.383 .049 tT1 .049 .019 -.035 -.045 .000 .045 .035 -.019 -.049 -.019 .035 .045 .000 -.045 -.035 -.019 Q S 3 .000 .924 .707 -.383 -1.000 -.383 .707 .924 .000 -.924 -.707 .383 1.000 .383 -.707 -.924 .159 tT1 .000 .147 .112 -.061 -.159 -.061 .112 .147 .000 -.147 -.112 .061 .159 .061 -.112 -.147 Z C 4 1.000 .000 -1.000 .000 1.000 .000 -;1.000 .000 1.000 .000 -1.000 .000 1.000 .000 -1.000 .000 .054 .054 .000 -.054 .000 .054 .000 -.054 .000 .054 .000 -.054 .000 .054 .000 -.054 .000 tT1 ~ S 4 .000 1.000 .000 -1.000 .000 1.000 .000 -1.000 .000 1.000 .000 -1.000 .000 1.000 .000 -1.000 .113 (5 .000 .113 .000 -.113 .000 .113 .000 -.113 .000 .113 .000 -.113 .000 .113 .000 -.113 C 5 1.000 -.383 -.707 .924 .000 -.924 .707 .383 -1.000 .383 .707 -.924 .000 .924 -.707 -.383 .056 Z ..., .056 -.021 -.040 .052 .000 -.052 .040 .021 -.056 .021 .040 -.052 .000 .052 -.040 -.021 tT1 S 5 .000 .924 -.707 -.383 1.000 -.383 -.707 .924 .000 -.924 .707 .383 -1.000 .383 .707 -.924 .083 il .000 .077 -.059 -.032 .083 -.032 -.059 .077 .000 -.077 .059 .032 -.083 .032 .059 -.077 :r: C 6 1.000 -.707 .000 .707 -1.000 .707 .000 -.707 1.000 -.707 .000 .707 -1.000 .707 .000 -.707 .057 Ẓ ... .057 -.040 .000 .040 -.057 .040 .000 -.040 .057 -.040 .000 .040 -.057 .040 .000 -.040 I:) .000 .707 -1.000 .707 .000 -.707 1.000 -.707 .000 .707 -1.000 .707 .000 -.707 1.000 -.707 C S 6 .061 .000 .043 -.061 .043 .000 -.043 .061 -.043 .000 .043 -.061 .043 .000 -.043 .061 -.043 tT1 [fJ C 7 1.000 -.924 .707 -.383 .000 .383 -.707 .924 -1.000 .924 -.707 .383 .000 -.383 .707 -.924 .056 .056 -.052 .040 -.021 .000 .021 -.040 .052 -.056 .052 -.040 .021 .000 -.021 .040 -.052 S 7 .000 .383 -.707 .924 -1.000 .924 -.707 .383 .000 -.383 .707 -.924 1.000 -.924 .707 -.383 .044 .000 .017 -.031 .041 -.044 .041 -.031 .017 .000 -.017 .031 -.041 .044 -.041 .031 -.017 Fig. 13-11. (Cont.) (C) Worksheet filled in for inverse Fourier transform. VI "'"
446 MUSICAL ApPLICATIONS OF MICROPROCESSORS mere 28 instead of the original 256! Although it is not immediately apparent how the redundancies can be identified in a general algorithm, one can use matrix theory and trigonometric identities and derive the FFT algorithm from this rype of table. A completely different way to look at the redundancy involves the concept of decimation. As an example, consider a sample record with N samples where N is even. Using the SFT, one could perform N2 operations and wind up with the spectrum. Now consider splitting the record into two smaller records with NI2 samples in each such that the even numbered samples go to one record and the odd numbered ones to go the other. If an SFT is performed on each record, it will require N 2 /4 operations for a total of only N 2 /2 operations. The trick is to combine the two resulting spectrums into one representing the true spectrum of the original record. This can be accomplished by duplicating the even sample spectrum ofN14 harmonics and adding with a progressive phase shift the odd sample spectrum, also duplicated, to yield N12 harmonics total. The combination requires N extra multiplications and additions. If N is divisible by 4, the decimation operation could be repeated to give (our records of N14 samples each. The SFT can be performed on each, requiring only N 2 /4 operations, and the four resulting spectrums combined in two stages to form the final spectrum. In fact, if N is a power of two, decimation can be repeated until there are NI2 subrecords, each having only two samples as in Fig. 13-12. Note the resulting scrambled order of the samples. This is called bit-reversed order because ifthe binary representation of the scrambled sample numbers is observed in a mirror, it will appear to be a simple ascending binary sequence. Since the discrete Fourier transform of two samples is trivial (the cosine component of the single harmonic is the sum of the sample values, while the sine component is the difference), most of the computation is combining the subspectra together in stages to form the final spectrum. The essence of the FFT, then, is complete decimation of the time samples followed by recombination of the frequency specrra to form the exact spectrum of the original sample set. Complex Arithmetic At this point in the description of the FFT we will be forced to commit a mathematical "copout" that is customary in such a discussion. In any case, it will not only simplify the discussion but will facilitate the development of an FFT program. The first step is to treat each harmonic, which has a cosine and a sine component, as a single complex number. Although the cosine component is normally called the "real" part and the sine component the "imaginary" part, both are quite real as far as a physical waveform is concerned. Therefore, it is best to forget about "real" and "imaginary" and
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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 7 Increa
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MUSIC SYNTHESIS PRINGPLES 9 VERY SM
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MUSIC SYNTHESIS PRINOPLES 11 repres
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MUSIC SYNTHESIS PRINCIPLES 13 The w
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MUSIC SYNTHESIS PRlNCIPLES 15 was c
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MUSIC SYNTHESIS PRlNOPlES 17 In act
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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 33 from
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MUSIC SYNTHESIS PRINCIPLES 35 ever,
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MUSIC SYNTHESIS PRINCIPLES 37 have
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MUSIC SYNTHESIS PRINCIPLES 39 No li
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MUSIC SYNTHESIS PRINCIPLES 41 cropr
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44 MUSICAL ApPLICATIONS OF MICROPRO
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60 MUSICAL ApPLICATrONS OF MICROPRO
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68 MUSICAL ApPLICAnONS OF MICROPROC
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82 MUSICAL ApPUCATIONS OF MICROPROC
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84 MUSICAL ApPUCATIONS OF MICROPROC
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vvv·(B) 86 MUSICAL ApPLICATIONS OF
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100 MUSICAL ApPLICAnONS OF MICROPRO
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102 MUSICAL ApPLICATIONS OF MICROPR
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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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Table 5-4. 68000 Instruction Set Su
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Table 5-4. 68000 Instruction Set Su
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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 201 12 ~ E 1-
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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 215 +20 +15 +1
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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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300 MUSICAl ApPLICATIONS OF MICROPR
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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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Page 407 and 408: 394 MUSICAL ApPLICATIONS OF MICROPR
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DIGITAL FILTERING 495 OUTPUT Fig. 1
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DIGITAL FILTERING 497 INPUT ~8~~~~~
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DIGITAL FILTERING 499 the turnover
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DIGITAL FILTERING 501 frequency is
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+ 3 1 oL-_-...l__---l__---L__---r-_
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DIGITAL FILTERING 505 When programm
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DIGITAL FILTERING 507 lowest freque
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DIGITAL FILTERING 509 Even with a p
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DIGITAL FILTERING 511 OUTPUT Fig. 1
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DIGITAL FILTERING 513 INPUT VARIABL
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DIGITAL FILTERING 515 INPUT (C) -(.
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DIGITAL FILTERING 517 NOTE: Xl, Y,
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DIGITAL FILTERING 519 AMPLITUDE, (d
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DIGITAL FILTERING 521 o ~ -10 ~ -20
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DIGITAL FILTERING 523 1.0 0.8 0.6
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DIGITAL FILTERING 525 When the outp
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16 Source-SignalAnalys,is One of th
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SOURCE-SIGNAL ANALYSIS 545 nOdB T I
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--------, I I ! II i I I I I , , i"
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SOURCE-SIGNAL ANALYSIS 549 3 IN -"
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SOURCE-SIGNAL ANALYSIS 551 Data Rep
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SOURCE-SIGNAL ANALYSIS 553 SIGNAL B
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SOURCE-SIGNAL ANALYSIS 555 -5° FH
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SOURCE-SIGNAL ANALYSIS 557 -5 -10 ~
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SOURCE-SIGNAL ANALYSIS 559 Since th
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SOURCE-SIGNAL ANALYSIS 561 o -5 -10
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SOURCE-SIGNAL ANALYSIS 563 o -5 -10
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SOURCE-SIGNAL ANALYSIS 565 overlapp
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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 571 componen
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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 761 i
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LOW-COST SYNTHESIS TECHNIQUES 763 0
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LOW-COST SYNTHESIS TECHNIQUES 765 S
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LOW-COST SYNTHESIS TECHNIQUES 767 c
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LOW-COST SYNTHESIS TECHNIQUES 769 O
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LOW-COST SYNTHESIS TECHNIQUES 771 +
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LOW-COST SYNTHESIS TECHNIQUES 773 F
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LOW-COST SYNTHESIS TECHNIQUES 775 F
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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 781 T
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LOW-COST SYNTHESIS TECHNIQUES 783 t
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LOW-COST SYNTHESIS TECHNIQUES 785 "
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LOW-COST SYNTHESIS TECHNIQUES 787 m
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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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Musical Applications of Microproces
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