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DIGITAL TONE GENERATION TECHNIQUES 453 each other. This is important because the actual number of multiplications required to evaluate the node pair can then be cut in half. After the first column of nodes has been processed, everything is back in storage in a partially transformed state, and processing of the next column of nodes can begin. When the rightmost column has been evaluated, the array contains the N complex harmonics. Note that for the first column the nodes that should be paired for calculation are adjacent. In the next column, the pair members are two apart, in the third column four apart, and so forth. Also note that the mass of crossing lines forms a group of two nodes in the first column, four nodes in the next column and so forth until there is only one mass in the final column. These and other symmetries discovered by examining the diagram should make the progression sufficiently clear so that generalization for other values ofN can be understood. Since the progression is consistent from column to column, it is clear that a general FFT subroutine can be written for any value of N that is a power of two. Table 13-1 shows the actual sequence of intermediate results corresponding to the diagram. The example time samples are of a seven-harmonic approximation to a sawtooth wave with a few degrees of phase shift included to make life interesting. Note that the spectrum output starts with the dc component in the zero array position, fundamental in position 1, etc. up to the seventh harmonic in position 7. The Nyquist component is in position 8 (the sine part is always zero), while higher array positions contain the first through the seventh harmonics again but in descending sequence and with the sign of the imaginary components (sines) flipped. Also note that the correct amplitudes of the frequency components have been multiplied by N/2 except for the dc component, which is N times too large. An FFT Subroutine in BASIC Figure 13-17 is an FFT program written in BASIC. Although BASIC and fast are contradictory terms, the program serves to illustrate the algorithm and to allow the reader to become familiar with it. The array of complex input samples is actually stored and handled as two arrays of normal numbers; D 1 being the real part and D2 being the imaginary part. Note that the array subscript starts at zero. When the transformation is complete, the spectrum is stored in the same two arrays. To insure that N is a power of two, the record size is specified by K, which is 10gzN and must be a positive integer. In the program itself, statements 3110 to 3125 scramble the naturally ordered time samples into bit-reversed order using the method illustrated earlier. N3 being the "for" loop index counts naturally from 1 to N. Nl in conjunction with statements 3114 to 3117 acts like a bit-reversed counter taking on successive values of 0, 8, 4, 12, 2, etc. (for the N= 16 case). Whenever N 1 is greater than N3, the samples pointed to by N 1 and N3 are swapped.
454 MUSICAL ApPLICATIONS OF MICROPROCESSORS 3000 REM FAST FOURIER TRANSFORM USING TIME DECOMPOSITION WITH 3001 REM INPUT BIT REVERSAL 3002 REM COMPLEX INPUT DATA IN ARRAY D1 (REAL PART) AND D2 (IMAGINARY 3003 REM PART). 3004 REM COMPUTATION IS IN PLACE, OUTPUT REPLACES INPUT 3005 REM K SPECIFIES NUMBER OF POINTS, K=LOG(2)N 3100 REM SCRAMBLE THE INPUT DATA INTO BIT REVERSED ORDER 3110 N=2**K 3111 N1=0 3112 N2=N-1 3113 FOR N3=1 TO N2 3114 N4=N 3115 N4=N4/2 3116 IF N1+N4>N2 GOTO 3115 3117 N1=N1-INT(N1/N4)*N4+N4 3118 IF N1
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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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Table 12·1. Design Data for Chebys
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~ o Table 12·1. (Cont.) Ripple = 1
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Page 432 and 433: DIGITAL TONE GENERATION TECHNIQUES
Page 456 and 457: Sample Number Harm. No. o 2 3 4 5 6
Page 458 and 459: Sample Spectrum Number 0 1 2 3 4 5
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Page 494 and 495: 14 DigitalFiltering In analog synth
Page 496 and 497: DIGITAL FILTERING 483 Input Samples
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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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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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