Fast Fourier Transforms on Motorola's Digital Signal Processors

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;************************************************************************** ; ; Split N/2 Complex FFT(Hn) for N real FFT(Fn) ; SPLIT56 macro IDATA,COEF,POINTS,ODATA SPLIT56 ident 1,0 ; ; ; Fi=0.5(Hi+Hn/2-i*)-0.5j(Hi-Hn/2-i*)W i=0,1,,,N-1 ; ; Bit reverse input, Normal order output ; This macro amplifies coefficients of FFT by 2. ; If absolute values of spectrum are desired, then scaling up factor is 2^(N-1), ; assuming inputs are scaled by 2^N before complex FFT. ; POINTS is the number of real data /2 ; COEF is twiddle factor location other than TF used in complex FFT (see sincosr) ; ; move #POINTS-1,n0 ;number of complex FFT input data -1 move #POINTS/2-1,n2 ;loop counter move #ODATA,r0 ;r0 ptr to Ar=Hi move #COEF-POINTS/2+1,r2 ;twiddle factor start location move r2,r6 ;r6 -> Wi lea (r0)+n0,r5 ;r5 ptr to Br & Bi move #IDATA,r3 ;r3 pointer for A’ move r3,r4 move n0,r1 ;r1 ptr for B’, r1=B move #POINTS/2,n0 move n0,n5 move m5,m3 ;m3 and m1 linear address move m5,m1 move #0,m0 ;bit reverse address move m0,m5 ;bit reverse address move x:(r0),b ;b=Ar0 move x:(r5),x1 y:(r0),a ;a=Ai0,x1=Br add a,b x:(r1)+,x0 ;b=Ar0+Ai0=DC, for ptr reason inc r1 subl b,a r3,r4 ;r4 ptr to temp location asl b y:(r1),a ;a=something asl a b,x:(r3)+ y:(r5),b ;a=Niquist=Ar0-Ai0, save DC,b=Bi move a,y:(r0)+n0 ;save Niq in y:ODATA temp, move y:(r0),y0 ;y0=Ai do n2,_end_split add y0,b y0,a a,y:(r1)- ;b=Ai+Bi=H2r,a=Ai, save prev. Bi’ subl b,a x:(r0),b b,y1 ;a=Ai-Bi=H1i, b=Ar, y1=H2r sub x1,b x:(r0)+n0,a a,y:(r4) ;b=Ar-Br=H2i,a=Ar again, ;save H1i temp,r0->nA subl b,a x:(r2)+,x1 y:(r6)+,y0 ;a=Ar+Br=H1r,x1=Wr,y0=Wi mac x1,y1,a b,x0 a,y:(r5) ;a=H1r+Wr*H2r,x0=H2i,save H1r temp macr y0,x0,a y:(r5)-n5,b ;a=H1r+Wr*H2r-Wi*H2i=Ar’, b=H1r subl a,b a,x:(r3) y:(r4),a ;b=H1r-(Wr*H2r-Wi*H2i)=Br’, ;a=H1i,save Ar’ mac -x1,x0,a b,x:(r1) y:(r5),b ;a=H1i-Wr*H2i,save Br’,b=nBi Figure B-2 Real FFT for DSP56001/2 (sheet 4 of 5) B-8 MOTOROLA
macr y1,y0,a y:(r4),y0 ;a=Wi*H2r-Wr*H2i+H1i=Ai’,y0=H1i again sub y0,a x:(r5),x1 a,y:(r3)+ ;a=Wi*H2r-Wr*H2i, x1=nBr,save Ai’ sub y0,a y:(r0),y0 ;a=Wi*H2r-Wr*H2i-H1i=Bi’,y0=nAi _end_split move y0,a a,y:(r1) ;save last Bi’,conjugate last Ai neg a #ODATA,r5 move #IDATA,r0 move a,y:(r4) move y:(r5),a move a,y:(r0) ;move Niq. back endm ;split56 Figure B-2 Real FFT for DSP56001/2 (sheet 5 of 5) MOTOROLA B-9
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Motorola’s High-Performance DSP T
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SECTION 1 Definition and History SE
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SECTION 5 Optimizing Performance of
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APPENDIX B Real-Valued Input FFT Ta
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Figure 4-3 Figure 4-4 Figure 4-5 Fi
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Table 8-1 Table 8-2 Table 8-3 Table
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When interpreted as an infinite sum
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2. Pattern-Based ⎯ Many problems
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In summary, the basic nature of the
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ecause: j2πfnT+j2πn e T - ⎛ ---
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2.2 Windowing and Windowing Effects
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xf () x(nT) 1 0.9 0.8 0.7 0.6 0.5 0
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Note that many textbooks simply def
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“Since there are two independent
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appropriately called the decimation
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x(0) x(4) Figure 3-4 Decimation-in-
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Binary Index 000 x(0) 001 010 011 1
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3.4 The Decimation-in- Frequency Ra
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DSP56001/2 and DSP96002 hardware fe
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4. Bit-reversed addressing mode 5.
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4.3 Complexity of a Radix-2 DIT FFT
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The data paths are 24 bits wide, th
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The kernel shown in Figure 4-4 exec
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;Alters Program Control Registers ;
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PORT A CLK 4 ADDRESS 32 Control 19
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;r0 ➨ A ;r1 ➨ B ;r4 ➨ C ;r5
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5 5 7 ADDRESS Sigma Delta Codec PI/
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limits its results to those bits. T
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flow in the FFT calculation is to s
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of the sine table, 256 points. Howe
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normal_order=output_pointer; bitrev
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A′ A CW c BW b DW c W b = + + ( +
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;r0->A,r4->B, r1->C, r6->D; ;r1->A
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Ar′ = Ar + Br + ( Dr + Cr) Ai′
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“Optimization saves . . . 2067 in
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plexity to practical complexity ref
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5.1.2 Optimization for Fast
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3. pass. This change results in lon
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5.2 Example of Optimization 5.2.1 F
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Pass 0 1 2 3 4 5 6 7 8 A B C D W=(1
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The fully optimized 1024-point comp
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6.1 Real-Valued Input FFT Algorithm
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The twiddle factors appear to be in
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X=A+jB Y=C+jD X’=A+C+j(D+B) - Y
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takes four points in and four point
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Z( k) = DFT[ z( n) ] = DFT[ x( n) +
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F( k) a real sequence with N points
Page 95 and 96: Eight multiplications, five additio
Page 97 and 98: further: 1. Since the input data is
Page 99 and 100: 6.4 The Goertzel Algorithm Previous
Page 101 and 102: The power of magnitude of the DFT c
Page 103 and 104: SSI or HI The double buffering is i
Page 105 and 106: “To implement Eqn. 7-1, a two dim
Page 107 and 108: F( k) where: N - 1 2c( k) ---------
Page 109 and 110: 7.2.2 Two Dimensional DCT A one dim
Page 111 and 112: facturer may offer a higher perform
Page 113 and 114: 8.2.1.1 Complex FFT on Floating-Poi
Page 115 and 116: 8.2.2 FFT on Fixed-Point DSPs As me
Page 117 and 118: “Motorola's family of digital sig
Page 119 and 120: APPENDIX A Fully Optimized Complex
Page 121 and 122: ;**********************************
Page 123 and 124: ;**********************************
Page 125 and 126: movem 2,m5 movem 2,m7 move #data,r0
Page 127 and 128: ; ------------ NORMAL RADIX-2 BUTTE
Page 129 and 130: nop nop jmp * end ; ; Sine-Cosine T
Page 131 and 132: ; Input signal for FFT rfft56.asm a
Page 133 and 134: move #POINTS/2-1,m0 ;modulo address
Page 135 and 136: ; r0->A,r1->B,r4->A’,r5->B’,r6-
Page 137 and 138: _inner_pass do n3,_end_grp ;do grou
Page 139 and 140: APPENDIX B Real-Valued Input FFT B.
Page 141 and 142: count set 0 dup points/2 dc @cos(@c
Page 143 and 144: B.2 Real FFT for DSP56001/2 ; ; Thi
Page 145: dup points/4 dc @cos(@cvf(count)*fr
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