Fast Fourier Transforms on Motorola's Digital Signal Processors

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x(0) x(1) x(2) x(3) x(4) x(5) x(6) x(7) Figure 3-9 Decimation-in-Frequency concept x(0) x(1) x(2) x(3) x(4) x(5) x(6) x(7) + + + + + + + + + - + - + - + - 1/N 2/N 3/N N/2 - point 3-8 MOTOROLA DFT N/2 - point DFT + + + + + + + + + + - + + + - + - + - + - + - + + + + + - + - Figure 3-10 Complete 8-point radix-2 DIF FFT 2/N 2/N + + + - k/N x(0) x(2) x(4) x(6) x(1) x(3) x(5) x(7) + + x(0) + - x(4) + + x(2) + - x(6) + + x(1) + - x(5) + + x(3) + - x(7) DIF Butterfly
3.4 The Decimation-in- Frequency Radix-2 <strong>Fast</strong> <strong>Fourier</strong> <strong>Transforms</strong> If Eqn. 2-10 is decomposed from the frequency domain, we can show the following equations exist: ( N⁄ 2) – 1 j X N ( 2k) ∑ [ xr () + xr ( + N⁄ 2) ] e r = 0 Eqn. 3-2 2πrk – --------------- N⁄ 2 = ( N⁄ 2) – 1 j X N ( 2k + 1) ∑ [ xr ()x – ( r + N ⁄ 2) ] e r = 0 2πrk 2 π r – --------------- – j ----------- N⁄ 2 N = e Eqn. 3-3 The decimation in frequency butterfly is shown in Figure 3-9. ■ MOTOROLA 3-9
Page 1 and 2: Motorola’s High-Performance DSP T
Page 3 and 4: SECTION 1 Definition and History SE
Page 5 and 6: SECTION 5 Optimizing Performance of
Page 7 and 8: APPENDIX B Real-Valued Input FFT Ta
Page 9 and 10: Figure 4-3 Figure 4-4 Figure 4-5 Fi
Page 11 and 12: Table 8-1 Table 8-2 Table 8-3 Table
Page 13 and 14: When interpreted as an infinite sum
Page 15 and 16: 2. Pattern-Based ⎯ Many problems
Page 17 and 18: In summary, the basic nature of the
Page 19 and 20: ecause: j2πfnT+j2πn e T - ⎛ ---
Page 21 and 22: 2.2 Windowing and Windowing Effects
Page 23 and 24: xf () x(nT) 1 0.9 0.8 0.7 0.6 0.5 0
Page 25 and 26: Note that many textbooks simply def
Page 27 and 28: “Since there are two independent
Page 29 and 30: appropriately called the decimation
Page 31 and 32: x(0) x(4) Figure 3-4 Decimation-in-
Page 33: Binary Index 000 x(0) 001 010 011 1
Page 37 and 38: DSP56001/2 and DSP96002 hardware fe
Page 39 and 40: 4. Bit-reversed addressing mode 5.
Page 41 and 42: 4.3 Complexity of a Radix-2 DIT FFT
Page 43 and 44: The data paths are 24 bits wide, th
Page 45 and 46: The kernel shown in Figure 4-4 exec
Page 47 and 48: ;Alters Program Control Registers ;
Page 49 and 50: PORT A CLK 4 ADDRESS 32 Control 19
Page 51 and 52: ;r0 ➨ A ;r1 ➨ B ;r4 ➨ C ;r5
Page 53 and 54: 5 5 7 ADDRESS Sigma Delta Codec PI/
Page 55 and 56: limits its results to those bits. T
Page 57 and 58: flow in the FFT calculation is to s
Page 59 and 60: of the sine table, 256 points. Howe
Page 61 and 62: normal_order=output_pointer; bitrev
Page 63 and 64: A′ A CW c BW b DW c W b = + + ( +
Page 65 and 66: ;r0->A,r4->B, r1->C, r6->D; ;r1->A
Page 67 and 68: Ar′ = Ar + Br + ( Dr + Cr) Ai′
Page 69 and 70: “Optimization saves . . . 2067 in
Page 71 and 72: plexity to practical complexity ref
Page 73 and 74: 5.1.2 Optimization for Fast
Page 75 and 76: 3. pass. This change results in lon
Page 77 and 78: 5.2 Example of Optimization 5.2.1 F
Page 79 and 80: Pass 0 1 2 3 4 5 6 7 8 A B C D W=(1
Page 81 and 82: The fully optimized 1024-point comp
Page 83 and 84: 6.1 Real-Valued Input FFT Algorithm
Page 85 and 86:
The twiddle factors appear to be in
Page 87 and 88:
X=A+jB Y=C+jD X’=A+C+j(D+B) - Y
Page 89 and 90:
takes four points in and four point
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Z( k) = DFT[ z( n) ] = DFT[ x( n) +
Page 93 and 94:
F( k) a real sequence with N points
Page 95 and 96:
Eight multiplications, five additio
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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
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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 and 146:
dup points/4 dc @cos(@cvf(count)*fr
Page 147 and 148:
macr y1,y0,a y:(r4),y0 ;a=Wi*H2r-Wr
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Fast Fourier Transforms on Motorola's Digital Signal Processors

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