Fast Fourier Transforms on Motorola's Digital Signal Processors

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Figure 2-2 shows the result of windowing a sine wave by a rectangular window. Windowing causes the following errors: 1. Leakage ⎯ Even though the input signal consists of a single-frequency component at 1000 Hz, the result clearly shows components at frequencies other than 1000 Hz. This is called the leakage effect: it appears as if energy has “leaked” from 1000 Hz to the rest of the spectrum. 2. Smoothing ⎯ Although the theoretical transform exhibits an infinitely narrow, and infinitely large peak at 1000 Hz, the actual peak has finite magnitude and exhibits finite width. It appears that the narrow peak has been “smeared” out in the frequency domain as a result of the windowing function in the time domain. This effect is appropriately termed the smoothing effect. 3. Ripple ⎯ The overall magnitude plot in Figure 2-2 shows an oscillatory character not present in the original <strong>Fourier</strong> transform: this is called the ripple effect. The origin of the ripple effect lies in the discontinuity (abrupt start and end) introduced in the signal by the window. Windows with more gradual transitions generally have lower sidelobes and less ripple. In general, a tradeoff exists between these different effects, and the advantages of an appropriate windowing function can be chosen for a specific application. For an excellent summary of existing windowing functions and their properties, see Reference 2. MOTOROLA 2-5
xf () x(nT) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9 -1 0.0026 0.0024 0.0022 0.002 0.0018 0.0016 0.0014 0.0012 0.001 0.0008 0.0006 0.0004 0.0002 0 Time Function 0 10 20 30 40 Sample number <strong>Fourier</strong> Transform Magnitude Ideal Actual 0 1 2 3 4 (Thousands) f(Hz) Figure 2-2 Windowing Effects When Windowing a Single Sine 2-6 MOTOROLA
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: 2.2 Windowing and Windowing Effects
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 and 34: Binary Index 000 x(0) 001 010 011 1
Page 35 and 36: 3.4 The Decimation-in- Frequency Ra
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
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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
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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) +
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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
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8.2.1.1 Complex FFT on Floating-Poi
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8.2.2 FFT on Fixed-Point DSPs As me
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“Motorola's family of digital sig
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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
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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.
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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
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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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