Fast Fourier Transforms on Motorola's Digital Signal Processors

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x(t) xf () 1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 Time Function 0.0 E+00 2.0E-04 4.0E-04 6.0E-048.0E-041.0E-031.2E-031.4E-031.6E-031.8E-03 2.0E-03 t(s) <strong>Fourier</strong> Transform Magnitude 1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 Ideal 0.3 0.2 T = 0.1 ms 0.1 0.0 0.5 ms 0.0 E+00 2.0 E+00 4.0 E+00 f (kHz) 6.0 E+00 8.0 E+00 Figure 2-1 <strong>Fourier</strong> Transform of a Rectangular Function MOTOROLA 2-3
2.2 Windowing and Windowing Effects The discussion of aliasing errors illustrates how the <strong>Fourier</strong> transform can be approximated by an infinite summation. In practice, the results need to be available within a finite time period, and the infinite summation must somehow be reduced to a finite summation. One obvious way to reduce the infinite summation is by simply truncating the sum in Eqn. 2-2 to N terms as: X˜ w f j2πfnT () T χnT ( )e – N– 1 = Eqn. 2-6 This truncation is frequently referred to as “windowing” because an infinite summation is viewed through a finite window. The resulting transform is called the windowed discrete-time <strong>Fourier</strong> transform (WDTFT). In mathematical terms, windowing is simply the multiplication of the signal by a window sequence of finite-length, w(n). In the simple case above, w(n)=1 for 0 ≤n ≤N – 1; otherwise, w(n)=0. Because of its rectangular shape, the window shown above is called the rectangular window. Unless the signal in question is of finite duration, this truncation will introduce other errors, resulting in a number of artifacts in the spectrum. To assess the effect of the windowing operation, a simple sine wave of the form: Eqn. 2-7 is sampled with a sampling frequency of 4000 Hz, and the windowed DTFT is computed with N=20. 2-4 MOTOROLA ∑ n=0 χ() t = sin( 2π1000t)
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: ecause: j2πfnT+j2πn e T - ⎛ ---
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 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
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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
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Eight multiplications, five additio
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further: 1. Since the input data is
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6.4 The Goertzel Algorithm Previous
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The power of magnitude of the DFT c
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SSI or HI The double buffering is i
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“To implement Eqn. 7-1, a two dim
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F( k) where: N - 1 2c( k) ---------
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7.2.2 Two Dimensional DCT A one dim
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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:
;**********************************
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;**********************************
Page 125 and 126:
movem 2,m5 movem 2,m7 move #data,r0
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; ------------ NORMAL RADIX-2 BUTTE
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nop nop jmp * end ; ; Sine-Cosine T
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; Input signal for FFT rfft56.asm a
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move #POINTS/2-1,m0 ;modulo address
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; r0->A,r1->B,r4->A’,r5->B’,r6-
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_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
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B.2 Real FFT for DSP56001/2 ; ; Thi
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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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