EE482: Digital Signal Processing Applications
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1 Professor Brendan Morris, SEB 3216, EE482: Digital Signal Processing Applications Spring 2014 TTh 14:305:45 CBC C222 Lecture 8 Frequency Analysis 14/02/18
2 2 Outline Fast Fourier Transform Butterfly Structure Implementation Issues
3 DFT Algorithm The Fourier transform of an analogue signal x(t) is given by: The Discrete Fourier Transform (DFT) of a discrete-time signal x(nt) is given by: Where: Chapter 19, Slide 3
4 DFT Algorithm If we let: then: Chapter 19, Slide 4
5 DFT Algorithm x[n] = input X[k] = frequency bins W = twiddle factors X(0) X(1) X(k) X(N) = x[0]w N0 + x[1]w 0*1 N + + x[n]w 0*(N) N = x[0]w N0 + x[1]w 1*1 N + + x[n]w 1*(N) N : = x[0]w N0 + x[1]w k*1 N + + x[n]w k*(n) N : = x[0]w N0 + x[1]w (N)*1 N + + x[n]w (N)(N) N Note: For N samples of x we have N frequencies representing the signal. Chapter 19, Slide 5
6 Performance of the DFT Algorithm The DFT requires N 2 (NxN) complex multiplications: Each X(k) requires N complex multiplications. Therefore to evaluate all the values of the DFT ( X(0) to X(N) ) N 2 multiplications are required. The DFT also requires (N)*N complex additions: Each X(k) requires N additions. Therefore to evaluate all the values of the DFT (N)*N additions are required. Chapter 19, Slide 6
7 Performance of the DFT Algorithm Can the number of computations required be reduced? Chapter 19, Slide 7
8 DFT FFT A large amount of work has been devoted to reducing the computation time of a DFT. This has led to efficient algorithms which are known as the Fast Fourier Transform (FFT) algorithms. Chapter 19, Slide 8
9 DFT FFT [1] x[n] = x[0], x[1],, x[n] Lets divide the sequence x[n] into even and odd sequences: x[2n] = x[0], x[2],, x[n-2] x[2n+1] = x[1], x[3],, x[n] Chapter 19, Slide 9
10 DFT FFT Equation 1 can be rewritten as: [2] Since: Then: Chapter 19, Slide 10
11 DFT FFT The result is that an N-point DFT can be divided into two N/2 point DFT s: N-point DFT Where Y(k) and Z(k) are the two N/2 point DFTs operating on even and odd samples respectively: Two N/2- point DFTs Chapter 19, Slide 11
12 DFT FFT Periodicity and symmetry of W can be exploited to simplify the DFT further: [3] Or: : Symmetry And: Chapter 19, Slide 12 : Periodicity
13 DFT FFT Symmetry and periodicity: W 8 4 W 8 5 W 8 3 W 8 6 W 8 2 W 8 7 W 8 0 =W 8 8 W 8 1 = W 8 9 W k+n/2 N = - W k N W k+n/2 N/2 =W k N/2 W k+4 8 = - W k 8 W k+8 8 =W k 8 Chapter 19, Slide 13
14 DFT FFT Finally by exploiting the symmetry and periodicity, Equation 3 can be written as: [4] Chapter 19, Slide 14
15 DFT FFT Y(k) and W N k Z(k) only need to be calculated once and used for both equations. Note: the calculation is reduced from 0 to N to 0 to (N/2-1). Chapter 19, Slide 15
16 DFT FFT Y(k) and Z(k) can also be divided into N/4 point DFTs using the same process shown above: Chapter 19, Slide 16 The process continues until we reach 2 point DFTs.
17 DFT FFT x[0] x[2] x[4] N/2 point DFT y[0] y[1] y[2] X[0] = y[0]+ z[0] X[1] = y[1]+w 1 z[1] x[n-2] y[n-2] x[1] x[3] x[5] N/2 point DFT z[0] z[1] z[2] W 1 X[N/2] = y[0]- z[0] X[N/2+1] = y[1]-w 1 z[1] x[n] z[n/2] Illustration of the first decimation in time FFT. Chapter 19, Slide 17
18 FFT Implementation To efficiently implement the FFT algorithm a few observations are made: Each stage has the same number of butterflies (number of butterflies = N/2, N is number of points). The number of DFT groups per stage is equal to (N/2 stage ). The difference between the upper and lower leg is equal to 2 stage. The number of butterflies in the group is equal to 2 stage. Chapter 19, Slide 18
19 FFT Implementation Example: 8 point FFT W 1 W 3 Decimation in time FFT: Number of stages = log 2 N Chapter 19, Slide 19 Number of blocks/stage = N/2 stage Number of butterflies/block = 2 stage
20 FFT Implementation Example: 8 point FFT (1) Number of stages: W 1 W 3 Decimation in time FFT: Number of stages = log 2 N Chapter 19, Slide 20 Number of blocks/stage = N/2 stage Number of butterflies/block = 2 stage
21 Stage 1 FFT Implementation Example: 8 point FFT (1) Number of stages: N stages = 1 W 1 W 3 Decimation in time FFT: Number of stages = log 2 N Chapter 19, Slide 21 Number of blocks/stage = N/2 stage Number of butterflies/block = 2 stage
22 Stage 1 FFT Implementation Stage 2 Example: 8 point FFT (1) Number of stages: N stages = 2 W 1 W 3 Chapter 19, Slide 22 Decimation in time FFT: Number of stages = log 2 N Number of blocks/stage = N/2 stage Number of butterflies/block = 2 stage
23 Stage 1 FFT Implementation Stage 2 Stage 3 Example: 8 point FFT (1) Number of stages: N stages = 3 W 1 W 3 Chapter 19, Slide 23 Decimation in time FFT: Number of stages = log 2 N Number of blocks/stage = N/2 stage Number of butterflies/block = 2 stage
24 Stage 1 FFT Implementation Stage 2 Stage 3 Example: 8 point FFT (1) Number of stages: N stages = 3 (2) Blocks/stage: Stage 1: W 1 W 3 Chapter 19, Slide 24 Decimation in time FFT: Number of stages = log 2 N Number of blocks/stage = N/2 stage Number of butterflies/block = 2 stage
25 Block 1 Stage 1 FFT Implementation Stage 2 Stage 3 Example: 8 point FFT (1) Number of stages: N stages = 3 (2) Blocks/stage: Stage 1: N blocks = 1 W 1 W 3 Chapter 19, Slide 25 Decimation in time FFT: Number of stages = log 2 N Number of blocks/stage = N/2 stage Number of butterflies/block = 2 stage
26 Block 1 Block 2 Stage 1 FFT Implementation Stage 2 Stage 3 Example: 8 point FFT (1) Number of stages: N stages = 3 (2) Blocks/stage: Stage 1: N blocks = 2 W 1 W 3 Chapter 19, Slide 26 Decimation in time FFT: Number of stages = log 2 N Number of blocks/stage = N/2 stage Number of butterflies/block = 2 stage
27 Block 1 Block 2 Block 3 Stage 1 FFT Implementation Stage 2 Stage 3 W 1 Example: 8 point FFT (1) Number of stages: N stages = 3 (2) Blocks/stage: Stage 1: N blocks = 3 W 3 Chapter 19, Slide 27 Decimation in time FFT: Number of stages = log 2 N Number of blocks/stage = N/2 stage Number of butterflies/block = 2 stage
28 Block 1 Block 2 Block 3 Stage 1 FFT Implementation Stage 2 Stage 3 W 1 Example: 8 point FFT (1) Number of stages: N stages = 3 (2) Blocks/stage: Stage 1: N blocks = 4 Block 4 W 3 Chapter 19, Slide 28 Decimation in time FFT: Number of stages = log 2 N Number of blocks/stage = N/2 stage Number of butterflies/block = 2 stage
29 Block 1 Stage 1 FFT Implementation Stage 2 Stage 3 W 1 Example: 8 point FFT (1) Number of stages: N stages = 3 (2) Blocks/stage: Stage 1: N blocks = 4 Stage 2: N blocks = 1 W 3 Chapter 19, Slide 29 Decimation in time FFT: Number of stages = log 2 N Number of blocks/stage = N/2 stage Number of butterflies/block = 2 stage
30 Block 1 Block 2 Stage 1 FFT Implementation Stage 2 Stage 3 W 1 Example: 8 point FFT (1) Number of stages: N stages = 3 (2) Blocks/stage: Stage 1: N blocks = 4 Stage 2: N blocks = 2 W 3 Chapter 19, Slide 30 Decimation in time FFT: Number of stages = log 2 N Number of blocks/stage = N/2 stage Number of butterflies/block = 2 stage
31 Block 1 Stage 1 FFT Implementation Stage 2 Stage 3 W 1 Example: 8 point FFT (1) Number of stages: N stages = 3 (2) Blocks/stage: Stage 1: N blocks = 4 Stage 2: N blocks = 2 Stage 3: N blocks = 1 W 3 Chapter 19, Slide 31 Decimation in time FFT: Number of stages = log 2 N Number of blocks/stage = N/2 stage Number of butterflies/block = 2 stage
32 Chapter 19, Slide 32 Stage 1 FFT Implementation Stage 2 Stage 3 W 1 W 3 Decimation in time FFT: Number of stages = log 2 N Number of blocks/stage = N/2 stage Number of butterflies/block = 2 stage Example: 8 point FFT (1) Number of stages: N stages = 3 (2) Blocks/stage: Stage 1: N blocks = 4 Stage 2: N blocks = 2 Stage 3: N blocks = 1 (3) B flies/block: Stage 1:
33 Chapter 19, Slide 33 Stage 1 FFT Implementation Stage 2 Stage 3 W 1 W 3 Decimation in time FFT: Number of stages = log 2 N Number of blocks/stage = N/2 stage Number of butterflies/block = 2 stage Example: 8 point FFT (1) Number of stages: N stages = 3 (2) Blocks/stage: Stage 1: N blocks = 4 Stage 2: N blocks = 2 Stage 3: N blocks = 1 (3) B flies/block: Stage 1: N btf = 1
34 Chapter 19, Slide 34 Stage 1 FFT Implementation Stage 2 Stage 3 W 1 W 3 Decimation in time FFT: Number of stages = log 2 N Number of blocks/stage = N/2 stage Number of butterflies/block = 2 stage Example: 8 point FFT (1) Number of stages: N stages = 3 (2) Blocks/stage: Stage 1: N blocks = 4 Stage 2: N blocks = 2 Stage 3: N blocks = 1 (3) B flies/block: Stage 1: N btf = 1 Stage 2: N btf = 1
35 Chapter 19, Slide 35 Stage 1 FFT Implementation Stage 2 Stage 3 W 1 W 3 Decimation in time FFT: Number of stages = log 2 N Number of blocks/stage = N/2 stage Number of butterflies/block = 2 stage Example: 8 point FFT (1) Number of stages: N stages = 3 (2) Blocks/stage: Stage 1: N blocks = 4 Stage 2: N blocks = 2 Stage 3: N blocks = 1 (3) B flies/block: Stage 1: N btf = 1 Stage 2: N btf = 2
36 Chapter 19, Slide 36 Stage 1 FFT Implementation Stage 2 Stage 3 W 1 W 3 Decimation in time FFT: Number of stages = log 2 N Number of blocks/stage = N/2 stage Number of butterflies/block = 2 stage Example: 8 point FFT (1) Number of stages: N stages = 3 (2) Blocks/stage: Stage 1: N blocks = 4 Stage 2: N blocks = 2 Stage 3: N blocks = 1 (3) B flies/block: Stage 1: N btf = 1 Stage 2: N btf = 2 Stage 3: N btf = 1
37 Chapter 19, Slide 37 Stage 1 FFT Implementation Stage 2 Stage 3 W 1 W 3 Decimation in time FFT: Number of stages = log 2 N Number of blocks/stage = N/2 stage Number of butterflies/block = 2 stage Example: 8 point FFT (1) Number of stages: N stages = 3 (2) Blocks/stage: Stage 1: N blocks = 4 Stage 2: N blocks = 2 Stage 3: N blocks = 1 (3) B flies/block: Stage 1: N btf = 1 Stage 2: N btf = 2 Stage 3: N btf = 2
38 Chapter 19, Slide 38 Stage 1 FFT Implementation Stage 2 Stage 3 W 1 W 3 Decimation in time FFT: Number of stages = log 2 N Number of blocks/stage = N/2 stage Number of butterflies/block = 2 stage Example: 8 point FFT (1) Number of stages: N stages = 3 (2) Blocks/stage: Stage 1: N blocks = 4 Stage 2: N blocks = 2 Stage 3: N blocks = 1 (3) B flies/block: Stage 1: N btf = 1 Stage 2: N btf = 2 Stage 3: N btf = 3
39 Chapter 19, Slide 39 Stage 1 FFT Implementation Stage 2 Stage 3 W 1 W 3 Decimation in time FFT: Number of stages = log 2 N Number of blocks/stage = N/2 stage Number of butterflies/block = 2 stage Example: 8 point FFT (1) Number of stages: N stages = 3 (2) Blocks/stage: Stage 1: N blocks = 4 Stage 2: N blocks = 2 Stage 3: N blocks = 1 (3) B flies/block: Stage 1: N btf = 1 Stage 2: N btf = 2 Stage 3: N btf = 4
40 FFT Implementation Stage 1 Stage 2 Stage 3 W 1 W 3 Start Index Input Index Twiddle Factor Index N/2 = 4 Chapter 19, Slide 40
41 FFT Implementation Stage 1 Stage 2 Stage 3 W 1 W 3 Start Index Input Index Twiddle Factor Index N/2 = 4 4/2 = 2 Chapter 19, Slide 41
42 FFT Implementation Stage 1 Stage 2 Stage 3 W 1 W 3 Start Index Input Index Twiddle Factor Index N/2 = 4 4/2 = 2 2/2 = 1 Chapter 19, Slide 42
43 FFT Implementation Stage 1 Stage 2 Stage 3 W 1 W 3 Start Index 0 Input Index 1 Twiddle Factor Index N/2 = 4 4/2 = 2 2/2 = 1 Indicies Used W Chapter 19, Slide 43 W 3
44 44 FFT Decimation in Frequency Similar divide and conquer strategy Decimate in frequency domain / / / / Divide into first half and second half of sequence / 2 / / / / / 2 1 / /
45 45 FFT Decimation in Frequency Structure Stage structure Full structure Bit reversal happens at output instead of input
46 46 Inverse FFT Notice this is the DFT with a scale factor and change in twidle sign Can compute using the FFT with minor modifications Conjugate coefficients, compute FFT with scale factor, conjugate result For real signals, no final conjugate needed Can complex conjugate twidle factors and use in butterfly structure
47 47 FFT Example Example 5.10 Sine wave with 50Hz sin 0,1,, Hz Frequency resolution of DFT? Δ / 2Hz Location of peak 50 Δ 25
48 48 Spectral Leakage and Resolution Notice that a DFT is like windowing a signal to finite length Longer window lengths (more samples) the closer DFT approximates DTFT Convolution relationship Corruption of spectrum due to window properties (mainlobe/sidelobe) Sidelobes result in spurious peaks in computed spectrum known as spectral leakage Obviously, want to use smoother windows to minimize these effects Spectral smearing is the loss in sharpness due to convolution which depends on mainlobe width Example 5.15 Two close sinusoids smeared together To avoid smearing: Frequency separation should be greater than freq resolution, /Δ
49 49 Power Spectral Density Parseval s theorem - power spectrum or periodogram Power spectral density (PSD, or power density spectrum or power spectrum) is used to measure average power over frequencies Computed for time-varying signal by using a sliding window technique Short-time Fourier transform Grab samples and compute FFT Must have overlap and use windows Spectrogram Each short FFT is arranged as a column in a matrix to give the time-varying properties of the signal Viewed as an image She had your dark suit in greasy wash water all year
50 50 Fast FFT Convolution Linear convolution is multiplication in frequency domain Must take FFT of signal and filter, multiply, and ifft Operations in frequency domain can be much faster for large filters Requires zero-padding because of circular convolution Typically, will do block processing Segment a signal and process each segment individually before recombining
51 51 Ex: FFT Effect of N Take FFT of cosine using different N values n = [0:29]; x = cos(2*pi*n/10); N1 = 64; N2 = 128; N3 = 256; X1 = abs(fft(x,n1)); X2 = abs(fft(x,n2)); X3 = abs(fft(x,n3)); F1 = [0 : N1-1]/N1; F2 = [0 : N2-1]/N2; F3 = [0 : N3-1]/N3; subplot(3,1,1) plot(f1,x1,'-x'),title('n = 64'),axis([ ]) subplot(3,1,2) plot(f2,x2,'-x'),title('n = 128'),axis([ ]) subplot(3,1,3) plot(f3,x3,'-x'),title('n = 256'),axis([ ]) Transforms all have the same shape Difference is the number of samples used to approximate the shape Notice the sinusoid frequency is not always well represented Depends on frequency resolution
52 52 Ex: FFT Effect of Number of Samples Select a large value of N and vary the number of samples of the signal n = [0:29]; x1 = cos(2*pi*n/10); % 3 periods x2 = [x1 x1]; % 6 periods x3 = [x1 x1 x1]; % 9 periods N = 2048; X1 = abs(fft(x1,n)); X2 = abs(fft(x2,n)); X3 = abs(fft(x3,n)); F = [0:N]/N; subplot(3,1,1) plot(f,x1),title('3 periods'),axis([ ]) subplot(3,1,2) plot(f,x2),title('6 periods'),axis([ ]) subplot(3,1,3) plot(f,x3),title('9 periods'),axis([ ]) Transforms all have the same shape Looks like sinc functions More samples makes the sinc look more impulse-like FFT with large N but fewer samples does zero-padding E.g. taking length N signal and windowing with box Multiplication in time is convolution in frequency
53 53 Spectrum Analysis with FFT and Matlab FFT does not directly give spectrum Dependent on the number of signal samples Dependent on the number of points in the FFT FFT contains info between 0, Spectrum must be below /2 Symmetric across 0axis, Use fftshift.m in Matlab n = [0:149]; x1 = cos(2*pi*n/10); N = 2048; X = abs(fft(x1,n)); X = fftshift(x); F = [-N/2:N/2]/N; plot(f,x), xlabel('frequency / f s')
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