Chapter 8 The Discrete Fourier Transform
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1 Chapter 8 The Discrete Fourier Transform
2 Introduction Representation of periodic sequences: the discrete Fourier series Properties of the DFS The Fourier transform of periodic signals Sampling the Fourier transform Fourier representation of finite-duration sequences: the discrete Fourier transform Properties of the DFT Linear convolution using the DFT The discrete cosine transform (DCT) 2018/9/18 DSP 2
3 What is Discrete Fourier Transform (DFT)? A linear transformation (matrix) Samples of the Fourier transform of an aperiodic (with finite duration) sequence Extension of Discrete Fourier Series (DFS) 2018/9/18 DSP 3
4 time continuous discrete aperiodic Fourier transform discrete-time Fourier transform continuous x(t) = (1/2p) X(jw)e jwt dw x[n] = (1/2p) p p X(e jw )e jwn dw X(jw) = x(t) e -jwt dt X(e jw ) = S x[n] e -jwn periodic Fourier series discrete Fourier transform discrete x p (t) = S X[k] e jkwt x[n] = (1/N)S k=0 N-1 X[k]W N -kn X[k] = (1/T) 0T x p (t) e -jkwt dt X[k] = S n=0 N-1 x[n]w N kn periodic aperiodic frequency 2018/9/18 DSP 4
5 2018/9/18 DSP 5
6 Representation of Periodic Sequences: The Discrete Fourier Series Consider a periodic sequence x p [n] with period N, so that x p [n] = x p [n + rn] for any integer values of n and r. Such a sequence can be represented by a Fourier series corresponding to a sum of harmonically related complex exponential sequences. The periodic complex exponentials e n e e n rn j(2 p / N ) kn k[ ] k[ ] Fourier series representation 1 N ~ ~ (2 / ) [ ] [ ] j p x n X k e N kn k 2018/9/18 DSP 6
7 Representation of Periodic Sequences: The Discrete Fourier Series The harmonically related complex exponentials are identical for values of k separated by N e n e e e e n j(2 p / N )( k ) (2 / ) (2 / ) [ ] ln n j p N kn j p N kn kln k[ ] The Fourier series representation contain only N of these complex exponentials ~ N 1 1 ~ x[ n] X[ k] e N k 0 j(2 p / N ) kn To obtain the sequence of Fourier series coefficients Xk [ ] 1 x[ n] e X[ k] e N N 1 ~ N 1 N 1 ~ j(2 p / N ) rn j(2 p / N )( k r) n n0 n0 k0 ~ 2018/9/18 DSP 7
8 Representation of Periodic Sequences: The Discrete Fourier Series Interchange the order of summation 1 x[ n] e X[ k] e N N 1 ~ N 1 ~ N 1 j(2 p / N ) rn j(2 p / N )( k r) n n0 k 0 n0 Orthogonality of the complex exponentials: 1 N N 1 n0 e j(2 p / N )( k r) n 1, k r mn, m an integer 0, Otherwise Equation (8.6) becomes N 1 ~ ~ j(2 p / N ) rn n0 x[ n] e X[ r] 2018/9/18 DSP 8
9 The Fourier series X[ k] are obtained from x[ n] ~ N 1 ~ X[ k] x[ n] e n0 The sequence ~ N 1 ~ n0 j(2 p / N ) kn X[ k N] x[ n] e ~ Xk [ ] is periodic with period N j(2 p / N )( k N ) n N 1 ~ ~ j(2 p / N ) kn j2pn x[ n] e e X[ k] n0 For convenience in notation WN e j(2 p / N) 2018/9/18 DSP 9
10 DFS analysis-synthesis pair The tilde in indicates a periodic signal. is periodic of period N. 2018/9/18 DSP 10
11 2018/9/18 DSP 11
12 2018/9/18 DSP 12
13 2018/9/18 DSP 13
14 Magnitude and phase of the FS 2018/9/18 DSP 14
15 Properties of the Discrete Fourier Series Periodic sequence (period N) DFS coefficient (period N) x p [n] X p [k] periodic with period N x p1 [n], x p2 [n] X p1 [k], X p2 [k] periodic with period N ax p1 [n] + bx p2 [n] ax p1 [k] + bx p2 [k] Linearity x p [n-m] W km N X p [k] Shift of a sequence W -ln N x p [n] X p [k-l] Shift of a DFS coefficient X [n] Nx [-k] Duality S N-1 m=0 x p1 [m]x p2 [n-m] X p1 [k]x p2 [k] Periodic convolution x p1 [n]x p2 [n] (1/N)S N-1 l =0 X p1 [l]x p2 [k-l] Periodic convolution x p *[n] X p *[-k] x p *[-n] X p *[k] Re{x p [n]} X pe [k] = ½ (X p [k] + X p *[-k]) jim{x p [n]} X po [k] = ½ (X p [k] X p *[k]) x pe [n] = ½ (x p [n] + x p *[-n]) Re{X p [k]} x pe [n] = ½ (x p [n] + x p [-n]) Re{X p [k]} when x[n] is real x po [n] = ½ (x p [n] x p *[-n]) jim{x p [k]) x po [n] = ½ (x p [n] x p [-n]) jim{x p [k]) when x[n] is real Symmetry properties for x p [n] real X p [k] = X p *[k] Re{X p [k]} = Re{X p [-k]} Im{X p [k]} = -Im{X p [-k]} X p [k] = X p [-k] X p [k] = -X p [-k] 2018/9/18 DSP 15
16 2018/9/18 DSP 16
17 2018/9/18 DSP 17
18 8.2.5 Periodic Convolution Let x [ n] and 1 2 ~ ~ denoted by X 1[ k] and X 2[ k],respectively. ~ ~ ~ X 3 k X 1 k X 2 k The periodic sequence ~ N 1 ~ ~ x3 n x1 m x2 n m m0 ~ x3 n [ ] is [ ] [ ] [ ] periodic convolution Periodic convolution is commutative, ~ N 1 ~ ~ x3 n x2 m x1 n m [ ] [ ] [ ] m0 x [ n] be two periodic sequences, each with period N and with discrete Fourier series coefficients [ ] [ ] [ ] 2018/9/18 DSP 18
19 Periodic Convolution Apply the DFS analysis equation (8.11) to obtain ~ N1 N1 ~ ~ kn X 3[ k] x1[ m] x2[ n m] WN n0 m0 Interchange the order of summation ~ N1 ~ N1 ~ kn X 3[ k] x1[ m] x2[ n m] WN m0 n0 ~ The inner sum on the index n is the DFS for x2[ n m] N 1 ~ ~ kn km x2[ n m] WN WN X 2[ k] n0 2018/9/18 DSP 19
20 Periodic Convolution ~ N1 ~ ~ N1 ~ ~ ~ ~ km km X 3[ k] x1[ m] WN X 2[ k] x1[ m] WN X 2[ k] X 1[ k] X 2[ k] m0 m0 In summary N 1 ~ ~ DFS ~ ~ m0 x [ m] x [ n m] X [ k] X [ k] The duality theorem ~ ~ ~ x3[ n] x1[ n] x2[ n] 1 [ ] [ ] [ ] N ~ N 1 ~ ~ X 3 k X 1 l X 2 k l l0 2018/9/18 DSP 20
21 Periodic Convolution Periodic convolution is different from aperiodic convolution. The sum is over the finite interval 0 m N-1. ~ The values of x2[ n m] in the interval 0 m N-1 repeat periodically for m outside of that interval. The multiplication of the DFS coefficients of two periodic sequences corresponds to a periodic convolution of the sequences. 2018/9/18 DSP 21
22 Example 8.4 Periodic Convolution 2018/9/18 DSP 22
23 2018/9/18 DSP 23
24 2018/9/18 DSP 24
25 The Fourier Transform of Periodic Signals If x[n] is periodic with period N and the corresponding DFSs are Xk [ ], then the Fourier transform of x[n] is defined to be the impulse train jw 2p 2pk X ( e ) X [ k] w ( ) N N k jw To show that X( e ) is a FT representation of the periodic sequence xn [ ] 1 2p 1 2p jw jwn 2p 2pk jwn X ( e ) e dw X [ k] ( w ) e dw 2p 0 2p 0 N N k Interchange the order of integration and summation 1 2p jw jwn 1 2p 0 X ( e ) e dw X [ k] N 1 N k N 1 k0 X[ k] e 2p 0 j(2 p / N ) kn 2p k ( w ) e N Only the impulses corresponding to k 0,1, 2,...( N -1) are included. jwn d w 2018/9/18 DSP 25
26 Although the Fourier transform of a periodic sequence does not converge in the normal sense, the introduction of impulses permits us to include periodic sequences formally within the framework of Fourier transform analysis. 2018/9/18 DSP 26
27 2018/9/18 DSP 27
28 Consider a finite-length signal x[n] such that x[n]=0 except in the interval 0 n N-1, and consider the convolution of x[n] with the periodic impulse train p[n] x[ n] x[ n] p[ n] x[ n] [ n rn] r x[ n rn] r The Fourier transform of x[n] is jw jw jw jw 2p 2pk X ( e ) X ( e ) P( e ) X ( e ) w [ ] N N r We conclude that X k X e X e r 2p 2pk X e N N j(2 p / N ) k ( ) w [ ] j(2 p / N ) k jw [ ] ( ) ( ) w(2 p / N) k 2018/9/18 DSP 28
29 Periodic sequence formed by repeating a finite-length sequence 2018/9/18 DSP 29
30 Example 8.6 Relationship between the Fourier series coefficients and the Fourier transform of one period 2018/9/18 DSP 30
31 Sampling the Fourier Transform 2018/9/18 DSP 31
32 Sampling the Fourier Transform 2018/9/18 DSP 32
33 Sampling the Fourier Transform 2018/9/18 DSP 33
34 Sampling the Fourier Transform 2018/9/18 DSP 34
35 Sampling the Fourier Transform 2018/9/18 DSP 35
36 2018/9/18 DSP 36
37 If x[n] has finite length and we take a sufficient number of equally spaced samples of its Fourier Transform ( a number greater than or equal to the length of x[n]), then x[n] is recoverable from 2018/9/18 DSP 37
38 2018/9/18 DSP 38
39 The Discrete Fourier Transform 2018/9/18 DSP 39
40 2018/9/18 DSP 40
41 2018/9/18 DSP 41
42 Sequence x[n] with N = 5. Periodic sequence of X[n] with period N = 5. Fourier series coefficients X k for periodic sequence. DFT magnitude of x[n]. 2018/9/18 DSP 42
43 Sequence x[n] with N = 10. Periodic sequence of x[n] with period N = 10. DFT magnitude DFT phase. 2018/9/18 DSP 43
44 Properties of the Discrete Fourier Transform Finite-length sequence (length N) N-points DFT (length N) x[n], x 1 [n], x 2 [n] X[k], X 1 [k], X 2 [k] a x 1 [n] + b x 2 [n] ax 1 [n] + bx 2 [n] Linearity X[n] Nx[((-k)) N ] Duality x[((n-m)) N ] W km N X[k] Circular shift of a sequence W -ln N x[n] X[((k-l)) N ] S N-1 m=0 x 1 [m]x 2 [((n-m)) N ] X 1 [k]x 2 [k] Circular convolution x 1 [n]x 2 [n] (1/N)S N-1 l=0 X 1 [l]x 2 [((k-l)) N ] x*[n] X*[((-k)) N ] x*[((-n)) N ] X*[k] Re{x[n]} X ep [k] = ½ {X[((k)) N ] + X*[((-k)) N ] jim{x[n]} X op [k] = ½ {X[((k)) N ] - X*[((-k)) N ] x ep [n] = ½ {x[n] + x*[((-n)) N ] Re{X[k]} x ep [n] = ½ {x[n] + x[((-n)) N ] Re{X[k]} when x[n] is real x op [n] = ½ {x[n] - x*[((-n)) N ] jim{x[k]} x op [n] = ½ {x[n] - x[((-n)) N ] jim{x[k]} when x[n] is real Symmetry properties, when x[n] is real X[k] = X*[((-k)) N ] Re{X[k]} = Re{X[((-k)) N ] x[((n)) N ] = x [n] is a periodic sequence. Im{X[k]} = -Im{X[((-k)) N ] X[((k)) N ] = X [k] is a periodic sequence. X[k] = X[((-k)) N ] 2018/9/18 {X[k]} = -{X[((-k)) N ]} DSP 44
45 Property of the DFT 2018/9/18 DSP 45
46 Example 8.8: Circular shift of a sequence 2018/9/18 DSP 46
47 2018/9/18 DSP 47
48 Duality Real finite-length sequence x[n]. Real part of corresponding DFT X[k]. Imaginary part of X[k]. Real part of the dual sequence X[n]. Imaginary part of X[n]. The DFT of X[n] = x[k]. 2018/9/18 DSP 48
49 2018/9/18 DSP 49
50 2018/9/18 DSP 50
51 Circular Convolution 2018/9/18 DSP 51
52 Circular Convolution 2018/9/18 DSP 52
53 Example 8.10: Circular convolution with a delayed impulse sequence 2018/9/18 DSP 53
54 Example: Circular convolution of two rectangular pulses with 5 values each. Correction of the result by using Zero padding with L zero values, N = 2L Result is not correct when it is compared with that obtained by linear convolution. 2018/9/18 DSP 54
55 Circular Convolution 2018/9/18 DSP 55
56 8.7 Linear Convolution Using the Discrete Fourier Transform Because the Fast Fourier Transform (FFT) algorithms are available, it is computationally efficient to consider implementing a Convolution of Two Finite-Length Sequences by the following procedure: Compute the N-point DFT X 1 [k] and X 2 [k] of the two sequences x 1 [n] and x 2 [n], respectively. Compute the product X 3 [k] = X 1 [k] X 2 [k] for 0 k N-1. Compute the sequence x 3 [n] = x 1 [n] IDFT of X 3 [k]. x 2 [n] as the Circular Convolution as Linear Convolution with Aliasing N 2018/9/18 DSP 56
57 Linear convolution of two finite-length sequences 2018/9/18 DSP 57
58 Linear convolution of two finite-length sequences x 1 [n] s length L points. x 2 [n] s length P points. x 3 [n] = S m=- x1[m]x2[n-m], has its maximum length L+P-1 points. Thus, x 3 [n] = x 1 [n] N x 2 [n], where N L+P /9/18 DSP 58
59 Example 8.12: Circular convolution as linear convolution with aliasing Time aliasing in the circular convolution of two finite-length sequences can be avoided if N L+P-1. It should be clear that if the circular convolution is of sufficient length relative to the lengths of the sequences x 1 [n] and x 2 [n], then aliasing with nonzero values can be avoided, in which case the circular and linear convolutions will be identical. 2018/9/18 DSP 59
60 Example of linear convolution of two finite-length sequences. 2018/9/18 DSP 60
61 2018/9/18 DSP 61
62 2018/9/18 DSP 62
63 Finite-length Implementing Linear impulse Time-Invariant response h[n] Systems and indefinite-length Using the DFT: signal x[n] to be filtered. Theoretically, we store the entire samples and then implement the convolution procedure using a DFT for a large number points which is generally impractical to compute. No filter samples can be computed until all the input samples have been collected. Generally, we would like to avoid such a large delay in processing. The solution of both problems is to use block convoltuion. 2018/9/18 DSP 63
64 Block Convolution Techniques All of samples will be segmented into section of appropriate length (L). Each section can then be convolved with the finite-length impulse response and the filtered sections fitted together in an appropriate way. Overlap-Add Method Overlap-Save Method 2018/9/18 DSP 64
65 2018/9/18 DSP 65
66 Overlap-add method is the procedure of decomposition of x[n] into nonoverlapping sections of length L and the result of convolving each section with h[n] which are overlapped and added to construct the output. 2018/9/18 DSP 66
67 2018/9/18 DSP 67
68 Overlap-save method is the procedure of decomposition of x[n] into overlapping sections of length L and the result of convolving each section with h[n] which the portions of each filtered section to be discarded in forming the linear convolution. 2018/9/18 DSP 68
69 Input signal L L L L L L P-1 zeros x 1 [n] x 2 [n] x 1 [n] P-1 zeros x 2 [n] P-1 zeros P-1 Overlap x 3 [n] x 3 [n] Output signal P-1 points add together Discard P-1 points Overlap-save method P-1 points add together Overlap-add method 2018/9/18 DSP 69
70 8.8 The Discrete Cosine Transform (DCT) 2018/9/18 DSP 70
71 8.8.1 Definition of the DCT The DCT is a transform in the form of Eqs. (8.147) and (8.148) with basis sequences that are cosines. The DFT involves an implicit assumption of periodicity The DCT involves implicit assumptions of both periodicity and even symmetry 2018/9/18 DSP 71
72 8.8.1 Definition of the DCT In the development of the DFT, we represented finite-length sequences by first forming periodic sequences from which the finite-length sequence can be uniquely recovered and then utilizing an expansion in terms of periodic complex exponentials. The DCT corresponds to forming a periodic, symmetric sequence from a finite-length sequence in such a way that the original finite-length sequence can be uniquely recovered. 2018/9/18 DSP 72
73 8.8.1 Definition of the DCT 2018/9/18 DSP 73
74 8.8.1 Definition of the DCT 1. x [ n] has period (2N-2)=6 and has even symmetry 1 about both n=0 and n=(n-1)=3 2. x [ n] has period 2N=8 and has even symmetry 2 about the "half sample" points n=-1/2 and 7/2 3. x [ n] has period 3 about both n=0 and n=8 4 4N=16 and has even symmetry 4. x [ n] has period 4N and has even symmetry about the "half sample" points n=-1/2 and n=(2n-1/2)=15/2 2018/9/18 DSP 74
75 8.8.2 Definition of the DCT-1 and DCT /9/18 DSP 75
76 2018/9/18 DSP 76
77 2018/9/18 DSP 77
78 2018/9/18 DSP 78
79 8.8.3 Relationship between the DFT and the DCT /9/18 DSP 79
80 Energy Compaction Property of the DCT /9/18 DSP 80
81 Energy Compaction Property of the DCT /9/18 DSP 81
82 Applications of the DCT The major application of the DCT-2 is in signal compression. The popularity of the DCT in signal compression is mainly due to its energy concentration property. 2018/9/18 DSP 82
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