Finite-length Discrete Transforms Chapter 5 Sections 5 2 Chapter 5, - PowerPoint PPT Presentation

Finite-length Discrete Transforms Chapter 5 Sections 5 2 Chapter 5, Sections 5.2 - 5.10 5 10 D I Dr. Iyad Jafar d J f

Outline Outline � The Discrete Fourier Transform (DFT) � Matrix Representation of DFT � Finite-length Sequences � Circular Convolution � DFT Symmetry Properties � DFT Theorems 2

The Discre The Discrete F e Fourier T urier Transf ansform (DFT) orm (DFT) � In practice, we usually deal with finite-length sequences, x[n], 0 ≤ n ≤ N -1 � N 1 N 1 � ∑ � � � j j n � In this case, the DTFT is X(e ) x[n] e � n 0 |X(e j ω )| …. …. ω 2 π 0 -2 π � However, X(e j ω ) is a periodic and continuous function of , ( ) p ω !! � How about processing of digital signals in � How about processing of digital signals in frequency domain ? 3

The Discre The Discrete F e Fourier T urier Transf ansform (DFT) orm (DFT) |X(e j ω ) | j …. …. ω ω 2 2 π 0 -2 π Δω � Alternatively, consider working with one period of X(e j ω ) ! y, g p ( ) Still, we have infinite points? � Take N equally-spaced samples from X(e j ω ) over the range [0, q y p p ( ) g [ , 2 π ] � Accordingly, the spacing between the samples is g y, p g p Δω = 2 π / N � And the normalized angular frequency for each sample is � And the normalized angular frequency for each sample is ω = 2 π k / N , 0 ≤ k ≤ N-1 4

The Discre The Discrete F e Fourier T urier Transf ansform (DFT) orm (DFT) � Mathematically, � � j X[k] X(e ) � 2 k �� N � N 1 � 2 kn ∑ ∑ � j N � � � x[n] e [ ] , , 0 k N-1 � n 0 � This is called the N-point Discrete Fourier Transform (DFT) (DFT) � Note that the DFT � Always exists since it is a finite sum! y � When computing the DFT, all samples in x[n] are used to compute one sample in X[k] to compute one sample in X[k] 5

The Discre The Discrete F e Fourier T urier Transf ansform (DFT) orm (DFT) Example 5.1: compute the 4-point DFT for x[n] = {2, 0, -2, 1} [ ] { , , , } 6

The Discrete F The Discre e Fourier T urier Transf ansform (DFT) orm (DFT) Example 5.2: compute the N-point DFT for ⎧ 1 , n = 0 � ⎨ x[n] [ ] ⎨ � � 0 , 1 n N-1 ⎩ 7

The Discrete F The Discre e Fourier T urier Transf ansform (DFT) orm (DFT) � The inverse N-point Discrete Fourier Transform (IDFT) is given by � N 1 � ∑ ∑ 2 kn 1 1 j j N � � � x[n] X[k] e , 0 n N-1 N � k 0 � Verification! V ifi i ! � N 1 � 2 kn ∑ ∑ 1 j N � � � x[n] X[k] e , 0 n N-1 N N � k 0 ⎛ ⎞ � � N 1 N 1 � � 2 km 2 kn ∑ ∑ ∑ ∑ 1 � j j � � � � � � ⎜ ⎜ ⎟ ⎟ x[n] x[n] x[m] e x[m] e e e , 0 0 n n N-1 N 1 N N ⎜ ⎜ ⎟ ⎟ N ⎝ ⎠ � � k 0 m 0 ⎛ ⎛ ⎞ ⎞ � � N 1 N 1 N 1 N 1 � � � 2 k(m n) 2 k(m n) ∑ ∑ 1 1 � j � � � ⎜ ⎟ x[n] x[m] e , 0 n N-1 N ⎜ ⎟ N ⎝ ⎠ � � m 0 k 0 � Equals 0 when m ≠ n and N x[n] x[n] 8 when m = n

Matrix R Matrix Representation of DFT presentation of DFT � The DFT is given by � N 1 � 2 kn ∑ ∑ � j j N � � � X[k] x[n] e , 0 k N-1 � n 0 � Let W = e -j2 π /N so the DFT becomes � Let W N = e j2 π /N , so the DFT becomes � N 1 ∑ ∑ � � � kn X[k] [ ] x[n] W , 0 [ ] , k N-1 N N � n 0 � In matrix form ⎡ ⎤ x[0] ⎢ ⎥ x[1] ⎢ ⎢ ⎥ ⎥ ⎡ ⎤ ⎢ ⎥ � ⎣ � k 2k 3k (N 1)k X[k] 1 W W W ... W x[2] ⎦ ⎢ N N N N ⎥ ... ⎢ ⎥ ⎢ ⎢ ⎥ ⎥ � x[N 1] ⎣ ⎦ 9

Matrix R Matrix Representation of DFT presentation of DFT � In general, the DFT can be written as n = N-1 n 0 n = 0 k = 0 ⎡ ⎤ ⎡ 1 1 1 1 ... 1 ⎤ x[0] ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ � 2 3 (N 1) 1 W W W ... W x[1] ⎢ ⎥ ⎢ ⎥ N N N N ⎢ ⎥ ⎢ ⎥ � � 2 4 6 2(N 1) X[k] 1 W W W ... W x[2] ⎢ ⎢ ⎥ ⎥ N N N N ⎢ ⎢ ⎥ ⎥ ⎢ ⎥ ... ... k = N-1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ � � � � � � x[N 1] N 1 2(N 1) 3(N 1) (N 1)(N 1) 1 W W W ... W ⎣ ⎦ ⎢ ⎥ ⎣ ⎣ ⎦ ⎦ N N N N X[k] = D x N where D N is called the DFT matrix 10

Matrix Representation of DFT Matrix R presentation of DFT � For the inverse DFT X[k] = D x X[k] D � � 1 1 1 1 D X[k] = D D x N N N N � 1 x[n] = D X[k] N where the inverse of D N is h h f D ⎡ ⎡ ⎤ ⎤ 1 1 1 1 ... 1 ⎢ ⎥ � � � � � 1 2 3 (N 1) 1 W W W ... W ⎢ ⎥ N N N N 1 ⎢ ⎢ ⎥ ⎥ � � � � � W � � 1 2 4 6 2(N 1) ( ) D D 1 1 W W W W W W ... W ⎢ ⎥ N N N N N N ⎢ ⎥ ... ⎢ ⎢ ⎥ ⎥ � � � � � � � � � (N 1) (N 1) 2(N 1) 2(N 1) 3(N 1) 3(N 1) (N 1)(N 1) (N 1)(N 1) 1 W W W ... W ⎢ ⎥ ⎣ ⎦ N N N N 11

The Discre The Discrete F e Fourier T urier Transf ansform (DFT) orm (DFT) Example 5.3: compute the N-point DFT for x[n] = {-2, 1, 7, 3} using matrix notation. [ ] { , , , } g 12

The Discre The Discrete F e Fourier T urier Transf ansform (DFT) orm (DFT) � Note 1. The N-point DFT is computed from the DTFT through sampling through sampling � � � j X[k] X[k] X(e ) X(e ) � 2 k 2 k �� N � Note 2. The DTFT for a length-N sequence can be computed from the DFT by taking infinite samples. This p y g p implies that we can take as many as M samples for the DFT given that M > N g 13

The Discre The Discrete F e Fourier T urier Transf ansform (DFT) orm (DFT) � Note 3. For a finite length sequence x[n], 0 ≤ n ≤ N-1, the IDFT is periodic with period N In other words if the IDFT is periodic with period N . In other words, if y[n] = IDFT(X[k]),then � ∑ � � � � y[n] x[n mN] , 0 n N - 1 ��� m Proof: let x[n] be a length-N sequence, 0 ≤ n ≤ N-1, andY[k] be the N-point DFT of x[n], i.e., � � ∑ kl Y[k] x[l] W N l ��� � � l Now, the N-point IDFT of Y[k] � M 1 ∑ 1 1 � � � � � kn q[n] Y[k] W , 0 k N 1 N N 14 � k 0

The Discre The Discrete F e Fourier T urier Transf ansform (DFT) orm (DFT) Proof - continued � ⎛ ⎛ � ⎞ ⎞ M 1 ∑ ∑ ∑ ∑ 1 1 � � � � � kl kl kn k ⎜ ⎜ ⎟ ⎟ q[n] x[l] W W , 0 k N 1 ⎜ ⎟ N N N ⎝ ⎠ � ��� k 0 l � ⎛ � ⎞ M 1 ∑ ∑ 1 � � � � � � k(n l) ⎜ ⎟ q[n] x[l] W , 0 k N 1 ⎜ ⎟ N N N ⎝ ⎝ ⎠ ⎠ ��� � l k 0 However, � M 1 M 1 ⎧ ⎧ ∑ 1 , l = n + mN 1 � � � ⎨ k(n l) W N 0 , otherwise N ⎩ � k 0 So, � ∑ ∑ ~ � � � � � � � � � q[n]= y[n] q[n] y[n] x[n mN] x[n mN] , 0 0 n n N 1 N 1 ��� m 15

The Discre The Discrete F e Fourier T urier Transf ansform (DFT) orm (DFT) � Note 4. To compute the M-point DFT for a length-N sequence it is required that M ≥ N to avoid time-domain sequence, it is required that M ≥ N to avoid time domain aliasing between the translated copies of the original time-domain sequence time domain sequence. Example 5.4: consider x[n] = {1 , -1 , 1 , 1}, N = 4. If p [ ] { , , , }, we compute the 4-point DFT X[k], then the IDFT, x[n] X[k] y[n] � ∑ ∑ � � � � � y[n] x[n mN] , 0 n N 1 ��� m � … y[n] [ ] + x[n+8] + x[n+4] + x[n] + x[n-4] + x[n-8] [ 8] [ 4] [ ] [ 4] [ 8] … 16

The Discre The Discrete F e Fourier T urier Transf ansform (DFT) orm (DFT) Example 5.4: Continued. 1.5 1 0.5 y[n] y[n] x[n+4] 0 1.5 -0.5 -1 1 1 -1.5 -4 -2 0 2 4 6 8 n 1.5 0.5 1 0.5 0 + x[n] 0 -0.5 -0.5 -1 1 -1.5 -1 -4 -2 0 2 4 6 8 n 1.5 -1.5 1 5 1 1 -4 -2 0 2 4 6 8 0.5 n x[n-4] 0 -0.5 -1 17 -1.5 -4 -2 0 2 4 6 8 n

The Discrete F The Discre e Fourier T urier Transf ansform (DFT) orm (DFT) Example 5.5: consider x[n] = {1 , -1 , 1 , 1}, N = 4. If we compute the 2-point DFT X[k], then the IDFT, p p [ ], , x[n] [ ] X[k] [ ] y[ ] y[n] � ∑ ∑ � � � � � � � y[n] [ ] x[n mN] , 0 [ N] 0 n N 1 N 1 ��� m � … y[n] + x[n+4] + x[n+2] + x[n] + x[n-2] + x[n-4] … 18

The Discre The Discrete F e Fourier T urier Transf ansform (DFT) orm (DFT) Example 5.5: Continued. 1.5 1 0.5 y[n] y[n] x[n+2] [ 2] 0 2.5 -0.5 -1 2 -1.5 -4 -2 0 2 4 6 8 1.5 n 1.5 1 0.5 + x[n] 1 0 -0.5 0.5 -1 -1.5 0 -4 -2 0 2 4 6 8 1.5 n 1 -0.5 0 5 -2 0 2 4 6 0.5 n x[n-2] 0 -0.5 -1 19 -1.5 -4 -2 0 2 4 6 8 n