Tutorial 9 The Discrete Fourier Transform (DFT) SIPC , Spring 2017 Technion, CS Department

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1 Tutorial 9 The Discrete Fourier Transform (DFT) SIPC , Spring 2017 Technion, CS Department

2 The DFT Matrix The DFT matrix of size M M is defined as DFT = 1 M W 0 0 W 0 W 0 W where W = e i2π M i = 1 and denotes complex conjugate The DFT matrix is symmetric and unitary, hence, its inverse is DFT = 1 M W W W 0 W i.e., DFT DFT = DFT DFT = I , CS Department, Technion

3 Representation of a Discrete Signal in the DFT Domain Uniform sampling of the continuous signal φ t provides us the discrete set of M 2N + 1 samples: φ N, φ N 1,, φ 0,, φ N 1, φ N that can be also arranged in a column vector as follows: φ = φ 0 φ 1 φ N φ N φ N 1 φ 1 Note the position of the negative-indexed samples , CS Department, Technion

4 Representation of a Discrete Signal in the DFT Domain The representation of the discrete signal φ is φ F = DFT φ or in a more explicit form: φ 0 F φ 1 F φ N F F φ N F φ N 1 F φ 1 = 1 M W W W 0 W φ 0 φ 1 φ N φ N φ N 1 φ 1 Note that the negative index k (for k = 1,, N) can be considered also as the positive index M k.

5 Representation of a Discrete Signal in the DFT Domain The DFT-domain representation is obtained via φ F = DFT φ Multiplying both sides of the equation by DFT, i.e., DFT φ F = DFT DFT φ and, as the DFT matrix is unitary, we get φ = DFT φ F which is the inverse DFT procedure: Given φ F it provides the signal-domain representation φ , CS Department, Technion

6 DFT Example #1: The Kronecker Delta Consider the following discrete signal of M samples: For n = 0,, M 1 : φ n = δ n,n0 1, for n = n 0 0, otherwise where n 0 0,, M 1 δ n,n0 is also known as the Kronecker delta, here shifted to n 0. The DFT of the above signal is φ F = 1 M W W W 0 W The n th entry = 1 M The n th column of the DFT matrix W 0 n 0 W 1 n 0 W n 0 = 1 M e i2π M 0 n 0 e i2π M 1 n 0 e i2π M n 0 Note the particular case of n 0 = 0.

7 DFT Example #2: Cosine Signal Consider the following discrete signal of M samples: For n = 0,, M 1 : φ n = cos 2πk 0 M n where k 0 0,, M 1 Recall that cos 2πk 0 M n = 1 2 ei2πk0 M n e i2πk 0 M n = 1 2 Wk 0n + W k 0n The k th component of the DFT-domain representation of the above signal is φ k F = 1 M W k n φ n = 1 M W k n 1 2 Wk 0n + W k 0n = = 1 M 1 2 W k n W k0n + 1 W k n W k 0n = 1 2 M 1 2 W k k 0 n W k+k 0 n

8 DFT Example #2: Cosine Signal Let us examine the expression W k k 0 n : For k = k 0 : W k k 0 n = W 0 n = 1 = M For k k 0 : W k k 0 n = W k k 0 n = W k k 0 M 1 W k k 0 1 Sum of a geometric series Recall that W = e i2π M [and that W 0, W 1,, W are the M roots (of order M) of unity]. Noting that W k k 0 M = W M k k 0 = e i2π M M k k 0 = e i2π k k0 = 1 implies For k k 0 : W k k 0 n = 0 W k k 0 n = M δ k,k0 M, for k = k 0 0, otherwise

9 DFT Example #2: Cosine Signal Using the last result W k k 0 n = M δ k,k0 M, for k = k 0 0, otherwise The following development justifies the correspondence between the negative index k 0 and the index M k 0 : W k+k 0 n = W k+k 0 M n = W k M k 0 n = M δ k,m k0 M, for k = M k 0 0, otherwise We develop the expression for the k th component of the DFT-domain representation of the cosine signal: φ k F = 1 M 1 2 W k k 0 n + 1 W k+k 0 n = 1 2 M 1 2 M δ k,k M δ k,m k 0 = = M 2 δ k,k 0 + M 2 δ k,m k 0

10 We are given a noisy image of size : I noisy r, n = I r, n + noise r, n The noise is harmonic and follows the formula: noise[ r, n] A cos 2 fn r r f = 1 8 pixels The amplitude, A, and the phase, φ, are random and independent for each line.

11 A rad

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13 The Image-Domain Smoothing Alternative

14 Alternatives: Smoothing vs Median (8 pixels) No noise but image is blurred

15 DFT of the noise in line r 1 Recall that M 256 and f, hence 8 r 32 noise A cos 2 fn A cos 2 n r r r n r M Then, since the signal is a shifted cosine function, its DFT is DFT noise r k M A e 2 r i r, k 32, 224 0, else Here we would like to handle frequencies 32 and 224 (recall that 224 can also be considered as -32).

16 Noisy signal in DFT domain Filtered signal in DFT domain Notch Filter: Attenuate Specific Frequencies

17 The noise was significantly removed. Original image was not fully restored We cannot restore the attenuated frequencies

18 Notch filter Smoothing filter of 8 pixels

19 Implementation Filter in freq. domain: Filter=ones(1,256); Filter(32+1)=0; Filter(224+1)=0; Notch filter in the DFT domain Filtration: For k=1:size(i,1), Y=fft(I(k,:)).*Filter; I(k,:)=ifft(Y); end