241 ENEE 02
HOMEWORK ASSIGNMENT 37
Due Mon 05/12
A FIR lter acts on the input sequence given by x[0 : 5] = 1 3 5 2 6 2
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;
x[n] = 0 for n < 0 and n > 5
to produce the output sequence given by y [0 : 11] = 2 3 0 10 23 27 61 51 54 14 2
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and y [n] = 0 for n < 0 and n > 11. (i) (2 pts.) What is the length of the lter coecient vector b? (As usual, it is assumed that the rst and last entries in b are nonzero.) (ii) (6 pts.) Use the FFT command in MATLAB to determine the coecient vector b. (Alternatively, b can be determined by solving a triangular system of linear equations.) (iii) (6 pts.) Without performing a convolution, determine the response of the lter to the input sequence x(1) [ ] whose nonzero entries are x(1) [0 : 11] = 1 3 5 2 6 2 1 3 5 2 6 2
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(iv) (6 pts.) Without performing a convolution, determine the response of the lter to the input sequence x(2) [ ] whose nonzero entries are x(2) [0 : 9] = 2 6 10 4 11 7 5 2 6 2
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Solved Examples S 37.1 (P 4.21 in textbook). Consider the FIR lter y [n] = x[n] 2x[n 1] x[n 2] + 4x[n 3] x[n 4] 2x[n 5] + x[n 6] The response of the lter to the input : x[0 3] = (x[] = 0 otherwise) is given by y [0 : 9] = 1 1 0 7 8 13 20 1 11 4
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a0 a1 a2 a3
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(y [] = 0 otherwise); while the response to x[0 : 3] = ([] = 0 otherwise) is given by x y [0 : 9] = 1 2 2 10 8 10 18 2 9 4
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c0 c1 c2 c3
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([] = 0 otherwise). Determine the response of the lter to y x[0 : 7] = ([] = 0 otherwise). x a0 a1 a2 a3 2c0 2c1 2c2 2c3
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S 37.2 (P 4.24 in textbook). Consider the following MATLAB code: a b A B C c = = = = = = [ 2 -1 0 -1 2 ]. ; [ 3 -2 5 1 -1]. ; fft(a,9); fft(b,9); A.*B; ifft(C);
(i) Without using MATLAB, determine the vector c. (ii) Without performing a new convolution, determine the vector e obtained by the following code: a d A D E e = = = = = = [ 2 -1 0 -1 2 ]. ; [-3 2 -5 -1 1 3 -2 5 1 -1]. ; fft(a,14); fft(d,14); A.*D; ifft(E);
S 37.3 (P 4.25 in textbook). The data le s7.txt contains a vector s of length L = 64. Let b= 2 1 3 1 3 1 3 1 2
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(i) Use the CONV function in MATLAB to compute the convolution c = b s. What is the length of c? (ii) Compute c using three convolutions of output length 25 = 32, followed by a summation. Your answer should be the same as for part (i).