ENEE241hw12

ENEE241hw12 - ENEE 241 HOMEWORK ASSIGNMENT 12 Due Tue 05/03 Problem 12A Consider the FIR filter given by the following input-output

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Unformatted text preview: ENEE 241 HOMEWORK ASSIGNMENT 12 Due Tue 05/03 Problem 12A Consider the FIR filter given by the following input-output relationship (note the missing coefficient): √ √ y [n] = x[n] + 3x[n − 1] − 3x[n − 3] − x[n − 4] , n∈Z (i) (3 pts.) Show that the input sequences defined for all n by x(1) [n] = 1 and x(2) [n] = (−1)n both result in output sequences which are identically equal to zero. (ii) (3 pts.) Write MATLAB code which computes and plots the amplitude and phase response of the filter at 1024 equally spaced frequencies in [0, 2π ). Submit the plots, properly labeled. (iii) (4 pts.) Express the filter’s frequency response in the form H (ej ω ) = je−j (ωM/2) F (ω ) where F (ω ) is a real-valued sum of sines. (iv) (5 pts.) The amplitude response plotted in (ii) above has six zeros at frequencies other than ω = 0 and ω = π . Determine the values of these frequencies analytically, using the result of part (iii) and the identity sin 2θ = 2 sin θ cos θ. (v) (5 pts.) Determine analytically the exact locations of the local maxima of the amplitude response |H (ej ω )| in the frequency range [0, π /2]. Do so by differentiating F (ω ) and using the identity cos 2θ = 2 cos2 θ − 1. Express your answers using cos−1 (ρ1 ) and cos−1 (ρ2 ), where ρ1 and ρ2 are exact (sums of rational numbers and square roots thereof). Problem 12B Consider the FIR filter with the following input-output relationship (note the missing coefficient): y [n] = x[n] + 3x[n − 1] + 3x[n − 3] + x[n − 4] , n∈Z (i) Determine the response y (i) [ · ] of the filter to each of the input signals given by the equations below (valid for all n ∈ Z). x(1) [n] = (3/4)n (2) x (3) x (4) x (5) x [n] = (−4/3) (2 pts.) n −n (2 pts.) [n] = 1 + 3 (2 pts.) [n] = cos(n(π /6) + 2.5) (3 pts.) −n [n] = 2 · cos(nπ /6) (4 pts.) (ii) (5 pts.) The filter above is connected in series (cascade) with a filter having input-output relationship y [n] = x[n] − 2x[n − 1] + x[n − 2] , n∈Z Determine the system function H (z ) of the two-filter cascade. (iii) (2 pts.) Write out the input-output relationship of the two-filter cascade. Solved Examples S 12.1 (P 4.5 in textbook). Consider the FIR filter y [n] = x[n] − 3x[n − 1] + x[n − 2] + x[n − 3] − 3x[n − 4] + x[n − 5] (i) Write MATLAB code which includes the function fft, and which computes the magnitude and phase response of the filter at 256 equally spaced frequencies between 0 and 2π (1 − 256−1 ). (ii) Express the frequency response of the filter in the form e−j αω F (ω ) where F (ω ) is a real-valued sum of cosines. (iii) Determine the response y [n] of the filter to the exponential input sequence ￿ ￿n 1 x[n] = , n∈Z 2 S 12.2 (P 4.6 in textbook). The MATLAB code a H A q = = = = [ 1 -3 5 -3 1 ].’ ; fft(a,500); abs(H); angle(H); computes the magnitude response A and phase response q of a FIR filter over 500 equally spaced frequencies in the interval [0, 2π ). (i) If x and y are (respectively) the input and output sequences of that filter, write an expression for y [n] in terms of values of x. (ii) Determine the output y of the filter when the input x is given by ￿ ￿n 1 x[n] = , n∈Z 3 (iii) Express the frequency response of the filter in the form e−j αω F (ω ) where F (ω ) is a real-valued sum of cosines. S 12.3 (P 4.4 in textbook). Consider the FIR filter whose input x and output y are related by y [n] = x[n] − x[n − 1] − x[n − 2] + x[n − 3] (i) Write out an expression for the system function H (z ). (ii) Express |H (ej ω )|2 in terms of cosines only. Plot |H (ej ω )| as a function of ω . (iii) Determine the output y [n] when the input sequence x is given by each of the following expressions (where n ∈ Z): • x[n] = 1 • x[n] = (−1)n • x[n] = ej πn/4 • x[n] = cos(π n/4 + φ) • x[n] = 2−n • x[n] = 2−n cos(π n/4) (In all cases except the third, your answer should involve real-valued terms only.) S 12.4 (P 4.8 in textbook). Consider two FIR filters with coefficient vectors b and c, where b= and c= ￿ ￿ 32123 1 −2 2 −1 ￿T ￿T (i) Determine the system function H (z ) of the cascade. Is the cascade also a FIR filter? If so, determine its coefficient vector. (ii) Express the amplitude response of the cascade as a sum of sines or cosines (as appropriate) with real-valued coefficients. S 12.5 (P 4.9 in textbook). Consider the FIR filter with coefficient vector b= ￿ 1111 ￿T Two copies of this filter are connected in series (cascade). (i) Determine the system function H (z ) of the cascade. Is the cascade also a FIR filter? If so, determine its coefficient vector. (ii) Determine the response y [n] of the cascade to the sinusoidal input sequence ￿ nπ ￿ x[n] = cos , n∈Z 2 ...
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This note was uploaded on 09/27/2011 for the course ENEE 241 taught by Professor Staff during the Spring '08 term at Maryland.

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