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**Unformatted text preview: **UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN Department of Electrical and Computer Engineering ECE 410 Digital Signal Processing Quiz Number 6 Thursday, November 13, 2008 Student Name: Solutions Section: Prof. Bresler / Prof. Jones NOTE: You may not use any calculators, cell phones, earphones (or other forms of electronic media) on this quiz. You may use three sheets of handwritten notes. Problem 1 (20 points) An FIR filter is characterized by 1 , 0 1, (a) (14 points) Show that its amplitude response is given by
/ sin and give an equation for the coefficients {c[n]}. (b) (6 points) This filter can not be a (i) highpass filter; (ii) bandstop filter. Mark the correct statement(s) and explain why. (a) This is a generalized linear phase FIR filter, characterized by an anti-symmetric impulse response with odd length. Thus, we have 1 2 1 2 Now, consider taking the DTFT 0 Now, we use a change of variables in the second summation by letting ′ 1 and we have 1 2 sin 1 2 Now, we can perform another change of variables to bring the result to the form shown in the question by letting
′ 2 2 1 2 sin 2 1 2 sin Now, let . Thus, the amplitude response is given by sin 2 for 1,2, … , where (b) . (i) This filter cannot possibly be a highpass filter because sin 0 0 or . (ii) This filter could be a bandstop filter because its only restrictions exist at NOTE: Notice that this filter cannot be a lowpass or highpass filter. 2 Problem 2 (30 points) For the following systems described by a difference equation, please determine if the filter is (i) FIR, has (ii) even or odd length, (iii) has even or odd symmetry, and what (iv) GLP type it is. Also answer the question that follow and explain you reasoning. (a) (10 points) 3 2 3 4 Question: (5 points) This filter can not be a (i) lowpass filter; (ii) highpass filter. Mark the correct statement(s) and explain why. (b) (10 points) 3 1 3 2 3 3 4 3 5 Question: (5 points) This filter can not be a (i) lowpass filter; (ii) highpass filter. Mark the correct statement(s) and explain why. (a) First of all, let’s take the z-transform of the system 3 3 We notice that the z-transform of the system is in polynomial form; thus, it’s an FIR filter. Taking the DTFT to find its frequency response, we have 3 3 By inspection, the impulse response is the following (starting from the n = 0 term) 3, 0, 1, 0, 3 Thus, we notice that the impulse response is symmetric and has odd length. This corresponds to a type I generalized linear phase FIR filter. (i) FIR (ii) odd length (iii) even symmetry (iv) Type I GLP Question (i, ii) We cannot rule out the possibilities that this filter is a lowpass or highpass filter because 0 and are nonzero. However, if the filter was still type I GLP, but had even length, then it couldn’t be a highpass filter (or bandstop). 3 (b) By inspection of the difference equation, we have the following impulse response: 1 3, , 3,3, 8 1 ,3 8 This is an anti-symmetric even length sequence, thus, it corresponds to a type II GLP FIR filter. (i) FIR (ii) even length (iii) odd symmetry (iv) Type II GLP Question (i, ii) Because it is a Type II GLP with even length, its amplitude response has the following form: 1 2 sin Thus, it’s clear that filter 0. 0 0, so this filter cannot be a lowpass filter. However, it could be a highpass 4 Problem 3 (30 points) Find an expression for the coefficients {hn }n =19 for a 20-length FIR lowpass filter n =0 for both Hamming and rectangular windows. The cutoff frequency should be ωc=π/2. (i) Rectangular ⎧ − j (19 ) ω 2 ⎪ Gd (ω ) = ⎨e ⎪0 ⎩
g [n]= | ω |≤ ω c otherwise 1 2π ωc −ωc ∫e
ωc −j (19 ) ω 2 e jωn dω
(19 ) )ω 2 1 g [n]= 2π −ωc ∫e j ( n− dω
⎞ ⎟ ⎟ ⎠ g[ n] = 19 − j ( n − )ω c ⎛ j ( n − 19 )ωc 1 1 2 ⎜e 2 − e 19 ⎞ ⎜ 2π ⎛ j⎜ n − ⎟ ⎝ 2⎠ ⎝ g[n]= 1 π (n −
g [n]= 19 ) 2 sin ((n − 19 )ω c ) 2 ωc 19 sinc((n − )ω c ) π 2
0 ≤ n ≤ 19 otherwise 19 π ⎧1 ⎪ sinc((n − ) ) hrect [n] = ⎨ 2 22 ⎪ 0 ⎩
(ii) Hamming ⎧⎛ ⎛π ⎛ 19 ⎞ ⎞ ⎛ 2πn ⎞ ⎞ 1 ⎪⎜ .54 − .46 cos⎜ ⎟ ⎟ sinc⎜ ⎜ n − ⎟ ⎟ ⎜ ⎟2 ⎜2 hham min g [n] = ⎨⎝ 2 ⎠⎟ ⎝ 19 ⎠ ⎠ ⎝⎝ ⎠ ⎪ 0 ⎩ 0 ≤ n ≤ 19 otherwise 5 Problem 4 (20 points) Given the phase response of a generalized linear-phase FIR filter as shown below, answer the questions that follow and clearly explain your answers. a. Is the filter (i) Type-I GLP, (ii) Type-II GLP, (iii) neither Type I or Type II GLP, or (iv) the given information is insufficient to make any of the preceding statements? b. Identify all frequencies at which |Hd(ω)|=0. Can you characterize the filter as being (i) possibly low-pass, (ii) possibly high-pass, (iii)neither high-pass or low-pass, or (iv) the given information is insufficient to make any of the preceding statements? c. Can the filter length be determined from the given information? If so, what is it? a. Type-I GLP There are jumps of π and 2 π and the slope is constant so the filter is GLP. The angle is 0 at 0 so the filter is type 1 GLP. b. Identify all frequencies at which |Hd(ω)|=0. This is where the phase jumps by π ω = π/2 ω = -π/2 c. N is odd and the filter is Type-I GLP so the possibilities are low-pass, high-pass, bandpass is another possibility. d. Must be Type-I GLP, even symmetric. The phase has a slope of 1. 1 = (N-1)/2 Filter Length = 3 6 ...

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