Digital Signal Processing I Final Exam Fall 8 ECE538 7 Dec.. 8 Cover Sheet Test Duration: minutes. Open Book but Closed Notes. Calculators NOT allowed. This test contains FIVE problems. All work should be done on blank 8.5 x white sheets of paper (NOT provided). Do not return this test sheet, just return your answer sheets.
Digital Signal Processing I Final Exam Fall 8 ECE538 DSP I 7 Dec. 8 Problem. pts] You are given the matrix inversion lemma: A B C D ] = A +A B(D CA B) CA A B(D CA B) (D CA B) CA (D CA B) ] A symmetric-toeplitz matrix is formed from the first three autocorrelation values of an AR process as r xx ] = /3, r xx ] = /6, and r xx ] = / as R 3 = /3 /6 / /6 /3 /6 / /6 /3 (a) Use the matrix inversion lemma to compute the inverse of R 3 in terms of the inverse of ] /3 /6 R = /6 /3 ] R = (b) Use the matrix inversion lemma to compute the solution a 3 = R 3 r 3 where: r 3 = /6 / / in terms of a = R r = / ] where: r = /6 / ] (c) What is the model order p of the AR process?
Problem. points] A signal xn] of length is broken up into two overlapping blocks of length 8, denoted x n] and x n], respectively, for the purposes of filtering with hn] = {,, } of length M = 3 via the overlap-save method. The last M = points of x n] are the first two points of x n]. The first M = zeros in x n] were added to get the overlap-save method started, as done in Fig. 7.3. in the text. x n] = {,,,,,,,} and x n] = {,,,,,,, } (a) y n] is formed by computing X (k) as an 8-pt DFT of x n], H(k) as an 8-pt DFT of hn], and then y n] as the 8-pt inverse DFT of Y (k) = X (k)h(k). Write out the values of y n] in sequence form (similar to how x n] and x n] are written out above.) (b) y n] is formed by computing X (k) as an 8-pt DFT of x n], H(k) as an 8-pt DFT of hn], and then y n] as the 8-pt inverse DFT of Y (k) = X (k)h(k). Write out the 8 values of y n] in sequence form. (c) Show how y n] and y n] are combined to form the full linear convolution yn] = xn] hn], via the overlap-save method. 3
Problem 3. points] A second-order digital filter is to be designed from an analog filter having two poles in the s-plane at.+.j and..j and two zeros at j 3 and j 3, via the bilinear transformation method characterized by the mapping s = z z + Note that.+.j = (/5)+j(/5) and j 3 = jtan(π/3) (a) Is the resulting digital filter (BIBO) stable? Briefly explain why or why not. (b) Denote the frequency response of the resulting digital filter as H(ω) (the DTFT of its impulse response). You are given that in the range < ω < π, there is only one value of ω for which H(ω) =. Determine that value of ω. (c) Draw a pole-zero diagram for the resulting digital filter. Give the exact locations of the poles and zeros of the digital filter in the z-plane. (d) Plot the magnitude of the DTFT of the resulting digital filter, H(ω), over π < ω < π. You are given that H() = 6. Be sure to indicate any frequency for which H(ω) =. Also, specifically note the numerical value of H(ω) for ω = π and ω = π. (e) Determine the difference equation for the resulting digital filter.
Problem. points] (a) Consider a causal FIR filter of length M = with impulse response hn] = un] un ] Provide a closed-form expression for the 6-pt DFT of hn], denoted H 6 (k), as a function of k. Simplify as much as possible. You are given the following four values: H 6 () = H 6 () = j H 6 (8) = H 6 () = +j (b) Consider the sequence xn] of length L = 6 below, equal to a sum of several finitelength sinewaves. ( ) π xn] = +cos n +cos(πn), n =,,...,5. y 6 n] is formed by computing X 6 (k) as a 6-pt DFT of xn], H 6 (k) as a 6-pt DFT of hn], and then y 6 n] as the 6-pt inverse DFT of Y 6 (k) = X 6 (k)h 6 (k). Express the result y 6 n] as a weighted sum of finite-length sinewaves similar to how xn] is written above. (c) For the remaining parts of this problem, hn] is now defined as the causal FIR filter of length M = below. hn] = ( ) n {un] un ]} Provide a closed-form expression for the 6-pt DFT of hn], denoted H 6 (k), as a function of k. Simplify as much as possible. (d) Consider again the sequence xn] of length L = 6 below. ( ) π xn] = +cos n +cos(πn), n =,,...,5. y 6 n] is formed by computing X 6 (k) as an 6-pt DFT of xn], H 6 (k) as a 6-pt DFT of hn], and then y 6 n] as the 6-pt inverse DFT of Y 6 (k) = X 6 (k)h 6 (k). Express the result y 6 n] as a weighted sum of finite-length sinewaves. (e) Consider the sequence pn] of length L = 6 below. pn] = δn]+δn ]+δn 8]+δn ] Is pn] = xn]? If they are equal, provide an explanation as to why they are equal. 5
Problem 5. points] x n] x n] xn] x n] Figure (a). Analysis Section of Two-Stage Tree-Structured Filter Bank. x n] G (ω) G (ω) x n] G (ω) yn] x n] G (ω) Figure (b). Synthesis Section of Two-Stage Tree-Structured Filter Bank. 6
x n] x n] G (ω) xn] x n] x n] G (ω) + x n] x n] G (ω) Figure (a). Analysis Filter Bank, M =. Figure (b). Synthesis Filter Bank, M =. This problem is about synthesizing an M=3 channel nonuniform PR filter bank from a two stage tree-structured PR filter bank. h n] = sin( π (n+.5)) π (n+.5) h n] = ( ) n h n] Also, g n] = h n] and g n] = h n]. The combination of the analysis filter pair, {H (ω),h (ω)}, and synthesis filter pair {G (ω),g (ω)}, form a two-channel PR filter bank. (a) Use Noble s First Identity to determine the analysis filters H m (ω), m =,,, so that the system in Fig. (a) is equivalent to the system in Fig. (a). (i) Plot the frequency response for each of the three filters H m (ω), m =,,. You can put all three plots on the same graph to save space. (ii) Write a time-domain expression for EACH of the three filters h m n], m =,,. (b) Next, use Noble s Second Identity to express each synthesis filter, G m (ω), m =,,, in terms of G (ω) and G (ω). (c) Plot F(ω) = i= H i (ω)g i (ω) over π < ω < π. + yn] 7