ECE538 Final Exam Fall 2017 Digital Signal Processing I 14 December Cover Sheet

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1 ECE58 Final Exam Fall 7 Digital Signal Processing I December 7 Cover Sheet Test Duration: hours. Open Book but Closed Notes. Three double-sided 8.5 x crib sheets allowed This test contains five problems. You must show all work for each problem to receive full credit. No. Topic(s) of Problem Points. OFDM. Noble s Identities. Perfect Reconstruction Filter Bank. DFT s and Time-Domain Aliasing 5. Z-Transform, Difference Eqns., Bilinear Transform

2 Problem. An OFDM signal is synthesized as a sum of four sinewaves of length as x[n] = b k e jπk n {u[n] u[n ]} k= Each of the four values b k is one of the four QPSK values listed in the symbol alphabet below. b k {+j, j, +j, j} k =,,, We know in advance that the signal we transmit will be convolved with a filter of length L =. Thus, we add a cyclic prefix of length which effective creates a sum of sinewaves of length 7 as prescribed below: x[n] = b k e jπ k n {u[n+] u[n ]} k= The sequence of length 7 above is convolved with the filter below of length h[n] = {8,,} = 8δ[n]+δ[n ]+δ[n ] This ultimately yields the following sequence of length 7+- = 9 y[n] = x[n] h[n] = {+j, +6j,+7j,5 j,7+j, +5j,+7j,+j,+j} Your task is to determine the numerical values of each of the four symbols: b, b, b, and b. Explain and show all the steps in determining your answers. Please lay your work out nicely with logical ordering. Your work will be more important than your final answers.

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6 Problem. [ points] Determine the impulse response h[n] in Figure.(b) so that the I/O relationship of the system in Figure.(b) is exactly the same as the I/O relationship of the system in Figure.(a). Plot the magnitude AND the phase (two separate plots) of the DTFT of h[n] over π < ω < π. Hint: Analyze the system of Figure.(a) in the frequency domain using one of Noble s Identities. = ( j) n sin( { π n) BP π n { Figure.(a). BP ( j) n sin( { πn) π n = =?? { Figure.(b). 6

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9 g [n] g [n] x[n] Σ y [n] g [n] g [n] Figure (a). Analysis Filter Bank. Figure.. For Problem on Next Page. Figure (b). Synthesis Filter Bank. x[n] + s [n] y [n] h -[n] y [n] x[n-] x[n-] + + s [n] s [n] Matrix x B Matrix x A y [n] y [n] - - y [n-] y [n-] y[n] interleaver (clockwise) x[n-] + s [n] y [n] - y [n-] Figure.. For Problem on Next Page. 9

10 Problem. Consider the M = channel Filter Bank in Figure. on the previous page. You are to determine an efficient implementation of the entire structure having the form in Figure.. Then determine whether it achieves Perfect Reconstruction with the following causal analysis filters, each of which is of length and starts at n =. Achieving Perfect Reconstruction means the output is the same as the input except for possibly an amplitude-scaling and an integer-delay. h [n] = {,,, } h [n] = {,,, } h [n] = {,,,} h [n] = {,,,} The first element in each vector corresponds to n =. The corresponding synthesis filters are defined below. The four analysis filters and the four synthesis filters are all real-valued, and the synthesis filters are the time-reverses of the corresponding analysis filters according to g k [n] = h k [ n], k =,,, (a) Draw the efficient implementation of the top branch of the analysis side where the h [n] is followed by decimation by a factor of. The inputs for the efficient implementation should be x[n], x[n ], x[n ], x[n ]. (b) Draw the efficient implementation of the top branch of the synthesis side where zero-inserts is followed filtering by g [n] = h [ n] followed by decimation by a factor of. The outputs for the efficient implementation should be y[n], y[n ], y[n ], y[n ] and the commutator for the interleaver should operate clockwise. (c) Since all of the analysis and synthesis filters are each of length, the polyphase components are each of length. Thus, we can set h + [n] = δ[n] h + [n] = δ[n] h + [n] = δ[n] h + [n] = δ[n] h [n] = δ[n] h [n] = δ[n] h [n] = δ[n] h [n] = δ[n] With this in mind, the matrices B and A operate as follows. z [n] z [n] z [n] z [n] = b b b b b b b b b b b b b b b b x[n] x[n ] x[n ] x[n ] y[n] y[n ] y[n ] y[n ] = Determine the matrices A and B in this efficient implementation. a a a a a a a a a a a a a a a a (d) Determine if the Filter Bank achieves Perfect Reconstruction. Explain your answer. () z [n] z [n] z [n] z [n] NOTE: The matrices A and B are real-valued matrices that have nothing to do with a -pt DFT matrix.

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14 Problem. Let x[n] be of length L =, i.e., x[n] = for n < and n and h[n] be of length M = 5, i.e., h[n] = for n < and n 5. Let X 6 (k) and H 6 (k) denote 6-point DFT s of x[n] and h[n], respectively. The 6-point inverse DFT of the product Y 6 (k) = X 6 (k)h 6 (k), denoted y 6 [n], produces the following values: n 5 y 6 [n] Let X 5 (k) and H 5 (k) denote the 5-point DFT s of the aforementioned sequences x[n] and h[n]. The 5-point inverse DFT of the product Y 5 (k) = X 5 (k)h 5 (k), denoted y 5 [n], produces the following values: n y 5 [n] Given y 6 [n] and y 5 [n], find the linear convolution of x[n] and h[n], i. e., list all of the numerical values of y[n] = x[n] h[n].

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17 Problem 5. 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 and j, via the bilinear transformation method characterized by the mapping s = z z + (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() =.8. 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. 7

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Digital Signal Processing I Final Exam Fall 2008 ECE Dec Cover Sheet

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