NAME: 11 December 2013 Digital Signal Processing I Final Exam Fall Cover Sheet
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1 NAME: December Digital Signal Processing I Final Exam Fall Cover Sheet Test Duration: minutes. Open Book but Closed Notes. Three 8.5 x crib sheets allowed Calculators NOT allowed. This test contains four problems. All work should be done on the blank pages provided. Your answer to each part of the exam should be clearly labeled.
2 Problem. [5 points] System and System defined in parts (a) and (b), respectively, will eventually be connected in parallel. We first analyze them individually. (a) Consider System below: System : y [n] =.9j y [n ].9j x[n] x[n ] (i) Determine the Transfer Function for System, denoted H (z). H (z) is the Z- Transform of the impulse response, h [n], for System, although you can find H (z) anyway you like. Identify the poles and zeros, and do a pole-zero plot. (ii) Determine the frequency response of System, denoted H (ω). H (ω) is the DTFT of the impulse response, h [n], for System, although you can find H (ω) anyway you like. Plot the magnitude H (ω) over π < ω < π. (iii) Determine the autocorrelation of the impulse response h [n]. Do a stem plot of r h h [l]. (b) Consider System below: System : y [n] =.9j y [n ].9j x[n]+x[n ] (i) Determine the Transfer Function for System, denoted H (z). H (z) is the Z- Transform of the impulse response, h [n], for System, although you can find H (z) anyway you like. Identify the poles and zeros, and do a pole-zero plot. (ii) Determine the frequency response of System, denoted H (ω). H (ω) is the DTFT of the impulse response, h [n], for System, although you can find H (ω) anyway you like. Plot the magnitude H (ω) over π < ω < π. (iii) Determine the autocorrelation of the impulse response h [n]. Do a stem plot of r h h [l]. (c) The overall system is formed from connecting System and System in parallel. (i) Determine the Transfer Function for the overall system, denoted H(z). H(z) is the Z-Transform of the impulse response, h[n], for the parallel combination of Systems and. You can find H(z) anyway you like. Identify the poles and zeros for the overall system, and do a pole-zero plot. (ii) Determine the frequency response of the overall system, denoted H(ω). H(ω) is the DTFTof the impulse response, h[n], for the overall system; you can find H(ω) anyway you like. Plot the magnitude H(ω) over π < ω < π. showing as much detail as possible. Point out any frequencies for which H(ω) =. (iii) Determine the output of the overall system, y[n], when the input is the DT signal below. The overall system is the parallel combination of Systems and. ( π x[n] = +cos n+ π +( ) ) n < n <
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6 Problem. ف Continuous-Time Fourier Transform (rads/sec): X(ω) = F{x(t)} = x(t)e jωt dt ف } { { } sin(wt) ω Continuous-Time Fourier Transform Pair (rads/sec): F = rect πt W where rect(x) = for x <.5 and rect(x) = for x >.5. Continuous-Time Fourier Transform Property: F{x (t)x (t)} = π X (ω) X (ω), where * denotes convolution, and F{x i (t)} = X i (ω), i =,. Relationship between DTFT & CTFT frequency variables in rads/sec: ω = ΩT s Relationship between DTFT and CTFT frequency variables in Hz: ω = π F F s, where F s = T s is the sampling rate in Hz Problem (a). The impulse response of an analog filter which passes up to the maximum frequency ω M = rads/sec is g a (t) = π sin ( (t π )) π(t π ) This impulse response is sampled at the Nyquist rate ω s = rads/sec., where ω s = π/t s such the time between samples is T s = π sec, to form the discrete-time filter g[n] = g a (nt s ) where: T s = π (a) Determine G(ω), the DTFT of g[n]. Plot both the magnitude G(ω) and the phase G(ω) (separate plots) over π < ω < π. (b) Consider the continuous-time signal x a (t) below. A discrete-time signal is created by sampling x(t) according to x[n] = x a (nt s ) for T s = π. Plot the magnitude of the DTFT of x[n], X(ω), over π < ω < π. { π sin(t) x a (t) = T s πt } sin(6t) πt (c) x[n] is passed through the filter g[n] to form the output y[n] = x[n] g[n] A reconstructed signal is formed from y[n] above according to the formula below. Determine a simple, closed-form expression for the reconstructed signal y r (t). y r (t) = ف ف = n y[n]h(t nt s ) where: T s = π and h(t) = sin(t) πt 6
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9 Problem. Consider a causal FIR filter of length M = 9 with impulse response as defined below: n+ sin [ π ( ف h p [n] = +l9)] π ( {u[n] u[n 9]} n+ +l9) ف = l (a) Determine the 9-pt DFT of h p [n], denoted H 9 (k), for k 8. You can EITHER write an expression for H 9 (k), OR list the numerical values: H 9 () =?, H 9 () =?, H 9 () =?, H 9 () =?, H 9 () =?, H 9 (5) =?, H 9 (6) =?, H 9 (7) =?, H 9 (8) =?. (b) Consider the sequence x[n] of length L = 9 below, equal to a sum of 9 finite-length sinewaves. 8 x[n] = e jkπ 9 n {u[n] u[n 9]} k= y 9 [n] is formed by computing X 9 (k) as an 9-pt DFT of x[n], H 9 (k) as a 9-pt DFT of h[n] and, finally, then y 9 [n] is computed as the 9-pt inverse DFT of Y 9 (k) = X 9 (k)h 9 (k). Express the result y 9 [n] as a weighted sum of finite-length sinewaves similar to how x[n] is written above. Next, consider a causal FIR filter of length M = 8 with impulse response as defined below: h p [n] = ف ف = l 6j sin[ π 8 (n+l8)] π(n+l8) sin [ π 8 (n+l8)] ( ) π sin π(n+l8) (n+l8) {u[n] u[n 8]} (c) Determine all 8 numerical values of the 8-pt DFT of h p [n], denoted H 8 (k), for k 7. List the values clearly: H 8 () =?, H 8 () =?, H 8 () =?, H 8 () =?, H 8 () =?, H 8 (5) =?, H 8 (6) =?, H 8 (7) =?. (d) Consider the sequence x[n] of length L = 8 below, equal to a sum of 8 finite-length sinewaves. 7 x[n] = e jkπ 8 n {u[n] u[n 8]} k= y 8 [n] is formed by computing X 8 (k) as an 8-pt DFT of x[n], H 8 (k) as an 8-pt DFT of h[n], and then y 8 [n] as the 8-pt inverse DFT of Y 8 (k) = X 8 (k)h 8 (k). Express the result y 8 [n] as a weighted sum of finite-length sinewaves similar to how x[n] is written above. 9
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13 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.
14 Problem. Consider the M = channel Filter Bank in Figure on the previous page. You are to determine whether it achieves Perfect Reconstruction with the following causal analysis filters, each of which is of length and starts at n =. h [n] = {,,,} h [n] = {,,,} h [n] = {,,,} h [n] = {,,, } The corresponding synthesis filters are defined below. The four analysis filters and the four synthesis filters are all real-valued. g k [n] = h k [ n], k =,,, In the space provided on the next three blank pages, you need to derive and clearly specify all filters and matrices in the computationally efficient implementation of this filter bank drawn in Figure on the previous page. All answers are real-valued quantities. NOTE: The matrices A and B are real-valued matrices that have nothing to do with a -pt DFT matrix. NOTE : After clearly specifying all the filters and matrices in Figure, determine if the Filter Bank achieves Perfect Reconstruction. Explain your answer.
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