Page 2 ECS707P (2018) Question 1: Transfer functions of digital systems [25 marks] (a) A linear digital filter is described by the following difference equation: y(n) = 2x(n) + 4 exp( j⇡/2)x(n 1) + y(n 1) + 0.75y(n 2) (1) Find all poles and zeros of this filter in the form a+ jb and plot them in a fully labelled pole-zero diagram. [7 marks] (b) A second filter is described by the following difference equation: y(n) = 2x(n) + 3x(n 1) + y(n 1) + 0.75y(n 2) (2) Calculate the impulse response h(n) for this digital filter for all integer values of n, and plot its values for n 2 {0, 1, 2, 3, 4}. [4 marks] (c) Define BIBO stability, in words and by giving a mathematical expression to define it. Is the system defined by equation (2) stable in a BIBO sense? Show your work and reasoning as part of your answer. [4 marks] (d) Draw a block diagram of the system defined by equation (2). [4 marks] (e) Find the region of convergence of the transfer function specified by equation (2). Discuss whether it shows the system to be stable or unstable. [3 marks] (f) Find the expression for the frequency response of the phase for a filter having the following transfer function, H(z) = z(z 2) z2 1 , |z| > 1. (You do not need to plot it nor calculate values at specific points.) [3 marks] ECS07P (2018) Page 3 Question 2: DFT, convolution and system properties [25 marks] (a) Calculate the inverse DFT of X(k) = {1, 3, 2, 2}. [8 marks] (b) (i) Predict the length and (ii) calculate the values of the following linear convolution product, using the “convolution machine” technique as covered in the lectures: { 2, 1, 1} ⇤ {1, 3, 2, 2}. [7 marks] (c) Define and explain the following system properties, with reference to the impulse response h(n) of a digital signal processor: 1. causality 2. memory 3. stability 4. time invariance [4 marks] (d) An N -point DFT needs to be designed that calculates the spectrum of a signal with a sampling rate fs = 15 kHz, such that the minimum guaranteed frequency resolution is f = 1.5 Hz. Determine: 1. the number of points that is needed by a general DFT algorithm to achieve this; 2. the actual resolution achieved by an FFT-type implementation of the DFT; 3. the maximum frequency in the spectrum of a baseband signal that can be represented by the sampled signal. (Hint: An FFT algorithm is a DFT with a power-of-two number of frequency samples.) [6 marks] Turn over Page 4 ECS707P (2018) Question 3: Digital filtering [25 marks] Consider two digital brickwall LTI filters, identified with letters A and B. • Filter A is a linear phase lowpass filter with a normalised cutoff frequency of 0.5⇡ radians/sample. Its gain is 1. • Filter B is a linear phase highpass filter with a normalised cutoff frequency of 0.4⇡ radians/sample. Its gain is 1. (a) Sketch the shape of the magnitude response of each filter over the normalised frequency range [ ⇡, ⇡]. [4 marks] (b) Draw the magnitude response of the filter that results from a cascade of filter A followed by B, assuming their phase delays are equal. [3 marks] (c) Draw the magnitude response of the filter that results from a cascade of filter B followed by A followed by B, assuming their phase delays are equal. [3 marks] (d) Consider uniformly sampling a real continuous white noise signal x(t) at a sampling rate of 1 kHz. You then pass the sampled signal through filter A followed by filter B, and finally reconstruct the output by ideal (sinc) interpolation, creating the continuous real signal xAB(t). In what frequency bands of ( 1000, 1000) Hz will xAB(t) have no energy? [10 marks] (e) Denote the frequency response of filters A and B by HA(ej!) and HB(ej!). Find the phase response of filter B such that the magnitude response of the filter that results from A and B in parallel is zero everywhere. (Hint: Recall that sending a phasor of frequency ! through filter A will produce the output |HA(ej!)|ej![n \HA(ej!)].) [5 marks] ECS707P (2018) Page 5 Question 4: Digital filtering, part two [25 marks] For some of the following questions, consider the digital structure shown below. (a) Show your work as you find the transfer function of this filter. Specify its region of convergence! [5 marks] (b) For what values of k1 will this filter be stable? [2 marks] (c) Choose a value k1 6= 0 for which the filter will be stable, and find and plot the poles and zeros of the filter. [4 marks] (d) What kind of filter is this, e.g., lowpass and FIR, lowpass and IIR, etc. Explain your answer. [4 marks] (e) Devise a different digital filter structure that has the same transfer function. [4 marks] (f) Describe any two methods for FIR digital filter design. [6 marks] End of questions Turn over Page 6 ECS707P (2018) Appendix: Useful formulae linear convolution of two sequences x[n], h[n] : (x ? h)[n] = 1X m= 1 x[m]h[n m] Z-transform of sequence x[n] : X(z) = 1X n= 1 x[n]z n DTFT of sequence x[n] : X(ej!) = 1X n= 1 x[n]e j!n DFT of length-N sequence x[n] : X[k] = N 1X n=0 x[n]e j2⇡nk/N inverse DFT of length-N sequence X[k] : x[n] = 1 N N 1X k=0 X[k]ej2⇡nk/N Bilinear transformation : s = 2 T 1 z 1 1 + z 1 sum of the first N -terms of a geometric series of ↵ : SN = N 1X n=0 ↵n = 1 ↵N 1 ↵ Appendix: Useful signals Kroneker delta sequence [n] = 8Step function µ[n] = 8 00, else Table 3: Properties of the z-Transform Property Sequence Transform ROC x[n] X(z) R x1[n] X1(z) R1 x2[n] X2(z) R2 Linearity ax1[n] + bx2[n] aX1(z) + bX2(z) At least the intersection of R1 and R2 Time shifting x[n− n0] z−n0X(z) R except for the possible addition or deletion of the origin Scaling in the ejω0nx[n] X(e−jω0z) R z-Domain zn0x[n] X ( z z0 ) z0R anx[n] X(a−1z) Scaled version of R (i.e., |a|R = the set of points {|a|z} for z in R) Time reversal x[−n] X(z−1) Inverted R (i.e., R−1 = the set of points z−1 where z is in R) Time expansion x(k)[n] = { x[r], n = rk 0, n “= rk X(zk) R1/k for some integer r (i.e., the set of points z1/k where z is in R) Conjugation x∗[n] X∗(z∗) R Convolution x1[n] ∗ x2[n] X1(z)X2(z) At least the intersection of R1 and R2 First difference x[n]− x[n− 1] (1− z−1)X(z) At least the intersection of R and |z| > 0 Accumulation ∑n k=−∞ x[k] 1 1−z−1X(z) At least the intersection of R and |z| > 1 Differentiation nx[n] −z dX(z)dz R in the z-Domain Initial Value Theorem If x[n] = 0 for n < 0, then x[0] = limz→∞X(z) Table 4: Some Common z-Transform Pairs Signal Transform ROC 1. δ[n] 1 All z 2. u[n] 11−z−1 |z| > 1 3. −u[−n− 1] 11−z−1 |z| < 1 4. δ[n−m] z−m All z except 0 (if m > 0) or ∞ (if m < 0) 5. αnu[n] 11−αz−1 |z| > |α| 6. −αnu[−n− 1] 11−αz−1 |z| < |α| 7. nαnu[n] αz −1 (1−αz−1)2 |z| > |α| 8. −nαnu[−n− 1] αz −1 (1−αz−1)2 |z| < |α| 9. [cosω0n]u[n] 1−[cosω0]z−1 1−[2 cosω0]z−1+z−2 |z| > 1 10. [sinω0n]u[n] [sinω0]z−1 1−[2 cosω0]z−1+z−2 |z| > 1 11. [rn cosω0n]u[n] 1−[r cosω0]z−1 1−[2r cos ω0]z−1+r2z−2 |z| > r 12. [rn sinω0n]u[n] [r sinω0]z−1 1−[2r cos ω0]z−1+r2z−2 |z| > r
