CITY UNIVERSITY LONDON. MSc in Information Engineering DIGITAL SIGNAL PROCESSING EPM746

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1 No: CITY UNIVERSITY LONDON MSc in Information Engineering DIGITAL SIGNAL PROCESSING EPM746 Date: 19 May 2004 Time: 09:00-11:00 Attempt Three out of FIVE questions, at least One question from PART B

2 PART A Question 1 (a) State the Nyquist sampling theorem and describe some of its implications in Digital Signal Processing. Illustrate your answer by considering an analogue signal with a spectrum which is flat in the frequency range 0 f 10 kh and then decreases linearly to ero between 10 kh f 20 kh, sampled at a rate of (i) f s = 40 kh and (ii) f s = 20 KH. Assuming that the sampled signal is reconstructed back to analogue form by an ideal D/A reconstructor, sketch the spectra of the sampled and reconstructed signals for each sampling rate and discuss the phenomenon of aliasing in each case. (b) Three wheels are rotating at a frequency of 1 H (clockwise), 5 H (clockwise) and 3 H (anticlockwise), respectively. (Here 1 H represents rotational motion of 1 revolution in 1 second). The three wheels are sampled by a strobe light flashing at a rate of f s = 4 H. Explain why all three wheels will appear to rotate at the same frequency (and in the same direction) when their motion is reconstructed from their optical samples and why this phenomenon is an example of optical aliasing. (c) Consider the following analogue audio signal, where t is measured in milliseconds: x(t) = cos(10πt) + cos(30πt) + cos(60πt) What is the frequency content of the signal? Which of its three components are audible? [Assume that the audible frequency range extends from 0 to 20 kh.] [3 marks] The signal is pre-filtered by an anti-aliasing analogue pre-filter with magnitude frequency response H(f) which has a flat pass-band up to 20 kh and attenuates at a rate of 40 db/decade beyond 20 KH. After pre-filtering, the signal is sampled at an audio rate of 40 KH. The resulting samples are immediately reconstructed using an ideal D/A reconstructor. Determine the output y a (t) of the reconstructor and compare it with the audible part of x(t). (You may ignore the effects of the phase response of the filter. If you prefer to work with octaves you may assume that 3 decades are approximately equal to 10 octaves). [7 marks] Question 2 (a) The impulse response of a ZOH reconstructor is given by h(t) = 1 for 0 t T, and h(t) = 0 for t > T and t < 0. By taking Laplace transforms show that the transfer function of the ZOH reconstructor is: H(s) = 1 e st s where T is the sampling interval. By substituting s = jω in the transfer function H(s) find the frequency response of the ZOH reconstructor. Hence show that: H(jω) = T sin(ωt/2) e jωt/2 ωt/2 2 of 6

3 Sketch the magnitude and phase frequency responses of H(jω) and compare them with the corresponding characteristics of the ideal reconstructor whose frequency response is given by: H ideal (jω) = T for ω π T be realised in practice? and H ideal(jω) = 0 for ω > π. Why can t the ideal reconstructor T [8 marks] (b) A stereo analogue audio signal (two channels) is sampled at a rate of 48 kh and each sample is quantised using a 16 bit A/D converter. If the range of the A/D converter is R = 10 volts find: (i) the maximum quantisation error (assuming rounding and that the analogue signal is within the range of the converter) and, (ii) the dynamic range of the A/D converter, defined as 20 log 10 (R/Q), where Q is the quantisation width. Assuming that the dynamic range of human hearing is approximately 100 db, explain why a 16-converter is adequate. Calculate how many Megabytes of hard disk space are required to store a stereo audio signal of 3-minute duration. (c) Describe briefly the stages of the successive-approximation A/D conversion process of an analogue voltage of 1.6 volts to its: offset-binary 4-bit representation, and two s complement 4-bit representation assuming that the full-scale range of the A/D converter is 8 volts. Question 3 Consider the FIR filter H() = h 0 + h h h h 0 4 where h 0, h 1 and h 2 are real parameters. (Note the symmetry of the parameters around the central coefficient h 2 ). Show that the frequency response of the filter can be written in the form H(e jω ) = A(ω)e 2jω, where A(ω) = 2h 0 cos 2ω + 2h 1 cos ω + h 2 is purely real. Hence show that the phase-response of the filter is a linear function of ω (with the possible exceptions of phase jumps ±π rads at frequencies where A(ω) changes sign). Hint: In your derivation of the frequency response of H() you may find the following identity useful: e jθ + e jθ = 2 cos θ. In an attempt to make H() a low-pass filter the following constraints are imposed on A(ω): A(0) = 1, A(π/2) = A(π) = 0. Show that these constraints specify completely the transfer function of the filter as: H() = n() d() = of 6

4 Show also that this filter has a frequency response H(e jω ) = A(ω)e 2jω, in which A(ω) = 1 cos ω(1+cos ω). Hint: In deriving the required expression for A(ω) you may find the following 2 identity useful: cos(2θ) = 2 cos 2 (θ) 1. By carefully sketching A(ω) in the frequency range 0 ω π verify that H() is indeed a low-pass filter. In your sketch indicate clearly the values of A(ω) at frequencies ω = 0, π 3, π 2, 2π 3 and π rads/sample. Show also the frequency at which A(ω) attains its minimum value and the corresponding value of A(ω). By factoring the numerator and denominator polynomials n() and d() of H() find the poles and eros of the filter and indicate clearly their location in the -plane. Using a geometric argument verify that the DC gain of the filter is H(1) = 1. Hint: In your attempt to factor n() note that the frequency response specifications A(π/2) = A(π) = 0 imply that the polynomials + 1 and are factors of n(). PART B Question 4 (a) Obtain from first principles the Z-transform of the following two discrete-time signals: h 1 (n) = r n cos(θn)u(n) and h 2 (n) = r n cos(θn)u( n 1) where u(n) denotes the unit-step signal u(n) = 1 for n 0 = 0 for n < 0 and r > 0 is a real parameter. In each case indicate clearly the region of convergence of your Z-transform, i.e. the range of values of for which your result is valid. Suppose that h 1 (n) and h 2 (n) are the unit pulse responses of two linear, time-invariant filters. Determine which of the two filters is causal and state conditions for each filter to be boundedinput-bounded-output (BIBO) stable. [2 marks] (b) A communications channel between the transmitted signal (x(n)) and received signal (y(n)) has transfer function where r > 1. H() = 2 2r cos θ + r 2 ( r cos θ) Find the range of values of r (in terms of θ) for which H() is BIBO stable. Sketch the pole-ero pattern of H() in the -plane. [2 marks] 4 of 6

5 Explain how an approximate Finite-Impulse-Response (FIR) channel de-convolution filter can be obtained by truncating the tail of the impulse response of the inverse filter, H 1 (), and delaying it by an appropriate number of samples D. [2 marks] It is required that the magnitude of the error between the received and reconstructed signals x(n) x(n D) should not exceed a specified level δ for all samples n. Assuming that the received signal satisfies y(n) 1 for all n show that a sufficient condition for this requirement is that (r 1)r D 1 δ. [8 marks] Question 5 (a) It is found that the signal at the output of a strain-gauge amplifier is corrupted by persistent noise at the mains frequency (50 H) and its higher harmonics at 100 H, 150 H and 200 H. It is proposed to improve the signal to noise ratio by designing a digital notch filter which removes these frequencies and the DC component of the signal. Suggest a suitable design, assuming that the useful bandwidth of the strain-gauge signal extends from 5 to 70 H. You need to select an appropriate sampling rate and indicate clearly how you have chosen the location of the poles and eros of your filter. [10 marks] (b) Using the bilinear-transformation method design a first-order low-pass digital filter operating at a sampling rate of 10 KH, whose 3 db cut-off frequency is 1 kh. Obtain the realisation of this filter in direct and canonical forms. [10 marks] 5 of 6

6 f(n), n 0 Table of Z-transforms F () δ(n) n ( 1) 2 n 2 (+1) ( 1) 3 a n a na n a ( a) 2 e at n e at ne at n e at ( e at ) 2 ( cos(ωt )) 2 2 cos(ωt )+1 sin(ωt ) 2 2 cos(ωt )+1 e at sin(ωt ) 2 2e at cos(ωt )+e 2aT ( e at cos ωt ) 2 2e at cos(ωt )+e 2aT cos(ωt n) sin(ωt n) e at n sin(ωt n) e at n cos(ωt n) External Examiners: Prof. P.M. Taylor, Prof. M. Cripps Internal Examiner: Dr G. Halikias 6 of 6

NAME: ht () 1 2π. Hj0 ( ) dω Find the value of BW for the system having the following impulse response.

NAME: ht () 1 2π. Hj0 ( ) dω Find the value of BW for the system having the following impulse response. University of California at Berkeley Department of Electrical Engineering and Computer Sciences Professor J. M. Kahn, EECS 120, Fall 1998 Final Examination, Wednesday, December 16, 1998, 5-8 pm NAME: 1.