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Semester I Examinations 2010/
Exam Code(s) 4 BN 121 , 4 BP Exam(s) Fourth Electronic Engineering Fourth Electronic and Computer Engineering
Module Code(s) EE Module(s) Digital Signal Processing
Paper No. 1 Repeat Paper No
External Examiner(s) Prof. G. W. Irwin Internal Examiner(s) Prof. G. Ó Laighin Dr. E. Jones
Instructions: Answer any three questions. All questions carry equal marks (20 marks).
Duration 2hrs No. of Pages 6 Department(s) Electrical & Electronic Engineering Course Co-ordinator(s) Dr. E. Jones
Requirements : MCQ Handout Statistical Tables Graph Paper Log Graph Paper Other Material Standard mathematical tables
(a) Write the difference equation of the discrete-time system shown in Figure 1, and hence determine the first five samples of the system impulse response. You may assume that the digital filter memory initial values are zero. [8 marks]
T T
x ( n ) y ( n )
T
T
Figure 1.
(b) A discrete-time system has a finite-duration impulse response that consists of the samples {1, 2, -1}, commencing at n = 0. Using time-domain convolution, calculate the response of the system to a finite-duration input signal that consists of the samples {1, 3, 4, 2}, also commencing at n = 0. Indicate in detail the calculations needed to determine y (3). [7 marks]
(c) A digital filter is described by the following difference equation:
y ( n )+ 0. 6 y ( n − 1 )= x ( n )+ 0. 5 x ( n − 1 )
Write the transfer function of the system, and hence, or otherwise, obtain an expression for the magnitude response of the system. What is the DC gain of the system? [5 marks]
(a) The transfer function of a first-order low-pass filter is described by the following equation:
z
H z
If the sampling rate is 8 kHz, determine the cut off frequency of the filter in Hz (you may assume the cut-off frequency is the “-3 dB frequency”). [5 marks]
(b) A digital filter has the following transfer function:
1 2
1
1 0. 3 0. 5
−
− +
z z
z H z
Determine the frequency response, and hence the phase response of the system. What is the value of the phase response at a frequency equal to one-quarter of the sampling frequency? [5 marks]
(c) An instrumentation application requires the removal of mains interference (50 Hz) from a signal that is sampled at a rate of 1 kHz. Design a notch filter suitable for removing this interference. You may assume that a notch of width 40 Hz will suffice. [5 marks]
(d) Using the pole-zero placement method, determine the transfer function of a digital resonator with the following characteristics:
(i) Sampling rate of 16 kHz (ii) Centre frequency of 4 kHz (iii) Bandwidth of 40 Hz (iv) DC gain of 1
Sketch the pole zero map of the filter, and give an expression for the difference equation. [5 marks]
(a) Describe what is meant by windowing in short-time spectral analysis, and discuss the trade-offs that exist when choosing a window for a particular application. [3 marks]
Give the equation for the Hanning window, and sketch the general form of the function in the time domain. [2 marks]
(b) Give two ways in which high-order digital filters may be implemented using first- order and second-order filter sections. Use equations to illustrate your answer, where appropriate. [3 marks]
(c) A 512-tap FIR digital filter is to be implemented at a sampling rate of 16.384 kHz using digital hardware. Calculate the number of multiplies required to process four seconds duration of a (real) input signal, if the filter is implemented using a standard transversal filter structure (you may assume that no implementation efficiencies are possible).
Also, calculate the saving in the number of multiplies required to process the same duration of input signal, if the filter is implemented using fast convolution in the frequency domain using a 512-point FFT and Inverse FFT. State any assumptions made in your analysis. You may ignore any computational overhead associated with overlapping the signal in the frequency-domain approach. [7 marks]
(d) Draw a block diagram of an oscillator that produces a cosine wave with a frequency of 1.6 kHz, at a sampling rate of 8 kHz. The amplitude of the cosine wave should be 1. Calculate the values of the digital filter coefficients, and the initial conditions for the oscillator, assuming that the first sample produced by the oscillator corresponds to a time index of n = 3. [5 marks]
