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Semester One Examination 2012- 2013
Exam Code(s) 4 BP, 4 BN, 4 BSE Exam(s) Fourth Year Electronic & Computer Engineering Fourth Year Electronic Engineering Fourth Year Energy Systems – Electrical
Module Code(s) EE409, EE409.i Module(s) Digital Signal Processing Digital Signal Processing I
Paper No. 1 Repeat Paper
External Examiner(s) Prof. G. W. Irwin Internal Examiner(s) Prof. G. Ó Laighin Dr. E. Jones
Instructions: Answer any three questions from four All questions carry 20 marks each
Duration 2 hours
No. of Pages 6 pages (including cover page)
Discipline 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 six samples of the system’s response to the following input:
x ( n ) = 0. 95 nu ( n )
You may assume zero initial conditions in the filter. [7 marks]
T T
x ( n ) y ( n )
T T
Figure 1
(b) A digital filter is described by the following difference equation:
y ( n )= x ( n )− 0. 4 x ( n − 1 )+ 0. 5 y ( n − 2 )
Write the transfer function of the system, and hence obtain an expression for the magnitude and phase responses of the system. What is the gain of the system at a frequency equal to one quarter of the sampling frequency? What is the phase response of the system at the same frequency? [7 marks]
(c) A discrete-time system has a finite-duration impulse response that consists of the samples {2, 3, -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 {3, 1, -1, 4, 3}, also commencing at n = 0. Indicate in detail the calculations needed to determine y (3). [6 marks]
(a) A digital filter has the following transfer function:
1 0. 5 1 0. 9 2
1 2 1 ( )
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-fifth of the sampling frequency? [5 marks]
(b) A microprocessor-based power quality meter is used to estimate third harmonic distortion in a voltage signal. This requires extracting the third harmonic of a 50 Hz mains voltage waveform using a resonator. Design a discrete-time resonator to carry out this function, for a sampling rate of 1000 Hz. The bandwidth of the resonator should be 30 Hz, and the gain of the resonator at DC should be equal to 1.
Sketch the pole zero map of the filter, and give an expression for the difference equation. [7 marks]
(c) A healthcare instrumentation application requires the removal of interference at 100 Hz and 150 Hz from an EMG signal. Design a digital filter with notches at the interference frequencies to achieve this objective. The signal is sampled at a frequency of 800 Hz, and notches of width 20 Hz are required. [8 marks]
(a) A fourth-order filter has the following transfer function:
z z z
z z z H z
Draw a block diagram of a cascade implementation of the filter using first and second- order sections, where the second-order section is Direct Form II. [4 marks]
(b) A signal processing application requires spectral analysis of a signal that has a sampling frequency of 32 kHz. The spectral analysis is to be carried out using short- time analysis with a window length of 40 msec, and with a resolution such that the frequency step is no greater than 20 Hz. Calculate the minimum number of samples that must be used to zero-pad each frame of the signal in order to achieve the desired frequency resolution, assuming that the FFT algorithm is used for spectral analysis. [4 marks]
(c) A linear-phase 256-tap FIR digital filter is to be implemented at a sampling rate of 8 kHz using digital hardware in an FPGA. Calculate the number of multiplies and additions required to process an eight-second duration of the (real) input signal, if the filter is implemented using a transversal filter structure. Use the fact that the filter is linear phase to reduce the number of multiplies required.
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 256-point FFT and Inverse FFT. Assume that windowing with a 256-point Hamming window is used, where the coefficients of the window are pre-computed. Assume overlap of 50% is used in the frequency-domain filtering approach in determining the number of frames; however, you may ignore any additional overhead associated with overlap-add processing of the FFT output. [7 marks]
(d) For an oscillator whose output is p ( n )=sin( n θ 0 ), where θ 0 is determined by the
coefficient b 1 , show that a small change in oscillation frequency ∆ f 0 is related to a change ∆ b 1 in the coefficient b 1 by the following expression:
s
s
f
f
f b f 0
1 0
4 πcos 2 π
where f 0 is the frequency of oscillation in Hz, and f (^) s is the sampling frequency. [5 marks]
