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Summer Examinations 2009-
Exam Code(s) 4BN121, 4BP
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 five questions from seven.
All questions carry equal marks (20 marks each)
Duration 3 hours
No. of Pages
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’s response, starting at sample index n = 0, to the following input signal:
x ( n ) 0. 9 u ( n )
n
where u ( n ) is the unit step function. You may assume that the digital filter memory initial values are zero. [7 marks] Figure 1 [Q1(a)] (b) Write an expression for the frequency response of the system in Figure 1, and hence derive an expression for the system magnitude response. What is the magnitude response of the system at a frequency equal to one sixth of the sampling frequency? [8 marks] (c) A discrete-time system has a finite-duration impulse response that consists of the samples {1, -2}, 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, -1}, also commencing at n = 0. [5 marks]
- (a) A wireless transmission channel can be modeled by the following difference equation: y ( n ) = x ( n ) + x ( n - M ) where M is a constant. Obtain expressions for the magnitude and phase response of the channel. For M = 13, and for a sampling rate of 100 kHz, calculate the magnitude and phase responses at a frequency of 20 kHz. [7 marks] [cont’d]
- (a) Using the Impulse Invariant Transformation, design a digital filter based on the following continuous-time transfer function:
s s
H s
Assuming that the sampling rate is chosen to be ten times the highest pole frequency in the analogue filter, calculate the digital filter coefficients, and write down the transfer function. [7 marks] (b) A simple first-order analogue filter is described by the following transfer function: c c
s
H s
where c is the cut off frequency in radians/s. Using the bilinear transformation, determine the transfer function of the digital equivalent of this filter, if the desired cutoff frequency is 2.5 kHz and the sampling frequency is 18 kHz. If pre-warping was not carried out, what would be the actual cut off frequency of the digital filter? [7 marks] (c) Using the window method, obtain an expression for the impulse response of a causal FIR low-pass filter, with a sampling rate of 4 kHz, a cutoff frequency of 1 kHz, and with a group delay of exactly 4 samples. Hint: start with the impulse response of an ideal low- pass filter. [6 marks]
- (a) In the system shown in Figure 2, x ( n ) is a band pass signal whose spectrum extends from 100 Hz to 200 Hz, with a sampling rate of 2 kHz. The carrier frequency of the modulator is 1 kHz, and the cutoff frequency of the low pass filter is 1 kHz.
2 X
Low pass
Filter
Upsampling Filtering
cos( n 0 )
x ( n )
u ( n )^ v ( n )
y ( n )
Figure 2 [Q5(a)] Sketch (one above the other) the magnitude of each of the spectra X ( θ ), U ( θ 1 ), V ( θ 1 ) and Y ( θ 1 ), against the corresponding digital frequency variable expressed in radians, and against the frequency in kHz. [13 marks] [cont’d]
(b) A signal with a sampling frequency of 10 kHz must be re-sampled to 8 kHz. Draw a block diagram of a system, using a combination of an up-sampler, a low-pass filter and a down- samplers, to achieve this. Indicate clearly the cut-off frequency of the low-pass filter (you may assume that the filter is “ideal”). If the order of the low-pass filter required is 24, calculate the number of multiplications and additions needed per second for the filtering operation. You should attempt to take advantage of any factors that may reduce the number of multiplications required. [7 marks]
- (a) A speech utterance has a duration of 2.0 seconds, and consists of the following segments: an initial period of silence 100 msec long; a period of unvoiced speech 400 msec long; another period of silence of duration 200 msec; a period of voiced speech 400 msec long; a second period of unvoiced speech 300 msec long; a seond period of voiced speech 400 msec long; a final silent period 200 msec long. Sketch the general form of: (i) The Energy contour (ii) The Zero-Crossing Count contour You may assume that the signal to noise ratio is very high, so background noise does not have significant energy. [4 marks] (b) LPC analysis of a speech signal shows that the first two formants have frequencies of 600 Hz and 1300 Hz, and bandwidths of 250 Hz and 750 Hz, respectively. The sampling frequency is 8 kHz. Assuming that each formant can be represented by a complex conjugate pole pair, calculate the coefficients of the corresponding fourth-order vocal tract transfer function. Note: it is sufficient to represent the transfer function as the cascade of two second- order transfer functions. [7 marks] (c) Explain what are meant by reflection coefficients in the context of the speech production model, and give an expression for their calculation. Under what circumstances would reflection coefficients be more useful than, say, LPC coefficients? If the ratio of the area between a section of vocal tract and the one preceding it is 1.3, calculate the reflection coefficient between the sections. [5 marks] (d) Give expressions for the Average Magnitude Difference Function (AMDF), and the Autocorrelation Function, for pitch analysis of speech. The AMDF extracted from a frame of voiced speech displays nulls at lag values of 63 and 126 samples. Estimate the pitch frequency if the sampling frequency is 8 kHz. [4 marks]
Table of useful z-Transforms Sequence z-Transform
- Unit sample d ( n ) 1 d ( n - k ) z - k
- Unit step u ( n ) z /( z -1)
- Exponential an^ u ( n ) z /( z - a )
- Sinusoidal sin( on ) u ( n )
z sin o
z^2 2 z cos o 1
cos( on ) u ( n )
z^2 z cos o
z^2 2 z cos o 1
- Unit ramp nu ( n )
z
( z 1)^2
- Product of ramp and signal nx ( n )
dz
dX z
z
- Sum of Series:
( 1 )^1
z
N
N z
z z z z
