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Cork Institute of Technology
Bachelor of Engineering (Honours) in Electronic Engineering –
Award
(NFQ – Level 8)
January 2006
Mathematics and Digital Signal Processing
(Digital Signal Processing)
(Time: 1.5 Hours)
Answer ONE question from EACH section. All questions carry equal marks.
Examiners: Dr. J. Connell Prof. G. Hurley Dr. S. Foley
Section A
Q1.
a) What is aliasing? What hardware is used to avoid it and comment on its effectiveness? Outline the rules of aliasing when it does take place. Comment on the spectral implications of the aliasing process. (7 marks)
b) An analogue signal is given by the expression: ) 3
() 3 5 cos( 440
x t = − π − t V.
It is sampled at f (^) s = 100 Hz. After how many samples will the sequence repeat itself? Calculate x ( 2 ). Write out an expression for a signal, x 1 (^) ( t ), lying in the spectral range 0 ≤ f < fs / 2 which would produce identical samples at f (^) s = 100 Hz. (8 marks)
c) Discuss the mechanism of quantisation in an ADC.
If x ( 2 )in b) was quantised in a 10 bit ADC with an input dynamic range of ± 10 V , calculate the corresponding binary number thus produced. Assume
that V 2
− + corresponds to 0000000001. (10 marks)
Q2.
a) A periodic digital signal is given as {…1 -4 -2 3 1 -4 -2….}.
↑ Using the Radix-2 FFT flow diagram, derive an expression for x ( n ). (15 marks)
b) Compare the number of complex multiplications and additions in a) with a straight implementation of the IDTFS expression. Hence write down general expressions for the number of complex multiplications and additions in an N- point Radix 2 FFT. (6 marks)
c) A second periodic digital signal is given as {…-4 -2 3 1 -4 -2 3….}.
↑ Comment on the nature of its spectrum relative to that of the signal in a). (4 marks)
Section B
Q3.
a) Comment on the amplitude/phase responses of even/odd order and symmetric/asymmetric impulse response filters. Hence specify their suitability, or otherwise, for LP, HP, BP, BR filtering operations. (12 marks)
b) Design an eighth order, nonrecursive filter to remove all frequencies below
1250Hz. The passband gain is to be H ( ej ω)= 1. The window function is Hamming. Assume the input signal is being sampled at 8kHz. (13 marks)
Q4.
a) A digital, recursive, second order filter is required to reject all frequencies between 1100Hz and 2000Hz in a signal. The design is to be based on the normalised, first order, lowpass, Butterworth expression and is to incorporate the Bilinear Transformation Method. The system sampling frequency is 8kHz.
Draw its implementation flow chart using the Direct Form II approach and hence write down the system algorithm(s). (16 marks)
