Computer Science-QA247

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Signal Processing

 

QUESTIONS
 

In all question you can use Matlab (or other similar tools) but all steps of your solutions should be explained and supported by the adequate equations.
 

1. (a) Evaluate the Discrete Time Fourier Transform (DTFT) of the sequence . Try to simplify the expression as much as possible. [0] 3, [1] 2, [2] 0, [3] 2, [4] 3 (ZERO otherwise)
 
(b) Evaluate the Discrete Time Fourier Transform (DTFT) of the sequence . Try to simplify the expression as much as possible. [2] 3, [1] 2, [0] 0, [1] 2, [2] 3 (ZERO otherwise)
 
(c) Plot amplitude and phase spectra of sequences from (a) and (b). Compare them and draw conclusions.
 
(d) For the sequence calculate its DTFT and Discrete Fourier Transform (DFT). Verify that DFT and DTFT are equal for the corresponding frequencies. [0] 2, [1] 1, [3] 1, [4] 2 (ZERO otherwise)
 

2. A causal signal has the following equation: x [n ]=1/3n U(n)

 
a) Evaluate the signal energy using the time-domain approach and using the frequency-domain approach. Are the results the same?
 

b) Estimate the percentage of the signal energy carried by the frequencies below 0.125Hz and by the frequencies below 0.25Hz.

 
3. Find and sketch the impulse responses of the systems defined by following difference equations
 

a) y[k]=1.5y[k-1]-0.5y[k-2]+0.5x[k-1]
b) y[k]=y[k-1]+2x[k-1]+x[k-2]
 
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4. Determine difference equations for the systems represented by the following two diagrams. Find the z-transfer functions of these systems and discuss their stability.
 

5. Evaluate the natural responses and the impulse responses for the systems defined by the following equations (use any technique you know). Assess stability of both systems.
 

a) y[k]=y[k-2]-x[k]+x[k-1]
b)y[k]=1.5y[k-1]-0.5y[k-2]+x[k]-x[k–2]

 
6. An analog signal is discretized in two different ways
 

i. Ts1 = 0.02, i.e. fs1 = 50Hz;

ii. Ts2 = 0.002, i.e. fs2 = 500Hz.
For both cases design a high-pass DIGITAL Butterworth filter of the second order with the cutoff frequency 20Hz.
Plot frequency characteristics of your filters to confirm that the filters meet the specification. Write the difference equations for the filters.

 
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