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GTU Electrical and Electronic Engineering (Semester 7)
Discrete Time Signal Processing
December 2014
Discrete Time Signal Processing
December 2014
1 (a)
Prove that LTI system is stable if its impulse response is absolutely summable. Determine the range of values of parameter "a" for which the LTI system with impulse response h(n)=anu(n) is stable.
7 M
1 (b)
Determine impulse response and step response of a causal and stable LTI system described by second-order difference equation \[ y(n)- \dfrac {1}{12}y(n-1)- \dfrac {1}{12} y(n-2)=x(n) \]
7 M
2 (a)
Explain following properties of Z transform.
(i) Time shifting (ii) Differentiation
(i) Time shifting (ii) Differentiation
7 M
Answer any two question from Q2 (b) or Q2 (c)
2 (b)
Using long division determine the inverse z-transform of the following: \[ X(z)= \dfrac {1}{1-\frac {3}{2}z^{-1}+\frac {1}{2}z^{-2}} \] when ROC: |z|>1.
7 M
2 (c)
Using partial-fraction expansion find the inverse z-transform of the following: \[ X(z)= \dfrac {1- \frac {1}{2}z^{-1}}{1-\frac {1}{4}z^{-2}} \ \ \ |z| > \dfrac {1}{2} \]
7 M
Answer any two question from Q3 (a), (b) or Q3 (c), (d)
3 (a)
Determine z-transform and ROC of the following sequences
i) x(n)=[3(2n)-4(3n)]u(n)
ii) x(n)=anu(n)
i) x(n)=[3(2n)-4(3n)]u(n)
ii) x(n)=anu(n)
7 M
3 (b)
Determine DTFT of the sequence given by
i) x(n)=u(n)-u(n-6)
ii) x(n)=anu(n)
i) x(n)=u(n)-u(n-6)
ii) x(n)=anu(n)
7 M
3 (c)
Explain following properties of discrete time Fourier transform, i) Convolution ii) Frequency differentiation.
7 M
3 (d)
Perform the circular convolution of two sequence.
x1(n) {1,1,2,2,}
x2(n)={1,2,3,4}
x1(n) {1,1,2,2,}
x2(n)={1,2,3,4}
7 M
Answer any two question from Q4 (a), (b) or Q4 (c), (d)
4 (a)
Derive the DFT of the sample data sequence x(n)={1,1,2,2,3,3}.
7 M
4 (b)
Explain decimation-in-time Radix-2 FFT algorithm.
7 M
4 (c)
Determine IDFT of X(k)={1,2,3,4}.
7 M
4 (d)
Explain Divide and Conquer Approach to Computation of the DFT.
7 M
Answer any two question from Q5 (a), (b) or Q5 (c), (d)
5 (a)
What are different specification required to design a low pass IIR digital filter? Compare IIR digital filter design using the Butterworth and Chebyshev approximations.
7 M
5 (b)
Explain Frequency Sampling method for FIR digital filter design.
7 M
5 (c)
Describe the design of discrete-time IIR filters using bilinear transformation method.
7 M
5 (d)
Compare the commonly used windowing techniques for FIR filter design.
7 M
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