Digital Signal Processing — Winter 2022

Represent the sequence x (n) = {0.5, 3, 0, 1) into a sum of weighted impulse sequences.

What is Nyquist criteria for sampling? What will happen if sampling process do not follow the Nyquist criteria?

The impulse response of a linear time-invariant system is h (n) = {1, 2, 1,0,1}. Determine by graphical method, response of the ↑ System to the input signal x (n) = {1, 2, 3, 1}.

Adiscrete-time signal x (n) is defined as: n 1+ , −3 ≤ n ≤ −13 x (n) = { 1, 0 ≤ n ≤ 0, elsewℎere Determine its values and sketch the signal x (n).

State and prove frequency shifting and convolution properties of z-transform.

Consider a system with impulse response 1 n ( ) , 0 ≤ n ≤ ℎ (n){ 0, elsewℎere Determine the input x (n) for 0 ≤ n ≤ 8 that will generate the output sequence y(n) = { 1,2,2.5,3,3,3,2,1,0, . . .}

A LTI system is characterized by the system function12 H (Z) = + 1 1−3z−1 1− z−12 Specify the ROC of H(z) and determine h(n) for the following conditions:

The system is stable. ii) The system is causal. iii) The system is anticausal.

Draw nature of Region of Convergence (ROC) in Z-plane for infinite duration causal, anticausal and two-sided sequence.

Determine the response of the system1 y (n) = y(n−1)− y (n−2)+x(n)66 to the input signal x (n) = 𝛿 (n)− 𝛿 (n−1).3

Obtain the direct form Iand direct form II structures for the following system: 1+0.875 z−1 H (Z) = (1+0.2 z−1 +0.9 z−2)(1−0.7 z−1)

Consider the signal x (n) = {−1,2,−3,2,−1} ↑ with Fourier transform X (𝜔). Compute ∫ 𝜋 |X(𝜔)|2 d𝜔. −𝜋

Compute the 4-point DFT of a discrete time sequence {1, 0, 2, 1}.

Determine the zeros for the following FIR systems and indicate whether the system is minimum phase, maximum phase, or mixed phase. H (Z) = 6+z−1 −z−21 H (Z) = 1−z−1 −6z−22 H (Z) = 1− z−1 − z−2322 H (Z) = 1+ z−1 − z−2433

Explain how circular convolution can be converted to linear convolution by zero padding.

Using decimation in time algorithm, compute 4-point DFT of the sequence x(n)={0, 1, 2, 3}

4-point DFT of a sequence x(n) is {1,0,1,0}. By using butterfly diagram as in DIT-FFT, find x(n).

Explain necessity of windowing in FIR filter design.

Find out H (Z) for the given H (S) = 2 using impulse S2+3S+2 invariance method. Assume T=1s.

Consider input sequence x (n) = {1,2,3} and impulse response of a systemℎ (n) = {1,1}. Find the linear convolution using graphical circular convolution method. Match result of same using tabulation/ matrix method.

Explain need of anti-aliasing filter in a down sampler.

Explain how to achieve sampling rate conversion by rational factor L/M. Where Land Mare constant.

List and explain the application of Multirate Signal Processing.

Explain how echo cancellation is achieved using adaptive filters.

Compare Up sampling and Down Sampling.

Explain pipelining and MAC architecture of Digital Signal Processor.