Digital Signal Processing — Summer 2018

For the following systems, determine whether it is memoryless, stable, causal, or time invariant.

y[n]= ∑2n x[k] (04) k=−∞ (ii) y[n]= x[n2] (03)

Perform the circular convolution of the following sequences using graphical method. x1[n] = {1 2,2} x2[n] = {1,2,3,4}

Adiscrete time signal x[n] is defined as, 1 −1 ≤ n ≤ 31 x[n]= { n = 42 0 otℎerwise

Sketch the sequence (ii) Sketch x[n-2] (iii) Sketch x[n] u[2-n] (iv) Sketch x[2n]

Consider the difference equation511 y[n]− y[n−1]+ y[n−2] = x[n−1]663

What are the impulse response, and frequency response for the causal LTI (04) system satisfying the difference equation?

What is the general form of the homogeneous solution of the difference (03) equation?

The continuous time signal x (t) = sin(20𝜋t)+cos(40𝜋t) c Is sampled with a sampling period Tto obtain the discrete-time signal 𝜋n 2𝜋n x[n] = sin( )+cos( )55

Determine a choice for Tconsistent with this information.

Is your choice for Tin part (a) unique? If so, explain why? If not, specify another choice of Tconsistent with the information given.

State and Prove the following properties of Z-Transform.

Time shifting

Differentiation of X(z)

Acausal LTI system has the system function 1+2z−1+z−2 H(z)=1 (1+ z−1)(1−z−1)2 Find the impulse response of the system, h[n]. OR1

(a) Consider two LTI systems connected in series. Show that the overall system response does not depend on the order of interconnection.

Show that Forward difference system is inverse system of One-sample delay system.

Find Z Transform and ROC of following signals.

List the properties of DFT. Prove any two properties.

Consider a causal LTI system Swith impulse response h[n] and system function (1−2z−1)(1−4z−1) H(z) =1 z(1− z−1)2

Draw a direct form II flow graph for the system S.

Draw the transposed form of the flow graph in Part (a).

Write short note on structures for Linear Phase FIR system.

Compute 4-point DFT X[k] of a sequence x[n] = {0,1,2,3}

Discuss Radix-2 Decimation In Time (DIT) FFT algorithm.

Design the symmetric FIR low pass filter for which desired frequency response is expressed as e−jw𝜏 for |w|≤ w H (w)= { c d 0 elsewℎere The length of the filter should be 7 and w = 1rad/sample. Make use of rectangular c window.

With help of block diagram explain architecture of TMS320C6XXX processor

Write short note on Design of IIR filter by the Bilinear Transformation Method.