Digital Signal Processing — Winter 2020

Compare the direct form-Iand II structures of an IIR systems, with M-zeros and N-poles.

Calculate the percentage saving in calculations in a 512- point radix-2 FFT, When compared in direct DFT.

Draw and explain architectural block diagram of TMSC6000 DSP processor.

Give any three properties of Butterworth lowpass filters.

Give the equation specifying Kaiser window. List the advantages of Kaiser window.

Design a single-pole lowpass digital filter with a 3-db bandwidth of 0.2π, using the bilinear transformation applied to the analog filter Ω c H(s) = s + Ω c Where Ω is the 3-dB bandwidth of the analog filter. c

List the application of an adaptive filter. Briefly explain any one of it.

Realize the following FIR system with minimum number of multipliers. h(n)={-0.5, 0.8, -0.5}

Determine all the FIR filters which are specified by the lattice parameters11 K = ,K = 0.6, K = −0.7 and K = 2 3.

Determine a direct-form realization for the following linear phase filter. ℎ(n) = {1,2,3,4,3,2,1}

Find the inverse DFT of Y(k)= {1,0,1,0}.

Derive the signal flow graph for the N= 16-point, radix-4 decimation-in-time FFT algorithm in which the input sequence is in normal order and the computations are done in place.1

Determine the inverse Fourier transform of X(ej𝜔) = 2𝜋𝛿(𝜔 − 𝜔 ), |𝜔 | ≤ 𝜋.00

Determine the inverse of the system with impulse response n1 ℎ(n) = ( ) u(n).2

Determine |H(𝜔)|2 for the system y(n) = −0.1y(n − 1) + 0.2y(n − 2) + x(n) + x(n − 1).

Determine the energy density spectrum of the signal x(n) = anu(n), − 1 < a < 1

Prove the Parseval’s relation ∞ 1 𝜋 ∑ x (n)x∗(n) = ∫ X (𝜔)X∗(𝜔)d𝜔 2𝜋 n=−∞ −𝜋

Determine the particular solution of the difference equation51 y(n) = y(n − 1) − y(n − 2) + x(n)66 When the forcing function x(n) = 2n,n ≥ 0 and zero elsewhere.

Find the z-transform of nanu(n).

Test the stability of the following systems. i. y(n) = cos[x(n)] ii. y(n) = x(- n -2 )

Find the response of the time invariant system with impulse response h(n) = {1,2,1,-1} to an input signal x(n) = {1,2,3,1}.

Determine the regions of convergence of right-sided, left- sided, and finite-duration two-sided sequences.

An analog ECG signal contains useful frequencies up to 100Hz. i. What is the Nyquist rate for this signal? ii. Suppose that we sample this signal at a rate of 250 samples/s. What is the highest frequency that can be represented uniquely at this sampling rate? Justify your answer.

Determine the inverse z-transform of1 X(z) 11.5z10.5z2 when ROC is |z| < 0.5 and |z| > 1 .