Q: Q.5. Explain the following: (a) Properties of the DFT [8] (b) DIT Algorithm [8]

Properties of the DFT include linearity, periodicity, circular shift, circular convolution, Parseval's theorem, and conjugate symmetry for real sequences; the Decimation-In-Time (DIT) FFT algorithm recursively splits an N-point DFT into two (N/2)-point DFTs of the even- and odd-indexed input samples, combining their results via butterfly computations using twiddle factors, reducing computational complexity from O(N-squared) to O(N*log2(N)).

Q: Q.5. Given x(n) = {1,2,3,4,4,3,2,1}. Find X(K) using DIF FFT Algorithm. [16]

For x(n)={1,2,3,4,4,3,2,1}, applying the DIF FFT algorithm's first-stage decomposition and completing the recursive computation gives X(k)={20, -5.8284-j2.4142, 0, -0.1716-j0.4142, 0, -0.1716+j0.4142, 0, -5.8284+j2.4142}.

Question 1

Q.1. Explain the following using suitable mathematical derivation and wave fo…

Accepted Answer

Decimation reduces the sampling rate by an integer factor M, causing the signal spectrum to stretch (requiring anti-aliasing filtering before downsampling), while interpolation increases the sampling rate by factor L, causing the signal spectrum to compress into repeated images (requiring anti-imaging filtering after zero-insertion).

Question 2

Q.1. (a) Obtain the two fold expanded signal y(n) of the input signal x(n) = …

Accepted Answer

The two-fold expansion (zero-insertion) operation inserts a single zero sample between each pair of consecutive original samples, illustrated with a general example sequence since the given x(n) formula is ambiguously stated; discrete-time processing of continuous-time signals is achieved via an equivalent A/D-processing-D/A chain.

Question 3

Q.2. Explain the following linear systems - (a) Minimum phase system [8] (b) …

Accepted Answer

A minimum phase system has all its poles and zeros located strictly inside the unit circle, giving it the smallest possible phase lag (group delay) among all systems sharing the same magnitude response; an all-pass system has a magnitude response of exactly unity at all frequencies, achieved through a specific pole-zero mirroring arrangement about the unit circle, used for phase correction/equalization without altering magnitude response.

Question 4

Q.2. (a) Determine H(z) and its poles and zeros if y(n) + (3/4)y(n-1) + (1/8)…

Accepted Answer

For y(n)+3/4*y(n-1)+1/8*y(n-2)=x(n)+x(n-1), H(z)=(1+z^-1)/(1+3/4*z^-1+1/8*z^-2) with a zero at z=-1 and poles at z=-0.25 and z=-0.5; for y(n)+1/2*y(n-1)=x(n)-x(n-1), the magnitude response |H(e^jw)|=|1-e^-jw|/|1+0.5e^-jw| evaluates to 0 at w=0, 0.5471 at w=pi/4, 1.2649 at w=pi/2, and 4.0 at w=pi, showing high-pass characteristic behavior.

Question 5

Q.3. (a) Determine direct form I and II for the second order filter given by …

Accepted Answer

The given second-order filter y(n)=2b*cos(w0)*y(n-1)-b^2*y(n-2)+x(n)-b*cos(w0)*x(n-1) has transfer function H(z)=[1-b*cos(w0)*z^-1]/[1-2b*cos(w0)*z^-1+b^2*z^-2], with Direct Form I using separate feedforward and feedback delay chains (3 total delays) and Direct Form II (canonic form) sharing a single delay chain (2 total delays, matching the maximum of numerator and denominator order); realization block diagrams and signal flow graphs both represent the same underlying system using adders, multipliers, and delays, or nodes and weighted branches respectively.

Question 6

Q.3. Obtain the cascade and parallel realisations for the system function giv…

Accepted Answer

For H(z)=[1+(1/4)z^-1]/[(1+(1/2)z^-1)(1+(1/2)z^-1+(1/4)z^-2)], the cascade realisation factors the transfer function into a first-order section combined with the zero, connected in series with a second-order all-pole section; the parallel realisation uses partial fraction expansion to express H(z) as a sum of a first-order term and a second-order term, connected in parallel with their outputs summed.

Question 7

Q.4. (a) For the analog transfer function H(s) = 1/[(S+1)(S+2)], determine H(…

Accepted Answer

For H(s)=1/[(S+1)(S+2)] with impulse invariance and T=1s, the resulting digital filter is H(z)=1/(1-0.3679z^-1)-1/(1-0.1353z^-1); applying the bilinear transformation to H(s)=2/[(S+1)(S+3)] with T=0.1s gives H(z)=(0.004141+0.008282z^-1+0.004141z^-2)/(1-1.643892z^-1+0.668737z^-2).

Question 8

Q.4. Design a digital Butterworth filter that satisfies the following constra…

Accepted Answer

For the given digital Butterworth filter design specifications (0.9<=|H(e^jw)|<=1 for 0<=w<=pi/2, |H(e^jw)|<=0.2 for 3*pi/4<=w<=pi, T=1s), the bilinear-transformation-based design procedure gives prewarped analog frequencies Omega_p=2.0 rad/s and Omega_s=4.8284 rad/s, required filter order N=3 (rounded up from the exact calculated value of 2.6255), and cutoff frequency Omega_c=2.5467 rad/s.

Question 9

Q.5. Explain the following: (a) Properties of the DFT [8] (b) DIT Algorithm [8]

Accepted Answer

Properties of the DFT include linearity, periodicity, circular shift, circular convolution, Parseval's theorem, and conjugate symmetry for real sequences; the Decimation-In-Time (DIT) FFT algorithm recursively splits an N-point DFT into two (N/2)-point DFTs of the even- and odd-indexed input samples, combining their results via butterfly computations using twiddle factors, reducing computational complexity from O(N-squared) to O(N*log2(N)).

Question 10

Q.5. Given x(n) = {1,2,3,4,4,3,2,1}. Find X(K) using DIF FFT Algorithm. [16]

Accepted Answer

For x(n)={1,2,3,4,4,3,2,1}, applying the DIF FFT algorithm's first-stage decomposition and completing the recursive computation gives X(k)={20, -5.8284-j2.4142, 0, -0.1716-j0.4142, 0, -0.1716+j0.4142, 0, -5.8284+j2.4142}.

Digital Signal Processing