Q: Q.5. Determine the 8 point DFT of the following sequence - x(n) = [1/2, 1/2, …

For the sequence x(n)=[1/2,1/2,1/2,1/2,0,0,0,0], the computed 8-point DFT gives X(0)=2, X(1)=0.5-j1.2071, X(2)=0, X(3)=0.5-j0.2071, X(4)=0, X(5)=0.5+j0.2071, X(6)=0, X(7)=0.5+j1.2071, exhibiting the conjugate symmetry expected for a real-valued input sequence.

Question 1

Q.1. (a) Explain Sampling theorem. Give necessary condition of sampling. [8]

Accepted Answer

The sampling theorem states that a continuous-time, band-limited signal with maximum frequency fm can be perfectly reconstructed from its samples if the sampling rate fs is at least twice fm (the Nyquist rate); continuous-time processing of discrete-time signals is achieved by converting the discrete signal to continuous-time via an ideal D/A converter, processing it with an analog LTI system, then converting the result back to discrete-time via an ideal A/D converter.

Question 2

Q.1. What are sampling rate alteration devices? How can we increase or decrea…

Accepted Answer

Sampling rate alteration devices are decimators (which reduce the sampling rate by an integer factor M, requiring anti-aliasing filtering before discarding samples) and interpolators/expanders (which increase the sampling rate by an integer factor L, requiring zero-insertion followed by anti-imaging filtering); non-integer rational-factor rate conversion is achieved by cascading interpolation and decimation with a shared, combined filter.

Question 3

Q.2. (a) Obtain a linear convolution of following two discrete time signals -…

Accepted Answer

Interpreting x(n) as the rectangular pulse [1,1,1] for n=0,1,2 (a formalized sum-of-shifted-impulses representation) and computing its self-convolution gives the triangular sequence y(n)={1,2,3,2,1} for n=0 through 4; linear convolution satisfies the properties of commutativity, associativity, distributivity over addition, and the identity property with the unit impulse.

Question 4

Q.2. Find out the particular solution for the following difference equations …

Accepted Answer

For the difference equation y(n)+3y(n-1)=x(n), the characteristic root is r=-3 (unstable), and for a step input the particular solution is a constant K=A/4; for y(n)+3y(n-1)-4y(n-2)=x(n), the characteristic roots are r=-4 and r=1, and since r=1 coincides with the step input's own natural mode, the particular solution takes the modified form yp(n)=K*n rather than a simple constant.

Question 5

Q.3. (a) Draw and explain block diagram representation for discrete time LTI …

Accepted Answer

Discrete-time LTI systems are represented using block diagrams built from three basic elements - adders, constant multipliers (scalar gains), and unit delay elements; the given transfer function H(z)=z^-1/[(1+(1/3)z^-1)(1-(1/4)z^-1)] is realized in Direct Form I (separate feedforward/feedback delay chains) and cascade form (factored into two first-order sections connected in series).

Question 6

Q.3. (a) What are IIR and FIR filters? Draw basic structures for them and exp…

Accepted Answer

IIR filters use feedback (recursive) structures giving theoretically infinite impulse response duration, achieving sharp frequency responses with fewer coefficients but risking instability and nonlinear phase; FIR filters use only feedforward structures giving finite impulse response duration, always stable and capable of exactly linear phase, but generally requiring higher order for comparable selectivity; digital filters offer advantages including precision, stability, and reconfigurability over analog filters, at the cost of bandwidth limitations and quantization effects.

Question 7

Q.4. Determine H(z) using impulse invariance method at 5Hz sampling frequency…

Accepted Answer

Applying the impulse invariance method to Ha(s)=1/[(s+1)(s+2)] at a 5Hz sampling frequency (T=0.2s), the partial-fraction expansion gives Ha(s)=1/(s+1)-1/(s+2), and mapping each pole via z=e^(-aT) yields H(z)=0.2*[1/(1-0.8187z^-1)-1/(1-0.6703z^-1)].

Question 8

Q.4. (a) Explain design technique of FIR filters using - (i) Rectangular wind…

Accepted Answer

FIR filters can be designed using windowing techniques - rectangular window (simplest, poorest sidelobe attenuation), Hamming window (tapered, substantially improved sidelobe suppression at the cost of wider transition band), and Kaiser window (adjustable shape parameter allowing flexible trade-off between transition width and stopband attenuation); Chebyshev Type I filters achieve equiripple passband behavior with monotonic stopband roll-off, providing a sharper transition than Butterworth for the same filter order at the cost of passband ripple.

Question 9

Q.5. (a) What is Discrete Fourier Transform? List out the properties of DFT. [6]

Accepted Answer

The Discrete Fourier Transform (DFT) converts a finite N-point discrete-time sequence into an N-point discrete frequency-domain representation by sampling the sequence's DTFT at N equally-spaced frequencies; key DFT properties include linearity, circular time/frequency shift, circular convolution, Parseval's theorem, and conjugate symmetry for real sequences. For x(n)=a^n*u(n), 0<=n<=N-1, the N-point DFT reduces via the geometric series formula to X(k)=(1-a^N)/(1-a*e^(-j2*pi*k/N)).

Question 10

Q.5. Determine the 8 point DFT of the following sequence - x(n) = [1/2, 1/2, …

Accepted Answer

For the sequence x(n)=[1/2,1/2,1/2,1/2,0,0,0,0], the computed 8-point DFT gives X(0)=2, X(1)=0.5-j1.2071, X(2)=0, X(3)=0.5-j0.2071, X(4)=0, X(5)=0.5+j0.2071, X(6)=0, X(7)=0.5+j1.2071, exhibiting the conjugate symmetry expected for a real-valued input sequence.

Digital Signal Processing