Q: Q.7. State the Sampling theorem.

The sampling theorem states that a band-limited signal with maximum frequency B Hz can be perfectly reconstructed from its samples if sampled at a rate fs ≥ 2B (the Nyquist rate).

Q: Q.8. What is meant by aliasing? How to avoid it?

Aliasing is the distortion where higher frequencies masquerade as lower frequencies when a signal is sampled below the Nyquist rate; it is avoided by sampling at fs ≥ 2B or by using an anti-aliasing low-pass filter before sampling.

Q: Q.9. List the basic characteristics of digital signal processor.

Digital signal processors feature fast, dedicated hardware multiply-accumulate (MAC) units, Harvard/modified-Harvard memory architecture for parallel instruction/data access, pipelining, and specialized addressing modes optimized for high-speed, real-time numeric computation.

Q: Q.10. Show that the following system is nonlinear and time invariant:

The system y(n+2)+2y(n)=x(n+1)+2 is nonlinear (due to the additive constant +2, violating homogeneity/superposition) but time-invariant (since its coefficients do not explicitly depend on n).

Question 1

Q.1. What is the need for multirate Signal processing?

Accepted Answer

Multirate signal processing (changing the sampling rate via decimation/interpolation) is needed to efficiently match differing sampling rates across systems, reduce computation/storage, and enable applications like sample-rate conversion and subband coding.

Question 2

Q.2. What conditions are to be satisfied by the impulse response of an FIR sy…

Accepted Answer

For linear phase, the FIR impulse response must be symmetric (h(n) = h(M-1-n)) or antisymmetric (h(n) = -h(M-1-n)) about its center.

Question 3

Q.3. Write computational efficiency of FFT over DFT.

Accepted Answer

Direct DFT computation requires O(N²) complex multiplications, while FFT reduces this to O(N log₂N), giving a dramatic speedup that grows with N (e.g., ~100x for N=1024).

Question 4

Q.4. Write some examples of multirate digital systems.

Accepted Answer

Examples of multirate digital systems include audio sample-rate converters, digital audio broadcasting, subband/wavelet coders (e.g., MP3, JPEG), software-defined radio, and oversampling ADCs/DACs (sigma-delta converters).

Question 5

Q.5. State two properties of DFT.

Accepted Answer

Two key DFT properties are linearity (DFT of a weighted sum equals the weighted sum of DFTs) and circular time-shifting (a circular shift in the time domain corresponds to multiplication by a complex exponential in the frequency domain).

Question 6

Q.6. Explain causality of a linear time invariant system.

Accepted Answer

A linear time-invariant system is causal if its output at any time n depends only on present and past input values, equivalently if its impulse response h(n) = 0 for all n < 0.

Question 7

Q.7. State the Sampling theorem.

Accepted Answer

The sampling theorem states that a band-limited signal with maximum frequency B Hz can be perfectly reconstructed from its samples if sampled at a rate fs ≥ 2B (the Nyquist rate).

Question 8

Q.8. What is meant by aliasing? How to avoid it?

Accepted Answer

Aliasing is the distortion where higher frequencies masquerade as lower frequencies when a signal is sampled below the Nyquist rate; it is avoided by sampling at fs ≥ 2B or by using an anti-aliasing low-pass filter before sampling.

Question 9

Q.9. List the basic characteristics of digital signal processor.

Accepted Answer

Digital signal processors feature fast, dedicated hardware multiply-accumulate (MAC) units, Harvard/modified-Harvard memory architecture for parallel instruction/data access, pipelining, and specialized addressing modes optimized for high-speed, real-time numeric computation.

Question 10

Q.10. Show that the following system is nonlinear and time invariant:

Accepted Answer

The system y(n+2)+2y(n)=x(n+1)+2 is nonlinear (due to the additive constant +2, violating homogeneity/superposition) but time-invariant (since its coefficients do not explicitly depend on n).

Question 11

Q.1. Compare direct form I and direct form II realization of IIR systems.

Accepted Answer

Direct Form I implements the numerator (zeros) and denominator (poles) sections as two separate cascaded delay chains, needing 2N delay elements, while Direct Form II combines them into a single shared delay chain, needing only N delay elements (canonical form).

Question 12

Q.2. Compute the DFTs of the sequence:

Accepted Answer

For x(n)=2^-n, n=0..7, computing the 8-point DFT via the DIT-FFT algorithm gives X(0)=1.9922, with the remaining X(k) values obtained through the standard 3-stage (log2(8)=3) butterfly decomposition using twiddle factors W8^k.

Question 13

Q.3. What is a Hamming window function? Obtain its frequency domain character…

Accepted Answer

The Hamming window is w(n) = 0.54 - 0.46cos(2πn/(M-1)) for 0≤n≤M-1, offering a good trade-off between main-lobe width and side-lobe attenuation (about -41 dB peak side-lobe), used to reduce spectral leakage in FIR filter design and spectral analysis.

Question 14

Q.4. Two causal discrete-time signals x[n] and y[n] are related by:

Accepted Answer

Since y[n] is the running sum of x[n], X(z)=(1-z⁻¹)Y(z); computing this gives x[n] = 2(n-2) for n≥3 and 0 otherwise, so x[2] = 0.

Question 15

Q.5. A signal x(t) = exp(-2πBt) u(t) is the input to an ideal low pass filter…

Accepted Answer

For x(t)=e^{-2πBt}u(t) passed through an ideal LPF of bandwidth B Hz, evaluating the mean-square error integral using Parseval's theorem gives the energy in the frequency content of x(t) beyond ±B Hz, which computes to approximately 0.0537/B.

Question 16

Q.6. A sequence x(n) with the Z-transform:

Accepted Answer

Since h(n)=2δ(n-3) makes the system act as an ideal scaled delay, y(n)=2x(n-3); reading x(1) from the given X(z) polynomial (which has no z^-1 term) gives x(1)=0, so the output at n=4 is y(4)=0.

Question 17

Q.7. Explain the difference between Butterworth and Chebyshev filter.

Accepted Answer

Butterworth filters have a maximally flat passband response with no ripple in either band but a gradual roll-off, while Chebyshev filters allow controlled ripple in the passband (Type I) or stopband (Type II) in exchange for a much sharper transition band for the same filter order.

Question 18

Q.1. By means of DFT and IDFT, determine the response of the FIR filter with …

Accepted Answer

Using DFT/IDFT-based (circular) convolution with adequate zero-padding to length 7, the linear convolution of h(n)={5,6,7} and x(n)={1,2,-1,5,6} gives the FIR filter output y(n) = {5, 16, 14, 33, 53, 71, 42}.

Question 19

Q.2. Determine a direct form realization for the following linear phase filte…

Accepted Answer

For the symmetric linear-phase filter h(n)={1,2,3,4,3,2,1} (M=7, odd length, Type I linear phase), the direct form realization exploits the symmetry h(n)=h(6-n) to fold the structure, combining input samples in pairs before multiplication and halving the number of multipliers from 7 to 4.

Question 20

Q.3. The desired frequency response of a low pass filter is:

Accepted Answer

Using the window method with M=7 and a rectangular window on the ideal lowpass response with cutoff 3π/4 and linear phase delay of 3 samples, the resulting FIR filter coefficients h(n) are obtained by sampling the ideal impulse response's sinc-based formula at n=0..6.

Question 21

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

Accepted Answer

Using the bilinear transformation design procedure — pre-warping the digital frequencies, finding the required analog Butterworth order N and cutoff Ωc from the given passband/stopband specs, then mapping back via s=(2/T)(1-z⁻¹)/(1+z⁻¹) — yields a digital Butterworth filter satisfying the given constraints.

Question 22

Q.5. Consider an FIR lattice filter with coefficients K1 = 0.65, K2 = 0.5, K3…

Accepted Answer

For an FIR lattice filter with reflection coefficients K1=0.65, K2=0.5, K3=0.9, converting to direct-form coefficients via the standard lattice-to-direct-form recursion gives the impulse response h(n) = {1, 1.475, 1.825, 0.9}, and the direct form structure is a simple 4-tap tapped-delay-line FIR realization using these coefficients.

Digital Signal Processing