Digital Signal Processing (EC-601) - Important Questions

Define/state stability, causality (and invertibility) properties of discrete-time LTI systems and how stability is determined from the impulse response.

Solve the second-order linear constant-coefficient difference equation for $y[n], n \ge 0$ given the input sequence and initial conditions.

Discuss Direct Form-I and Direct Form-II implementations of discrete-time systems and their main differences.

Determine whether the system $y[n]=0.6y[n-1]+x[n]$ is stable and causal with justification.

Explain the significance of pole-zero analysis for discrete-time systems, including effects on stability and frequency response.

Explain discrete-time systems and their classification with examples.

Determine the frequency response $H(e^{j\omega})$ of an LTI system with impulse response $h[n]=(0.5)^n u[n]$.

Implement a given second-order difference equation using Direct Form-I and Direct Form-II structures with block diagrams.

Determine the impulse response of the FIR feedforward system $y[n] = \alpha_1 x[n] + \alpha_2 x[n-1] + \alpha_3 x[n-2] + \alpha_4 x[n-3]$.

Describe the typical standard sequences and how they are represented.

Explain the different representations of discrete-time signals.

Find the impulse response of the recursive IIR system $y(n)-2y(n-1)+4y(n-2)=x(n)+x(n-1)$ with zero initial conditions.

Represent the system described by the given state-space equations using a signal flow graph.

Determine the Z-transform of exponential sequences including $x[n] = -\alpha^n u[-n-1]$, $x[n]=\alpha^n u[n]+b^n u[-n-1]$, and $x[n]=(1/2)^n u[n]$.

Prove and explain the convolution in time and scaling in time-domain properties of the Z-transform.

Use properties of the z-transform to find the z-transform of a modulated exponential sequence.

Find the inverse Z-transform of $X(z)=\frac{z^{-4}}{z-1}+z^{-6}+\frac{z^{-3}}{z+0.5}$.

Discuss the advantages of analyzing discrete-time LTI systems in the z-domain compared to the time-domain with examples.

Describe the concept of region of convergence of a rational z-transform.

Determine the inverse Z-transform via partial fraction expansion of $H(z)=\frac{z(z+2.0)}{(z-0.2)(z+0.6)}$.

Determine the inverse Z-transform of $X(z)=\frac{1+z^{-1}}{1+\frac{1}{3}z^{-1}}$.

Determine the inverse Z-transform of $X(z)=\frac{z^2}{(z-a)^2}$ with ROC $|z|>a$.

Describe the z-transform and its properties.

Find the transfer function of the given signal flow graph of the system.

Determine the 4-point circular convolution of two given length-4 sequences.

State and prove four properties of the DFT (e.g., linearity, circular time/frequency shifting, time reversal).

Explain the concept and derivation/expression of the Discrete Fourier Series (DFS) and its properties.

Compute the 2D DFT of the following $2\times 2$ image matrix: $$\begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix}$$

Compute the N-point DFT of the given finite-length sequence $x(n)$.

Compute the DFS coefficients for the periodic signal $x[n] = \{1, 2, 3, 4\}$ with period 4.

Illustrate and explain the properties of the Discrete Fourier Series.

Define the DFT and explain its important properties.

Derive and prove the properties of the Discrete Fourier Series (including time shifting, time reversal, convolution).

Find the DFT of the infinite sequence $x(n) = (1/2)^n u(n)$.

Compute the 8-point DFT using the radix-2 decimation-in-frequency FFT algorithm showing intermediate stages.

Determine the DFT of a given sequence using the Radix-2 FFT algorithm with detailed steps.

Explain the Decimation-in-Time FFT algorithm and how it decomposes the DFT computation.

For a sequence of length $N = 6$ (where $N = 2\times 3$), describe how the FFT can be applied using the Cooley-Tukey algorithm. Perform the computation for $x[n] = \{1, 2, 3, 4, 5, 6\}$.

What is the Fast Fourier Transform (FFT) and how does it improve the efficiency of computing the DFT, with example?

Perform a single stage of the DIT-FFT algorithm for $x[n]=\{1,2,3,4,5,6,7,8\}$ and illustrate the decomposition and computation.

What is FFT? Derive the expression for the FFT / Radix-2 FFT algorithm.

Determine the 4-point IDFT of $\{1,-j,0,j\}$ using the DIT-FFT algorithm.

Compute the 8-point DFT/FFT of $x[n]=n+1$ (i.e., $\{1,2,\dots,8\}$) using the DIT-FFT algorithm.

Explain the decimation-in-frequency (DIF) FFT algorithm.

Characterize IIR filters and compare advantages and disadvantages of IIR filters versus FIR filters.

Compare and contrast the impulse invariant transformation and bilinear transformation methods for IIR filter design.

Design a low-pass FIR filter with given cutoff frequency and length using the rectangular window method, give coefficients and frequency response.

Design a high-pass IIR filter using the bilinear transformation method from a Butterworth analog prototype.

Explain the design techniques for FIR filters.

Convert a given analog filter $H(s)$ to a digital filter using the impulse invariant technique.

Outline the steps involved in the design of FIR filter using Hanning / Hamming window.

Write the magnitude and phase functions of an FIR filter with symmetric impulse response and odd length N.

Write short notes on DFT, Bilinear transformation method, Rectangular window and block diagram.

Explain different realization structures for FIR systems.

Write a short note on any two

Write a short note on any two.

Write short notes on any two of the following: