MSC Exam Digital Signal Processing Jan 2022

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Questions and Answers

What is the primary characteristic of the output function y[n] in the discrete-time system S?

It results in a static output regardless of the input.

It has no relation to the input signal x[n].

It is linearly independent of the input signal x[n].

It includes a multiplicative term engaged with the input signal. (correct)

Is the given discrete-time system S stable?

Yes, because the output always remains bounded. (correct)

No, because the output can diverge for bounded inputs.

No, due to the nature of the ejω0n term.

Yes, stability is determined solely by the value of α.

Which property indicates that the output of system S solely depends on the present input and not on past inputs?

Causal (correct)

Stable

Time-invariant

Linear

What is necessary for the system S to be considered linear?

It must satisfy the property of superposition. Signup and view all the answers

Which condition must the sampling interval T1 meet to avoid aliasing when converting continuous-to-discrete time?

It should be greater than the Nyquist rate. Signup and view all the answers

To sketch the Fourier transform X(ejω) of x[n] with a sampling interval T1 = 2ms, which element is crucial?

The band-limited nature of the input signal spectrum. Signup and view all the answers

What is the condition for the system to be considered time-invariant?

The system's response should remain unchanged with shifts in input. Signup and view all the answers

How can the Fourier transforms of y[n] and x[n] be related in this system?

By a function S(ejω) that modifies the input's transform. Signup and view all the answers

What is the nature of the system given by the difference equation y[n] = x[n] − x[n − 1] + y[n − 1] − y[n − 2]?

Causal linear time-invariant system Signup and view all the answers

Which condition must the system satisfy to have a stable inverse?

The poles of the transfer function must lie inside the unit circle. Signup and view all the answers

In the context of the given causal system, what would be the result of applying the N-point inverse DFT?

A sequence representing the linear convolution of x1[n] and x2[n] Signup and view all the answers

What is the minimal N needed to ensure no aliasing occurs when obtaining x̃[n] from x1[n] and x2[n]?

N must be equal to the sum of the lengths of x1[n] and x2[n] minus one Signup and view all the answers

Which of the following statements is true regarding the linear convolution of two sequences?

It can be computed without the need for finite-length sequences under certain conditions. Signup and view all the answers

Given the system's difference equation, what type of stability assessment should be conducted?

Analyze the characteristic equation for pole locations Signup and view all the answers

What would be an appropriate method for finding the causal inverse of the system defined by the difference equation?

Use recursive equations based on the original difference equation Signup and view all the answers

How can you determine if the system has a unique inverse?

By finding the system's impulse response and ensuring it is not periodic Signup and view all the answers

What condition must be met for x̃[n] to equal x[n] in the context of linear convolution?

N must be large enough to prevent aliasing. Signup and view all the answers

What is the minimum size of the DFT necessary for linear convolution without aliasing?

It must be the sum of lengths of x1[n] and x2[n] minus one. Signup and view all the answers

For which values of n will the equality x̃[n] = x[n] hold true?

For n = 0, 1,..., N - 1. Signup and view all the answers

What denotes the Fourier Transform pair for the unit impulse function δ[n]?

1, the whole complex plane. Signup and view all the answers

What is the z-transform representation for the sequence a^n u[n] where |a| < 1?

1/(1 - a z^{-1}). Signup and view all the answers

What effect does a larger DFT size have on the sampling of linear convolution?

It improves time-domain resolution. Signup and view all the answers

Which condition is necessary for the z-transform of the sequence -a^n u[-n - 1]?

The ROC must be |z| > |a|. Signup and view all the answers

What is the Fourier Transform representation of cos(ω0 n + φ)?

π(δ(ω-ω0) + δ(ω+ω0)). Signup and view all the answers

Study Notes

Examination Information

Examination is part of a degree program
Governed by College Regulations and academic board
MSC Examination for Fundamentals of Digital Signal Processing
January 2022
Time allowed: Two hours
Answer all questions
Answer each question in a separate answer book
Write question number on the cover
Fourier and Z-transform pairs provided on pages 5 and 6 of the paper

Questions One to Five

Discrete-time system S, input x[n], output y[n]
y[n] = S{x[n]} = e^jω₀nx[n] + a
ω₀ and a are real constants
0 < |a| < ∞, 0 < |ω₀| < π
Stability, causality, linearity, time invariance of the system
Fourier transforms of y[n] and x[n] related as Y(e^jω) = S(e^jω)X(e^jω)

Questions Six to Eight

Discrete-time processing of continuous-time signals
Input signal x(t) with spectrum X(jΩ)
Band-limited to Ω_N = 1000π rad/s
Sampling interval T₁ to avoid aliasing
Sketch X(e^jω) for T₁ = 2ms
Find y_c(t) given H(e^jω) and T₂ = T₁

Question Nine to Eleven

Causal linear time-invariant system
Difference equation: y[n] = x[n] - x[n-1] + 0.5y[n-1] - 0.25y[n-2]
Input x[n], output y[n] Assess stability of the inverse system, find a causal inverse, determine if the system has a unique inverse

Question Twelve to Fourteen

Sequences x₁[n] and x₂[n]
x₁[n] = 2δ[n-2] + 3δ[n-3] + δ[n-4]
x₂[n] = 2δ[n] + δ[n-1] + δ[n-2]
Inverse DFT of X[k] = X₁[k]X₂[k]
Identify N for linear convolution between x₁[n] and x₂[n] without aliasing
Minimal size N and range of n for x[n] and x[n] Numerical values of x[n], n=0, 1, ..., N-1

Fourier Transform Pairs

Provided on page 6 of the document, various pairs listed
Includes δ[n], u[n], cos(ω₀n), and sin(ω₀n), others

Z-transform Pairs

Provided on page 7 of the document, various pairs listed
- Includes δ[n], u[n], cos(ω₀n), sin(ω₀n), and others

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Description

Test your knowledge on the fundamentals of digital signal processing with this MSC examination from January 2022. The quiz covers discrete-time systems, stability, causality, Fourier transforms, and more. Make sure to review the provided Fourier and Z-transform pairs for a comprehensive understanding.