MSC Exam Digital Signal Processing Jan 2022

A system is considered stable if every bounded input signal produces a bounded output signal.

A system is causal if the output at any time instance depends only on present and past input values, not on future input values.

A system is linear if it satisfies the principle of superposition, meaning the output response to a sum of inputs is equal to the sum of the responses to individual inputs.

A system is time-invariant if a time shift in the input signal results in the same time shift in the output signal.

A system is linear time-invariant (LTI) if it satisfies both linearity and time-invariance properties. The frequency response of an LTI system is represented by the function S(ejω).

To avoid aliasing, the sampling frequency (fs) should be at least twice the highest frequency component (ΩN) in the continuous-time signal.

The Fourier transform of a discrete-time signal x[n] is denoted by X(ejω). It represents the frequency spectrum of the signal.

For a continuous-time signal xc(t) with maximum frequency component ΩN, the sampling interval T1 should satisfy: fs = 1/T1 ≥ 2ΩN/2π, or T1 ≤ π/ΩN.

The minimal size of the DFT, N, required to ensure that samples of the linear convolution of x1[n] and x2[n], denoted as x[n], are contained within the DFT output, x̃[n], without aliasing.

The range of indices 'n' for which the DFT output, x̃[n], corresponds exactly to the original discrete-time signal, x[n], after performing linear convolution of x1[n] and x2[n].

The numerical values of the DFT output, x̃[n], for a specific range of 'n' values, from 0 to N-1. These values represent the frequency domain representation of the convolution result.

The convolution of two discrete-time signals, x1[n] and x2[n], also denoted by x[n], is a process where each element of the output signal is calculated by summing the products of the corresponding elements of the two input signals.

The Discrete Fourier Transform (DFT) of a discrete-time signal x[n] is a mathematical operation that transforms the signal into its frequency domain equivalent, denoted by x̃[n].

Aliasing occurs when the sampling frequency of a signal is too low, resulting in the higher frequency components of the signal being incorrectly interpreted as lower frequencies, leading to distortion in the sampled signal.

The size of the DFT determines the number of frequency components that can be represented by the transform. Ideally, the size of the DFT should be large enough to provide adequate frequency resolution and avoid aliasing.

A system is considered stable if its output remains bounded for all bounded inputs. To determine stability, analyze the system's impulse response and its behavior in response to bounded inputs.

The inverse of a system is another system that undoes the effects of the original system. It takes the output from the original system and produces its original input. A system has an inverse if and only if it is invertible, which means its input can be uniquely determined from its output.

A causal system is one where the output at any given time depends only on the present and past inputs. This means the output does not depend on future inputs.

The impulse response of a system describes its output when the input is an impulse function, which is a brief spike at time zero. The impulse response provides valuable information about a system's behavior and is often used in signal processing and system analysis. It serves as a concise representation of how a system reacts to a sudden input. Understanding the impulse response can help us gain insight into the system's characteristics, including its stability, frequency response, and overall behavior.

The N-point DFT (Discrete Fourier Transform) is a mathematical tool used to analyze the frequency content of a discrete-time signal. It decomposes a signal into a sum of sinusoids at different frequencies. The N-point DFT operates on a sequence of N samples and produces a corresponding sequence of N complex numbers that represent the frequency components of the signal.

Linear convolution is a fundamental operation in signal processing that combines two signals to produce a new signal reflecting the interaction between them. It involves shifting one signal and multiplying it with the other, then summing the products for each shift. Linear convolution is used to emulate the time-domain interaction of two systems or signals, providing insight into how these signals would behave in real-world scenarios. Understanding linear convolution is crucial for various applications, such as filter design, system analysis, and audio signal processing.

Aliasing occurs when the sampling rate of a signal is too low, leading to the distortion of the original signal's frequency content. In essence, high-frequency components in the signal become indistinguishable from lower-frequency components, causing a loss of fidelity in the sampled representation. To avoid aliasing, the sampling rate must be at least twice the highest frequency present in the signal. Understanding aliasing is essential for accurate signal representation and analysis in various applications, including audio and image processing, digital communication, and control systems.