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A continuous-time signal, x(t), has a Fourier transform denoted by X (w), and is expressed as: a) b) c) (i) (ii) (iii) (iv) 0, X (w) = w² 0, w<-Wm if-wm ≤ w

A continuous-time signal, x(t), has a Fourier transform denoted by X (w), and is expressed as: a) b) c) (i) (ii) (iii) (iv) 0, X (w) = w² 0, w<-Wm if-wm ≤ w

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
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A continuous-time signal, x(t), has a Fourier transform denoted by X (w), and is expressed as: a) (i) b) c) d) (ii) (iii) (iv) 0, w² 0, Write down the definition of the Fourier transform, and calculate the value of the integral: X (w) = (i) (ii) w<-wm if-wm ≤w<wm w > wm f x(t) dt I = Using the definition of the inverse Fourier transform, calculate the value of x(0); i.e. x(t) evaluated at t = 0. You may use the identity: In this part of the question, you do not need to find the full expression for x(t). Determine whether energy or power provides the most suitable measure of the size of the time-domain signal, x(t), justifying your answer. Find an expression for the signal, x(t), written in terms of t and trigonometric functions of t. Show all your working. Using the appropriate form of Parseval's theorem, calculate the value of the measure chosen to find the size of the signal, x(t). 2x [ 2²₁² dx = ² ( ²² - ² + ²) ec² с What is the Nyquist frequency, wn, for the signal x(t)? The signal x(t) is now sampled at a rate of ws. + constant If we wn, plot the resulting spectrum of the sampled signal, labelling carefully key frequencies. If ws< Wn, plot the resulting spectrum of the sampled signal, explaining key characteristics of the resulting graph.
Transcribed Image Text:A continuous-time signal, x(t), has a Fourier transform denoted by X (w), and is expressed as: a) (i) b) c) d) (ii) (iii) (iv) 0, w² 0, Write down the definition of the Fourier transform, and calculate the value of the integral: X (w) = (i) (ii) w<-wm if-wm ≤w<wm w > wm f x(t) dt I = Using the definition of the inverse Fourier transform, calculate the value of x(0); i.e. x(t) evaluated at t = 0. You may use the identity: In this part of the question, you do not need to find the full expression for x(t). Determine whether energy or power provides the most suitable measure of the size of the time-domain signal, x(t), justifying your answer. Find an expression for the signal, x(t), written in terms of t and trigonometric functions of t. Show all your working. Using the appropriate form of Parseval's theorem, calculate the value of the measure chosen to find the size of the signal, x(t). 2x [ 2²₁² dx = ² ( ²² - ² + ²) ec² с What is the Nyquist frequency, wn, for the signal x(t)? The signal x(t) is now sampled at a rate of ws. + constant If we wn, plot the resulting spectrum of the sampled signal, labelling carefully key frequencies. If ws< Wn, plot the resulting spectrum of the sampled signal, explaining key characteristics of the resulting graph.
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