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4.10. (a) Use Tables 4.1 and 4.2 to help determine the Fourier transform of the following signal: sin t = t (b) Use Parseval's relation and the result of the previous part to determine the nu- merical value of A 2oulay or sin t dt TTt

4.10. (a) Use Tables 4.1 and 4.2 to help determine the Fourier transform of the following signal: sin t = t (b) Use Parseval's relation and the result of the previous part to determine the nu- merical value of A 2oulay or sin t dt TTt

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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4.10. (a) Use Tables 4.1 and 4.2 to help determine the Fourier transform of the following 4.10. (a) Use Tables 4.1 and 4.2 to help determine the Fourier transform of the followi. signal: 2 sin t udeupe zlearlid/2 x(t) = t TTt (b) Use Parseval's relation and the result of the previous part to determine the nu- merical value of 4. A = pulay or on sin t dt oy se Tt
Transcribed Image Text:4.10. (a) Use Tables 4.1 and 4.2 to help determine the Fourier transform of the following 4.10. (a) Use Tables 4.1 and 4.2 to help determine the Fourier transform of the followi. signal: 2 sin t udeupe zlearlid/2 x(t) = t TTt (b) Use Parseval's relation and the result of the previous part to determine the nu- merical value of 4. A = pulay or on sin t dt oy se Tt
In the preceding sections and in the problems at the end of the chapter, we have o 4.6 TABLES OF FOURIER PROPERTIES AND OF BASIC FOURIER TRANSFORM PAIRS D s vellmie In Table 4.2, we have assembled a list of many of the basic and important Fourier ered some of the important properties of the Fourier transform. These are summarized in transform pairs. We will encounter many of these repeatedly as we apply the tools of Table 4.1, in which we have also indicated the section of this chapter in which each prop eg01 The Continuous-Time Fourier Transform Tables of Fourier Properties and of Basic Fourier Transform Pairs EE mmot Sec. 4.6 328 Chap.4 SE 329 hvol od tedi wollct BASIC FOURIER TRANSFORM PAIRS r2o molens libbs nl Signal consid Fourier series coefficients Fourier transform (if periodic) Z akeikwor 27 2 ar8(w - kwo) on ( erty has been discussed. k=-∞ k=-0 ak as we apply the tools of transform pairs. We will encounter many of these repeatedly 2T8(w – wo) aj = 1 ak = 0, otherwise TABLE 4.1 PROPERTIES OF THE FOURIER TRANSFORM Fourier transform T[8(w – wo) +8(w + wo)] aj = a-j = Aperiodic signaloh cos wot SAig9mo Section sgbaed lsabr odj Property ak = 0, otherwise x(t) y(t) X(jw) Y(jw) porl sin wot [8(@ – wo) – 8(w + wo)] aj = -a-j = gino sif lo ak = 0, otherwise at 1o yageupant a boisv Linearity Time Shifting Frequency Shifting Conjugation Time Reversal ax(t) + by(t) x(t – to) ejao! x(t) aX(jw) + bY(jw) - julo X(jw) ao = 1, a = 0, k# 0 this is the Fourier series representation for any choice of T >0 4.3.1 x(t) = 1 2m 8(w) %3D e X(j(@ – wo)) X(-jw) X(-jw) ja 4.3.2 4.3.6 sdt tu omua x'(t) x(-t) 4.3.3 Periodic square wave 1, |2| < T, 0, T, < \e| < } 4.3.5 too 2 sin kwoT, Time and Frequency Scaling Convolution (kwoT sinc T 4.3.5 x(at) 8(@ – kwo) sin kwoT, - k and k=-∞ TT 4.4 x(t) * y(t) X(jw)Y(jw) x(t + T) = x(t) ntt thobiaod gailuffiedu? 9adbeis 2 8(t – nT) wup 1sl easqans 4.5 Multiplication x(t)y(t) X(je)Y(j(@ – 0))do 2Tk -8 4.3.4 Differentiation in Time jwX(jw) bo for all k T ak T k=-∞ T n= -∞ ai nwoda 1sdr 4.3.4 Integration x(t)dt X(jw) + mX(0)8(@) 2 sin wT, hitlor ow jw a bus x(jw) gorg frotulovm | 0, > T, x(t)- 541 4.3.6 Differentiation in d racod ela al mataya ad S Da da X(ja) = X(-jw) Frequency sin Wt 1, l이 < W X(jo) =0. lo > W Tt giRe{X(j@)} = Re{X(-ju)} Im{X(j@)} = -Ia{X(-ju) |X(j@)| = |X(- j@)I *X(jw) = -*X(-jw) X(jw) real and even bonoles ei aidT x(t) real 4.3.3 Conjugate Symmetry for Real Signals 8(t) 1 1 + T8(@) jw u(t) 4.3.3 Symmetry for Real and x(t) real and even Even Signals Symmetry for Real and Odd Signals ot ou 4.3.3 8(t- to) e juio x(t) real and odd L S 4.3.3 Even-Odd Decompo- x() = Ev{x(1)} [x(1) real] e al u(t), Re{a} > 0 1 %3D [x(t) real] Re{X(jw)} jIm{X(j@)} a + jw sition for Real Sig- *() = O{x(t)} [x(t) real] nals 1 te du(t), Re{a} > 0 -at (a + jw)? 4.3.7 Parseval's Relation for Aperiodic Signals (n-1)! e-at u(t), 1 Refa} > 0 (a + jw)" dt = |X(j@)fdw
Transcribed Image Text:In the preceding sections and in the problems at the end of the chapter, we have o 4.6 TABLES OF FOURIER PROPERTIES AND OF BASIC FOURIER TRANSFORM PAIRS D s vellmie In Table 4.2, we have assembled a list of many of the basic and important Fourier ered some of the important properties of the Fourier transform. These are summarized in transform pairs. We will encounter many of these repeatedly as we apply the tools of Table 4.1, in which we have also indicated the section of this chapter in which each prop eg01 The Continuous-Time Fourier Transform Tables of Fourier Properties and of Basic Fourier Transform Pairs EE mmot Sec. 4.6 328 Chap.4 SE 329 hvol od tedi wollct BASIC FOURIER TRANSFORM PAIRS r2o molens libbs nl Signal consid Fourier series coefficients Fourier transform (if periodic) Z akeikwor 27 2 ar8(w - kwo) on ( erty has been discussed. k=-∞ k=-0 ak as we apply the tools of transform pairs. We will encounter many of these repeatedly 2T8(w – wo) aj = 1 ak = 0, otherwise TABLE 4.1 PROPERTIES OF THE FOURIER TRANSFORM Fourier transform T[8(w – wo) +8(w + wo)] aj = a-j = Aperiodic signaloh cos wot SAig9mo Section sgbaed lsabr odj Property ak = 0, otherwise x(t) y(t) X(jw) Y(jw) porl sin wot [8(@ – wo) – 8(w + wo)] aj = -a-j = gino sif lo ak = 0, otherwise at 1o yageupant a boisv Linearity Time Shifting Frequency Shifting Conjugation Time Reversal ax(t) + by(t) x(t – to) ejao! x(t) aX(jw) + bY(jw) - julo X(jw) ao = 1, a = 0, k# 0 this is the Fourier series representation for any choice of T >0 4.3.1 x(t) = 1 2m 8(w) %3D e X(j(@ – wo)) X(-jw) X(-jw) ja 4.3.2 4.3.6 sdt tu omua x'(t) x(-t) 4.3.3 Periodic square wave 1, |2| < T, 0, T, < \e| < } 4.3.5 too 2 sin kwoT, Time and Frequency Scaling Convolution (kwoT sinc T 4.3.5 x(at) 8(@ – kwo) sin kwoT, - k and k=-∞ TT 4.4 x(t) * y(t) X(jw)Y(jw) x(t + T) = x(t) ntt thobiaod gailuffiedu? 9adbeis 2 8(t – nT) wup 1sl easqans 4.5 Multiplication x(t)y(t) X(je)Y(j(@ – 0))do 2Tk -8 4.3.4 Differentiation in Time jwX(jw) bo for all k T ak T k=-∞ T n= -∞ ai nwoda 1sdr 4.3.4 Integration x(t)dt X(jw) + mX(0)8(@) 2 sin wT, hitlor ow jw a bus x(jw) gorg frotulovm | 0, > T, x(t)- 541 4.3.6 Differentiation in d racod ela al mataya ad S Da da X(ja) = X(-jw) Frequency sin Wt 1, l이 < W X(jo) =0. lo > W Tt giRe{X(j@)} = Re{X(-ju)} Im{X(j@)} = -Ia{X(-ju) |X(j@)| = |X(- j@)I *X(jw) = -*X(-jw) X(jw) real and even bonoles ei aidT x(t) real 4.3.3 Conjugate Symmetry for Real Signals 8(t) 1 1 + T8(@) jw u(t) 4.3.3 Symmetry for Real and x(t) real and even Even Signals Symmetry for Real and Odd Signals ot ou 4.3.3 8(t- to) e juio x(t) real and odd L S 4.3.3 Even-Odd Decompo- x() = Ev{x(1)} [x(1) real] e al u(t), Re{a} > 0 1 %3D [x(t) real] Re{X(jw)} jIm{X(j@)} a + jw sition for Real Sig- *() = O{x(t)} [x(t) real] nals 1 te du(t), Re{a} > 0 -at (a + jw)? 4.3.7 Parseval's Relation for Aperiodic Signals (n-1)! e-at u(t), 1 Refa} > 0 (a + jw)" dt = |X(j@)fdw
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