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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 (1)x rag TTt = t ole in the (b) Use Parseval's relation and the result of the previous part to determine the nu merical value of sin t dt A = TTt

4.10. (a) Use Tables 4.1 and 4.2 to help determine the Fourier transform of the following signal: sin t (1)x rag TTt = t ole in the (b) Use Parseval's relation and the result of the previous part to determine the nu merical value of sin t dt A = 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
Section: Chapter Questions
Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...
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(a) Comes out ot this but new help getting there

?(??)= 1/2?  { ?,−2<? <0
                     −?, 0<? <2
                       0,⁡⁡⁡|?|>2

(b) comes out to ?= 1/(2?^2)  but new help with that too

4.10. (a) Use Tables 4.1 and 4.2 to help determine the Fourier transform of the following signal: sin t (1)x drao (b) Use Parseval's relation and the result of the previous part to determine the nu- merical value of m We dt sin t A 2oulay or TTt
Transcribed Image Text:4.10. (a) Use Tables 4.1 and 4.2 to help determine the Fourier transform of the following signal: sin t (1)x drao (b) Use Parseval's relation and the result of the previous part to determine the nu- merical value of m We dt sin t A 2oulay or TTt
In Table 4.2, we have assembled a list of many of the basic and important Foure ered some of the important properties of the Fourier transform. These are summarized transform pairs. We will encounter many of these repeatedly as we apply the tools In the preceding sections and in the problems at the end of the chapter, we have consid Table 4.1, in which we have also indicated the section of this chapter in which each p 4.6 TABLES OF FOURIER PROPERTIES AND OF BASIC FOURIER TRANSFORM PAIRS The Continuous-Time Fourier Transform The Continuous-Time Fourier Transform Tables of Fourier Properties and of Basic Fourier Transform PairsEE Chap ol Sec. 4.6 328 329 TABLE 4.2 BASIC FOURIER TRANSFORM PAIRS Signal Fourier transform Fourier series coefficients (if periodic) prop E ai8(w - koro) erty has been discussed. apply the tools transform pairs. We will encounter many of these repeatedly as we el 2m8(w - wo) a, = 1 a, - 0, otherwise TABLE 4.1 PROPERTIES OF THE FOURIER TRANSFORM Aperiodic signal a la in (8(w - w) + 8(a + we)) Fourier transform cos wof di = a- Property ouy ao Section a, = 0, otherwise i agband lasbT X(jw) Yja) () sin wof (8( - wo) - 8(w + ao)] ai = -a-= st le vansipait a, - 0, otherwise --- axt) + by(t) x(t - to) d = 1, a, - 0, k0 (this is the Fourier series representation for any choice of T >0 aX(ja) + bY(jw) e Julo X(jw) X(J(@ - wo)) X'(- ja) X(- ju) 1v ja laa Lincarity Time Shifting 4.3.1 4.3.2 4.3.6 4.3.3 4.3.5 x(t) = 1 2n 6(w) Frequency Shifting Conjugation Time Reversal x'(r) x(-1) Periodic square wave 1, <T, x(1) = 0. T, < |r| s } * 2 sin kwgT kanT sinc 4.3.5 Time and Frequency Scaling x(at) -8(w - kan) sin kwT and km 4.4 X(jo)Y(ja) Convolution x(1) • y() x(t + T) = x(t) b 4.5 Multiplication (1)y(t) Differentiation in Time x(1) dt * 81 - nT) 2nk a7 for llk 4.3.4 jwX(jw) all k A1maseve a 1 4.3.4 Integration ( 1, e|< T, x(t) 0, >T x()dr ia Xjw) + TX(0)8(a) 2 sin wT,is bos su 4.3.6 Differentiation in Frequency baa d j X(jw) dw - ba tx(1) sin Wt (1, l < W (o. k) > W reted astal l labloun X(jw) = X'(- ju) X(ja) = RefX(j@)} = Re(X(- jo) Im(X(jw)} = -Sm{X(= joll X(j) = |X(-jaw) *X(jw) = -X(-jas) X(jw) real and even pleasi T n x(t) real 4.3.3 Conjugate Symmetry for Real Signals 8(1) %3D + 8(w) ja u(t) 4.3.3 Symmetry for Real and Even Signals Symmetry for Real and Odd Signals x(t) real and even 4.3.3 8(t - to) x(f) real and odd A e PEA 54.3.3 X(jw) purely imaginary and od 1 Even-Odd Decompo- x,() = Evfx(1)) [x(t) real) RefX(jw) e"u(t), Refa} >0 a+ ja sition for Real Sig- x,(1) = Od{x(1)} [x(t) real] jám(X(jw)} nals te u(t), Refa} >0 (a + jw) -1e"u(t), Refa) > 0 4.3.7 Parseval's Relation for Aperiodic Signals (a + ja) 2m
Transcribed Image Text:In Table 4.2, we have assembled a list of many of the basic and important Foure ered some of the important properties of the Fourier transform. These are summarized transform pairs. We will encounter many of these repeatedly as we apply the tools In the preceding sections and in the problems at the end of the chapter, we have consid Table 4.1, in which we have also indicated the section of this chapter in which each p 4.6 TABLES OF FOURIER PROPERTIES AND OF BASIC FOURIER TRANSFORM PAIRS The Continuous-Time Fourier Transform The Continuous-Time Fourier Transform Tables of Fourier Properties and of Basic Fourier Transform PairsEE Chap ol Sec. 4.6 328 329 TABLE 4.2 BASIC FOURIER TRANSFORM PAIRS Signal Fourier transform Fourier series coefficients (if periodic) prop E ai8(w - koro) erty has been discussed. apply the tools transform pairs. We will encounter many of these repeatedly as we el 2m8(w - wo) a, = 1 a, - 0, otherwise TABLE 4.1 PROPERTIES OF THE FOURIER TRANSFORM Aperiodic signal a la in (8(w - w) + 8(a + we)) Fourier transform cos wof di = a- Property ouy ao Section a, = 0, otherwise i agband lasbT X(jw) Yja) () sin wof (8( - wo) - 8(w + ao)] ai = -a-= st le vansipait a, - 0, otherwise --- axt) + by(t) x(t - to) d = 1, a, - 0, k0 (this is the Fourier series representation for any choice of T >0 aX(ja) + bY(jw) e Julo X(jw) X(J(@ - wo)) X'(- ja) X(- ju) 1v ja laa Lincarity Time Shifting 4.3.1 4.3.2 4.3.6 4.3.3 4.3.5 x(t) = 1 2n 6(w) Frequency Shifting Conjugation Time Reversal x'(r) x(-1) Periodic square wave 1, <T, x(1) = 0. T, < |r| s } * 2 sin kwgT kanT sinc 4.3.5 Time and Frequency Scaling x(at) -8(w - kan) sin kwT and km 4.4 X(jo)Y(ja) Convolution x(1) • y() x(t + T) = x(t) b 4.5 Multiplication (1)y(t) Differentiation in Time x(1) dt * 81 - nT) 2nk a7 for llk 4.3.4 jwX(jw) all k A1maseve a 1 4.3.4 Integration ( 1, e|< T, x(t) 0, >T x()dr ia Xjw) + TX(0)8(a) 2 sin wT,is bos su 4.3.6 Differentiation in Frequency baa d j X(jw) dw - ba tx(1) sin Wt (1, l < W (o. k) > W reted astal l labloun X(jw) = X'(- ju) X(ja) = RefX(j@)} = Re(X(- jo) Im(X(jw)} = -Sm{X(= joll X(j) = |X(-jaw) *X(jw) = -X(-jas) X(jw) real and even pleasi T n x(t) real 4.3.3 Conjugate Symmetry for Real Signals 8(1) %3D + 8(w) ja u(t) 4.3.3 Symmetry for Real and Even Signals Symmetry for Real and Odd Signals x(t) real and even 4.3.3 8(t - to) x(f) real and odd A e PEA 54.3.3 X(jw) purely imaginary and od 1 Even-Odd Decompo- x,() = Evfx(1)) [x(t) real) RefX(jw) e"u(t), Refa} >0 a+ ja sition for Real Sig- x,(1) = Od{x(1)} [x(t) real] jám(X(jw)} nals te u(t), Refa} >0 (a + jw) -1e"u(t), Refa) > 0 4.3.7 Parseval's Relation for Aperiodic Signals (a + ja) 2m
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