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Q2. if r(t) has the Fourier transform X(w) = jw+b, find V(w) for the following: a) v(t) = x(5t - 4) b) v(t) = t²x(t) c) v(t) = x(t)ei21

Q2. if r(t) has the Fourier transform X(w) = jw+b, find V(w) for the following: a) v(t) = x(5t - 4) b) v(t) = t²x(t) c) v(t) = x(t)ei21

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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Q2. if r(t) has the Fourier transform X(w) = jw+b, find V (w) for the following: a) v(t) = x(5t - 4) b) v(t) = t²x(t) c) v(t) = x(t)ei21
Transcribed Image Text:Q2. if r(t) has the Fourier transform X(w) = jw+b, find V (w) for the following: a) v(t) = x(5t - 4) b) v(t) = t²x(t) c) v(t) = x(t)ei21
TABLE 3.1 Properties of the Fourier Transform Property Linearity Right or left shift in time Time scaling Time reversal Multiplication by a power of t Multiplication by a complex exponential Multiplication by sin (wot) Multiplication by cos(wot) Differentiation in the time domain Integration in the time domain Convolution in the time domain Multiplication in the time domain Parseval's theorem Special case of Parseval's theorem Duality Transform Pair/Property ax(t) + bv(t)→aX(w) + bV(w) x(tc)X(w)e-jose x (at) < > ² x ( =) a > X a x(−1)→X(-w) = X(w) a>0 d" t"x(t) → j" don X (w) n = 1, 2, ... dw" x(t)ejat →X(w - wo) wo real x(t) sin(wt) ↔ { [X(w + w) − X(w − w)] x(t) cos(wit) → [X(w + wo) + X(w − w)] din X(t) → (jw)"X(w) n = 1, 2,... 1 [x(x) dx → — X (w) + 7X (0)8(w) jw x(t) *v(t) →X(w)V(w) x(t)v(t) → __X(w)*V(w) [x(1)v(1) dt = 1 [~_* x²(1) dt = 2 / 2TT X(t)→2πx(-w) [XV (oo) das 1X (w) ² do -xx TABLE 3.2 Common Fourier Transform Pairs 1, -∞ < t < ∞ → 2π8 (W) −0.5 + u(t) ↔ - jw 1 jw u(t) <> πδ(ω) + 8(t) →1 8(tc)e-jwc, c any real number 1 jw + b² e-btu(t). ejut → 2πd (w - wo), wo any real number τω 2π P:(t)→7 sinc- b>0 7 sinc→2πp, (w) 27 2|t| τω (1 - 2!!!) p.,(1)→ sinc²(e) T sinc²(# )+2π(1 - 2!!)P, (c) 2 T cos(wot) →T[8(w + wo) + d(w = wo)] cos(wat + 0)→π[e¯jªs(w + wo) + ejªs(w - wo)] sin(wot) → jπ[8(w + wo) − 8(w - wo)] sin(wat + 0) → jπ[e¯jªs(w + wo) — ejªts (w - wo)]
Transcribed Image Text:TABLE 3.1 Properties of the Fourier Transform Property Linearity Right or left shift in time Time scaling Time reversal Multiplication by a power of t Multiplication by a complex exponential Multiplication by sin (wot) Multiplication by cos(wot) Differentiation in the time domain Integration in the time domain Convolution in the time domain Multiplication in the time domain Parseval's theorem Special case of Parseval's theorem Duality Transform Pair/Property ax(t) + bv(t)→aX(w) + bV(w) x(tc)X(w)e-jose x (at) < > ² x ( =) a > X a x(−1)→X(-w) = X(w) a>0 d" t"x(t) → j" don X (w) n = 1, 2, ... dw" x(t)ejat →X(w - wo) wo real x(t) sin(wt) ↔ { [X(w + w) − X(w − w)] x(t) cos(wit) → [X(w + wo) + X(w − w)] din X(t) → (jw)"X(w) n = 1, 2,... 1 [x(x) dx → — X (w) + 7X (0)8(w) jw x(t) *v(t) →X(w)V(w) x(t)v(t) → __X(w)*V(w) [x(1)v(1) dt = 1 [~_* x²(1) dt = 2 / 2TT X(t)→2πx(-w) [XV (oo) das 1X (w) ² do -xx TABLE 3.2 Common Fourier Transform Pairs 1, -∞ < t < ∞ → 2π8 (W) −0.5 + u(t) ↔ - jw 1 jw u(t) <> πδ(ω) + 8(t) →1 8(tc)e-jwc, c any real number 1 jw + b² e-btu(t). ejut → 2πd (w - wo), wo any real number τω 2π P:(t)→7 sinc- b>0 7 sinc→2πp, (w) 27 2|t| τω (1 - 2!!!) p.,(1)→ sinc²(e) T sinc²(# )+2π(1 - 2!!)P, (c) 2 T cos(wot) →T[8(w + wo) + d(w = wo)] cos(wat + 0)→π[e¯jªs(w + wo) + ejªs(w - wo)] sin(wot) → jπ[8(w + wo) − 8(w - wo)] sin(wat + 0) → jπ[e¯jªs(w + wo) — ejªts (w - wo)]
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