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Time convolution: if x₁(t) ® x2(t) = ƒ‰ ×₁(T)x2(t − t) dɩ then x1(t) + x2(t) ↔ X₁(w)X₂(w) Example find the inverse Fourier transform for X(w) : using time convolution property. = 1 (a+jw)² Solution: X(w) = X₁(w)X2(w) → x(t) = F¯¹(X₁(w)X2(w)) = x₁ (t) ® x2(t) 1 1 ×1₁(t) = F-1 { a + jw } = e−at u(t) X2(t) = F-1 (+) = e-at u(t) (a , \a + jw x(t)= f% X1(t)x2(t − z)dt = % e-atu(t)e-a(t-1)u(t – z)dt "d₁ = e¯at f√² d₁ = te¯ªt t>0 = te˜at u(t) =

Time convolution: if x₁(t) ® x2(t) = ƒ‰ ×₁(T)x2(t − t) dɩ then x1(t) + x2(t) ↔ X₁(w)X₂(w) Example find the inverse Fourier transform for X(w) : using time convolution property. = 1 (a+jw)² Solution: X(w) = X₁(w)X2(w) → x(t) = F¯¹(X₁(w)X2(w)) = x₁ (t) ® x2(t) 1 1 ×1₁(t) = F-1 { a + jw } = e−at u(t) X2(t) = F-1 (+) = e-at u(t) (a , \a + jw x(t)= f% X1(t)x2(t − z)dt = % e-atu(t)e-a(t-1)u(t – z)dt "d₁ = e¯at f√² d₁ = te¯ªt t>0 = te˜at u(t) =

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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Time convolution: if x₁(t) ® x2(t) = ƒ‰ ×₁(T)x2(t − t) dɩ then x1(t) + x2(t) ↔ X₁(w)X₂(w) Example find the inverse Fourier transform for X(w) : using time convolution property. = 1 (a+jw)² Solution: X(w) = X₁(w)X2(w) → x(t) = F¯¹(X₁(w)X2(w)) = x₁ (t) ® x2(t) 1 1 ×1₁(t) = F-1 { a + jw } = e−at u(t) X2(t) = F-1 (+) = e-at u(t) (a , \a + jw x(t)= f% X1(t)x2(t − z)dt = % e-atu(t)e-a(t-1)u(t – z)dt "d₁ = e¯at f√² d₁ = te¯ªt t>0 = te˜at u(t) =
Transcribed Image Text:Time convolution: if x₁(t) ® x2(t) = ƒ‰ ×₁(T)x2(t − t) dɩ then x1(t) + x2(t) ↔ X₁(w)X₂(w) Example find the inverse Fourier transform for X(w) : using time convolution property. = 1 (a+jw)² Solution: X(w) = X₁(w)X2(w) → x(t) = F¯¹(X₁(w)X2(w)) = x₁ (t) ® x2(t) 1 1 ×1₁(t) = F-1 { a + jw } = e−at u(t) X2(t) = F-1 (+) = e-at u(t) (a , \a + jw x(t)= f% X1(t)x2(t − z)dt = % e-atu(t)e-a(t-1)u(t – z)dt "d₁ = e¯at f√² d₁ = te¯ªt t>0 = te˜at u(t) =
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