Evaluate the following continuous-time convolution integrals: (a) y(t) = (u(t) - u(t - 2)) * u(t) (b) y(t) = e3u(t) * u(t + 3) (c) y(t) = cos(mt)(u(t + 1) − u(t − 1)) * u(t) - (d) y(t) = (u(t + 3) − u(t − 1)) * u(−t + 4) - - (e) y(t) = (tu(t) + (10 − 2t)u(t - 5) - (10 t)u(t - 10)) * u(t) (f) y(t) = 2t² (u(t + 1) − u(t− 1)) *2u(t + 2) - (g) y(t) = cos(πt) (u(t+1) - u(t-1)) (u(t + 1) − u(t − 1)) (h) y(t) = cos(2πt)(u(t+1) – u(t− 1)) * e¯¹u(t) - (i) y(t) = (28(t + 1) + 8(t − 5)) * u(t − 1) (j) y(t) = (8(t + 2) + 8(t− 2)) * (tu(t) + (102t)u(t - 5) - (10 t)u(t - 10)) * - (k) y(t) = e^u(t) * (s(t + 2) - u(t)) (1) y(t) = e¹u(t) *Σp-o(²) ³8(t - 2p) (m) y(t) = (28(t) P-0 + 8(t− 2)) * -o (¹) ³8(t - p) (n) y(t) = eu(t) * e¹u(t) * (o) y(t) = u(t) h(t), where h(t) = [ et<0 le-³tt≥ 0
Evaluate the following continuous-time convolution integrals: (a) y(t) = (u(t) - u(t - 2)) * u(t) (b) y(t) = e3u(t) * u(t + 3) (c) y(t) = cos(mt)(u(t + 1) − u(t − 1)) * u(t) - (d) y(t) = (u(t + 3) − u(t − 1)) * u(−t + 4) - - (e) y(t) = (tu(t) + (10 − 2t)u(t - 5) - (10 t)u(t - 10)) * u(t) (f) y(t) = 2t² (u(t + 1) − u(t− 1)) *2u(t + 2) - (g) y(t) = cos(πt) (u(t+1) - u(t-1)) (u(t + 1) − u(t − 1)) (h) y(t) = cos(2πt)(u(t+1) – u(t− 1)) * e¯¹u(t) - (i) y(t) = (28(t + 1) + 8(t − 5)) * u(t − 1) (j) y(t) = (8(t + 2) + 8(t− 2)) * (tu(t) + (102t)u(t - 5) - (10 t)u(t - 10)) * - (k) y(t) = e^u(t) * (s(t + 2) - u(t)) (1) y(t) = e¹u(t) *Σp-o(²) ³8(t - 2p) (m) y(t) = (28(t) P-0 + 8(t− 2)) * -o (¹) ³8(t - p) (n) y(t) = eu(t) * e¹u(t) * (o) y(t) = u(t) h(t), where h(t) = [ et<0 le-³tt≥ 0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Evaluate the following continuous-time convolution
integrals:
(a) y(t) = (u(t) − u(t − 2)) * u(t)
(b) y(t) = e³tu(t) * u(t + 3)
(c) y(t) = cos(mt)(u(t + 1) − u(t − 1)) * u(t)
-
(d) y(t) = (u(t + 3) − u(t− 1)) * u(-t + 4)
-
-
(e) y(t) = (tu(t) + (10 − 2t)u(t - 5)
− (10 – t)u(t – 10)) * u(t)
-
(f) y(t) = 2t² (u(t + 1) − u(t − 1)) *2u(t + 2)
-
(g) y(t) = cos(πt)(u(t + 1)
- u(t-1)) (u(t + 1) - u(t-1))
*
(h) y(t) = cos(2mt)(u(t + 1)
- u(t-1)) * e¯¹u(t)
-
(i) y(t) = (28(t + 1) + 8(t − 5)) * u(t − 1)
(j) y(t) = (8(t + 2) + 8(t − 2)) * (tu(t)
-
+ (102t)u(t - 5)
- (10 t)u(t-10))
(k) y(t) = e^'u(t) * (s(t + 2) - u(t))
(1) y(t) = e¯¹¹u(t) * Σp-o(²)²s(t - 2p)
(m) y(t) = (28(t)
+ 8(t − 2)) + Σo (²)²s(t − p)
*
(n) y(t) = eu(t) * e¹u(t)
(o) y(t) = u(t)
*
h(t), where h(t) = {
e²t t < 0
e-³t t≥ 0
-3r
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