Problem 1 (5 pts each). Evaluate the following integrals involving s(t).(a)(t)e-jwt dt; (Q1)+∞(b) to 8(2t - 3) sin 4t dt; (Q2)-∞(c) f± 8(t + 3)e¯dt; (Q3)(d) ſ± 8(1 − t)(t³ + 4)dt; (Q4)(e) et cos(t-5)] 8(t − 3)dt; (Q5)-∞

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Answered: Problem 1 (5 pts each). Evaluate the…

Introductory Circuit Analysis (13th Edition)

13th Edition

ISBN:9780133923605

Author:Robert L. Boylestad

Publisher:Robert L. Boylestad

Chapter1: Introduction

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2) zero. a) Determine which integrals would evaluate to be zero and provide the reason that they would be zero or not T/2 2 cos(9w,t) dt -T/2 T/2 b) 2 cos(9w,t) cos(9w₁t) dt T/2 c) cos(7w,t) cos(6w,t) dt T/2 T/2 d) cos(6w,t) sin(6w,t) dt
Two rectangular pulses with amplitude A and pulse-width T are to be convolved by each other. h(t) x(1)| A A T h(t) * x(t) = [ h(1)x(t – A)d2 The graphical interpretation of the convolution integral has been illustrated below for five different values of t (the amount of time shift before multiplication and integration). (a) (b) x(t-X) h(a) x(t-) h() t = 0 t = T/2 -T+t -T+t (c) x(t-X) (d) x(t-A) h(a) h(a) t = T t = 3T/2 -Tft -T+t in. (e) x(t-A) h(A) t = 2T -T+t Parameters: A = 4 T = 2
2) Consider the following integrals. Determine if they would evaluate to be zero on non zero. If zero, state the reason why. T a) cos(8w,t) cos(7w,t) dt -T/2 T/2 cos (8w,t)cos (8w,t)dt c) cos (8w,t)sin(8w,t)dt T/2 d) sin (6w,t) sin(8w,t)dt -T/2
(b) Based on the system in Figure QI (b), given that x(t) = 2u(t + 2) – u(t) – u(t – 2) and y(t) = 2u(t + 1.5) + (2t – 2)u(t) – 2t. u(t – 2) Express the signal g(t) using a single analytical expression with the aid of unit step function, u(t). x(t) 2 ► y(t) g(t)
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1. Sketch the following CT functions: (a) g(t) = 2u(4 --t) (b) x(t)=5 sgn(t – 4) (c) g(t) = 1+ sgn(4-t) (d) x(t)= 68(3t +9) (e) y(t) = 5 sine(t / 2) (f) f(t)=-sinc(2(1+1)) (g) x(t) = u(t+ 1/2)r(1/2-1) (h) h(t) = 4sinc(21)sgn(-t) 2. (a) A continuous time signal x(t) is shown in Fig. 1(a). Sketch each of the following signals: (i) x(t – 2) (ii) x(1–t) (iii) x(2t + 2) (iv) x(2 -- t/3) (b) For the signal h(t) depicted in Fig. 1(b), sketch each of the following signals: (1) (1/2)h(t)u(t) + h(-t)u(t) (ii) h(t/2)8(t +1) (iii) h(t)[u(t + 1) – u(t – 1)} x(t) h(t) 2 1 3 -1 1 2 -2 -1 2 -1 Fig. 1(a) Fig. 1(b) 3. Sketch the following DT signals: (a) x[n] = 2 cos(2m/6)– 3sin(m/3) (b) g[n] = (3/4)" (c) x[n] = 2(0.9)" sin(2m/4) 4. A discrete time signal x[n] is shown in Fig.2. Sketch and label carefully each of the following signals: (a)x[n-2] (b)x[4 – n] (c)x[2n] (d)x[2n+1] (e)x[n]u{2 – n] (f)x[n–1]S[n– 3] (g)(1/2)x[n]+ (1/2)(-1)" x[n] (h)x[n°] x[n] -3 -2 -1 0 12 3 4 6 7 8 9 10 11 12 n->
The following waveform is sampled at a random time t over a period T = 1.5. Let X = {sample valuc of v(t)}. Find (c) P{0 < X < 3}. v(t) 0.5 1.0 1.5 2.0 2.5
4) Sketch the following signals - a) w(t) = u(t) — u(t – 1) where u(t) is the unit step function. b) p(t) = sin(2πt)w(t) - P(t - 2n) 8(t = n) d) n=-∞ e) a[n], where the (complex) discrete time signal ∞ f) Za[n] g) x(t)]², where r(t) = 2-¹jt h) Zx(t) a[n] = j n 2 n‡0 n = 0
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Problem 1 (5 pts each). Evaluate the following integrals involving s(t). (a)(t)e-jwt dt; (Q1) +∞ (b) to 8(2t - 3) sin 4t dt; (Q2) -∞ (c) f± 8(t + 3)e¯dt; (Q3) (d) ſ± 8(1 − t)(t³ + 4)dt; (Q4) (e) et cos(t-5)] 8(t − 3)dt; (Q5) -∞