electrical assignmnet signal & system

Full Name: Lab Section:

ECE 3500 (Fall 2015) – HW #4 Due Date: Oct. 1, 2015

Homework learning objectives: By the end of this homework, you should be able to: • Practice manipulating the complex exponentials • Analyze periodic signals using the Fourier series • Synthesize periodic signals using the Fourier series

Question #1: (6 pts) Convert complex exponentials into sines and cosines with Euler’s Formula:

e−jθ = cos(θ)− j sin(θ) , e+jθ = cos(θ) + j sin(θ)

(a) x1(t) = e −jt + e+jt

(b) x2(t) = 1 j

( e−2jt −e+2jt

) (c) x3(t) = (2 + j)e

+jπt + (2− j)e−jπt − (2−3j)e+j4πt − (2 + 3j)e−j4πt

Question #2: (6 pts) Convert cosines and sines into complex exponentials with Euler’s Formula:

cos(θ) = 1

2

( e+jθ + e−jθ

) , sin(θ) =

1

2j

( e+jθ −e−jθ

) (a) x1(t) = cos(2πt)

(b) x2(t) = cos(2πt) + 3 sin(4πt)

(c) x3(t) = cos(3πt) + 5 sin(6πt) + 2 cos(6πt)

Question #3: (8 pts) Determine the Fourier Series coefficients for the following signals. For these signals, use the trigonometric form of the Fourier series, i.e.,

x(t) = a0 + ∞∑ k=1

ak cos(kω0t) +

∞∑ k=1

bk sin(kω0t)

a0 = 1

T0

∫ T0

x(t)dt , ak = 2

T0

∫ T0

x(t) cos(kω0t)dt , k ≥ 1

bk = 2

T0

∫ T0

x(t) sin(kω0t)dt , k ≥ 1

where ω0 is the fundamental angular frequency of our periodic signal ω0 = 2π/T0.

(a) x1(t) = 3 sin(4πt) + 2

(b) x2(t) = 3 sin(4t) + 5 cos((5/3)t)

(c) x3(t) = 3 sin(4πt) + 6 cos(6πt) + 5 cos(7πt)

(d) x4(t) = 4 cos(5πt + π/4) [hint: you may need to use a trigonometric identity]

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Question #4: (6 pts) Prove the following statements using the trigonometric form of the Fourier Series shown in the previous question.

(a) If bk = 0 for all k and x(t) 6= 0, then x(t) is even. (b) If ak = 0 for all k and x(t) 6= 0, then x(t) is odd. (c) Any continuous-time, periodic signal x(t) can be written as x(t) = xe(t) + xo(t), where xe(t) is

an even signal and xo(t) is an odd signal.

Question #5: (8 pts) Determine the Fourier Series coefficients ck for the following signals. For these signals, use the complex exponential form of the Fourier series, i.e.,

x(t) = ∞∑

k=−∞ cke

jkω0t , ck = 1

T0

∫ T0

x(t)e−jkω0tdt

where ω0 is the fundamental angular frequency of our periodic signal ω0 = 2π/T0.

(a) x1(t) = 3 sin(4πt) + 2

(b) x2(t) = 3 sin(4t) + 5 cos((5/3)t)

(c) x3(t) = 3 sin(4πt) + 6 cos(6πt) + 5 cos(7πt)

(d) x4(t) = 4 cos(5πt + π/4) [hint: you may need to use a trigonometric identity]

Question #6: (8 pts) Given the following Fourier Series coefficients ck (for the complex exponential form of the Fourier Series), determine the corresponding periodic signal x(t). Write the result as a real-valued function if possible (i.e., not as complex exponentials).

(a) c0 = 2, c1 = −j, c−1 = j, and all other ck = 0. Assume ω0 = 2π. (b) c0 = 0, c1 = −j + 2, c−1 = j + 2, and all other ck = 0. Assume ω0 = 1. (c) c0 = −3,c2 = −2j, c−2 = 2j, c3 = 1, c−3 = 1, and all other ck = 0. Assume ω0 = π. (d) c0 = 2,c2 = j, c−2 = −j, c3 = −2− j, c−3 = −2 + j, and all other ck = 0. Assume ω0 = 2π.

Question #7: (5 pts) Compute Fourier Series coefficients ck (for the complex exponential form of the Fourier Series) for the signal x(t) below. Assume the pattern continues for −∞ < t < ∞.

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