Homework 6: Systems and Signals

Homework 6
ECE 102: Systems and Signals

1. An LTI causal system is described by the equation:
dy(t)
dt +
Z 3t
0
y(t − τ )τ
2
e
3−τ
dτ =
dx(t)
dt − x(t), t > 0
y(0) = 0, x(t) = y(t) = 0 for t < 0
(a) Find H(s)
(b) The periodic signal | cos2
(t)| + 1 is applied to the system. Find the Fourier coefficients of
the output y˜(t).
(c) Find the Fourier series coefficients of z(t) = ˜y(t − 3) ∗ y˜(2t)
2. Find the Fourier transform of the following signals:
(a) x1(t) = (
t
2
, 0 ≤ t < 1
0, otherwise
(b) x2(t) = [u(t + 2) − u(t − 2)] cos(100t) + 1
(c) x3(t) = R t
−∞ cos(5(t − σ))δ(σ − 2)dσ
3. Consider the time-domain real signal x(t) with a Fourier Transform X(ω), where x(t) is shown
below.
(a) Find X(0) and R ∞
−∞ X(ω)dω
(b) Compute R ∞
−∞ |X(ω)|
2dω
(c) Compute R ∞
−∞ X(ω)Y (ω)dω, where Y (ω) = 2 sin(ω)
ω
e
j2ω
Hint: For any real signals f(t) and g(t), we have:
Z ∞
−∞
f(t)g(−t)dt =
1
2π
Z ∞
−∞
F(ω)G(ω)dω
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(d) Sketch the inverse Fourier transform of Re{X(ω)}.
Note: You can answer all parts without explicitly evaluating X(ω)
4. Consider a signal x(t) with Fourier transform X(jω). Suppose you are given the following facts:
i. x(t) is real and non-negative.
ii. F
−1{(1 + jω)X(jω)} = Ae−2tu(t), where A is a constant.
iii. R ∞
−∞ |X(jω)|
2dω = 2π
Determine a closed form expression for x(t) and find the constant A.
5. Consider the following signal:
x(t) = e
−t
sin(2πt)u(t) + δ(t − 2)
(a) Use the Laplace transform to find the Fourier transform X(Ω) of signal x(t)
(b) Compute amplitude and phase spectrum.
(c) Use MATLAB to plot amplitude and phase spectrum for Ω ∈ [−10, 10].
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