1. An LTI causal system is described by the equation:
dy(t)
t
dx(t)
+
y(t − τ)τ2e3−τ dτ =
− x(t), t > 0
dt
0
dt
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:
t2, 0 ≤ t < 1
(a) x1(t) =
0, otherwise
(b) x2(t) = [u(t + 2) − u(t − 2)] cos(100t) + 1
(c) x3(t) = −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 −∞∞ X(ω)dω
(b) Compute −∞∞ |X(ω)|2dω
(c) Compute
∞
X(ω)Y (ω)dω, where Y (ω) =
2 sin(ω)
ej2ω
−∞
ω
Hint: For any real signals f(t) and g(t), we have:
∞
1 ∞
−∞ f(t)g(−t)dt =
−∞ F (ω)G(ω)dω
2π
1
(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. −∞∞ |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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