EED-201 (Signals and Systems): Assignment 3
1. A continuous-time signal x(t) is obtained at the output of an ideal low-pass filter with cutoff
frequency ωc= 1000π. If impulse-train sampling is performed on x(t), which of the following
sampling periods would guarantee that x(t) can be recovered from its sampled version using
an appropriate low-pass filter?
i) T= 0.5×10−3ii) T= 2 ×10−3iii) T= 10−4.
2. Let x(t) be a signal with Nyquist rate ωo. Determine the Nyquist rate for each of the
following signals:
i) x(t) + x(t−1) ii) dx(t)
dt iii) x2(t) iv) x(t) cos(ω0t).
3. Determine the Fourier transform for −π≤ω < π in the case of each of the following periodic
signals:
i) sin π
3n+π
4ii) 2 + cos π
6n+π
8.
4. An LTI system with impulse response h1[n] = 1
3nu[n] is connected in parallel with another
causal LTI system with impulse response h2[n]. The resulting parallel interconnection has
the frequency response:
Hejω =−12+5e−jω
12−7e−jω+e−j2ω.
Determine h2[n].
5. Compute the Fourier transform of each of the following signals: (use properties where needed)
i) x[n] = 1
3|n|u[−n−2] ii) x[n] = 1
2|n|cos π
8(n−1)iii) x[n] = sin(πn/5)
πn cos 7π
2n
iv) x(t) = 1 + cos πt, |t| ≤ 1
0,|t|>1v) x(t) = P+∞
n=−∞ e−|t−2n|vi) d
dt {u(−2−t)+u(t−
2)}.
6. Find the signal x(t) or x[n] corresponding to the following Fourier transforms:
i) Xejω = cos2ω+ sin23ω(discrete)
ii) Xejω =P∞
k=−∞(−1)kδω−π
2k(discrete)
iii) X(jω) = 2 sin[3(ω−2π)]
(ω−2π)(continuous)
iv) X(jω) = 2[δ(ω−1) −δ(ω+ 1)] + 3[δ(ω−2π) + δ(ω+ 2π)] (continuous)
v) X(jω) = (sin2(3ω))cos ω
ω2(continuous).
7. The input and the output of a stable and causal LTI system are related by the differential
equation
d2y(t)
dt2+ 6dy(t)
dt + 8y(t)=2x(t)
i) Find the impulse response of the system.
ii) What is the response of this system if x(t) = te−2tu(t)?
8. Impulse-train sampling of x[n] is used to obtain:
g[n] = P∞
k=−∞ x[n]δ[n−kN ].
If X(ejω ) = 0 for 3π/7≤ |ω| ≤ π, determine the largest value for the sampling interval N
which ensures that no aliasing takes place while sampling x[n].
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