Homework #7 Signals & Systems 

ECE102 Homework #7
Signals & Systems 
100 points total.
1. (16 points) Bandpass sampling
The figure below shows the Fourier transform of a real bandpass signal, i.e., a signal whose
frequencies are not centered around the origin. We want to sample this signal. Let Fs in Hz
represent the sampling frequency.
(a) (8 points) One option is to sample this signal at the Nyquist rate. Then to recover the
signal, we pass its sampled version through a low pass filter. What is the Nyquist rate
of this signal? What is the cutoff frequency of the low pass filter?
(b) (8 points) Since the signal might have high frequency components, Nyquist rate for this
signal can be high. In other words, we need to have a lot of samples of the signal, which
means that the sampling scheme is costly. It turns out that for this type of signal,
we can sample it at a sampling frequency lower that the Nyquist rate and we can still
recover the signal, however in this case, we will use a bandpass filter. To see this, we
have the following two options for the sampling frequency:
• Fs = 2 Hz;
• Fs = 2.5 Hz;
For each case, draw the spectrum of the signal after sampling it. To recover the signal,
which Fs can we use? How we should choose the frequencies of the bandpass filter?
What is the minimum Fs we can use and still recover the signal?
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2. (24 points) Sampling with imperfect sampler
Imperfections in a sampler cause characteristic artifacts in the sampled signal. In this problem
we will look at the case where the sample timing is non-uniform, as shown below: The
sampling function f(t) has its odd samples delayed by a small time τ .
(a) (6 points) Write an expression for f(t) in terms of two uniformly spaced sampling
functions.
(b) (6 points) Find F(jω), the Fourier transform of f(t). Express the impulse trains as
sums, and simplify.
(c) (6 points) Find F(jω), for the case where τ = 0, and show that this is what you expect.
(d) (6 points) Assume the signal we are sampling has a Fourier transform
Sketch the Fourier transform of the sampled signal. Include the baseband replica, and
the replicas at ω = ±π. Assume that τ is small, so that e
jωτ ' 1 + jωτ . .
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3. (20 points) Laplace Transform
(a) (12 points) Find the Laplace transforms of the following signals and determine their
region of convergence.
i. f(t) = te−at(sin ω0t)
2u(t)
ii. f(t) =



0, 0 ≤ t < 1
1, 1 ≤ t < 2
e
−3(t−2) 2 ≤ t
(b) (8 points) The Laplace transform of a causal signal x(t) is given by
X(s) = 1
s
2 + 2s + 5
, ROC: Re{s} > −1
Which of the following Fourier transforms can be obtained from X(s) without actually determining the signal x(t)? In each case, either determine the indicated Fourier
transform or explain why it cannot be determined.
i. F{x(t)e
−t}
ii. F{x(t)e
3t}
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4. (20 points) Inverse Laplace Transform
Find the inverse Laplace transform f(t) for each of the following F(s): (f(t) is a causal
signal)
(a) (6 points) F (s) = e
−s
(s + 3)
(s − 1)2
(s − 2)
(b) (6 points) F (s) = s + 4
s
3 + 4s
(c) (8 points) F (s) = 1
(s + 1)(s
2 + 2s + 2)
5. LTI system (20 points)
Assume a causal LTI system S1 is described by the following differential equation:
d
2y(t)
dt2
+ 3
dy(t)
dt + 2y(t) = ax(t), y(0) = 0, y0
(0) = 0
where a is a constant. Moreover, we know that when the input is e
t
, the output of the system
S1 is
1
2
e
t
.
(a) (6 points) Find the transfer function H1(s) of the system. (The answer should not be
in terms of a, i.e., you should find the value of a).
(b) (8 points) Find the output y(t) when the input is x(t) = u(t).
(c) (6 points)The system S1 is linearly cascaded with another causal LTI system S2. The
system S2 is given by the following input-output pair:
S2 input : u(t) → output : r(t)
Find the overall impulse response.
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