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Category: ECE 102

Description

5/5 - (6 votes)

1. (12 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) (2 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) (10 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 recover the

signal. 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?

2. (18 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:

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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) (Optional) 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

(e) (Optional) If we know g(t) is real and even, can we recover g(t) from the non-uniform

samples g(t)f(t)?

3. (21 points) Laplace Transform

(a) (15 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

−2(t−3) 2 ≤ t

iii. f(t) = (

sin(2πt), 0 ≤ t < 1

0, otherwise

(b) (6 points) The Laplace transform of a causal signal x(t) is given by

X(s) = 1

s

2 + 2s + 5

, ROC: Re{s} > −1

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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}

4. (15 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) F (s) = e

−s

(s + 1)

(s − 2)2

(s − 3)

(b) F (s) = s + 4

s

3 + 4s

(c) F (s) = 1

(s + 1)(s

2 + 2s + 2)

5. (19 points) LTI system

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

. (Note: this is not a causal exponential, e

tu(t). Rather, you should consider using

the eigenfunction property of LTI systems.)

(a) (7 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) (5 points) Find the output y(t) when the input is x(t) = u(t).

(c) (7 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) − u(t − 1) → output : r(t) − 2r(t − 1) + r(t − 2)

Find the overall impulse response.

6. (15 points) Feedback systems

The response to a remote manipulator can be modeled by this system:

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x(t) is the position we request, and y(t) is the position of the manipulator. Its impulse

response is a decaying exponential function with a time constant of 1 s (the time constant

of e

−λtu(t) is 1/λ), which is too slow to be practically usable. In order for a manipulator to

feel immediate and interactive, we would like the response time to be no more than 100 ms.

(a) (5 points) Find the step response of the system, and plot it.

(b) (5 points) To speed up the response, we add a feedback loop around the system, along

with a gain stage:

Find the transfer function of this system.

(c) (5 points) Choose a such that the time constant is 100 ms. Solve for the step response,

and plot it on the same graph as part (a).

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