This document discusses continuous and discrete signal processing. It provides examples of problems and their answers related to Fourier transforms and filters. Specifically, it contains: - An example problem about the Fourier transform of a signal x(t) and properties about x(t) and its transform X(jΩ). - A problem about passing an impulse through an ideal low-pass filter and sketching the output signal y(t). - A problem determining the transfer function and frequency response of a causal filter with impulse response h(t) = 5e−3t. It also finds the -6 dB cutoff frequency.

1. SIGNAL PROCESSING
Continuous and Discrete
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2. Problem 1: (The following is taken from the Signal Processing PhD Quals written
exam for January 2007. Note: this is not the complete exam.) Assume we have a
signal x(t) with a Fourier transform X(jΩ) given by
where X0 is some real valued number, W is a real valued positive number, and Ω is
specified in units of radians/second.
(a) What is the value of x(t) at t = 0?
(b) For an arbitrary t, what is the relationship between x(t) and x(−t)?
(c) What is the value of ʃ ∞ −∞ x(t)dt?
(d) What is the value of ʃ ∞ −∞ |x(t)| 2 dt?
Answer: These solutions are all based on the elementary properties of the Fourier
transform (see the class handout).
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3. (b) Using the symmetry properties, we note that X ( j!) is real, therefore x(!t) = x(t),
that is they are complex conjugates.
(c) This one is a little tricky! We use the property that
BUT note that there is a singularity at ! = 0. The question is: what is the value of X (
j0) ? The problem statement specifies that 0 X ( j!) = X for 0 " !
On the other hand if you approximate the step discontinuity with a smooth function
(say erf()) around ! = 0, you can argue that the value of X ( j0) = 0.75X0 , or
So the answer is dependent on your assumption about the discontinuity! (d) From
Parseval’s theorem
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4. Problem 2: An impulse δ(t) is passed through an ideal low-pass filter with
frequency response H(jΩ) = 1 for |Ω| < Ωc and H(jΩ) = 0 otherwise. Find and
sketch y(t), the output of the filter. Is this a causal filter?
Answer:
If and impulse is passed through the filter, we obtain the impulse response h(t) F
{H( j!}
The filter is acausal.
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5. Problem 3: After measurement and curve-fitting it is determined that a causal
signal processing filter has an impulse response h(t) = 5e−3t for t > 0.. What is the
filter’s (a) transfer function, and (b) its frequency response function? Determine
the -6 dB cutoff frequency, that is the frequency at which the output amplitude is
one half of the low frequency response
Answer: h (t) = 5e- 3t Let’s compute the Fourier transform of h(t)
Note: lower limit in integral is 0 because a real filter is a causal system.
a) The transfer function can be found by taking the Laplace transform, which can
be viewed as a Fourier transform where jω is replaced by s=σ+jω .
b) The frequency response is given by H(jω) computed previously.
c) We find the cut-off frequency by solving:
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