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Homework 3 Solution, ECE438, Fall 2014
Question 1
Let x(t) be a continuous-time signal and let y[n]=x(nT) be a sampling of that signal with period T>0. We would like to interpolate the samples (i.e., "connect the dots") in order to try to recover x(t).
a) Derive a formula for a band-limited interpolation of the samples (i.e., an expression for a continuous signal z(t) in terms of the samples y[n]). (Do not simply write down the formula; show how to derive it.)
Solution
Sampling the signal produces copies of the spectrum. The spectrum of y[n] can be express in cycles/sec as:
$ \begin{align} Y(f) = \frac{1}{T_s} \sum_{k=-\infty}^{\infty}X(f-k\frac{1}{T}) \end{align} $
To filter out the copies, and just preserve the one at baseband, we can use the filter
$ H_r(f)=\begin{cases} T \mbox{, where } |f| \leq \frac{1}{2T_s} \\ 0 \mbox{, else}\end{cases} $
$ h_r(t) = sinc\left ( \frac{t}{T} \right ) $
Performing the convolution in the time domain
$ \begin{align} z(t) &= comb_{T_s} \{ x(t) \} \ast sinc \left ( \frac{t}{T} \right ) \\ &=\left ( \sum_{n=-\infty}^{\infty} x(nT) \delta (t-nT) \right )\ast sinc \left ( \frac{t}{T} \right ) \\ &=\sum_{n=-\infty}^{\infty} x(nT) sinc \left (\frac{t-nT }{T}\right) \\ \end{align} $
b) Show that your interpolation is equal to the original signal at all sample points.
Solution
Looking at the mth sample point
$ \begin{align} z(mT) &=\sum_{n=-\infty}^{\infty} x(nT) sinc \left (\frac{mT-nT }{T}\right) \\ &=\sum_{n=-\infty}^{\infty} x(nT) sinc \left (m-n \right) \mbox{, where } sinc(m-n) = \begin{cases} 1 \mbox{, where } m=n \\ 0 \mbox{, else}\end{cases} \\ &= x(mT) \end{align} $
c) Under what circumstances is your interpolation equal to the original signal x(t)? Explain.
Solution
The original signal, x(t), must be bandlimited, so that X(f) = 0 for all f>1/2T. Otherwise, there will be aliasing.
Question 2
Again, we consider a continuous-time signal x(t) and a sampling y[n]=x(nT) of that signal.
a) Write a formula for a piece-wise constant interpolation of the samples.
b) Derive the relationship between the Fourier transform of the interpolation you wrote in 2a) and the Fourier transform of x(t). (Do not simply write down the formula; show how to derive it.)
c) Is the interpolation you wrote in 2a) band-limited? Explain.
Question 3
Let
$ x(t)=7 \text{sinc } ( \frac{t-5}{2} ). $
a) Obtain the Fourier transform X(f) of the signal and sketch the graph of |X(f)|.
b) What is the Nyquist rate $ f_0 $ for this signal?
c) Let
$ T = \frac{1}{3 f_0}. $
Write a mathematical expression for the Fourier transform $ X_s(f) $ of
$ x_s(t)= \text{ comb}_T \left( x(t) \right). $
Sketch the graph of $ |X_s(f)| $.
d) Let
$ T = \frac{1}{5 f_0}. $
Write a mathematical expression for the Fourier transform $ {\mathcal X}_d(\omega) $ of $ x_d[n]= x(nT) $ and sketch the graph of $ |{\mathcal X}_d(\omega)| $.
Discussion
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