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Homework 3, ECE438, Fall 2015, Prof. Boutin
Hard copy due in class, Friday September 18, 2015.
The first goal of this homework is to to further understand the relationship between a signal and a sampling of that signal, viewed in the frequency domain. This time, we are looking at signals beyond pure frequencies. The second goal of this homework is to learn different ways to reconstruct a signal.
Question 1
Consider the signal $ x(t)=5 \text{sinc } ( \frac{t-7}{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}{4 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{2}{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)| $.
Question 2
Let x(t) be a continuous-time signal and consider a sampling y[n]=x(nT) of that signal.
a) Write a formula for a zero-order hold reconstruction 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.
d) Show that your interpolation is equal to the original signal at all sample points.
e) Under what circumstances is your interpolation equal to the original signal x(t)? Explain.
Question 3
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.)
b) Show that your interpolation is equal to the original signal at all sample points.
c) What is the relationship between the Fourier transform of the interpolation you wrote in 3a) and the Fourier transform of x(t)? (Simply write down the formula; no need to show how to derive it.)
c) Under what circumstances is your interpolation equal to the original signal x(t)? Explain.
Hand in a hard copy of your solutions. Pay attention to rigor!
Presentation Guidelines
- Write only on one side of the paper.
- Use a "clean" sheet of paper (e.g., not torn out of a spiral book).
- Staple the pages together.
- Include a cover page.
- Do not let your dog play with your homework.
Discussion
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