Practice Question 4, ECE438 Fall 2010, Prof. Boutin
Frequency domain view of filtering.
Note: There is a very high chance of a question like this on the final.
Define a signal x(t) and take samples every T (using a specific value of T). Store the samples in a discrete-time signal z[n]. Obtain a mathematical expression for the Fourier transform of x(t) and sketch it. Obtain a mathematical expression for the Fourier transform of y[n] and sketch it.
Let's hope we get a lot of different signals from different students!
Post Your answer/questions below.
I thought I would start with a function that had a simple F.T.
$ x(t) = \delta(t), T=1 $
$ \begin{align} z[n] &= x_T[n] \\ &= \delta(t+T) \end{align} $
Fourier Transform of x(t) = 1
$ y[n] = x(t)*z[n] $ <-- is this correct?
I only solved the general form for this problem.
$ \begin{align} \mathcal{F}(x(t)) = \int_{-\infty}^{\infty} x(t)e^{-j\omega t} \end{align} $
$ \begin{align} z[n] = comb_T(x(t)) \end{align} $
I'm not sure what y[n] is equal to. I'm assuming that y[n] is the same as z[n]. Then the FT of y[n] is
$ \begin{align} Y(e^{j\omega}) = \frac{1}{T}rep_\frac{1}{T}(X(e^{j\omega})) \end{align} $
- Mike Wolfer
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