Lecture 21 Blog, ECE301 Spring 2011, Prof. Boutin
Monday February 28, 2011 (Week 8) - See Course Schedule.
In the first part of today's lecture, we finished discussing the properties of the continuous-time Fourier transform. We then used these properties to obtain a simple expression for the frequency response of a causal LTI system defined by a differential equation.
In the second part of the lecture, we began covering chapter 5, which is about the discrete-time Fourier transform. We gave the definition for the discrete-time Fourier transform and its inverse. We pointed out the linearity property of the discrete-time Fourier transform. We also observed that the discrete-time Fourier transform is periodic with period $ 2 \pi $ and gave a mathematical proof of this fact. We then considered the problem of obtaining an expression for the Fourier transform of a periodic discrete-time signal. We noticed that the summation formula diverges in this case. However, using the linearity property of the discrete-time Fourier transform, we were able to reduce the problem to that of finding the Fourier transform of a discrete-time complex exponential. We then "guessed" what the Fourier transform of a discrete-time complex exponential should be. Our guess was somewhat wrong because it was not periodic with period $ 2 \pi $. We fixed that problem by adding an infinite number of shifted copies (shifted by $ 2 \pi k $, with k integer)of our signal.
Action items before the next lecture:
- Read Sections 5.4, 5.5 and 5.6 in the book.
- Solve the following practice problems on computing the Fourier transform of a discrete-time signal:
Relevant Rhea Pages
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Next: Lecture 22
