![]() In particular, it is the Fourier transform of the impulse response. And the change in complex amplitude, which corresponds to the frequency response, in fact is what led to the definition of the Fourier transform. In other words, if you have a complex exponential into a linear time-invariant system, the output is a complex exponential. ![]() And as you recall, and as is the same for continuous-time, the reason that we picked complex exponentials was because of the fact that they are eigenfunctions of linear time-invariant systems. And, of course, there is the corresponding analysis equation. Now, just as in continuous-time, in discrete-time the Fourier transform corresponded to a representation of a sequence as a linear combination of complex exponentials. So what we want to talk about is generalizing the Fourier transform, and what this will lead to in discrete-time is a notion referred to as the z-transform. PROFESSOR: In the last several lectures, we've talked about a generalization of the continuous-time Fourier transform and a very similar strategy also applies to discrete-time, and that's what we want to begin to deal with in today's lecture. To make a donation, or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. The following content is provided under a Creative Commons license.
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