Lecture 8: Transforms We Have Met and Loved

We have studied this year a variety of transforms:

Laplace transforms

which are useful for system analysis, including transient and stability analysis. By evaluating at we explored also the concept of frequency response.

Z-transforms

which are the transform appropriate for discrete-time systems. Like the Laplace transform, it can be used for transient analysis, stability analysis, and, by evaluating at we get the concept of frequency response.

Fourier series

which are used to provide a representation of periodic signals. This has some application to circuit analysis for periodic signals, and leads, by taking a limit to signals of increasingly longer period, to the Fourier transform. We also saw that we can take the idea of series representations of functions and use a variety of other basis functions for other useful representations.

Fourier transforms

which can be used to examine frequency response of signals. By means of their properties, we are also lead to consider concepts such as modulation. Fourier transforms do not really address the stability issues that Laplace transforms do, nor can they be used as conveniently for transient analysis. However, by not starting at , they simplify some other issues.

The Discrete-time Fourier Transform

is to the Z-transform what the Fourier transform is to the Laplace transform. That is, we have an exact frequency component representation of signals that are not periodic by evaluating a (possibly two-sided) Z-transform at The DTFT is the study of this set of lecture notes.

The Discrete Fourier Transform (DFT)

can be used to compute a transform of a finite-length discretely-sampled set of data. The DFT can be used for computational signal analysis, and its implementation in the form of the FFT is very common. However, because the signal is truncated in time and in frequency, it does not provide an exact frequency analysis (although there are techniques to get close enough in practice).