The Discrete Time Fourier Transform - Synthesis Lectures on Signal Processing - page 2

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The entire software package should be stored in a single folder on the user's computer, and the full
file name of the folder must be placed on the MATLAB or LabVIEW search path in accordance with the
instructions provided by the respective software vendor (in case you have encountered this notice before,
which is repeated for convenience in each chapter of the topic, the software download only needs to be
done once, as files for the entire series of four volumes are all contained in the one downloadable folder).
See Appendix A for more information.

1.3 INTRODUCTION TO TRANSFORM FAMILIES
The chief differences among the transforms mentioned below involve whether they 1) operate on con-
tinuous or discrete time signals, 2) provide continuous or discrete frequency output, and 3) use constant
unity-amplitude correlators (in the case of the Fourier family of transforms), or dynamic (decaying,
steady-state, or growing correlators) in the case of the Laplace and z - transforms.
The following table summarizes the main characteristics of a number of well-known transforms
with respect to the following categories: Input Signal Domain (continuous C or discrete D signal), Output
(Frequency) Domain (produces continuous C or discrete D frequency output), and Correlator Magnitude
(constant, unity magnitude for Fourier-based transforms, or variable magnitudes (decaying, growing, or
constant unity) for Laplace Transform and z -Transform, accordingly as e − σt or

n , respectively.

| z |

Transform Input Output Correlator Mag.

e − σt

Laplace Transform C C

Fourier Transform C C 1

Fourier Series C D 1

Discrete Time Fourier Transform D C 1

Discrete Fourier Series D D 1

Discrete Fourier Transform D D 1

n

z -transform D C | z |

For purposes of discrete signal processing, what is needed is a numerically computable repre-
sentation (transform) of the input sequence; that is to say, a representation which is itself a finite but
complete representation that can be used to reenter the time domain, i.e., reconstruct the original signal.
For transforms that are not computable in this sense, samples of the transform can be computed. Of all
the transforms discussed in the following section of the chapter, only the Discrete Fourier Series (DFS)
and Discrete Fourier Transform (DFT) are computable transforms in the sense mentioned above.
The use of dynamic correlators results in a transform that is an algebraic expression that implicitly
or explicitly contains information on the system poles and zeros. The system response to signals other
than steady-state, unity amplitude signals can readily be determined, although such transforms can also
be evaluated to produce the same result provided by the Fourier Transform (in the case of the Laplace
Transform) or the DTFT, DFS, and DFT (in the case of the z -Transform). Thus, the Laplace Trans-
form and z -Transform are more generalized transforms having great utility for system representation
and computation of response to many types of signals from the continuous and discrete time domains,
respectively.
While this topic is concerned chiefly with discrete signal processing, we give here a brief discussion
of certain continuous time transforms (Laplace, Fourier, Fourier Series) to serve as background or points