文档介绍：Chapter 5. The Discrete Fourier Transform
Gao Xinbo
School of ., Xidian Univ.
xbgao@
./teach/matlabdsp/
Review 1
The DTFT provides the frequency-domain (w) representation for absolutely summable sequences.
The z-transform provides a generalized frequency-domain (z) representation for arbitrary sequences.
Two features mon:
Defined for infinite-length sequences
Functions of continuous variable (w or z)
From the putation viewpoint, these two features are troublesome because one has to evaluate infinite sums at uncountably infinite frequencies.
Review 2
To use Matlab, we have to truncate sequences and then evaluate the expression at finitely many points.
The evaluation were obviously approximations to the exact calculations.
In other words, the DTFT and the z-transform are not putable transform.
Introduction 1
Therefore we turn our attention to a putable transform.
It is obtained by sampling the DTFT transform in the frequency domain (or the z-transform on the unit circle).
We develop this transform by analyzing periodic sequences.
From FT analysis we know that a periodic function can always be represented by a bination of harmonically plex exponentials (which is form of sampling).
This give us the Discrete Fourier Series representation.
We extend the DFS to finite-duration sequences, which leads to a new transform, called the Discrete Fourier Transform.
Introduction 2
The DFT avoids the two problems mentioned above and is a putable transform that is suitable puter implementation.
The putation of the DFT for long sequences is prohibitively time consuming.
Therefore several algorithms have been developed to pute the DFT.
These are collectively called fast Fourier transform (or FFT) algorithms.
The Discrete Fourier Series
Definition: Periodic sequence
N: the fundamental period of the sequences
From FT analysis we know that the periodic functions can be synthesized as a bination plex exponentials whose frequencies are multiples (or harmonics