of the spectral lines depends on the number of data samples. This issue will be further
discussed in Section 7.3.2.
Since the DFT coefficient X(k) is a complex variable, it can be expressed in polar form
where the DFT magnitude spectrum is defined as
and the phase spectrum is defined as
These spectra provide a complementary way of representing the waveform, which
clearly reveals information about the frequency content of the waveform itself.
7.1.2 Important Properties of DFT
The DFT is important for the analysis of digital signals and the design of DSP systems.
Like the Fourier, Laplace, and z-transforms, the DFT has several important properties
that enhance its utility for analyzing finite-length signals. Many DFT properties are
similar to those of the Fourier transform and the z-transform. However, there are some
differences. For example, the shifts and convolutions pertaining to the DFT are circular.
Some important properties are summarized in this section. The circular convolution
property will be discussed in Section 7.1.3.
If fxng and fyng are time sequences of the same length, then
DFTaxn byn aDFTxn bDFTyn aXk bYk,
where a and b are arbitrary constants. Linearity is a key property that allows us to
compute the DFTs of several different signals and determine the combined DFT via the
summation of the individual DFTs. For example, the frequency response of a given
system can be easily evaluated at each frequency component. The results can then be
combined to determine the overall frequency response.
FAST FOURIER TRANSFORM AND ITS APPLICATIONS