TABLE OF CONTENTS At this point we have discussed the design and implementation of digital filters. In the process we have investigated the characteristics of the input and output signals in both the time and frequency domains. It is time now to investigate a more direct relationship between the time domain and frequency domain for discrete time systems. We will begin by discussing the discrete time version of the Fourier transform. After the discrete Fourier transform (DFT) discussion, we will learn about the more computational efficient version called the fast Fourier transform (FFT). The C code for the FFT will be developed, and finally, we will take a look at one method of using the FFT in linear filtering.
The Discrete Fourier Transform (DFT) can be used to compute the frequency content of any discrete time signal. Consider first (9.1), where ? is periodic with a period of 2 ?. (Remember that a radian frequency of 2 ? in the z-plane is equivalent to the sampling frequency ( F s) in the s-plane.) In (9.1), x(n) represents the time domain signal, which has an infinite number of samples. The spectrum that will result from sampling an analog signal will actually be many replicas of the analog spectrum spaced at multiples of the sampling frequency, as shown in Figure 5.2 in Chapter 5. We will be able to select just one of these spectrums by using a filter at the output of...
