Objectives:
This chapter investigates discrete Fourier transform (DFT) and fast Fourier transform (FFT) and their properties; introduces the DFT/FFT algorithms to compute signal amplitude spectrum and power spectrum; and uses the window function to reduce spectral leakage. Finally, the chapter describes the FFT algorithm and shows how to apply it to estimate a speech spectrum.
In time domain, representation of digital signals describes the signal amplitude versus the sampling time instant or the sample number. However, in some applications, signal frequency content is very useful otherwise than as digital signal samples. The representation of the digital signal in terms of its frequency component in a frequency domain, that is, the signal spectrum, needs to be developed. As an example, Figure 4.1 illustrates the time domain representation of a 1,000-Hz sinusoid with 32 samples at a sampling rate of 8,000 Hz; the bottom plot shows the signal spectrum (frequency domain representation), where we can clearly observe that the amplitude peak is located at the frequency of 1,000 Hz in the calculated spectrum. Hence, the spectral plot better displays frequency information of a digital signal.
The algorithm transforming the time domain signal samples to the frequency domain components is known as the discrete Fourier transform, or DFT. The DFT also establishes a relationship between the time domain representation and the frequency domain representation. Therefore, we can apply the DFT to perform frequency analysis of a time...
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