Discrete Fourier Transform and Signal Spectrum - Digital Signal Processing: Fundamentals and Applications - page 88

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Sample number n

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0 500 1000 1500 2000 2500 3000 3500 4000

Frequency (Hz)

FIGURE 4.1

Example of the digital signal and its amplitude spectrum.

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 domain sequence. In addition, the DFT is widely used in
many other areas, including spectral analysis, acoustics, imaging/video, audio, instrumentation, and
communications systems.
To be able to develop the DFT and understand how to use it, we first study the spectrum of periodic
digital signals using the Fourier series. (There is a detailed discussion of the Fourier series in
Appendix B.)

4.1.1 Fourier Series Coefficients of Periodic Digital Signals
Let us look at a process in which we want to estimate the spectrum of a periodic digital signal xðnÞ
sampled at a rate of f s Hz with the fundamental period T 0 ¼ NT , as shown in Figure 4.2 , where there
are N samples within the duration of the fundamental period and T ¼ 1 =f s is the sampling period. For
the time being, we assume that the periodic digital signal is band limited such that all harmonic
frequencies are less than the folding frequency f s = 2 so that aliasing does not occur.
According to Fourier series analysis (Appendix B), the coefficients of the Fourier series expansion
of the periodic signal xðtÞ in a complex form are