Discrete-Time Fourier Transform

In this chapter, Discrete-Time Fourier Transform, the version of the Fourier analysis that is more suitable for theoretical analysis of signals and systems is presented. It is derived starting with the DFT definition, resulting in the DTFT and its inverse. In the time domain, the input to the DTFT is infinite length samples of discrete signals. The corresponding transform, in the frequency domain, is a periodic continuous spectrum. Although it is usually presented assuming a sampling interval of 1 second, it can be easily extended for any sampling interval. The DFT is the samples of the DTFT spectrum at equal intervals on the unit-circle. The properties of the DTFT are presented, which are very useful in the analysis of signals and systems. The transfer function concept is the ratio of the DTFT of the output of a system to the DTFT of the input applied. Using the transfer function, the zero-state output of systems is computed for specific systems. The Hilbert transform, which is useful in developing complex signals with one-sided spectrum, is presented. The digital differentiator, which approximates the derivative, is described. Finally, the approximation of the DTFT and its inverse by the DFT and IDFT is presented with examples.