Sampling and Discrete Time Fourier Transform

Cuthbert Nyack

The first step in obtaining a signal for DSP is to sample an analog signal eg instrumentation, audio or video signal. Technically this usually involves a sample and hold and A/D converter. Here we will only be concerned with signals obtained at a constant sampling rate. One of the characteristic of sampled signals is that they are discrete in the time domain. This discreteness gives rise to a periodic spectrum in the frequency domain with a period equal to the sampling frequency. Notice that the situation here is the "opposite" of the Fourier Series where the signal is periodic in the time domain and the spectrum is discrete in the frequency domain.
Because the sampled signal is discrete in time, it is convenient to refer to its spectrum as the Discrete Time Fourier Transform (DTFT). This transform is intermediate between the Fourier Transform and the DFT and FFT.
The usual proceedure for finding the DTFT is to find the Z transform and replace Z with e^jwt. This is equivalent to finding the Z transform along the unit circle in the Z plane. For a finite number of samples x(0), x(1), x(2) ... x(n), the same result can be obtained by forming the expression x(0) + x(1)e^-jwt + x(2)e^-j2wt + ... + x(n)e^-jnwt.

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