Z Transform, Fourier Transform and the DTFT.

Cuthbert Nyack
The relation between the Z, Laplace and Fourier transform is illustrated by the above equation. It shows that the Fourier Transform of a sampled signal can be obtained from the Z Transform of the signal by replacing the variable z with ejwT.

This proceedure is equivalent to restricting the value of z to the unit circle in the z plane. Since the signal is discrete and the spectrum is continuous, the resulting transform is referred to as the Discrete Time Frequency Transform (DTFT).

The DTFT of a signal is usually found by finding the Z transform and making the above substitution.
The exponential time function and its Z transform is shown above. Making the above substitution into the Z Transform gives the expression below for the Fourier Transform of the sampled exponential function.
Expanding the complex exponent gives:-

The magnitude of the Fourier Transform is given by.
and the phase by:-

Depending on the value of wT the magnitude of the spectrum has maxima and minima. The maxima occurs at frequencies:-
and the minima at:-
The periodicity of the maxima and minima is given by the sampling frequency:-

The analog signal and its frequency spectrum is shown above and its magnitude and phase are given below.

For small T and w, the DTFT can be written as:-
The factor T arises because the DTFT refers to samples while the FT refers to areas.

The applet above shows the DTFT of the sampled damped exponent and the FT of the continuous damped exponent. In the plot the magnitude of the sampled signal spectrum is multiplied by T for it to agree with the analog spectrum at low frequencies.

eg parameters:-
(0.3, 0.25, 0.5, 1.11, NA...)
This shows
(a) the Fourier transform in close agreement with the DTFT at low frequencies.
(b) The real part of the spectrum has even symmetry about half the sampling frequency.
(b) The imaginary part of the spectrum has odd symmetry about half the sampling frequency.
(c) The spectrum of the DTFT is periodic in the frequency domain with period equal to the sampling frequency while the spectrum of the FT decreases continuously with frequency.

eg parameters:-
(0.3, 0.05, 0.25, 15.0, NA...)
This shows significant difference exists between the DTFT and the FT at half the sampling frequency and this difference increases with the sampling time. This is an example of aliasing.

For a discrete sampled signal, the DTFT spectrum is periodic and continuous.

This is the opposite of the Fourier Series where the signal is periodic and continuous while the spectrum is discrete and non periodic. Because of the periodicity and the symmetry about half of the sampling frequency, only the portion of the spectrum between 0 and half the sampling frequency is used. This also means that for the spectrum of the sampled signal to approach the spectrum of the analog signal within the range 0 to ws/2 then the analog spectrum should not extend beyond ws/2. Alternatively this can be stated as:-

The sampling frequency should be greater than twice the highest frequency in the signal spectrum.

Generally the spectrum of the sampled signal becomes closer to that of the analog signal at low frequencies as the sampling frequency is increased.
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© 2006 Cuthbert A. Nyack.