# 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 e^{jwT}.
##
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
w_{s}/2
then the analog spectrum should not extend
beyond w_{s}/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.