DTFT Time Limited Step.
Cuthbert Nyack
The Z transform of the first N samples of a unit step
function is given by the following equation.
Substituting
Z = ejwT gives
Z(w), the DTFT below.
For Comparison the Fourier Transform G(w) of a pulse
f(t) = 1 for 0 £ t £ t is given by.
Comparing the FT with the DTFT shows that the FT has one maximum at
w = 0, while the DTFT has maxima at
wT/2 = np
or w = n 2p/T
= nws.
The applet below shows how the frequency spectrum of
the limited step varies as the number of samples N changes. The spectrum
is compared with the Fourier Transform of a pulse with the same width.
The applet shows the periodicity of Z(w) compared with G(w). The spectrum of
Z(w) also shows some aliasing for
frequencies < ws/2
One of the problems with sampling the time limited step is that
it has spectral components at higher than half of the sampling
frequency. The applet below shows one way in which this effect can
be reduced. Here the first and last sample A(0, N - 1) are reduced in amplitude
to 'smooth' out the edges. As shown below, the real part,
imaginary part and magnitude of the spectrum at half the
sampling frequency can be made negligible by varying A(0,N-1).
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© 2000 Cuthbert A. Nyack.