Fourier Transform, Impulse Response of Ideal Filter,
An ideal low pass filter has unity gain over a finite frequency range
with a possible phase change. The transfer function below has a gain
of unity over tbe range - wc
to + wc. The phase changes from
to - f
over the passband of the filter.
The output from the filter is the inverse Fourier Transform of the
filter transfer function and is given by:-
The output response is shown in the applet below. By using
2 impulses the concept of channel capacity can also be
demonstrated. One impulse and its response is shown in
purple. This impulse is fixed.
The second impulse and its response
is shown in green.
The sum of the responses is shown in red and
the sum of the squares of each response is shown
in yellow. Applet
demonstrates the effect of bandwidth(in rad/s) and phase(in rad)
on the responses. The horizontal axis for time with HGain = 1 is
from -1.25 to +3.75.
In optics, the Rayleigh criterion is used to determine the resolving
power of optical instruments. According to this criterion, the images
of 2 points of light can be distinguished provided the diffraction
patterns produced are separated by a certain minimum distance which
is taken to be the distance from the maximum to the first zero. In the
applet below the yellow curve, which is the sum of the squares of each response
plays the same role as intensity in optics. By changing the delay of
the green impulse and monitoring the yellow curve this criterion
can be "illustrated". Assuming a communication channel has a low
pass filter transfer function, then this shows that the capacity
is twice the bandwidth in Hz.
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COPYRIGHT © 1996 Cuthbert Nyack.