Fourier Transform, Impulse Response of Ideal Filter, Channel Capacity.

Cuthbert Nyack
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 + f 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.