The impulse response for the RC frequency shape is given as
(2.4)
It is clear when we set 0, we obtain the impulse response of the ideal brick wall filter. This
response is a slowly decaying impulse response. Also, if we set 1, then this will have the fastest
decaying impulse response, but we have sacrificed occupied bandwidth, specifically twice the
Nyquist bandwidth, to obtain this feature.
Decomposition Tool. There is still, however, another potential issue with the Nyquist theorem in
that transmit impulse responses are used. Let us, for example, assume that the source is a non-returnto-
zero (NRZ) pulse stream. If we simply ignore the impulse stream and send the NRZ through the
Nyquist filter, we will see ISI present on the transmit signal. Let us explain this through the use of
Fig. 2.7. Here, NRZ pulses are sent to the Nyquist brick wall filter. The NRZ pulses can be represented
as an impulse stream convolved with the NRZ pulse. Next we perform the Fourier transform
and arrive with a sampled version of the (sin x)/x function.
Here we see that we can basically decompose the NRZ data stream into an impulse response stream
convolved with a (sin x)/x filter. In comparing this signal to that required by the Nyquist criteria, we
see that the (sin x)/x operation is the additional block.
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