Nyquist Channels (Filters)
The principle channel types of interest are the cosine, raised
cosine, and sine. The brick wall response is also of interest
since it provides the basic tool needed to evaluate the other
channel types.
Nyquist Communications Channel Categories
Class

Channel
Name

Channel Response _{ }

Impulse
Notation f[d]

Radix

1

Ideal LPF

_{1}

1

2

1

Cosine
[Duobinary]

_{ }

1,1

3

2

Raised
Cosine

_{ }

1,2,1

5

3


_{ }
_{ }

2,1,1

5

4

Sine
[Modified
Duobinary]

_{ }

1,0,1

3

5


_{
}

1,0,2,0,1

5

The output of each channel type is naturally slightly different.
Brick Wall Filter (Ideal LPF)
The ideal low pass channel has an infinite rolloff at
the cutoff frequency. This is of course not technically achievable.
_{ }
(Some textbooks take a more rigorous approach and include
negative frequencies. From an engineering perspective, the idea
of LPFs having a negative frequency response is not particularly
meaningful. For a more thorough discussion of the Fourier Transform,
please see Appendix 3)
The time domain response (also called the impulse response)
is given by:
_{ }
A plot of this function resembles:
This function is the familiar sync or sampling function
and forms the basic time domain element used to analyze Mary pulses.
Sinc Pulse

The Sinc envelope can be created by directly
implementing the mathematical expression 

The sinc function can also be approximated by a FIR Filter.
As the number of taps increase, the resolution increases,
but so does the delay.

It should be noted, that if a time domain pulse of this
exact shape were created, it would have an ideal cutoff in the frequency
domain. Time domain pulses of this exact type however, are not practical,
since the leading and trailing tails never completely vanish.
The data input stream 1110100010001111110 produces the
following output response:
Brick Wall Response (Ideal LPF)
Since the impulse response of an ideal LPF consists of
one sinc pulse, it is sometimes written as f[d] = 1.
If two d pulses
separated by _{ } occur
at the filter input, the peak of the second sync response will occur
at a zero crossing of the first response. This suggests that at
that precise moment, it is possible to distinguish between both
pulses even though a great deal of overlap or ISI has occurred.
By normalizing the bandwidth to unity, it can be observed
that the maximum bit rate with no ISI is:
Controlling ISI forms the bases of Mary signaling theory
and allows the Nyquist rate to be exceeded. If a transmitted pulse
waveform consists of sinc components, it is possible to separate
the sinc components at the receiver, thus exceeding the Nyquist
rate.
