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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

[Duo-binary]

1,1

3

2

Raised

Cosine

1,2,1

5

3

 

2,1,-1

5

4

Sine

[Modified

Duo-binary]

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 roll-off 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 M-ary 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 M-ary 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.

For More Research

rf Design April 2002 http://images.industryclick.com/files/4/0402Gentile50.pdf
Window Functions http://en.wikipedia.org/wiki/Window_function
Intersymbol Interference http://en.wikipedia.org/wiki/Nyquist_ISI_criterion