Nyquist ISI criterion

Key focus: As per Nyquist ISI criterion, to achieve zero intersymbol interference (ISI), samples must have only one non-zero value at each sampling instant.

Consider an equivalent baseband communication system model with baud rate ( is the symbol period) shown in Figure 1, where the transmitter, channel and the receiver are represented as band-limited filters.

The entire combination of filters can be represented as the following Fourier transform pair

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Given the noise term , the output of the receiving filter is given as

where, is the overall impulse response of the system due to an impulse at its input. Normalizing (where the signal of interest lies), at the symbol sampling instants ,

where, is the symbol of interest at sampling instant, is the noise at that sampling instant and the remaining terms are contributions from other symbols – representing intersymbol interference. For nullifying the ISI terms, with an impulse of unit value applied at to the combined filters , the samples of the at the output of the filter combination should be 1 at the sampling instant and zero at all other sampling instants . This is called Nyquist criterion for zero ISI. In frequency domain, the frequency-shifted replicas of the overall system transfer function should add up to a constant value.

It can be readily seen that, in time domain, the simplest signal that naturally avoids ISI is the rectangular pulse of width but it consumes infinite bandwidth. On the other hand, the signal that avoids ISI with the least amount of bandwidth is a sinc pulse of bandwidth . The next section walks thorough the various choices of pulse shapes that are available for avoiding or mitigating ISI.

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Topics in this chapter

Pulse Shaping, Matched Filtering and Partial Response Signaling ● Introduction ● Nyquist Criterion for zero ISI ● Discrete-time model for a system with pulse shaping and matched filtering  □ Rectangular pulse shaping  □ Sinc pulse shaping  □ Raised-cosine pulse shaping  □ Square-root raised-cosine pulse shaping ● Eye Diagram ● Implementing a Matched Filter system with SRRC filtering  □ Plotting the eye diagram  □ Performance simulation ● Partial Response Signaling Models  □ Impulse response and frequency response of PR signaling schemes ● Precoding  □ Implementing a modulo-M precoder  □ Simulation and results