Nyquist ISI criterion

by

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 F_{sym}=1/T_{sym} (T_{sym} is the symbol period) shown in Figure 1, where the transmitter, channel and the receiver are represented as band-limited filters.

An equivalent communication system model
Figure 1: An equivalent communication system model

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

Nyquist criterion Fourier transform pairs

This article is part of the book Wireless Communication Systems in Matlab, ISBN: 978-1720114352 available in ebook (PDF) format (click here) and Paperback (hardcopy) format (click here).

Given the noise term n(t), the output b(t) of the receiving filter is given as

b(t)=\sum_{k=-\infty}^{\infty}a_k h(t-kT_{sym})+n(t) \quad\quad\quad (2)

where, h(t)=h_T(t) \ast h_C(t) \ast h_R(t) is the overall impulse response of the system due to an impulse at its input. Normalizing h(0)=1 (where the signal of interest lies), at the symbol sampling instants t=mT_{sym},

b(mT_{sym})=a_m + \underbrace{ \sum_{ \substack{k=-\infty \ k \neq m} }^{\infty} a_k h(mT_{sym}-kT_{sym})}_{\text{ISI terms}} + n(mT{sym}) \quad\quad\quad\quad\quad\quad (3)

where, a_m is the symbol of interest at mT_{sym} sampling instant, n(mT_{sym}) 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 t=0 to the combined filters h(t), the samples of the h(t) at the output of the filter combination should be 1 at the sampling instant t=0 and zero at all other sampling instants kT_b (k \neq 0). This is called Nyquist criterion for zero ISI. In frequency domain, the frequency-shifted replicas of the overall system transfer function H(f)=H_T(f)H_C(f)H_R(f) should add up to a constant value.

\sum_{k=-\infty}^{\infty} H \left(f+\frac{k}{T_{sym}} \right)= T_{sym} \quad\quad \forall f \quad\quad (4)

It can be readily seen that, in time domain, the simplest signal that naturally avoids ISI is the rectangular pulse of width T_{sym}=1/F_{sym} 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 F_{sym}/2. The next section walks thorough the various choices of pulse shapes that are available for avoiding or mitigating ISI.

Rate this article: PoorBelow averageAverageGoodExcellent (18 votes, average: 3.33 out of 5)

Books by the author

Wireless Communication Systems in Matlab
Wireless Communication Systems in Matlab
Second Edition(PDF)
PoorBelow averageAverageGoodExcellent (180 votes, average: 3.62 out of 5)

Digital modulations using Python
Digital Modulations using Python
(PDF ebook)
PoorBelow averageAverageGoodExcellent (134 votes, average: 3.56 out of 5)

digital_modulations_using_matlab_book_cover
Digital Modulations using Matlab
(PDF ebook)
PoorBelow averageAverageGoodExcellent (136 votes, average: 3.63 out of 5)
Hand-picked Best books on Communication Engineering
Best books on Signal Processing

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

2 thoughts on “Nyquist ISI criterion”

Post your valuable comments !!!