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 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

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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.

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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

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