ISI and two practical methods to minimize it

ISI

Intersymbol interference (ISI) is the interference caused at the sampling instant of a symbol by the spread-out tails of neighbouring symbols. A band-limited, dispersive channel stretches each pulse beyond its own interval, so the sample becomes

$$y(kT_b) = \mu a_k + \mu\sum_{n\ne k}a_n\,p\big((k-n)T_b\big) + n(kT_b)$$

The sum term is ISI; it reduces the eye opening and causes errors even without noise.

Method 1: Pulse shaping with raised cosine filters

The transmit and receive filters are designed so that the overall pulse satisfies the Nyquist criterion, $p(nT_b) = 0$ for $n \ne 0$. The ideal sinc pulse is not realizable, so the raised cosine spectrum is used:

• Roll-off factor $\alpha$ ($0 \le \alpha \le 1$); bandwidth $B_T = \frac{R_b}{2}(1 + \alpha)$.
• $p(t) = \text{sinc}(t/T_b)\dfrac{\cos(\pi\alpha t/T_b)}{1 - 4\alpha^2t^2/T_b^2}$ has zeros at all other sampling instants.
• Tails decay as $1/t^3$, so small timing errors cause little ISI.
• It is usually split into root raised cosine filters at the transmitter and the receiver (which also gives matched filtering).

Example: $R_b = 9600$ bps with $\alpha = 0.5$ needs $B_T = 4800 \times 1.5 = 7200$ Hz.

Method 2: Equalization

An equalizer is a filter at the receiver whose response cancels the channel distortion, so that the channel plus equalizer is close to a Nyquist pulse. The usual form is the transversal (tapped delay line) equalizer:

x(t)->[Tb]--->[Tb]--->[Tb]---> ...
   |      |       |       |
  c-N   c-N+1    ...     cN    (tap gains)
   |      |       |       |
   +------+---(sum)-------+---> y(t) to detector

$$y(kT_b) = \sum_{n=-N}^{N} c_n\,x\big((k-n)T_b\big)$$

• Zero-forcing equalizer: the $2N+1$ taps are chosen so that $y = 1$ at $k = 0$ and $y = 0$ at the $N$ nearest sampling points on each side ($2N$ points in all), forcing ISI to zero there.
• Adaptive equalizer: the taps are adjusted continuously (LMS algorithm) using a training sequence or past decisions, so it tracks time-varying channels such as telephone and mobile links.

(Other methods: correlative coding such as duobinary, and choosing the best sampling instant from the eye pattern.)