ISI; solutions for zero ISI; duobinary with precoder

ISI

Intersymbol interference (ISI) is the overlap of pulse tails from neighbouring symbols at the sampling instant of the present symbol, caused by the limited bandwidth and dispersion of the channel. It causes decision errors even without noise.

Two solutions for zero ISI

1. Nyquist pulse shaping: use an overall pulse with $p(nT_b) = 0$ for $n \ne 0$, e.g. the raised cosine pulse (bandwidth $\frac{R_b}{2}(1+\alpha)$).
2. Correlative (partial response) coding: e.g. duobinary signalling, which adds a known, controlled ISI that the receiver removes, allowing the minimum bandwidth $R_b/2$ with practical filters.

(Adaptive equalization is a third common method.)

Duobinary encoding with precoder

Duobinary signalling: each output sample is the sum of the present and previous binary levels:

$$c_k = a_k + a_{k-1}, \qquad a_k = \pm 1$$

The equivalent filter is a delay-and-add followed by an ideal low-pass filter:

$$H(f) = 2\cos(\pi fT_b)\,e^{-j\pi fT_b}, \quad |f| \le \frac{1}{2T_b}$$ $$h(t) = \frac{T_b^2\,\sin(\pi t/T_b)}{\pi t\,(T_b - t)}$$

$H(f)$ falls smoothly to zero at $f = 1/(2T_b)$, so it is realizable. The output has three levels: $-2, 0, +2$.

Problem: without precoding, decoding uses $\hat a_k = c_k - \hat a_{k-1}$, so one error propagates to later bits.

Precoder: form $d_k = b_k \oplus d_{k-1}$ (modulo-2) before level mapping ($d = 1 \to +1$, $d = 0 \to -1$). Then

• $c_k = \pm 2$ when $d_k = d_{k-1}$, i.e. $b_k = 0$;
• $c_k = 0$ when $d_k \ne d_{k-1}$, i.e. $b_k = 1$.

Decision rule: $|c_k| < 1 \Rightarrow \hat b_k = 1$, $|c_k| > 1 \Rightarrow \hat b_k = 0$. Each bit is found from its own sample, so there is no error propagation.

b_k -->(XOR)---+--> level map --> a_k --+------------>(+)--> [ideal LPF] --> c_k
         ^     | d_k                    |              ^
         |     v                        |              |
         +--[Tb delay]                  +--[Tb delay]--+

Example: input $b_k = 1101001$, initial $d_0 = 1$.

($a_0 = +1$.) The decoded data equal the input.