ISI; duobinary with precoder for 0010110

Intersymbol interference

Intersymbol interference (ISI) is the overlap of the tails of neighbouring pulses at the sampling instant of the present pulse. It arises because a band-limited channel spreads each pulse beyond its own bit interval. The sample at $t = kT_b$ becomes

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

where the middle term is ISI. ISI closes the eye pattern and causes errors even without noise.

Duobinary coding with precoder

Duobinary (class I partial response) signalling deliberately adds a known amount of ISI from the previous bit so that data at rate $R_b$ can be sent in the minimum bandwidth $R_b/2$ with a realizable filter.

$$c_k = a_k + a_{k-1}, \qquad a_k = \pm 1$$ $$H(f) = 1 + e^{-j2\pi fT_b} = 2\cos(\pi fT_b)\,e^{-j\pi fT_b}, \quad |f| \le \frac{1}{2T_b}$$

The output has three levels (−2, 0, +2).

Need for precoder: without it, the receiver decodes $\hat a_k = c_k - \hat a_{k-1}$, so one wrong decision spreads to all later bits (error propagation).

Precoder: $d_k = b_k \oplus d_{k-1}$; level map $d_k = 1 \to +1$, $d_k = 0 \to -1$. Then $c_k = 0$ when $b_k = 1$ and $c_k = \pm 2$ when $b_k = 0$.

Decision rule: $|c_k| < 1 \Rightarrow \hat b_k = 1$; $|c_k| > 1 \Rightarrow \hat b_k = 0$.

b_k -->(XOR)---+--> level map --> a_k --+------------>(+)--> [LPF, B = Rb/2] --> c_k
         ^     | d_k                    |              ^
         |     v                        |              |
         +--[Tb delay]                  +--[Tb delay]--+

Illustration for 0010110

Initial precoder bit assumed $d_0 = 1$ ($a_0 = +1$).

The decoded sequence 0010110 equals the input, and each bit is decided from its own sample only.