ISI; modified duobinary for 10110011

Inter Symbol Interference (ISI)

ISI is the overlapping of neighbouring pulses in a digital signal, so that at the sampling instant of one symbol the received value also contains the tails of other symbols. It appears because a band-limited channel spreads each pulse in time (a pulse limited in bandwidth cannot be limited in time).

The receiver output at the sampling instant $t_i = iT_b$ is

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

The first term is the wanted symbol, the second is ISI, the third is noise.

Causes: limited channel bandwidth, non-ideal filters, multipath and dispersion in cables or fibres, timing errors at the receiver.

Effects: wrong decisions even without noise; higher bit error rate; reduced eye opening and noise margin; sensitivity to timing jitter.

 pulse k-1     pulse k     pulse k+1
     /\         /\          /\
 ___/  \_     _/  \_      _/  \___
         \___/      \____/
        tail of k-1 adds to k

Remedies: Nyquist pulse shaping (raised cosine), correlative coding (duobinary, modified duobinary), and equalization.

Modified duobinary coding

Modified duobinary introduces a controlled ISI between a symbol and the one two bit intervals before it:

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

$H(0) = 0$, so the spectrum has no DC component. The output takes three levels (−2, 0, +2). A precoder $d_k = b_k \oplus d_{k-2}$ is used so that each bit is decoded from one sample: $|c_k| = 2 \Rightarrow 1$, $c_k = 0 \Rightarrow 0$.

Illustration for 10110011

Assume $d_{-1} = d_0 = 0$; level mapping $d = 1 \to +1$, $d = 0 \to -1$.

The decoded output 10110011 matches the input.