Reducing ISI; modified duobinary with precoder

Methods to reduce Intersymbol Interference

1. Nyquist pulse shaping (raised cosine filtering): shape the overall response so that each pulse has zero value at all other sampling instants. The raised cosine spectrum with roll-off $\alpha$ meets the Nyquist criterion, needs bandwidth $B = \frac{R_b}{2}(1+\alpha)$, and its tails decay as $1/t^3$, so timing errors cause little ISI.
2. Correlative (partial response) coding: add a controlled, known amount of ISI (duobinary, modified duobinary) so that the signal fits in the minimum bandwidth $R_b/2$ with realizable filters; the receiver removes the known ISI.
3. Equalization: a transversal (tapped delay line) filter after the channel cancels the channel's distortion. Zero-forcing equalizers force ISI to zero at the sampling points; adaptive (LMS) equalizers track changing channels.
4. Eye pattern monitoring and correct sampling: sample at the instant of maximum eye opening using good timing recovery.
5. Reducing the signalling rate or using more channel bandwidth, so pulses spread less.

Modified duobinary coding

Modified duobinary is a correlative coding scheme that correlates the present symbol with the symbol two bit intervals earlier:

$$c_k = a_k - a_{k-2}, \qquad a_k = \pm 1$$

Transfer function:

$$H(f) = 1 - e^{-j4\pi fT_b} = 2j\sin(2\pi fT_b)\,e^{-j2\pi fT_b}, \quad |f| \le \frac{1}{2T_b}$$

Impulse response: $h(t) = \text{sinc}\left(\frac{t}{T_b}\right) - \text{sinc}\left(\frac{t-2T_b}{T_b}\right)$.

$|H(f)|$ is zero at $f = 0$ and at $f = 1/(2T_b)$, so the signal has no DC component (useful for transformer-coupled lines and SSB) and is easy to filter. The output has three levels: $-2, 0, +2$.

Precoder: to avoid error propagation, the data are precoded as $d_k = b_k \oplus d_{k-2}$, then $a_k = +1$ if $d_k = 1$ and $-1$ if $d_k = 0$. 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 --+------------->(+)--> c_k
         ^     | d_k                    |               ^ (-)
         |     v                        |               |
         +--[2Tb delay]                 +--[2Tb delay]--+

Illustration for input 01001101

Initial precoder bits assumed $d_{-1} = d_0 = 0$ (so $a_{-1} = a_0 = -1$).

The decoded sequence 01001101 equals the input. Each bit is decided from its own sample, so a single error does not propagate.