Correlative coding; minimizing error in duobinary

Correlative (partial response) coding introduces a controlled, known amount of ISI between successive symbols so that the data can be sent at rate $R_b$ through the minimum Nyquist bandwidth $R_b/2$ using realizable filters. The receiver removes the known ISI. The most common type is duobinary signalling.

Duobinary signalling

"Duobinary" means doubling the transmission capacity of a straight binary system. The binary input $b_k$ is mapped to $a_k = \pm 1$ and passed through a delay-and-add circuit followed by an ideal low-pass filter:

$$c_k = a_k + a_{k-1}$$

a_k ---+------------>(+)--> [ideal LPF  ] --> c(t)
       |              ^     [|f|<1/2Tb  ]
       +--> [delay Tb]+

Transfer function: the delay-and-add has $1 + e^{-j2\pi fT_b}$; with the ideal Nyquist filter $H_N(f) = 1$ for $|f| \le 1/(2T_b)$:

$$\begin{aligned} H_I(f) &= \left(1 + e^{-j2\pi fT_b}\right)H_N(f) \\ &= \left(e^{j\pi fT_b} + e^{-j\pi fT_b}\right)e^{-j\pi fT_b} \\ &= 2\cos(\pi fT_b)\,e^{-j\pi fT_b}, \quad |f| \le \frac{1}{2T_b} \end{aligned}$$

and zero elsewhere. The magnitude is a half cosine that falls smoothly to zero at $f = 1/(2T_b)$, so it is much easier to approximate than the brick-wall filter.

Impulse response: sum of two sinc pulses one bit apart:

$$h_I(t) = \frac{\sin(\pi t/T_b)}{\pi t/T_b} + \frac{\sin[\pi(t-T_b)/T_b]}{\pi(t-T_b)/T_b} = \frac{T_b^2\,\sin(\pi t/T_b)}{\pi t\,(T_b - t)}$$

$h_I(t)$ equals 1 at $t = 0$ and $t = T_b$ and is zero at all other sampling instants; its tails decay as $1/t^2$, so it is less sensitive to timing errors.

Output levels: $c_k \in \{-2, 0, +2\}$. Detection without precoding: $\hat a_k = c_k - \hat a_{k-1}$.

 |H(f)|
  2 +---.
    |     '.
    |       '.
    |         '.
  0 +-----------+----> f
    0        1/(2Tb)

Minimizing error in duobinary coding

The main error source is error propagation: since each decision uses the previous decision, one error causes further errors. It is minimized by precoding: $d_k = b_k \oplus d_{k-1}$ at the transmitter. Then $c_k = 0$ means $b_k = 1$ and $c_k = \pm 2$ means $b_k = 0$, so each bit is decoded from its own sample (threshold $|c_k| = 1$) and an error does not spread. Proper thresholds and noise filtering also help.