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In particular, I have a signal coming from a twisted pair of arbitrary length with a differential encoded biphase line code. What should be a correct approach to timing recovery in this case?

Currently I am trying to implement a baud-rate timing recovery algorithm using a Mueller&Muller TED (timing error detector), loop filter (average over 120 symbols) and counter-divider with adjustable initial value as a digital VCO. The source code for current implementation of the algorithm in python is here. For TED I use the following timing function:

$$ \frac{(a_k^2 - 5)}{16}(a_k x_{k-1} - a_{k-1} x_k ) $$

For simulation purposes I have sampled versions of the signal in half-duplex mode (i.e. without echo). Signal samples look like this for two different channel lengths:

signal from short line

signal from short line

For these signals the timing recovery algorithm converges to the phase near signal's zero crossings. This does not allow to correctly determine the original bit sequence (e.g. "+1 -1 +1 +1 -1 +1 ...") and I would like it to converge to the maximum eye opening, i.e. somewhere near the peaks.

I've managed to get this by adding a $(1 - aD)$ filter, where $a$ is $0,75$ for the first signal sample and $1,325$ for the second one. But because a channel has an arbitrary length, the shape of a signal is unknown during design phase and this filter with fixed coefficients seems like a bad solution. Moreover, adding this filter was purely an empirical solution and is not backed up by any math.

I am very new to timing recovery and would appreciate any advice.

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  • $\begingroup$ I've posted what I've come to as an answer. But I will appreciate validation of its correctness and/or additional advice or alternative solutions. $\endgroup$
    – megasplash
    Jan 13 at 15:11
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A good option for timing function for such signal happens to be the canonical $(a_k x_{k-1} - a_{k-1} x_k )$ with additional condition to calculate its value only when $a_k$ equals $a_{k-1}$ (i.e. when "1" is preceded by "1" or "-1" is preceded by "-1"). As a result, no additional digital filter is needed.

The reason for this condition came from observations of S-curve for this timing function (see M. Rice «Chapter 8 Symbol Timing Synchronization» in "Digital communications: a discrete-time approach"). Having timing function (timing error function) $e(k)$ written as

$$e(k)=a(k) x(kT+\tau) - a(k-1) x((k-1)T+\tau),$$

where $T$ - is a symbol period, and $\tau$ - is a phase error,

the S-curve is an expected value of $e(k,\tau)$:

$$g(\tau)=E\{e(k,\tau)\}.$$

For the signal samples the proper S-curve with a single zero crossing was obtained only for "11", "00", "1100" and "0011" symbol patterns. The shorter symbol pattern (i.e. $a_k=a_{k-1}$) was selected to simplify further implementation.

Here is an example plot of a signal sample and values of $e(k)$ as a function of a phase (which vaguely represents $g(\tau)$), for different symbol patterns.

Timing function and phase

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