Learn SDR
Harvey Mudd College
Lesson 18 — Symbol Timing Recovery and Synchronization
Even if the clocks in the transmitter and receiver happened to have exactly the same frequency, they will inevitably have an arbitrary phase offset, which means that sampling at the receiver won’t line up with the symbol times on the transmitter. So, after using the automatic gain controller to get the signal level right and the frequency-locked loop to perform the gross frequency correction, we now have to correct for the slowly drifting phase offset, interpolate between samples to sample effectively at the moment when the no ISI property of the raised-cosine pulses allows us to measure the current pulse without contamination from the other pulses, and then correct for an absolute phase offset.
Strategy
The primary problem we will address in this lesson is determining the proper timing of the symbol. Imagine that we use four samples per symbol, so we sample at four equally spaced times during the symbol pulse. Because clocks drift, the odds that we will have one of these samples at the proper time are vanishingly small. We need to interpolate between samples to “read” the symbol at the optimum time. How can we do this?
The strategy we will employ takes full advantage of the Nyquist-Shannon sampling theorem. Recall the gist of this important insight: provided that we sample a bandwidth-limited signal at or above its Nyquist frequency—meaning that we take at least two samples per cycle of the maximum frequency that is present in the signal—the discrete samples have all the information needed to perfectly reconstruct the continuous signal. Amazing but true!
So, how do we use this theorem to our advantage here? We will use a strategy that allows us to precompute a number of filters that will interpolate at many places in the symbol and use an error signal to select the best filter to use. Even better, the filters will apply a root-raised-cosine filter so that the output of the filter is the time-shifted and corrected signal that most closely corresponds to the symbol that was original transmitted.
To develop an error signal we can use to home in on the correct timing, consider the eye diagram shown in Fig. 1. Precisely at the time marked 0, the intersymbol interference disappears and only the values +1 and –1 will be measured. Unfortunately, we may be sampling a bit before this time or a bit after.
Figure 1 — Schematic illustration of the paths in an eye diagram for binary phase-shift keying. To keep bandwidth modest, each symbol extends many periods before and after the symbol we wish to transmit, which leads to many different paths for transition from the previous symbol at either –1 or +1 to the next (also at either –1 or +1).
Consider what happens if we sample a bit too early, as illustrated in Fig. 2. The product of the signal value \(x\) and its time derivative \(\dot{x}\) (the slope) is positive for both transition from –1 to +1 and from +1 to –1. If we take the average of \(x \dot{x}\) across all paths, we will get a positive value that indicates we should shift our timing a bit later.
Figure 2 — When we sample too early (at the time indicated by the vertical dashed line), we can learn something from the two sloped paths that represent transitions from +1 to –1 and from –1 to +1. The product of the signal value \(x\) and its time derivative \(\dot{x}\) (the slope) is positive for both types of transition.
We can make a similar deduction for the case when we sample a bit too late, as illustrated in Fig. 3.
Figure 3 — When we sample too late (at the time indicated by the vertical dashed line), the two sloped paths again agree: the product \(x \dot{x}\) is negative for both types of transition.
So, we will take \begin{equation} e = \langle x \dot{x} \rangle \end{equation} to yield an error signal we can use to adjust the sample timing, where the angle brackets indicate a time average. For this signal, the roughly horizontal paths that represent two consecutive symbols of the same value contribute nothing on average, but the two diagonal paths produce a signal that grows larger the farther from the correct timing we are and is positive when we need to move later and negative when we need to move earlier. Note, however, that we need data that has transitions. Without them, the error signal disappears.
It is not clear to me why \(\langle \mathrm{sign}(x) \dot{x} \rangle\) should be linear with timing error, however, since the slope along the diagonal paths is roughly constant.
Armed with this error signal, we now face several challenges:
- How do you compute the derivative with sparse samples?
- How do you interpolate?
- How do you use the error signal to move closer to the right timing?
As is our custom, we will use convolution by digital filters to manage the first two of these tasks. First, we will apply a root-raised-cosine filter to the raw data \(r\) to produce the signal \(x\): \begin{equation} x = r * p_{\rm\scriptsize RRC} \end{equation}
To get the derivative, we could use a derivative filter
\begin{equation} \dot{x} = x * d = (r * p_{\rm\scriptsize RRC}) * d = r * (p_{\rm\scriptsize RRC} * d) \end{equation}
To interpolate, we leverage the Nyquist-Shannon sampling theorem and upsample the signal by spacing each real value in \(r\) with \(N - 1 = 31\) zeros and then passing the result through an appropriate low-pass filter to produce the smoothed output. In fact, we’ll generate a bank of \(N\) filters, each designed to interpolate at a different place between the original samples of \(r\). This is called a polyphase filter bank.
Loop settings of the Symbol Sync block
| Parameter | Value | Why |
|---|---|---|
| Timing error detector | Signal times Slope (ML) | the \(x \dot{x}\) detector described above; works for BPSK and QPSK |
| Samples per symbol | sps = 16 |
|
| Expected TED gain | ted_gain = 1.1*alpha/sps (0.034 here) |
see below |
| Loop bandwidth | 0.02 | normalized to the symbol rate; the loop settles in about 200 symbols (3 ms here) |
| Damping factor | 1.0 | critically damped |
| Maximum deviation | 0.1 | the clock period may only wander by 0.1 samples from sps; the real clocks differ by parts per million, so this is generous. (The default of 1.5 lets the period estimate wander far more than any real clock error.) |
| Output samples per symbol | 1 | one clean sample per symbol out |
| Interpolating resampler | Polyphase Filterbank, MF with rcc_taps |
the 32-arm root-raised-cosine filter bank does the matched filtering and the interpolation |
Expected TED gain. The loop filter is designed assuming the error detector produces ted_gain units of error per sample of timing offset. Measuring the detector’s S-curve in a simulation for root-raised-cosine pulses gives, to about 10%,
\begin{equation}
\texttt{ted_gain} \approx 1.1\,\alpha\,A^2 / \texttt{sps}
\end{equation}
where \(A\) is the signal amplitude going in (the AGC just before this block sets it to its reference, 1.0). Like the FLL bandwidth, this is an expression in the flowgraph (1.1*alpha/sps), so it stays right when you change alpha, sps or the sample rate. It holds for sps of 8 and up (at sps = 4 the constant drops to about 0.8); QPSK is about 6% lower, which doesn’t matter. GNU Radio’s default of 1.0 is 30 times too large for this flowgraph: the loop then has 30 times less gain than designed, which makes it about 5 times slower and badly under-damped (it rings for hundreds of symbols). (The GNU Radio examples use a loop bandwidth of 0.045; with the gain set correctly, 0.02 gives the same settling time with less timing jitter, so that is what the flowgraphs use.)
The Skip Head block before the Symbol Sync throws away the first second of samples so the FLL has already locked before the timing loop starts.
Flowgraphs
Running: after FLL, Symbol Sync and Costas loop the BPSK constellation (right) is two clean points:
All GNURadio flowgraphs are at https://github.com/gallicchio/learnSDR