Jim, KG4SGP just sent me an email asking about how timing recovery (timing estimation) works in the FDMDV modem. In fdmdv.c, this is performed by the rx_est_timing() function.
At the receiver, we need to “sample” the PSK modem signal at just the right time. A timing estimator looks at the received signal and works out the best time to sample. Timing estimation is a major part of a PSK demodulator. if you get the timing estimate wrong, the scatter diagram looks worse and you get more bit errors.
The basic idea is we pass the modem signal through a non-linearity. The non-linearity could be the absolute function (i.e. a rectifier), a square law, or in analog-land a diode. This strips the phase information from the signal leaving the amplitude (envelope of the original signal) bobbing up and down at the symbol rate. Turns out that the phase of this envelope signal is related to the timing offset of the PSK signal. The phase can be extracted using a single point Discrete Fourier Transform (DFT).
Here’s an example using some high school trig functions. Consider a simple BPSK signal that consists of alternating symbols ….,-1,+1,-1,+1….. Once filtered, this will look something like a sine wave at half the symbol rate, r(t)=cos(Rs(t-T)/2), where T is the timing offset, and Rs is the symbol rate. So if Rs=50 symbols/s (50 baud), r(t) would be a sine wave at 25 Hz, with some time offset T.
If we square r(t) we get s(t)=r(t)*r(t) = 0.5 + 0.5*cos(Rs(t-T)), using the trig identify cosacosb=(cos(a-b)+cos(a+b))/2. The second term of s(t) is a sine wave of frequency Rs, with phase = RsT. So if we perform a single point DFT at frequency Rs on s(t), the phase will be related to the timing offset.
That’s pretty much what happens in rx_est_timing(). We use the parallel QPSK signals, and a nice long window of modem samples to get a good estimate in the presence of noise and frequency selective fading.