Considering a typical scenario where there might exist a small phase offset between local oscillator between the transmitter and receiver.
Figure 1 : Transmitter receiver with constant phase offset
In such cases, it might be desirable to estimate and track the phase offset such that the performance of the receiver does not degrade.
A simple first order digital phase locked loop for tracking the constat phase offset can be as
Assuming that the transmitter signal gets rotated by a constant phase , the received signal . In a simple no-noise case, assuming that the phase offset is small (and the signal gets decoded correctly), the estimate of phase offset is,
Typically, a first order phase locked loop which converges to is used for facilitating synchronous demodulation.
Figure 2 :
First order digital phase locked loop (PLL)
(adapted from Fig 5.7 of [Mengali])
The estimate from each sampling instant is accumulated to form the estimate . This estimated phase is removed from the received samples to generate . The parameteris a non-zero positive constant in the range controls the rate of convergence of the loop. Higher value of indicates faster convergence, but is more prone to noise effects. Lower value of is less noisy, but results in slower convergence.
Assuming , the phase estimate at the output of the filter is
Substituting, , then
A simple Matlab code to simulate this can be as follows:
% random +/-1 BPSK source xn = 2*(rand(1,1000) >0.5) - 1; % introducing a phase offset of 20 degrees phiDeg = 20; phiRad = phiDeg*pi/180; yn = xn*exp(j*phiRad); % first order pll alpha = 0.01; phiHat = 0; for ii = 1:length(yn) yn(ii) = yn(ii)*exp(-j*phiHat); % demodulating circuit xHat = 2*real(yn(ii)>0) -1 ; phiHatT = angle(conj(xHat)*yn(ii)); % accumulation phiHat = alpha*phiHatT + phiHat; % dumping variables for plot phiHatDump(ii) = phiHat; end plot(phiHatDump*180/pi,'r.-')
Figure 3: Convergence of for two values.