First order digital PLL for tracking constant phase offset

Considering a typical scenario where there might exist a small phase offset between local oscillator between the transmitter and receiver.

tx_rx_phase_offset.gif

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.

first order pll

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.-')

phiHat_n_alpha

Figure 3: Convergence of for two values.

Reference:

[1] Synchronization Techniques for Digital Receivers (Applications of Communications Theory) by Umberto Mengali

14 thoughts on “First order digital PLL for tracking constant phase offset

  1. i tried this an worked. however, i would appreciate help in understanding what happened here…

    % demodulating circuit
    xHat = 2*real(x_n(ii)>0) – 1 ;
    phiHatT = angle(conj(xHat)*x_n(ii)) – angle(conj(xHat)*xn(ii));
    if phiHatT pi
    phiHatT = -2*pi + phiHatT
    end

    1. % demodulating circuit
      phiHatT = angle(x_n(ii)) – angle(xn(ii));
      if phiHatT pi
      phiHatT = -2*pi + phiHatT
      end

      works too…

  2. i got it to work, but i still dont understand quite well what i said before. are you projecting an X axis or something like that? i really want to know. thank you.

    look:

    clc
    clear all;

    % sine wave
    Tp = 1/60;
    T = Tp/128;
    Fp = 1/Tp;
    Fs = 1/T;
    w = 2*pi*Fp;
    t = [0:T:809*T];
    xn = sin(w*t);
    %al simular con e^jwt siempre quedara una parte representada en coord imaginarias, por lo k habria k hacer unas modificaciones para que se detecte cuando no sea error de fase y sea causa de la misma onda.
    %sin(wt+phi) = (e^jwt-e^-jwt)*e^phi/(2j)

    % introducing a phase offset
    phiDeg = -100;
    phiRad = phiDeg*pi/180;
    yn = xn*exp(j*phiRad);
    % se puede representar un desfase cualquiera como e^j(wt+phi) ?

    % first order pll
    alpha = 0.05;
    phiHat = 0;
    x_n = zeros(length(t));

    for ii = 1:length(t)
    x_n(ii) = yn(ii)*exp(-j*phiHat);

    % demodulating circuit
    xHat = 2*real(x_n(ii)>0) – 1 ;
    phiHatT = angle(conj(xHat)*x_n(ii)) – angle(conj(xHat)*xn(ii));

    % accumulation
    phiHat = alpha*phiHatT + phiHat;
    % dumping variables for plot
    phiHatDump(ii) = phiHat;
    end
    %plot(t,xn);
    %hold on;
    %plot(t,yn);
    plot(t,phiHatDump*180/pi,’r.-‘);

  3. Im trying to use this for a sine wave in the input. problem is it only works for 90 and -90. what can i do about it? i dont quite understand how you get the angle with
    % demodulating circuit
    xHat = 2*real(yn(ii)>0) -1 ;
    phiHatT = angle(conj(xHat)*yn(ii));

      1. Hi,

        Thanks fr the reply Krishna. In the previous comment by Venkat, he describes quite similar scenario to my program. How can I check the accurancy of the code?

        I have another question, I am working on low SNR environment. Does these phase lock loop work on low snr region?

        Thanks in advance,

  4. Hello Krishna,
    I have referred your example on first order PLL for constant phase tracking.It was very useful.
    https://dsplog.com/2007/06/10/first-order-digital-pll-for-tracking-constant-phase-offset/
    But in the example a complex carrier is being used at the transmitter and receiver.Practically , when I use a cosine carrier at Tx. and cosine,sine carriers at Rx.(as in costas loop) , can the same loop filter be used ?
    I have tried to simulate the above situation in the following script but was unable to estimate the phase. Please help me……….

    % MODULATION

    clc;
    clear all;
    close all;

    [b,a] = butter(1,0.0156,’low’); % low pass filter to remove
    nterm_i = 0; % double frequency component
    dterm_i = 0;
    nterm_q = 0;
    dterm_q = 0;

    n_sym = 10; % number of symbols
    fs = 12.8e6;
    t = 0:1/fs:100e-6 – 1/fs;
    fc = fs/8; % carrier freuency
    theta = 70*pi/180; % phase offset
    r = 0.1e6; % symbol rate
    oversamp = fs/r;
    sym = randint(n_sym,1)*2-1;
    in = 0;
    for ind=1:1:n_sym
    tmp(1:oversamp) = sym(ind);
    in = [in tmp];
    end
    tx = in(2:end);
    tx_carrier = cos(2*pi*fc*t + theta);

    tx_out = tx.*tx_carrier;

    % first order pll
    alpha = 0.05;
    phiHat = 0;

    for ii = 1:1:length(t)

    % Remove carrier
    rx_i(ii) = tx_out(ii)*cos(2*pi*fc*t(ii) + phiHat);
    rx_q(ii) = tx_out(ii)*sin(2*pi*fc*t(ii) + phiHat);

    % low-pass IIR filter for I-channel
    iir_in_i(ii) = rx_i(ii);
    iir_out_i(ii) = b(1)*iir_in_i(ii)+ nterm_i + dterm_i;
    nterm_i = b(2)*iir_in_i(ii);
    dterm_i = a(2)*iir_out_i(ii);

    % low-pass IIR filter for Q-channel
    iir_in_q(ii) = rx_q(ii);
    iir_out_q(ii) = b(1)*iir_in_q(ii)+ nterm_q + dterm_q;
    nterm_q = b(2)*iir_in_q(ii);
    dterm_q = a(2)*iir_out_q(ii);

    % demodulating circuit
    xHat = 2*(iir_out_i(ii)>0) -1 ; % symbol estimate
    phiHatT =angle(conj(xHat)*rx_i(ii)); % phase error estimate angle(iir_out_i(ii) + i*iir_out_q(ii));%

    % accumulation
    phiHat = alpha*phiHatT + phiHat; % phase accumulator output
    % dumping variables for plot
    phiHatDump(ii) = phiHat;
    end

    plot(phiHatDump*180/pi,’-‘);

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