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.**

**Reference:**

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{ 14 comments… read them below or add one }

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

% demodulating circuit

phiHatT = angle(x_n(ii)) – angle(xn(ii));

if phiHatT pi

phiHatT = -2*pi + phiHatT

end

works too…

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.-’);

it still wont work for angles above 90 degrees….

@Ananias: There are two assumptions

a) phase is constant

b) slicing of the received symbols is needed to remove the effect of modulation

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));

@Ananias: Was the phase constant in the sinusoidal input?

Hi Krishna,

Can this code be used to recover the carrier ? Carrier signal is in 1GHz range.

Thank you

@Dinu: For recovering the carrier, we might need a second order loop. Will write about it soon.

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,

@Dinu: This piece of code does only phase tracking and not frequency tracking. I have not tried checking the performance in the low SNR region.

thanks

Hello Krishna,

I have referred your example on first order PLL for constant phase tracking.It was very useful.

http://www.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,’-’);

@venkat: Does your transmit signal contain phase information? Did the code work in no noise case?