# It’s just a “Phase Noise”–So don’t miss it!

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This blog tries to explain Phase noise and its effects in OFDM based communication systems like 5G-NR.

In wireless communication systems there is notion of carrier wave (or carrier). This carrier is modulated with signal that needs to be transmitted. suppose signal that needs to be transmitted is x(t) and carrier wave is A*cos(w0(t)). In real world, carrier wave is represented as A*cos(w0(t) + phi(t)). where phi(t) is phase noise. Because of this, in practical systems, we get not only x(t) around the w0(t) but we also see side-bands or spurs.

When we see noise spectrum of an oscillator. There are regions in which the flicker noise, 1/f noise dominates and other regions where the white noise from sources such as shot noise and thermal noise dominate.

Let us understand this by using an example in Octave. Below given is Octave script that is doing following tasks:

• Generating a sine wave signal and plotting it.
• Adding white noise to the phase of the signal and plotting it.
• Adding 1/f noise (flicker noise) to the phase of the signal and plotting it.

clear all;
close all;
sigma = 1.2;
fsHz = 655360;
dt = 1/fsHz;
t = 0:dt:500*dt;
%t = 0:0.01:10;
signal0  = cos(2*pi*4*t) + sin(2*pi*4*t);
signal1 = cos(2*pi*4*t+sigma *randn(1,length(t))) + ...
sin(2*pi*4*t+sigma *randn(1,length(t)));;
% Generate 1/f noise (AWGN noise added over time)
noise = randn(1,length(t));
for i = 2 : length(t)
noise(i) = noise(i) + noise(i-1);
end
signal2 = signal0 + noise;
% signal0 = original signal
% signal1 = signal0 + white noise added to phase
% signal2 = signal0 + 1/f noise added to phase

faxis = linspace(-fsHz/2,fsHz/2,length(t));
subplot(3, 1, 1);
plot(faxis/1000,fftshift(abs(fft(signal0))),'b','linewidth',1.5);
grid on;
title('subplot-1: original signal no noise');
xlabel('Frequency (KHz)')
faxis = linspace(-fsHz/2,fsHz/2,length(t));

subplot(3, 1, 2);
plot(faxis/1000,fftshift(abs(fft(signal1))),'r','linewidth',1.5);
grid on;
title('subplot-2: signal + AWGN added to phase');
xlabel('Frequency (KHz)')
faxis = linspace(-fsHz/2,fsHz/2,length(t));

subplot(3, 1, 3);
plot(faxis/1000,fftshift(abs(fft(signal2))),'k','linewidth',1.5);
grid on;
title('subplot-3: signal + 1/f noise added to phase');
xlabel('Frequency (KHz)')
• Subplot-3 represents region where (1/f) noise dominates.Instead of a pure delta function, there is broadening of the spectrum near the carrier frequency.
• Subplot-2 represents region where white noise dominates. There is fluctuations of the spectrum far from the carrier frequency.

OFDM baseband signal model considering phase noise

Ref: R1-163984

When the mismatch of oscillator frequencies between transmitter and receiver occurs, the frequency difference implies a shift of the received signal spectrum at the baseband. In OFDM, this creates a misalignment between the bins of FFT and the peaks of the sinc pulses of the received signal. This breaks orthogonality between the subcarriers so that results in a spectral leakage between them. Each subcarrier interferes with every other (although the effect is dominant between adjacent subcarriers), and as there are many subcarriers this is a random process equivalent to Gaussian noise. Thus, this frequency offset lowers the SINR of the receiver. An OFDM receiver will need to track and compensate phase noise.

The base-band received signal in the presence of only phase noise, assumed that there is no additive white Gaussian noise (AWGN), is given as the following equation:

where the transmitted signal is multiplied by a noisy carrier exp(jθ[n]).

The received signal is passed through the FFT in order to obtain the symbol transmitted on the m-th subcarrier in the OFDM symbol as follows:

Since the first term of the right hand side in (2) (i.e., mean of exp(jθ[n]) during one OFDM symbol duration) does not depend on subcarrier index m, it is called common phase error (CPE). This term causes common phase rotation in constellations of received symbols. The CPE can be estimated from the reference signals and removed.

And the second term corresponds to the summation of the information of the other sub-carriers each multiplied by some complex number which comes from an average of phase noise with a spectral shift. The result is also a complex number that is added to each sub-carrier’s useful signal and has the appearance of Gaussian noise. It is normally known as inter-carrier interference (ICI) or loss of orthogonallity.

Hence the phase noise can have two main impacts: one is that each subcarrier can be affected by a Common Phase Error (CPE) , which appears as the multiplication of the complex channel gain equally across all subcarriers; the other is the Inter-Carrier Interference (ICI) , which results in loss of orthogonality between subcarriers assuming OFDM waveform.

The ICI due to phase noise creates a fuzzy constellation as shown in Figure below:

~Peace

Dheeraj

## One thought on “It’s just a “Phase Noise”–So don’t miss it!”

1. Rakesh Samala says:

Hi Sir,
If there is small CFO(~15HZ) prsent along with awgn noise, the constellation looks very similar to the figure above shown. So, how do we know whether it is caused by phase noise or CFO?

Comments are closed.