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EVM with phase noise

Posted By Krishna Sankar On July 9, 2012 @ 6:30 am In Analog | 9 Comments

The previous post on phase noise discussed about finding the root mean square phase noise for a given phase noise profile [1]. In this post let us discuss about the impact of phase noise on the error vector magnitude (evm) of a transmit symbol.

## Error Vector Magnitude due to constant phase offset

Consider a system model introducing a constant phase offset$\phi$ and thermal noise $n$ as shown in the figure.

Figure : System model with phase noise and thermal noise

The received symbol $y$is,

$y=e^{j\phi}x + n$,

where

$\phi$is the phase distortion in radians,

$x$ is the transmit symbol and

$n$ is the contribution due to thermal noise

Expanding into real and imaginary components,

$\underbrace{$\begin{array}{rr}y_{re}\\y_{im}\end{array}$}_Y=\underbrace{$\begin{array}{rr}\cos\phi & -\sin\phi\\\sin\phi &\cos\phi\end{array}$}_\Phi\underbrace{$\begin{array}{rr}x_{re}\\x_{im}\end{array}$}_X+\underbrac{$\begin{array}{rr}n_{re}\\n_{im}\end{array}$}_N$.

Representing in matrix algebra,

$Y=\Phi X+N$

The power of the error vector is,

$err^2=$\Phi X+N-X$^T$\Phi X+N-X$$.

Finding the average error over many realizations,

$\begin{array}{lll}E\{err^2\}&=&E\{$\Phi X+N-X$^T$\Phi X+N-X$\}\\&=&E\{X^T\Phi^T\Phi X+X^T\Phi^TN-X^T\Phi^TX+\\&&N^T\Phi X +N^TN -N^TX\\ &&-X^T\Phi X -X^TN +X^TX\}\end{array}$.

The individual terms can be simplified as,

i) $E\{X^T\Phi^T\Phi X\}=E\{X^TX\}=E_s$

as $\Phi^T\Phi=I$.

ii) $E\{X^T\Phi^TN\}=E\{N^T\Phi X\}=E\{N^TX\}=E\{X^TN\}=0$

as $X$ and$N$ are uncorrelated.

iii) $\begin{array}{lll}E\{-X^T\Phi^TX -X^T\Phi X\}&=&-E\{X^T$$\Phi^T+\Phi$$X\}\\&=&-E\{X^T\cos\phi X\}\\&=&-E_s\cos\phi\end{array}$.

iv) $E\{N^TN\}=N_0$, the variance of the noise.

Applying (i), (ii) , (iii) and (iv), the error term simplifies to

$\begin{array}{lll}E\{err^2\}&=&2E_s -2E_s\cos\phi+N_0\\&=&\frac{N_0}{E_s}+2-2\cos\phi\end{array}$.

## EVM due to random phase offset

The above equation derives the evm when the system is affected by a constant phase offset $\phi$.   Assume that the phase $\phi$is Gaussian distributed with zero mean and variance $\phi_{rms}^2$ radians^2 having a probability density function as,

$p(\phi)=\frac{1}{\sqrt{2\pi\phi^2_{rms}}}e^{-\frac{\phi^2}{2\phi^2_{rms}}}$.

The conditional error power for a given phase  is,

$\begin{array}{lll}evm^2$$\phi$$&=&\frac{N_0}{E_s}+2-2\cos\phi\end{array}$.

Computing the average over all realization of phase,

$\begin{array}{lll}evm^2$$\phi$$&=&\frac{N_0}{E_s}+2-2\int_{-\infty}^{\infty}\cos\phi p$$\phi$$d\phi\\&=&\frac{N_0}{E_s}+2-2\frac{1}{\sqrt{2\pi \phi^2_{rms}}}\int_{-\infty}^{\infty}\cos\phi e^{-\frac{\phi^2}{2\phi^2_{rms}}}d\phi\end{array}$.

The integral term is,
$\frac{1}{\sqrt{2\pi \phi^2_{rms}}}\int_{-\infty}^{\infty}\cos\phi e^{-\frac{\phi^2}{2\phi^2_{rms}}}d\phi=e^{-\frac{\phi^2_{rms}}{2}}$

(Note : proof will be discussed in another post)

Then the error vector power is,

$\Large{\begin{array}{lll}evm^2&=&\frac{N_0}{E_s}+2-2e^{-\frac{\phi^2_{rms}}{2}}\end{array}}$

and the error vector magnitude is,

$\Large{\begin{array}{lll}evm&=&\sqrt{\frac{N_0}{E_s}+2-2e^{-\frac{\phi^2_{rms}}{2}}\end{array}$

Using Taylor series, and assuming that the $\phi_{rms}^2$ is small,

$e^x = 1 + x + \frac{x^2}{2!} + ...$,

$\Large{\begin{array}{lll}evm&\simeq&\sqrt{\frac{N_0}{E_s}+\phi^2_{rms}}\end{array}$

Figure : Example constellation plot (Es/N0=30dB, $\phi_{rms}$=5 degrees)

## Matlab/Octave example

Attached script computes the evm of a QPSK modulated symbol versus Es/N0 for different values of rms phase noise.

% Script for simulating the error vector magnitude (evm) of a QPSK
% modulated symbol affected by phase noise and thermal noise
% ----------------------------------------------------------

clear;close all;
N = 10^5 % number of bits or symbols

Es_N0_dB = [15:3:40]; % multiple Eb/N0 values
phi_rms_deg_vec = [0:1:5];

for ii = 1:length(Es_N0_dB)
for jj = 1:length(phi_rms_deg_vec)
% Transmitter
ip_re = rand(1,N)>0.5; % generating 0,1 with equal probability
ip_im = rand(1,N)>0.5; % generating 0,1 with equal probability
s = 1/sqrt(2)*(2*ip_re-1 + j*(2*ip_im-1)); % QPSK modulation

n = 1/sqrt(2)*[randn(1,N) + j*randn(1,N)]; % thermal noise
phi = phi_rms_deg_vec(jj)*(pi/180)*randn(1,N); % phase noise

y = s.*exp(j*phi) + 10^(-Es_N0_dB(ii)/20)*n;

% error vector
error_vec = (y-s);

evm(ii,jj)  = error_vec*error_vec';
theory_evm(ii,jj) =  10^(-Es_N0_dB(ii)/10) + 2 - 2*exp(-(phi_rms_deg_vec(jj)*pi/180).^2/2);
end
end

figure;
plot(Es_N0_dB,10*log10((evm/N)),'s-');
hold on;grid on
plot(Es_N0_dB,10*log10(theory_evm),'+-');
xlabel('Es/N0,dB');
ylabel('error vector magnitude, dB');
title('EVM vs Es/N0 for different phase noise rms values');
legend('0deg rms','1deg rms','2deg rms','3deg rms','4deg rms','5deg rms')

Figure : EVM vs Es/N0 for different values of rms phase noise

## Summary

As a quick rule of thumb, for a system  with rms phase noise of 1 degree, the evm due to phase noise alone is -35.16dB and rises by 6dB per octave (or 20dB per decade).

The phase noise profile used in this simulations assumes a Gaussian distributed flat spectrum, which is not the case in typical phase noise profiles. The EVM with a classical phase noise profile will be discussed in another post.

## References

[1] Georgiadis, A.; , “Gain, phase imbalance, and phase noise effects on error vector magnitude,” Vehicular Technology, IEEE Transactions on , vol.53, no.2, pp. 443- 449, March 2004
doi: 10.1109/TVT.2004.823477
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1275708&isnumber=28551 [2]