The previous post on phase noise discussed about finding the root mean square phase noise for a given phase noise profile. 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 and thermal noise as shown in the figure.

**Figure : System model with phase noise and thermal noise**

The received symbol is,

,

where

is the phase distortion in radians,

is the transmit symbol and

is the contribution due to thermal noise

Expanding into real and imaginary components,

.

Representing in matrix algebra,

The power of the error vector is,

.

Finding the average error over many realizations,

.

The individual terms can be simplified as,

i)

as .

ii)

as and are uncorrelated.

iii) .

iv) , the variance of the noise.

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

.

## EVM due to random phase offset

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

.

The conditional error power for a given phase is,

.

Computing the average over all realization of phase,

.

The integral term is,

(Note : proof will be discussed in another post)

Then the **error vector power** is,

and the **error vector magnitude** is,

Using Taylor series, and assuming that the is small,

,

**Figure : Example constellation plot (Es/N0=30dB, =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 % Noise addition 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

D id you like this article? Make sure that you do not miss a new article
by subscribing to RSS feed OR subscribing to e-mail newsletter.
* Note: Subscribing via e-mail entitles you to download the free e-Book on BER of BPSK/QPSK/16QAM/16PSK in AWGN.*

{ 0 comments… add one now }