**- DSP log - http://www.dsplog.com -**

Transmit spectrum with phase noise

Posted By __Krishna Sankar__ On August 28, 2012 @ 6:07 am In __Analog__ | __No Comments__

The earlier posts on phase noise discussed about phase noise in oscillators ^{[1]}, conversion of phase noise profile to jitter ^{[2]} and the impact of phase noise on the error vector magnitude (evm) ^{[3]}. This post discuss the impact of phase noise on the spectrum of the transmit waveform. A simple random QPSK modulated symbols, oversampled and passed through a root raised cosine filtering is used for the simulation.

Consider a simple system model having both phase noise and thermal noise as show below.

**Figure: System model with phase 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 The phase is Gaussian distributed with zero mean and variance radians^2 having a probability density function as, . From the post on phase noise on the error vector magnitude (evm) ^{[3]}, we know that the

Es/N0 = 40dB

Phase Noise (deg, rms) |
EVM, dB |

0 | -40.00 |

1 | -33.93 |

2 | -28.80 |

3 | -25.46 |

4 | -23.03 |

5 | -21.13 |

** **

**Table : Resultant EVM with phase noise and thermal noise**

Further, from the post on phase noise in oscillators ^{[1]}, it is also known that the phase noise will cause skirts in the spectrum of the carrier.

**Figure : Spectrum on the transmit waveform with random QPSK symbols, root raised cosine filtering (oversampling by 4), Es/N0 = 40dB and different RMS phase noise.** **Observations**As can be seen from the above plot, the phase noise elevates the out-of band emissions in the transmit spectrum (from -40dB in the no phase noise case to -21.13dB for the 5 degree rms phase) So it is important to have a lower phase noise to meet the spectral mask specification in addition to minimize the impact on error vector magnitude.

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

% Script for simulating the transmit spectrum of a QPSK % modulated symbol affected by phase noise and thermal noise % ---------------------------------------------------------- clear;close all; N = 10^5; % number of symbols os = 4; % oversampling factor Es_N0_dB = 40; phi_rms_deg_vec = [0:1:5]; % root raised cosine filter t_by_Ts = [-4:1/os:4]; beta = 0.5; ht = (sin(pi*t_by_Ts*(1-beta)) + 4*beta*t_by_Ts.*cos(pi*t_by_Ts*(1+beta)))./(pi*t_by_Ts.*(1-(4*beta*t_by_Ts).^2)); ht((length(t_by_Ts)-1)/2+1) = 1 -beta + 4*beta/pi; ht([-os/(4*beta) os/(4*beta)]+(length(t_by_Ts)-1)/2+1) = beta/sqrt(2)*((1+2/pi)*sin(pi/(4*beta))+(1-2/pi)*cos(pi/(4*beta))); ht = ht/sqrt(os); 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 % Pulse shaping s_os = [s ; zeros(os-1,length(s))]; s_os = s_os(:).'; s_os = conv(ht,s_os); s_os = s_os(1:os*N); % Thermal and Phase Noise addition n = 1/sqrt(2)*[randn(1,N*os) + j*randn(1,N*os)]; % thermal noise phi = phi_rms_deg_vec(jj)*(pi/180)*randn(1,N*os)*sqrt(os); % phase noise y = s_os.*exp(j*phi) + 10^(-Es_N0_dB(ii)/20)*n; % computing the transmit spectrum [Pxx1(jj,:) W2 ] = pwelch(y,[],[],1024,'twosided'); % matched filtering y_mf_out = conv(y,fliplr(ht)); y_mf_out = y_mf_out(length(ht):os:end); % error vector error_vec = (y_mf_out-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([-512:511]/1024,10*log10(fftshift(Pxx1))); xlabel('frequency, Hz'); ylabel('amplitude, dB'); legend('0 deg rms','1 deg rms', '2 deg rms', '3 deg rms', '4 deg rms', '5 deg rms'); title('spectrum Es/N0 = 40dB, root raised cosine filtering and different rms phase noise'); axis([-0.5 0.5 -50 5]); grid on;

Article printed from DSP log: **http://www.dsplog.com**

URL to article: **http://www.dsplog.com/2012/08/28/transmit-spectrum-phase-noise/**

URLs in this post:

[1] phase noise in oscillators: **http://www.dsplog.com/2012/06/08/oscillator-phase-noise/**

[2] conversion of phase noise profile to jitter: **http://www.dsplog.com/2012/06/22/phase-noise-psd-to-jitter/**

[3] phase noise on the error vector magnitude (evm): **http://www.dsplog.com/2012/07/09/evm-phase-noise/**

[4] click here to SUBSCRIBE : **http://www.feedburner.com/fb/a/emailverifySubmit?feedId=1348583&loc=en_US**

Click here to print.

Copyright © 2007-2012 dspLog.com. All rights reserved. This article may not be reused in any fashion without written permission from http://www.dspLog.com.