In typical wireless system simulations, there is a need to model the phase noise profile of the local oscillator. For eg, the phase noise profile of the oscillator can be of the shape described in the post on Phase Noise Power Spectral Density to Jitter. While looking around for example Matlab code, found two references [1, 2] which uses the approach of defining the phase noise profile in frequency domain, and then using ifft() to convert to the time domain samples. This post gives a brief overview of the modeling and provides an example Matlab/Octave code.

## Modeling

a) Assume a system with sampling frequency Hz and having samples. In frequency domain we can define in steps of Hz.

b) Consider a phase noise profile defined as follows :

freq |
PSD, dBc/Hz |

0 | -65 |

1kHz | -65 |

10kHz | -95 |

100kHz | -115 |

1MHz | -125 |

10MHz | -125 |

** Table : Example phase noise profile**

From the phase noise profile, using linear interpolation (in log10 of the frequency axis), to find the phase noise power spectral density for frequencies from in steps of Hz.

c) Generate a white Gaussian noise sample of length and scale it with the phase noise power spectral density

(Scaling by is to normalize the resolution bandwidth to unity)

d) Use ifft() to find the time domain samples.

e) On the rel samples obtained from step (d), take to form the time domain phase noise samples.

**Note :**

When x is small, .

## Example Matlab script

% Script for simulating the an example phase noise profile % ---------------------------------------------------------- clear all; close all; fs_Hz = 20e6; N = 10^5; nIter = 100; % phase noise profile psd_f_hz = [ 0 1e3 1e4 1e5 1e6 10e6]; psd_val_dbc_per_hz = [-65 -65 -95 -115 -125 -125]; % defining the frequency vector freq_v_Hz = [0:N/2]/N*fs_Hz; delta_f = fs_Hz/N; slope = [psd_val_dbc_per_hz(2:end) - psd_val_dbc_per_hz(1:end-1) ]./... (log10(psd_f_hz(2:end)) - log10(psd_f_hz(1:end-1))); constant = 10.^(psd_val_dbc_per_hz(1:end-1)/10).* ... (psd_f_hz(1:end-1).^(-slope/10)); integral = constant.*(psd_f_hz(2:end).^(1+slope/10) - ... psd_f_hz(1:end-1).^(1+slope/10) )./(1+slope/10); %% finding the rms jitter % finding index with slope == -10 idx = find(slope==-10); integral(idx) = constant(idx).*(log(psd_f_hz(idx+1)) - log(psd_f_hz(idx))); rms_jitter_radians = sqrt(2*integral); integrated_jitter_radians = sqrt(2*sum(integral)) % interpolating the phase noise psd values psd_ssb_dB = -Inf*ones(1,N/2+1); % from [0:N/2] for ii=1:length(psd_f_hz)-1 [tt1 fl_idx ] = (min(abs(psd_f_hz(ii) - freq_v_Hz))); [tt2 fr_idx ] = (min(abs(psd_f_hz(ii+1) - freq_v_Hz))); fvec = [freq_v_Hz(fl_idx):delta_f:freq_v_Hz(fr_idx)]; pvec = slope(ii)*log10(fvec+eps) + psd_val_dbc_per_hz(ii) - slope(ii)*log10(psd_f_hz(ii)+eps); psd_ssb_dB(fl_idx:fr_idx) = pvec; end % forming the full vector from [-N/2:-1 0:N/2-1 ]/N*fs_Hz psd_dB = -Inf*ones(1,N); psd_dB([-N/2:-1]+N/2+1) = psd_ssb_dB([N/2+1:-1:2]); psd_dB([0:N/2-1]+N/2+1) = psd_ssb_dB(1:N/2); psd_linear = 10.^(psd_dB/20); for (jj = 1:nIter) % defining frequency vector phase_noise_freq = 1/sqrt(2)*(randn(1,N) + j*randn(1,N)); phase_noise_freq_scale = N*sqrt(delta_f)*phase_noise_freq; phase_noise_freq_psd = phase_noise_freq_scale .*psd_linear; % converting to time domain phase_noise_td = ifft(fftshift(phase_noise_freq_psd)); pn_td = exp(j*(sqrt(2)*real(phase_noise_td))); % for estimating jitter and plotting pn_without_carrier = (pn_td - 1); est_jitter_pwr_radians(jj) = mean(pn_without_carrier.*conj(pn_without_carrier)); hF = 1/(N*sqrt(delta_f))*fft(pn_without_carrier,N); hFPwr(jj,:) = hF.*conj(hF); end est_integrated_jitter_radians = sqrt(mean(est_jitter_pwr_radians)); title_str = sprintf('Phase noise profile, est jitter %2.5f radians (expected %2.5f radians)', est_integrated_jitter_radians, integrated_jitter_radians); figure semilogx( [-N/2:N/2-1]/N*fs_Hz, 10*log10(fftshift(mean(hFPwr))), 'r^-' ); hold on;grid on; semilogx([0:N/2]/N*fs_Hz,psd_ssb_dB,'mp-'); semilogx(psd_f_hz,psd_val_dbc_per_hz,'bs-'); legend('est-freq-response','original','interpolated'); xlabel('freq, Hz'); ylabel('dBc/Hz'); axis([1 10e6 -140 -50]); title(title_str);

**Figure : Example phase noise profile (expected and simulated)**

## Summary

The above approach seems to allow a way to model an arbitrary phase noise power spectral density. However, the fact that this approach needs a large ifft() of length can potentially slow down the simulation.

## References

[1] Phase Noise by Alex Bar-Guy, 27 Oct 2005 (Updated 08 Dec 2005) http://www.mathworks.com/matlabcentral/fileexchange/8844-phase-noise

[2] Baseband-equivalent phase noise model, Submitted by Markus Nentwig on Dec 18 2011

http://www.dsprelated.com/showcode/246.php

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{ 13 comments… read them below or add one }

Just confused with your method to model phase noise. Could you please talk about the math behind this modeling and why this modeling is good? Thanks,

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If Ashish still needing the code for Radix-4 and if you if you can convey her, the code is the folowing:

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Innitialize variables.

t = 1:1:256;

x = sin(2*pi*0.35*t)+sin(2*pi*0.38*t);

x1 = x;

n = length(x);

t = log(n)/log(4);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Radix-4 Algorithm

for q = 1:t

L = 4^q;

r = n/L;

Lx = L/4;

rx = 4*r;

y = x;

for j = 0:Lx-1

for k = 0:r-1

a = y(j*rx + k + 1);

b = exp(-i*2*pi*j/L)*y(j*rx + r + k + 1);

c = exp(-i*2*pi*2*j/L)*y(j*rx + 2*r + k + 1);

d = exp(-i*2*pi*3*j/L)*y(j*rx + 3*r + k + 1);

t0 = a + c;

t1 = a – c;

t2 = b + d;

t3 = b – d;

x(j*r + k + 1) = t0 + t2;

x((j + Lx)*r + k + 1) = t1 – i*t3;

x((j + 2*Lx)*r + k + 1) = t0 – t2;

x((j + 3*Lx)*r + k + 1) = t1 + i*t3;

end

end

end

@Roberto: Thanks!

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a good site for knowing concepts of DSP…i’m working on a project ” SOFTWARE BASED GPS RECEIVER FOR PSUEDOLITE APPLICATIONS “…..

I am trying to change acquistion algorithm, in which i need a MATLAB code for RADIX-4 FFT and RADIX-16 FFT…could u please give the code for it????????

@Ashish: Thanks. Sorry I do not have the code.

great work Krishna, thank you for sharing!