LTPDA Toolbox™

contents

Cross coherence estimates

Description

The LTPDA method ao/cohere estimates the cross-coherence of time-series signals, included in the input aos following the Welch's averaged, modified periodogram method [1]. Data are windowed prior to the estimation of the spectra, by multiplying it with a spectral window object, and can be detrended by a polinomial of time in order to reduce the impact of the border discontinuities. The window length is adjustable to shorter lenghts to reduce the spectral density uncertainties, and the percentage of subsequent window overlap can be adjusted as well.

Syntax

    
    b = cohere(a1,a2,pl)
  

a1 and a2 are the 2 aos containing the input time series to be evaluated, b is the output object and pl is an optional parameters list.

Parameters

• 'Nfft' - number of samples in each fft [default: length of input data] Notice: analyzing a single segment produces as a result an object full of 1! A string value containing the variable 'fs' can also be used, e.g., plist('Nfft', '2*fs')
• 'Win' - the window to be applied to the data to remove the discontinuities at edges of segments. [default: taken from user prefs].
  The window is described by a string with its name and, only in the case of Kaiser window, the additional parameter 'psll'.
  For instance: plist('Win', 'Kaiser', 'psll', 200).
• 'Olap' - segment percent overlap [default: -1, (taken from window function)]
• 'Order' - order of segment detrending
  • -1 - no detrending
  • 0 - subtract mean [default]
  • 1 - subtract linear fit
  • N - subtract fit of polynomial, order N
• 'Navs' - number of averages. If set, and if Nfft was set to 0 or -1, the number of points for each window will be calculated to match the request. [default: -1, not set]
• 'Times' - interval of time to evaluate the calculation on. If empty [default], it will take the whole section.
• 'Type' - type of scaling of the coherence function. Choose between:
• 'C' - Complex Coherence Sxy / sqrt(Sxx * Syy) [default]
• 'MS' - Magnitude-Squared Coherence (abs(Sxy))^2 / (Sxx * Syy)

As an alternative to setting the number of points 'Nfft' in each window, it's possible to ask for a given number of coherence estimates by setting the 'Navs' parameter, and the algorithm takes care of calculating the correct window length, according to the amount of overlap between subsequent segments.

The function makes cross-coherence estimates between the 2 input aos. If passing two identical objects or linearly combined signals, the output will be 1 at all frequencies. The same will happen if analyzing only a single window.

Algorithm

The algorithm is based in standard MATLAB's tools, as the ones used by pwelch. The standard deviation of the mean is computed as [2]

Example

Evaluation of the cross-coherence of two time-series represented by: a low frequency sinewave signal superimposed to white noise and a linear drift, and a low frequency sinewave signal at the same frequency, phase shifted and with different amplitude, superimposed to white noise.

    
    % parameters
    nsecs = 5000;
    fs = 10;
    nfft = 1000;
    
    % build first signal components
    x1 = ao(plist('waveform','sine wave','f',0.1,'A',1,'nsecs',nsecs,'fs',fs,'yunits','m')) 
    x2 = ao(plist('waveform','noise','type','normal','nsecs',nsecs,'fs',fs,'yunits','m')) 
    x3 = ao(plist('tsfcn', 't','nsecs',nsecs,'fs',fs,'yunits','m'));
    
    % add components
    x = x1 + x2 + x3;
    
    % build second signal components
    y1 = ao(plist('waveform','sine wave','f',0.1,'A',2,'nsecs',nsecs,'fs',fs,'phi',90));
    y2 =  4*ao(plist('waveform','noise','type','normal','nsecs',nsecs,'fs',fs));
    
    % add components and set units
    y = y1 + y2;
    y.setYunits('V');

    % compute coherence
    pl = plist('win','BH92','nfft',nfft, 'order',1);
    Cxy  = cohere(x,y,pl);
    
    %plot
    iplot(Cxy);
  

References

1. P.D. Welch, The Use of Fast Fourier Transform for the Estimation of Power Spectra: A Method Based on Time Averaging Over Short, Modified Periodograms, IEEE Trans. on Audio and Electroacoustics, Vol. 15, No. 2 (1967), pp. 70 - 73.
2. G.C. Carter, C.H. Knapp, A.H. Nuttall, Estimation of the Magnitude-Squared Coherence Function Via Overlapped Fast Fourier Transform Processing , IEEE Trans. on Audio and Electroacoustics, Vol. 21, No. 4 (1973), pp. 337 - 344.

Cross-spectral density estimates

Transfer function estimates

©LTP Team