MSPEC is the jSpectral module of jLab.

  MSPEC  Multitaper power and cross spectra.
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    MSPEC implements spectral and cross-spectral analysis using the 
    multitaper method for real or complex-valued data. 
 
    MSPEC is to be run after calling SLEPTAP to compute the multitapers.
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    One-sided power spectrum
 
    [F,S]=MSPEC(X,PSI) returns the one-sided power spectrum of X at 
    positive frequencies using data tapers PSI. 
   
    Input:   
        X  --  M x N matrix containing N length M time series
      PSI  --  M x K matrix of K data tapers
  
    Output:
        F  --  [M/2] nonnegative *radian* frequencies
        S  --  [M/2] x N one-sided power spectrum matrix
                 
    In the above, [M/2] means: M/2 if M is even, and (M+1)/2 if M is odd. 
    See FOURIER for the calculation of the Fourier frequencies.
 
    The spectra matrices are averages over the K "eigenspectra" computed 
    with each of the K tapers, as discussed in Park et al. JGR 1987.
 
    The one-sided spectrum S is normalized such that its sum over all 
    frequencies F, (1/2/pi)*SUM(S,1)*DF where DF is the frequency 
    increment, approximates the signal variance; see discussion below.
 
    MSPEC(...,'detrend') detrends the data before computing the spectra.
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    Cross-spectra of real-valued data
    
    MSPEC can be used to compute the cross-spectrum of two real-valued
    time series or sets of time series.
 
    [F,SXX,SYY,SXY]=MSPEC(X,Y,PSI);   --- For cross-spectra
         
    Input:
        X  --  M x N matrix containing N length M time series
        Y  --  M x N matrix containing N length M time series
      PSI  --  M x K matrix of K data tapers
  
    Output:
        F  --  M/2 nonnegative frequencies
      SXX  --  [M/2] x N one-sided spectra of X
      SYY  --  [M/2] x N one-sided spectra of Y
      SXY  --  [M/2] x N one-sided cross spectra of X and Y 
 
    See TWOSPECPLOT for plotting SXX and SYY simultaneously.
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    Rotary spectra of complex-valued data
 
    MSPEC can also the so called "rotary spectra" of complex-valued 
    time series or sets of time series.
 
    [F,SPP,SNN,SPN]=MSPEC(Z,PSI);   --- For rotary spectra of Z=X+iY
 
    Input:   
        Z  --  M x N matrix containing N length M time series
      PSI  --  M x K matrix of K data tapers
    
    Output:   
         F  --  M/2 nonnegative frequencies
       SPP  --  [M/2] x N positively rotating power spectrum matrix
       SNN  --  [M/2] x N negatively rotating power spectrum matrix  
       SPN  --  [M/2] x N rotary cross spectral matrix      
 
    Note that the rotary spectra are defined such that SXX+SYY=SPP+SNN.
   
    The rotary spectra SPP and SNN are normalized such that the sum of SPP
    over all frequencies plus that of SNN approximates the variance of Z. 
 
    See TWOSPECPLOT for plotting SPP and SNN simultaneously.
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    Sample rate
 
    [F,S]=MSPEC(DT,...) specifies the sample interval to be used in the
    calculation of the frequency array F. DT defaults to unity.
 
    Spectral values depend linearly upon the sample rate in order that the 
    integral of the spectra over frequency approximate the variance.
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    Periodogram
 
    MPSEC can be used to form the naive spectral estimator, known as the
    periodogram. Although this is not generally a good way to estimate the
    spectrum, it can be useful as a comparision.
 
    MSPEC(X,[]) or MSPEC(X,Y,[]) with PSI empty uses the default, or boxcar
    taper, normalized to unit energy. This returns the periodogram.  
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    Normalizations
 
    By default, MSPEC uses *radian* frequency as in cos(f t).  Optionally
    MSPEC(,...,'cyclic') will use *cyclic* frequency, as in cos(2 pi f t).
 
    MSPEC is normalized to approximately recover the time series variance. 
    For the MSPEC periodogram, this recovery is exact, although the 
    expressions are complicated somewhat by the use of one-sided spectra.  
 
    Real-valued data
 
    [F,S]=MSPEC(DT,X,[]) where X is a real-valued time series of length M
    recovers the variance of X, STD(X,1).^2, as follows:
 
      (1/2/pi)*(F(2)-F(1))*SUM(S(2:end))               -- M odd
      (1/2/pi)*(F(2)-F(1))*(SUM(S(2:end-1))+S(end)/2)  -- M even
 
    Note that the zero frequency is omitted in the summation, and for even 
    time series length, the power at the Nyquist S(end) must be divided by 
    two to avoid double-counting by the one-sided spectrum.  The "1" in the 
    argument of STD forces STD to use an N rather than N-1 normalization. 
 
    Complex-valued data
 
    [F,SPP,SNN]=MSPEC(DT,Z,[]) where Z is a complex-valued time series of 
    length M recovers the variance of Z, STD(Z,1).^2, as follows:
 
      (1/2/pi)*(F(2)-F(1))*(SUM(SPP(2:end))+SUM(SNN(2:end)))   -- M odd
      (1/2/pi)*(F(2)-F(1))*(SUM(SPP(2:end))+SUM(SNN(2:end-1))) -- M even
 
    Again the modification for even M prevents the power at the Nyquist
    from being double-counted.  This modification is necessary because the 
    negative rotary spectrum duplicates the Nyquist when M is even.  
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    Cross-spectra of complex-valued data
 
    To compute the cross-spectra of two complex-valued time series or sets 
    of time series Z1 and Z2, run MSPEC repeatedly.
 
    [F,SP1P1,SP2P2,SP1P2]=MSPEC(Z1,Z2,PSI);  
    [F,SN1N1,SN2N2,SN1N2]=MSPEC(CONJ(Z1),CONJ(Z2),PSI);  
 
    The first call returns the spectra and cross-spectra of Z1 and Z2 at
    positive frequencies, while the second returns their spectra and cross-
    spectra at negative frequencies.  Finally
 
    [F,SP1P1,SN2N2,SP1N2]=MSPEC(Z1,CONJ(Z2),PSI);  
 
    returns the cross-spectra between the positive rotary components of Z1 
    and the negative components of Z2.
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    Adaptive spectra
 
    MSPEC(...,LAMBDA,'adaptive'), where LAMBDA contains the eigenvalues of
    the tapers as computed by SLEPTAP, alternately uses the "adaptive"
    multitaper method of Thomson (1982).
  
    This implementation follows that of Park et al. (1987a), JGR.
 
    For cross-spectra or for rotary spectra, the weights appearing in the
    adaptive spectra are derived for the total spectrum of each signal 
    compoment, that is for SXX+SYY or SPP+SNN as appropriate.  Then the
    separate spectra and co-spectra are computed using identical weights.
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    Cell array input / output
 
    MSPEC generates cell array output given cell array input.
 
    Let's say one has P different time series, X1, X2,..., XP.  Put these 
    into a cell array X{1}=X1, X{2}=X2, ..., X{P}=XP, and then use
    '[psi,lambda]=sleptap(cellength(x))' to make a cell array of tapers.
 
    [F,S]=MSPEC(X,PSI) then returns cell arrays F and S corresponding 
    to the Fourier frequencies and spectra of the P arrays.  
 
    The other argument forms given above also work.  In particular, 
    specifiying the sample time through MPSEC(DT,...) works, with DT either
    a scalar or an array of the same length as the cell array X.
 
    The spectra can then be plotted with CELLPLOT(F,S), or TWOSPECPLOT for
    a pair of output spectra.
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    Parallelization
 
    MSPEC(..., 'parallel') when the input fields X, X and Y, or Z are cell
    arrays, parallelizes the spectral estimation by looping over the cells
    with a PARFOR loop.  This requires Matlab's Parallel Computing Toolbox.
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    Example 
 
    The example at the top of this help file shows clockwise (left) and 
    counterclockwise (right) rotary spectra from moored current meter  
    measurements of the ocean currents in the Labrador Sea.
    
    The periodogram is in gray, and blue and red are multitaper spectra 
    with P=4 and P=32, respectively.  The local Coriolis frequency is 
    marked with a dashed line.  Tidal and inertial peaks are apparent.
 
    The main point of this figure is to show that increasing P increases
    the degree of frequency-domain smoothing.   
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    'mspec --t' runs some tests.
    'mspec --f' generates the above sample figure from Bravo mooring data.
 
    See also: SLEPTAP, HERMFUN, MTRANS, MSVD, TWOSPECPLOT.
 
    Usage   [f,s]=mspec(x,psi);    
            [f,s]=mspec(dt,x,psi);     
            [f,spp,snn,spn]=mspec(z,psi);     
            [f,sxx,syy,sxy]=mspec(x,y,psi);
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    This is part of JLAB --- type 'help jlab' for more information
    (C) 2000--2015 J.M. Lilly --- type 'help jlab_license' for details        

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