MVFREQZ multivariate frequency response [S,h,PDC,COH,DTF,DC,pCOH,dDTF,ffDTF,pCOH2,PDCF,coh,GGC,Af,GPDC] = mvfreqz(B,A,C,N,Fs) [...] = mvfreqz(B,A,C,f,Fs) INPUT: ======= A, B multivariate polynomials defining the transfer function a0*Y(n) = b0*X(n) + b1*X(n-1) + ... + bq*X(n-q) - a1*Y(n-1) - ... - ap*Y(:,n-p) A=[a0,a1,a2,...,ap] and B=[b0,b1,b2,...,bq] must be matrices of size Mx((p+1)*M) and Mx((q+1)*M), respectively. C is the covariance of the input noise X (i.e. D'*D if D is the mixing matrix) N if scalar, N is the number of frequencies if N is a vector, N are the designated frequencies. Fs sampling rate [default 2*pi] A,B,C and D can by obtained from a multivariate time series through the following commands: [AR,RC,PE] = mvar(Y,P); M = size(AR,1); % number of channels A = [eye(M),-AR]; B = eye(M); C = PE(:,M*P+1:M*(P+1)); N number of frequencies for computing the spectrum Fs sampling rate in [Hz] f vector of frequencies (in [Hz]) OUTPUT: ======= S power spectrum h transfer functions, abs(h.^2) is the non-normalized DTF  PDC partial directed coherence  DC directed coupling COH coherency (complex coherence)  DTF directed transfer function pCOH partial coherence dDTF direct Directed Transfer function ffDTF full frequency Directed Transfer Function pCOH2 partial coherence - alternative method GGC a modified version of Geweke's Granger Causality [Geweke 1982] !!! it uses a Multivariate AR model, and computes the bivariate GGC as in [Bressler et al 2007]. This is not the same as using bivariate AR models and GGC as in [Bressler et al 2007] Af Frequency transform of A(z), abs(Af.^2) is the non-normalized PDC  PDCF Partial Directed Coherence Factor  GPDC Generalized Partial Directed Coherence [9,10] see also: FREQZ, MVFILTER, MVAR REFERENCE(S):  H. Liang et al. Neurocomputing, 32-33, pp.891-896, 2000.  L.A. Baccala and K. Samashima, Biol. Cybern. 84,463-474, 2001.  A. Korzeniewska, et al. Journal of Neuroscience Methods, 125, 195-207, 2003.  Piotr J. Franaszczuk, Ph.D. and Gregory K. Bergey, M.D. Fast Algorithm for Computation of Partial Coherences From Vector Autoregressive Model Coefficients World Congress 2000, Chicago.  Nolte G, Bai O, Wheaton L, Mari Z, Vorbach S, Hallett M. Identifying true brain interaction from EEG data using the imaginary part of coherency. Clin Neurophysiol. 2004 Oct;115(10):2292-307.  Schlogl A., Supp G. Analyzing event-related EEG data with multivariate autoregressive parameters. (Eds.) C. Neuper and W. Klimesch, Progress in Brain Research: Event-related Dynamics of Brain Oscillations. Analysis of dynamics of brain oscillations: methodological advances. Elsevier.  Bressler S.L., Richter C.G., Chen Y., Ding M. (2007) Cortical fuctional network organization from autoregressive modelling of loal field potential oscillations. Statistics in Medicine, doi: 10.1002/sim.2935  Geweke J., 1982 J.Am.Stat.Assoc., 77, 304-313.  L.A. Baccala, D.Y. Takahashi, K. Sameshima. (2006) Generalized Partial Directed Coherence. Submitted to XVI Congresso Brasileiro de Automatica, Salvador, Bahia.  L.A. Baccala, D.Y. Takahashi, K. Sameshima. Computer Intensive Testing for the Influence Between Time Series, Eds. B. Schelter, M. Winterhalder, J. Timmer: Handbook of Time Series Analysis - Recent Theoretical Developments and Applications Wiley, p.413, 2006.  M. Eichler On the evaluation of informatino flow in multivariate systems by the directed transfer function Biol. Cybern. 94: 469-482, 2006.