INSTMOM Univariate and multivariate instantaneous moments. [A,OMEGA,UPSILON]=INSTMOM(X), where X is an analytic signal, computes the amplitude A, instantaneous *radian* frequency OMEGA, and instantaneous bandwidth assuming a unit sample rate. X is an array with the first dimension being "time". Thus, X can be a matrix of analytic signals oriented as column vectors, or a 2- or 3-D wavelet transform such as output by WAVETRANS. The output arrays are the same size as X. The instantaneous frequency, bandwidth, and curvature are defined as A = abs X OMEGA = d/dt Im ln X = d/dt arg X UPSILON = d/dt Re ln X = d/dt ln abs X where i=SQRT(-1) as usual. INSTMOM(X,DIM) computes the moments along dimension DIM, instead of the default of computing the moments along the rows (DIM=1). For details, see Lilly & Olhede (2010), "Bivariate instantaneous frequency and bandwidth", IEEE Trans. Sig. Proc., 58 (2), 591--603. _____________________________________________________________________ Sample interval INSTMOM(DT,...) uses sample interval DT, where DT is a scalar, for computing time derivatives. DT=1 is the default. _____________________________________________________________________ Joint instantaneous moments INSTMOM can also calculate the joint instananeous moments of multivariate signals, as defined in Lilly and Olhede (2010). [JA,JOMEGA,JUPSILON]=INSTMOM(X1,X2,...,XN,DIM) returns the *joint* instantaneous moments calculated across the N signals X1,X2,... XN, based on the univariate instantaneous moments along dimension DIM. The joint instantaneous amplitude JA is the root-mean-square of the component amplitudes across dimension JDIM, while JOMEGA is power- weighted average of the component instantaneous frequencies. For details and for the definition of the joint instantaneous bandwidth JUPSILON, see Lilly and Olhede (2010). [JA,JOMEGA,JUPSILON]=INSTMOM(X,DIM,JDIM) also works, where the joint instantaneous moments are calculated across dimensions JDIM of X. The joint instantaneous moments JA, JOMEGA, and JUPSILON then have the same size as X, except along dimension JDIM where they have only one entry. Note that DIM is no longer optional when JDIM is used. _____________________________________________________________________ Instantaneous curvature INSTMOM can also return the next-higher order instantaneous moment, which is more rarely encountered. [A,OMEGA,UPSILON,XI]=INSTMOM(X) returns the instantaneous curvature XI, defined as XI = d^2/dt^2 abs X / abs X + i d^2/dt^2 arg X = UPSILON^2 + d/dt UPSILON + i d/dt OMEGA Similarly [JA,JOMEGA,JUPSILON,JXI]=INSTMOM(X,DIM,JDIM) returns the joint instantaneous curvature JXI for the multivariate signal X. For details on the univariate and joint instantaneous curvature, see Lilly and Olhede (2012a), "Analysis of modulated multivariate oscillations", IEEE Trans. Sig. Proc., 60 (2), 600--612. _____________________________________________________________________ Boundary conditions The first and last points must be treated differently, as the central difference is not defined there. Three different methods can be used. INSTMOM(...,STR) specifies the method: STR= 'endpoint' (the default), 'periodic', or 'nans'. See VDIFF for details. _____________________________________________________________________ 'instmom --f' generates some sample figures. Usage: [a,om]=instmom(x); [a,om,up,xi]=instmom(dt,x); [a,om,up,xi]=instmom(dt,x,dim); [a,om,up,xi]=instmom(dt,x1,x2,x3,x4,dim); [a,om,up,xi]=instmom(dt,x,dim,jdim); __________________________________________________________________ This is part of JLAB --- type 'help jlab' for more information (C) 2007--2015 J.M. Lilly --- type 'help jlab_license' for details