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HARMINV(1) harminv HARMINV(1) NAME harminv - extract mode frequencies from time-series data SYNOPSIS harminv [OPTION]... [freq-min-freq-max]... DESCRIPTION harminv is a program designed to solve the problem of "harmonic inver- sion": given a time series consisting of a sum of sinusoids ("modes"), extract their frequencies and amplitudes. It can also handle the case of exponentially-decaying sinusoids, in which case it extracts their decay rates as well. harminv is often able to achieve much greater accuracy and robustness than Fourier-transform methods, essentially because it assumes a spe- cific form for the input. It uses a low-storage "filter-diagonalization method" (FDM), as de- scribed in V. A. Mandelshtam and H. S. Taylor, "Harmonic inversion of time signals," J. Chem. Phys. 107, 6756 (1997). See also erratum, ibid 109, 4128 (1998). INPUT harminv reads in a sequence of whitespace-separated real or complex numbers from standard input, as well as command-line arguments indicat- ing one or more frequency ranges to search, and outputs the modes that it extracts from the data. (It preferentially finds modes in the fre- quency range you specify, but may sometimes find additional modes out- side of that range.) The data should correspond to equally-spaced time intervals, but there is no constraint on the number of points. Complex numbers in the input should be expressed in the format RE+IMi (no whitespace). Otherwise, whitespace is ignored. Also, comments be- ginning with "#" and extending to the end of the line are ignored. A typical invocation is something like harminv -t 0.02 1-5 < input.dat which reads a sequence of samples, spaced at 0.02 time intervals (in ms, say, corresponding to 50 kHz), and searches for modes in the fre- quency range 1-5 kHz. (See below on units.) OUTPUT harminv writes six comma-delimited columns to standard output, one line for each mode: frequency, decay constant, Q, amplitude, phase, and er- ror. Each mode corresponds to a function of the form: amplitude * exp[-i (2 pi frequency t - phase) - decay t] Here, i is sqrt(-1), t is the time (see below for units), and the other parameters in the output columns are: frequency The frequency of the mode. If you don't recognize that from the expression above, you should recall Euler's formula: exp(i x) = cos(x) + i sin(x). Note that for complex data, there is a dis- tinction between positive and negative frequencies. decay constant The exponential decay constant, indicated by decay in the above formula. The inverse of this is often called the "lifetime" of the mode. The "half-life" is ln(2)/decay. Q A conventional, dimensionless expression of the decay lifetime: Q = pi |frequency| / decay. Q, which stands for "quality fac- tor", is the number of periods for the "energy" in the mode (the squared amplitude) to decay by exp(-2 pi). Equivalently, if you look at the power spectrum (|Fourier transform|^2), 1/Q is the fractional width of the peak at half maximum. amplitude The (real, positive) amplitude of the sinusoids. The amplitude (and phase) information generally seems to be less accurate than the frequency and decay constant. phase The phase shift (in radians) of the sinusoids, as given by the formula above. error A crude estimate of the relative error in the (complex) fre- quency. This is not really an error bar, however, so you should treat it more as a figure of merit (smaller is better) for each mode. SPURIOUS MODES Typically, harminv will find a number of spurious solutions in addition to the desired solutions, especially if your data are noisy. Such so- lutions are characterized by large errors, small amplitudes, and/or small Q (large decay rates / broad linewidths). You can omit these from the output by the error/Q/amplitude screening options defined be- low. By default, modes with error > 0.1 and Q < 10 are automatically omit- ted, but it is likely that you will need to set stricter limits. UNITS The frequency (and decay) values, both input and output, are specified in units of 1/time, where the units of time are determined by the sam- pling interval dt (the time between consecutive inputs). dt is by de- fault 1, unless you specify it with the -t dt option. In other words, pick some units (e.g. ms in the example above) and use them to express the time step. Then, be consistent and use the inverse of those units (e.g. kHz = 1/ms) for frequency. Note that the frequency is the usual 1/period definition; it is not the angular frequency. OPTIONS -h Display help on the command-line options and usage. -V Print the version number and copyright info for harminv. -v Enable verbose output, printed to standard output as comment lines (starting with a "#" character). Also, any "#" comments in the input are echoed to the output. -T Specify period-ranges instead of frequency-ranges on the command line (in units of time corresponding to those specified by -t). The output is still frequency and not period, however. -w Specify angular frequencies instead of frequencies, and output angular frequency instead of frequency. (Angular frequency is frequency multiplied by 2 pi). -n Flip the sign of the frequency (and phase) convention used in harminv. (The sign of the frequency is only important if you have complex-valued input data, in which case the positive and negative frequency amplitudes can differ.) -t dt Specify the sampling interval dt; this determines the units of time used throughout the input and output. Defaults to 1.0. -d d Specify the spectral "density" d to search for modes, where a density of 1 indicates the usual Fourier resolution. That is, the number of basis functions (which sets an upper bound on the number of modes) is given by d times (freq-max - freq-min) times dt times the number of samples in your dataset. A maximum of 300 is used, however, to prevent the matrices from getting too big (you can force a larger number with -f, below). Note that the frequency resolution of the outputs is not limited by the spectral density, and can generally be much greater than the Fourier resolution. The density determines how many modes, at most, to search for, and in some sense is the density with which the bandwidth is initially "searched" for modes. The default density is 0.0, which means that the number of basis functions is determined by -f (which defaults to 100). This of- ten corresponds to a much larger density than the usual Fourier resolution, but the resulting singularities in the system matri- ces are automatically removed by harminv. -f nf Specify a lower bound nf on the number of spectral basis func- tions (defaults to 100), setting a lower bound on the number of modes to search for. This option is often a more convenient way to specify the number of basis functions than the -d option, above, which is why it is the default. -f also allows you to employ more than 300 basis functions, but careful: the computation time scales as O(N nf) + O(nf^3), where N is the number of samples, and very large matrices can also have degraded accuracy. -s sort Specify how the outputs are sorted, where sort is one of fre- quency/error/Q/decay/amplitude. (Only the first character of sort matters.) All sorts are in ascending order. The default is to sort by frequency. -e err Omit any modes with error (see above) greater than err times the largest error among the computed modes. Defaults to no limit. -E err Omit any modes with error (see above) greater than err. De- faults to 0.1. -F Omit any modes with frequencies outside the specified range. (Such modes are not necessarily spurious, however.) -a amp Omit any modes with amplitude (see above) less than amp times the largest amplitude among the computed modes. Defaults to no limit. -A amp Omit any modes with amplitude (see above) less than amp. De- faults to no limit. -Q q Omit any modes with |Q| (see above) less than q. Defaults to 10. BUGS Send bug reports to S. G. Johnson, stevenj@alum.mit.edu. AUTHORS Written by Steven G. Johnson. Copyright (c) 2005 by the Massachusetts Institute of Technology. harminv June 4, 2005 HARMINV(1)
NAME | SYNOPSIS | DESCRIPTION | INPUT | OUTPUT | SPURIOUS MODES | UNITS | OPTIONS | BUGS | AUTHORS
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