Provided by: harminv_1.4.2-1build1_amd64 
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 inversion": 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 specific form for the input.
It uses a low-storage "filter-diagonalization method" (FDM), as described 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 indicating one or more frequency ranges to search, and outputs the modes that
it extracts from the data. (It preferentially finds modes in the frequency range you specify, but may
sometimes find additional modes outside 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 beginning 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 frequency 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 error. 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
distinction 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 factor", 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) frequency. 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 solutions 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 below.
By default, modes with error > 0.1 and Q < 10 are automatically omitted, 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 sampling interval dt (the time between consecutive inputs). dt is by
default 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 often corresponds to a much larger density than the usual Fourier
resolution, but the resulting singularities in the system matrices are automatically removed by
harminv.
-f nf Specify a lower bound nf on the number of spectral basis functions (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 frequency/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. Defaults 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. Defaults 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)