harminv - Online in the Cloud

PROGRAM:

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.

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