HARMINV(1)harminvHARMINV(1)NAMEharminv - extract mode frequencies from time-series data
SYNOPSISharminv [OPTION]... [freq-min-freq-max]...
DESCRIPTIONharminv 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
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).
INPUTharminv 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
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 fre‐
quency range 1-5 kHz. (See below on units.)
OUTPUTharminv 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 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
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 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
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 sys‐
tem matrices 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.
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)
