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Harmonic Inversion of Time Signals by the Filter Diagonalization Method (FDM), implemented by Steven G. Johnson, Massachusetts Institute of Technology.

Harminv is a free program (and accompanying library) to solve the problem of "harmonic inversion." Given a discrete, finite-length signal that consists of a sum of finitely-many sinusoids (possibly exponentially decaying), it determines the frequencies, decay constants, amplitudes, and phases of those sinusoids.

It can, in principle, provide much better accuracy than
straightforward FFT based methods, essentially because it assumes a
specific form for the signal. (Fourier transforms, in contrast, attempt
to represent *any* data as a sum of sinusoidal components.)

We use a low-storage **"filter diagonalization method"**
(FDM) for finding the sinusoids near a given frequency interval,
described in:

- V. A. Mandelshtam and H. S. Taylor, "Harmonic inversion of time signals," J. Chem. Phys., vol. 107, no. 17, p. 6756-6769 (Nov. 1 1997). See also erratum, ibid, vol. 109, no. 10, p. 4128 (Sep. 8 1998).

This kind of spectral analysis has wide applications in many areas of
physics and engineering, as well as other fields. For example, it could
be used to extract the vibrational or "eigen" modes of a system from its
response to some stimulus, and also their rates of decay in dissipative
systems. FDM has been applied to analyze, e.g., NMR experimental data
and numerical simulations of quantum mechanics or electromagnetism.
In general, it is useful when you know on physical grounds that your
system consists of a **small number of decaying & oscillating
modes in the bandwidth of interest,** plus a limited amount of
noise, and is **not appropriate to analyze completely arbitrary
waveforms.**

The latest version is Harminv 1.4.1, which can be downloaded in source-code form at:

What's new in each version is described in the Harminv release notes. Harminv is distributed under
the GNU GPL and comes
with **NO WARRANTY** (see the license for more details).
Development sources can be found on GitHub:

It would be courteous of you to **cite Harminv** and its
author in any publication for which you find it useful, in addition to
citing a Mandelshtam reference (either the one above or the review
article below).

To install Harminv, please see our installation instructions.

If you use Debian GNU/Linux or Ubuntu, you can use the Debian package of harminv, packaged by Loic Le Guyader.

A Python interface to Harminv was developed by Aaron O'Leary: pharminv.

Please file bug reports or feature requests as harminv Github issues.

Most people will use Harminv via the stand-alone `harminv`

program as described in its manual. To
briefly summarize, it takes a sequence of numbers (real or complex) from
standard input and a range of frequencies to search and outputs the
frequencies it finds.

It is also possible to call Harminv as library from a C/C++ program.

The input for harminv should just be a list of numbers (real or complex), one per line, as described in the harminv man page.

You can use the program `sines`

, in the harminv source
directory, to test harminv and to generate example inputs. The sines
program generates a signal consisting of a sum of decaying sinuoids with
specified complex frequencies. For example,

`./sines 0.1+0.01i 0.08+0.001i`

generates 10000 data points consisting of a signal with complex frequencies 0.1+0.01i and 0.08+0.001i, with amplitudes 1 and 2 respectively, sampled at time intervals dt=1.0. If we input this data into harminv, it should be able to extract these frequencies, decay rates, and amplitudes.

`./sines 0.1+0.01i 0.08+0.001i | harminv 0.05-0.15`

The output should be something like:

```
frequency, decay constant, Q, amplitude, phase, error
0.08, 1.000000e-03, 251.327, 2, 3.14159, 1.064964e-16
0.1, 1.000000e-02, 31.4159, 1, -4.31228e-15, 2.265265e-15
```

as expected. Note that we have to pass harminv a range of frequencies
to search, here 0.05-0.15, which shouldn't be too large and should
normally not include 0. In most cases, one would also specify the
sampling interval to harminv via `harminv -t <dt>`

, but
in this case we don't need to because `-t 1.0`

is the
default.

Run `./sines -h`

to get more options.

Essentially, FDM works by considering the time-series to be the autocorrelation function of a fictitious dynamical system, such that the problem of finding the frequencies and decay constants is re-expressed as the problem of finding the eigenvalues of the complex-symmetric time-evolution operator of this system. The key point is that, if you are only interested in frequencies within a known band-limited region, the matrix elements of this operator can be expressed purely in terms of Fourier transforms (or, really, z transforms) of your time-series. Then, one can simply diagonalize a small matrix (size proportional to the bandwidth and the number of frequencies) to find the desired result.

In general, for M data points and J frequencies, the time required is O(M J + J^3). The main point of the algorithm is not so much speed, however, but the effective solution of a very ill-conditioned fitting problem. (Even closely-spaced frequencies and/or weak decay rates can be resolved much more reliably by FDM than by straightforward fits of the data or its spectrum.)

See also:

Rongqing Chen and Hua Guo, "Efficient calculation of matrix elements in low storage filter diagonalization," J. Chem. Phys., vol. 111, no. 2, p. 464-471(Jul. 8 1999).

Michael R. Wall and Daniel Neuhauser, "Extraction, through filter-diagonalization, of general quantum eigenvalues or classical normal mode frequencies from a small number of residues or a short-time segment of a signal. I. Theory and application to a quantum-dynamics model," J. Chem. Phys., 102, no. 20, p. 8011-8022 (May 22 1995).

J. Chen and V. A. Mandelshtam, "Multiscale filter diagonalization method for spectral analysis of noisy data with nonlocalized features," J. Chem. Phys. 112 (10), 4429-4437 (2000).

V. A. Mandelshtam, "FDM: the filter diagonalization method for data processing in NMR experiments," Progress in Nuclear Magnetic Resonance Spectroscopy 38, 159-196 (2001). (

*Review article.*)V. A. Mandelshtam, "On harmonic inversion of cross-correlation functions by the filter diagonalization method," J. Theoretical and Computational Chemistry 2 (4), 497-505 (2003).