spectrum1d(1) compute auto- [and cross- ] spectra from one [or two] timeseries.


spectrum1d [ x[y]file ] -Ssegment_size] [ -C[xycnpago] ] [ -Ddt ] [ -Nname_stem ] [ -V ] [ -W ] [ -bi[s][n] ] [ -bo[s][n] ]


spectrum1d reads X [and Y] values from the first [and second] columns on standard input [or x[y]file]. These values are treated as timeseries X(t) [Y(t)] sampled at equal intervals spaced dt units apart. There may be any number of lines of input. spectrum1d will create file[s] containing auto- [and cross- ] spectral density estimates by Welch's method of ensemble ' averaging of multiple overlapped windows, using standard error estimates from Bendat and Piersol.

The output files have 3 columns: f or w, p, and e. f or w is the frequency or wavelength, p is the spectral density estimate, and e is the one standard deviation error bar size. These files are named based on name_stem. If the -C option is used, up to eight files are created; otherwise only one (xpower) is written. The files (which are ASCII unless -bo is set) are as follows:

Power spectral density of X(t). Units of X * X * dt.
Power spectral density of Y(t). Units of Y * Y * dt.
Power spectral density of the coherent output. Units same as ypower.
Power spectral density of the noise output. Units same as ypower.
Gain spectrum, or modulus of the transfer function. Units of (Y / X).
Phase spectrum, or phase of the transfer function. Units are radians.
Admittance spectrum, or real part of the transfer function. Units of (Y / X).
(Squared) coherency spectrum, or linear correlation coefficient as a function of frequency. Dimensionless number in [0, 1]. The Signal-to-Noise-Ratio (SNR) is coh / (1 - coh). SNR = 1 when coh = 0.5.


ASCII (or binary, see -bi) file holding X(t) [Y(t)] samples in the first 1 [or 2] columns. If no file is specified, spectrum1d will read from standard input.
segment_size is a radix-2 number of samples per window for ensemble averaging. The smallest frequency estimated is 1.0/(segment_size * dt), while the largest is 1.0/(2 * dt). One standard error in power spectral density is approximately 1.0 / sqrt(n_data / segment_size), so if segment_size = 256, you need 25,600 data to get a one standard error bar of 10%. Cross-spectral error bars are larger and more complicated, being a function also of the coherency.


Read the first two columns of input as samples of two timeseries, X(t) and Y(t).
  Consider Y(t) to be the output and X(t) the input in a linear system with noise. Estimate the optimum f requency response function by least squares, such that the noise output is minimized and the coherent outpu t and the noise output are uncorrelated. Optionally specify up to 8 letters from the set { x y c n p a g o } in any order to create only those output files instead of the default [all]. x = xpower, y = ypower, c = cpower, n = npower, p = phase, a = admit, g = gain, o = coh.
dt Set the spacing between samples in the timeseries [Default = 1].
name_stem Supply the name stem to be used for output files [Default = "spectrum"].
Selects verbose mode, which will send progress reports to stderr [Default runs "silently"].
Write Wavelength rather than frequency in column 1 of the output file[s] [Default = frequency, (cycles / dt)].
Selects binary input. Append s for single precision [Default is double]. Append n for the number of columns in the binary file(s). [Default is 2 input columns].
Selects binary output. Append s for single precision [Default is double].


Suppose data.g is gravity data in mGal, sampled every 1.5 km. To write its power spectrum, in mGal**2-km, to the file data.xpower, try

spectrum1d data.g -S256 -D1.5 -Ndata

Suppose in addition to data.g you have data.t, which is topography in meters sampled at the same points as data.g. To estimate various features of the transfer function, considering data.t as input and data.g as output, try

paste data.t data.g | spectrum1d -S256 -D1.5 -Ndata -C


Bendat, J. S., and A. G. Piersol, 1986, Random Data, 2nd revised ed., John Wiley & Sons.
Welch, P. D., 1967, "The use of Fast Fourier Transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms", IEEE Transactions on Audio and Electroacoustics, Vol AU-15, No 2.