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# spectrum1d

spectrum1d - Compute auto- [and cross- ] spectra from one [or two] time-series

## Synopsis

spectrum1d [ table ] -Ssegment_size] [ -C[xycnpago] ] [ -Ddt ] [ -L[h|m] ] [ -N[+]name_stem ] [ -b[i|o][ncol][type][w][+L|+B] ] [ -W ] [ -b[i|o][ncol][type][w][+L|+B] ] [ -f[i|o]colinfo ] [ -g[a]x|y|d|X|Y|D|[col]z[+|-]gap[u] ] [ -h[i|o][n][+c][+d][+rremark][+rtitle] ] [ -icols[l][sscale][ooffset][,...] ]

Note: No space is allowed between the option flag and the associated arguments.

## Description

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:

name_stem.xpower
  Power spectral density of X(t). Units of X * X * dt.
name_stem.ypower
  Power spectral density of Y(t). Units of Y * Y * dt.
name_stem.cpower
  Power spectral density of the coherent output. Units same as ypower.
name_stem.npower
  Power spectral density of the noise output. Units same as ypower.
name_stem.gain
  Gain spectrum, or modulus of the transfer function. Units of (Y / X).
name_stem.phase
  Phase spectrum, or phase of the transfer function. Units are radians.
name_stem.admit
  Admittance spectrum, or real part of the transfer function. Units of (Y / X).
name_stem.coh
  (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.

## Required Arguments

-Ssegment_size]
  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.

## Optional Arguments

table
  One or more ASCII (or binary, see -bi) files holding X(t) [Y(t)] samples in the first 1 [or 2] columns. If no files are specified, spectrum1d will read from standard input.
-C[xycnpago]
  Read the first two columns of input as samples of two time-series, 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 frequency response function by least squares, such that the noise output is minimized and the coherent output 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.
-Ddt
  dt Set the spacing between samples in the time-series [Default = 1].
-L
  Leave trend alone. By default, a linear trend will be removed prior to the transform. Alternatively, append m to just remove the mean value or h to remove the mid-value.
-N[+]name_stem
  name_stem Supply the name stem to be used for output files [Default = “spectrum”]. To place all the computed output columns in a single table, use -N+.

-V[level] (more ...)
  Select verbosity level [c].
-W
  Write Wavelength rather than frequency in column 1 of the output file[s] [Default = frequency, (cycles / dt)].

-bi[ncols][type] (more ...)
  Select binary input. [Default is 2 input columns].

-bo[ncols][type] (more ...)
  Select binary output. [Default is 2 output columns].

-f[i|o]colinfo (more ...)
  Specify data types of input and/or output columns.

-g[a]x|y|d|X|Y|D|[col]z[+|-]gap[u] (more ...)
  Determine data gaps and line breaks.

-h[i|o][n][+c][+d][+rremark][+rtitle] (more ...)
  Skip or produce header record(s).

-icols[l][sscale][ooffset][,...] (more ...)
  Select input columns.

-^
  Print a short message about the syntax of the command, then exits (NOTE: on Windows use just -).
+
  Print an extensive usage (help) message, including the explanation of any module-specific option (but not the GMT common options), then exits.
-?
  Print a complete usage (help) message, including the explanation of options, then exits.
--version
  Print GMT version and exit.
--show-datadir
  Print full path to GMT share directory and exit.

## Ascii Format Precision

The ASCII output formats of numerical data are controlled by parameters in your gmt.conf file. Longitude and latitude are formatted according to FORMAT_GEO_OUT, whereas other values are formatted according to FORMAT_FLOAT_OUT. Be aware that the format in effect can lead to loss of precision in the output, which can lead to various problems downstream. If you find the output is not written with enough precision, consider switching to binary output (-bo if available) or specify more decimals using the FORMAT_FLOAT_OUT setting.

## Examples

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, use

>     gmt 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, use

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

## See Also

gmt, grdfft

## References

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.

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