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.
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
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.