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Frequency uncertainty of the Fourier transform (FFT) for a record length

Source: R/fft_uncertainty.R
fft_frequency_uncertainty.Rd

Calculator for the frequency uncertainty of a Fourier based spectrum. The practical resolution of an FFT is the bin spacing, the inverse of the record length (\(\Delta f = 1/T\)). The smallest frequency uncertainty that is possible according to the Heisenberg-Gabor uncertainty principle (\(\Delta t \Delta f \ge 1/(4\pi)\)) with \(\Delta t = T\) is \(\Delta f = 1/(4\pi T)\), a factor \(4\pi\) smaller. The relative uncertainty at a frequency f is \(\Delta f / f = 1/(f T)\). For a record of N samples with sampling frequency fs, \(T = N / f_s\).

It is the Fourier counterpart of wavelet_uncertainty, which gives the uncertainty of the wavelet transform. Units are free: with the record length in metres (or kyr) the frequencies are in cycles per metre (or per kyr).

Usage

fft_frequency_uncertainty(
  T_length = NULL,
  N = NULL,
  fs = 1,
  f = NULL,
  period = NULL
)

Arguments

T_length

Length of the record \(T\) in time or depth (numeric, > 0; a vector gives one row per value), in any unit (e.g. metres or kyr); the frequencies of the output are in cycles per that unit. If NULL, it is computed as N / fs, so either T_length or N must be given. Default=NULL.

N

Number of samples of the record (numeric, > 0); only used when T_length is NULL, together with fs, to compute T_length = N / fs. Default=NULL.

fs

Sampling frequency, i.e. samples per unit of time or depth (numeric, > 0; the inverse of the sampling interval); only used with N. Default=1.

f

Frequency or frequencies of interest in cycles per unit of T_length (numeric vector, > 0). If given, the output also contains the relative uncertainties and the period ranges at these frequencies. NULL gives only the absolute uncertainties. Default=NULL.

period

Period(s) of interest in the unit of T_length (numeric vector, > 0), an alternative to f (converted with f = 1 / period); ignored when f is given. Default=NULL.

Value

A data frame with the record length, the practical uncertainty df_practical, the Heisenberg-Gabor limit df_gabor, their ratio and, when f or period is given, the frequency, the relative uncertainties (rel_practical, rel_gabor) and the period ranges implied by f - df and f + df for both definitions.

References

Gabor, D. (1946). Theory of communication. Part 1: The analysis of information. Journal of the Institution of Electrical Engineers - Part III 93, 429-441.

Examples

# 2 s record, 100 Hz: practical 0.5 Hz, Heisenberg-Gabor limit 0.040 Hz, 0.5 percent
fft_frequency_uncertainty(T_length = 2, f = 100)
#>   T_length df_practical   df_gabor    ratio   f rel_practical    rel_gabor
#> 1        2          0.5 0.03978874 12.56637 100         0.005 0.0003978874
#>   period period_min_practical period_max_practical period_min_gabor
#> 1   0.01          0.009950249           0.01005025      0.009996023
#>   period_max_gabor
#> 1       0.01000398
# 1000 samples every 0.5 kyr and the 405 kyr eccentricity cycle
fft_frequency_uncertainty(N = 1000, fs = 1 / 0.5, period = 405)
#>   T_length df_practical     df_gabor    ratio           f rel_practical
#> 1      500        0.002 0.0001591549 12.56637 0.002469136          0.81
#>    rel_gabor period period_min_practical period_max_practical period_min_gabor
#> 1 0.06445775    405             223.7569             2131.579         380.4754
#>   period_max_gabor
#> 1          432.904