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Instantaneous Amplitude, Phase and Frequency

computeInstantaneousAttributes.Rd

Computes, for each column of a matrix of modes, the analytic signal and its instantaneous amplitude, continuous phase and frequency.

Usage

computeInstantaneousAttributes(modes, samplingInterval)

Arguments

modes

Finite double matrix, samples by modes, with at least three rows; it may have zero columns. Attributes other than dim do not reach the outputs.

samplingInterval

Sampling interval \(\Delta t\) in seconds, a finite number greater than zero.

Details

The analytic signal \(z = c + i \mathcal{H} c\) of each column is computed with the discrete Fourier transform, multiplying each bin by \(1 + \mathrm{sgn}(\omega)\): positive frequencies are doubled, negative ones removed, and the zero bin and, for an even number of samples, the Nyquist bin keep the factor 1, so \(\mathrm{Re}\, z = c\). The transform implies the periodic extension of the record. The amplitude is \(a = |z|\). Within each run of samples with \(|z| > 0\), the phase starts at the principal value of \(\arg z\) and adds the increments \(\arg(z_{n+1} \bar z_n)\) taken in \((-\pi, \pi]\); a zero of \(z\) leaves its sample undefined (NA) and the next run starts again at its principal value. The frequency is $$f_n = \frac{\phi_{n+1} - \phi_{n-1}}{4 \pi \Delta t}$$ in Hz, the central difference of \(\omega = \phi'\); it is NA where one of the three phases is, and each border repeats its neighbour. These local undefined values raise no condition.

Value

A list of three matrices, samples by modes:

amplitude

Instantaneous amplitude, in the units of the modes.

phase

Continuous phase in radians, NA at zeros of the analytic signal.

frequency

Instantaneous frequency in Hz, NA where the phase is undefined.

See also

buildHilbertSpectrum, decomposeEMD, estimateMaskFrequency

Examples

SamplingInterval <- 1 / 1000
Time <- seq_len(1000) * SamplingInterval
# Three tones of 2, 24 and 288 Hz as columns.
Tones <- cbind(cos(2 * pi * 2 * Time), 0.25 * cos(2 * pi * 24 * Time),
               0.0625 * cos(2 * pi * 288 * Time))
OUT <- computeInstantaneousAttributes(Tones,
                                      samplingInterval = SamplingInterval)
apply(OUT$amplitude, 2, range)
#>      [,1] [,2]   [,3]
#> [1,]    1 0.25 0.0625
#> [2,]    1 0.25 0.0625
apply(OUT$frequency, 2, range)
#>      [,1] [,2] [,3]
#> [1,]    2   24  288
#> [2,]    2   24  288