BANDPOWER

Purpose:

Computes the mean squared power of a series within a frequency range.

Syntax:

BANDPOWER(x, fs, frange)

Alternate Syntax:

BANDPOWER(Pxx, f, frange, "psdflag")

Returns:

A scalar or array, the series power over the frequency range.

Example:

W1: 1..100

W2: {bandpower(w1)}

W3: {sum(powspec(w1))}

W4: {mean(w1 * w1)}

W2 == 3383.5

W3 == 3383.5

W4 == 3383.5

W2, W3 and W4 compute the mean squared power of the series in W1.

W2 and W3 perform identical calculations in the frequency domain.

W4 performs the mean squared time domain calculation.

This form of BANDPOWER computes the sum of the power spectrum computed by POWSPEC. From Parseval's theorem, the sum of the power spectrum terms equals the mean of the series squared,  i.e.:

sum(powspec(x)) == mean(x*x)

Example:

W1: 1..100

W2: {bandpower(w1, {0.0, 0.5})}

W3: {sum(powspec(hamming(w1, 3)))}

W2 == 2779.693386

W3 == 2779.693386

Computes the series power over the entire frequency range. In this case, BANDPOWER multiplies the input series by a HAMMING window corrected for the mean squared amplitude. BANDPOWER returns the same result as summing the power spectrum as computed by POWSPEC.

Example:

W1: 1..100

W2: {bandpower(w1, rate(w1), {0.0, 0.5})}

W3: {sum(powspec(hamming(w1, 3)))}

W2 == 2779.693386

W3 == 2779.693386

Same as above, except the sample rate is explicit.

Example:

W1: 1..100

W2: {bandpower(w1, {0.1, 0.3})}

W2 == 0.624722

Computes the series power of W1 over the frequency range 0.1 Hz to 0.3 Hz.

Example:

W1: gsin(1000, 1/1000, 20)^2

W2: gsqrwave(10000, 1/10000, 100)

W3: {bandpower(W1, {30, 50})}

W4: {bandpower(W2, {1000, 2000})}

W5: ravel(w1, w2)

W6: {bandpower(W5, ravel({30, 50}, {1000, 2000}))}

W3 == 0.125000

W4 == 0.005386

W6 == {{0.12500, 0.005386}}

W1 and W2 contain two different series with different sample rates.

W3 computes the series power of W1 between 30 Hz and 50 Hz.

W4 computes the series power of W2 between 1000 Hz and 2000 Hz.

W5 combines the data in W1 and W2 into a single two column series.

W6 computes the series power of each column of W5, producing the same result as the individual computations in W3 and W4.

Example:

W1: seedrand(1);gsin(1000, 0.001, 100)^2+gnorm(1000, 0.001)*0.2

W2: psd(hamming(w1, 3))

W3: {bandpower(w1, {150, 300})}

W4: {bandpower(w2, {150, 300}, "psd")}

W3 == 0.144707

W4 == 0.144707

W1 contains a squared 100 Hz sine wave sampled at 10 kHz with additive normally distributed random noise. The random number generated is set with a seed of 1 for consistent repeated results.

W2 computes the frequency domain power spectrum of W1. The data is multiplied by a HAMMING window with mean squared amplitude correction.

W3 computes the series power of W1 between 150 Hz and 300 Hz.

W4 computes the series power of W1 by summing the power spectrum in W2 between 150 Hz and 300 Hz.

W3 and W4 produce the same result.

Example:

W1: seedrand(1);gsin(1000, 0.001, 100)^2+gnorm(1000, 0.001)*0.2

W2: powspec(hamming(w1, 3))

W3: {bandpower(w1, rate(w1), {150, 300})}

W4: {bandpower(w2, xvals(w2), {150, 300}, "powspec")}

W3 == 0.144707

W4 == 0.144707

Same as above, except the sample rate and frequencies are explicit and the input to W4 is the power spectrum in W2.

Remarks:

BANDPOWER computes the series power by summing the appropriate bands of the PSD as calculated by PSD or the power spectrum as calculated by POWSPEC.

If a frequency range is provided, the input series is multiplied by a mean squared amplitude corrected Hamming window, otherwise all frequencies of the un-windowed power spectrum are summed.

The input is considered a frequency domain PSD if psdflag is "psd". If psdflag is "powspec", the input is considered a power spectrum as computed by POWSPEC.

See Also:

HAMMING

MEANFREQ

MEDFREQ

POWSPEC

PSD