Geometric mean of complex numbers.
Syntax
gm = gmean(X)
Description
gmean makes an attempt to eliminate the effect of phase wrapping when calculating the geometric mean of complex numbers. For matrices, gmean(X) is a row vector containing the complex geometric mean value of each column.
The complex geometric mean has important applications in the averaging of nonparametric estimates of complex transfer function values (see [1], [2], and "Solutions for Some Special Cases" in the User's Guide).
Example
x = -ones(100,2) + 0.1*randn(100,2) + j*0.01*randn(100,2);
mfalse1 = prod(x).^(1/100), mfalse2 = prod(x.^(1/100)),
mx = gmean(x)
Algorithm
The absolute value of the geometric mean is calculated from the arithmetic mean of the logarithms of the absolute values; the phase is the phase of the arithmetic mean value of the complex numbers.
References
[1] .J. Schoukens and R. Pintelon, "Measurement of Frequency Response Functions in Noisy Environments," IEEE Trans. on Instrumentation and Measurement, Vol. 39, No. 6, pp. 905-909, Dec. 1990.
[2] R. Pintelon, J. Schoukens and J. Renneboog, "The Geometric Mean of Power (Amplitude) Spectra Has a Much Smaller Bias than the Classical Arithmetic Averaging," IEEE Trans. on Instrumentation and Measurement, Vol. 37, No. 2, pp. 213-218, June 1988.
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