Estimate empirical transfer functions and periodograms.
Syntax
g = etfe(z)
g = etfe(z,M,N,T)
Description
etfe estimates the transfer function g of the general linear model
The matrix z contains the output-input data z = [y u], where y and u are column vectors. The routine works only for single-input, single-output systems.
For a time series, z = y. Then g is returned as the periodogram of y.
g is given in frequency function format (see freqfunc), with the estimate of 
at the frequencies
w = [1:N]/N*pi/T
The default values of N and T are 128 and 1, respectively. N must be a power of two.
When M is specified other than the default value M = [ ], a smoothing operation is performed on the raw spectral estimates. The effect of M is then similar to the effect of M in spa. This can be a useful alternative to spa for narrowband spectra and systems, which otherwise require large values of M.
When etfe is applied to time series, the corresponding spectral estimate is normalized in the way that is defined in "Some Special Topics" on page 3-68. Note that this normalization may differ from the one used by spectrum in the Signal Processing Toolbox.
Examples
Compare an empirical transfer function estimate to a smoothed spectral estimate:
ge = etfe(z);
gs = spa(z);
bodeplot([ge gs])
Algorithm
The empirical transfer function estimate is computed as the ratio of the output Fourier transform to the input Fourier transform, using fft. The periodogram is computed as the normalized absolute square of the Fourier transform of the time series.
The smoothed versions (M less than the length of z) are obtained by applying a Hamming window to the output FFT times the conjugate of the input FFT, and to the absolute square of the input FFT, respectively, and subsequently forming the ratio of the results. The length of this Hamming window is equal to the number of data points in z divided by M, plus one.
See Also
freqfunc, spa
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