Implementation Issues with Subspace Based Estimation Algorithms

MUSIC and ESPRIT are considered the two main subspace based estimation algorithms. They both require the theoretical or known correlation matrices. However, in reality, the problem starts often from a sequence of measured data. Therefore, three issues must be resolved from the given data sequence for subspace based estimation algorithms.

Correlation matrix and singular value decomposition
It is sufficient to simply use the unnormalized estimate of the correlation matrix,

$R_x=X^HX.$

Define $X1$ as the $X$ with the first column removed. The principal component can be obtained by the following,

$a^1=-X_1^{+M}x_0$

Where $x0$ is the first column vector of $X$ , and $X_1^{+M}$ is the rank $M$ pseudoinverse of the data matrix, or,

$X_1^{+M}=X_1^H\left(X_1X_1^H\right)^{-1}$

The above procedure simplifies the singular value decomposition greatly.

Number of Signals

In most common situations the number of signals present is not known. The second issue is how to decide the number of signals from the given sequence.

When the signal power is much larger than the noise power, estimating the number of signals is usually not a problem. Akaike information criterion (AIC) can often help.

AIC is defined in terms of the likelihood function for the observations,

$AIC\left(k\right)=-2ln{f_x}\left(x,\theta\right)+2k$

Where $k$ is the number of free parameters represented by $\theta$ . If the observation vector is Gaussian with zero mean, AIC can be further simplified to compute the ratio of geometric over arithmetic means.

Pseudospectrum

Spectrum is more easily computed utilizing FFT, then pseudospectrum should also be computed in similar ways. For MUSIC, the spectrum can be written

$P\left(\omega\right)=\frac{1}{\sum_{i=M+1}^{N}\left|w^He_i\right|^2}$

the denominator can also be simplified with FFT.

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Implementation Issues with Subspace Based Estimation Algorithms