Magdalena Salazar-Palma - Modern Characterization of Electromagnetic Systems and its Associated Metrology

Finally, {[ X y ] + [ ]} can be defined as

(1.39)

After multiplying each term in (1.39)by we get the following

(1.40)

The right‐hand side cancels and we are left with

(1.41)

From (1.31)and (1.41)we can solve for the model coefficient a as

(1.42)

The vector v xyis the first n elements of the n +1‐th columns of the right singular matrix V , of [ X y ] and v yyis the n + 1‐th element of the n + 1 columns of V . The best approximation of the model is then given by

(1.43)

This completes the total least squares solution.

This first chapter provides the mathematical fundamentals that will be utilized later. The principles presented are the basis of low‐rank modelling, singular value decomposition and the method of total least squares.

1 1 L. I. Scharf and D. Tufts, “Rank Reduction for Modeling Stationary Signals,” IEEE Transactions on Acoustics, Speech, and Signal Processing, Vol. ASSP‐35, pp. 350–355, 1987.

2 2 S. Haykin, Adaptive Filter Theory, Prentice Hall, Upper Saddle River, NJ, Third Edition, 1996.

3 3 G. Strang, Introduction to Linear Algebra, Cambridge Press, Wellesley, MA, Fifth Edition, 2016.

4 4 G. Golub and C. Van Loan, Matrix Computations, Johns Hopkins Studies in the Mathematical Sciences, The Johns Hopkins University Press, Baltimore, MD, 2013.

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