Selected Topics in Applied Signal Processing - Digital Signal Processing: An Introduction with MATLAB and Applications - page 158

x ð n Þ¼ X

M

h ð k Þ y ð n k Þ ð 3 : 46 Þ

k ¼ 0

Often it is convenient to write the above equation in matrix form as follows:

x ð n Þ¼ hy T ð n Þ¼ y ð n Þ h T ð 3 : 47 Þ

where

h ¼½ h ð 0 Þ h ð 1 Þ h ð 2 Þ; ... ; h ð M Þ;

which is the impulse response vector, and

y ð n Þ¼½ y ð n Þ y ð n 1 Þ y ð n 2 Þ; ... ; y ð n M Þ;

which is the observed signal vector at the time instant n. The error is given by:

e ð n Þ¼ d ð n Þ x ð n Þ ð 3 : 48 Þ

and the mean-squared error (MSE) is given by:

e mse ¼Ef½ e ð n Þ 2 g¼Ef½ d ð n Þ x ð n Þ 2 g¼Ef½ d ð n Þ X

M

h k y ð n k Þ 2 g 3 : 49 Þ

k ¼ 0

To minimize e mse it is necessary that:

o e mse
oh j

¼ 0 8 j ð 3 : 50 Þ

where for simplicity, the designation h j = h(j) is made. From Eqs. 3.47 , 3.49 , and
3.50 it is possible to write:

o e mse
oh j

¼E 2e ð n Þ: o e ð n Þ
oh j

ð 3 : 51 Þ

¼ 2 Ef e ð n Þ y ð n j Þg
¼ 2 Ef y ð n j Þ½ d ð n Þ y ð n Þ h T g ¼ 0; j ¼ 0 ; 1 ; ... ; M

Now Eq. 3.51 can be written in matrix form as follows:

2 Ef y T ð n Þ½ d ð n Þ y ð n Þ h T g ¼ 0
) Ef y T ð n Þ d ð n Þg Ef y T ð n Þ y ð n Þ h T g ¼ 0
) R yd ¼ R yy h T

where R yy is the observed signal autocorrelation, while R yd is the observed signal
cross-correlated with the desired signal. Hence, the optimal filter coefficients (in
the MSE sense) are given by: