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Additional resources for Adaptive Digital Filters, 2nd Edition (Signal Processing and Communications)
E. Dudgeon and R. M. , 1984. Y. Grenier, ‘‘Time Dependent ARMA Modeling of Non Stationary Signals,’’ IEEE Transactions ASSP-31, 899–911 (August 1983). L. R. Rabiner and R. W. , 1978. 3 Correlation Function and Matrix The operation and performance of adaptive ﬁlters are tightly related to the statistical parameters of the signals involved. Among these parameters, the correlation functions take a signiﬁcant place. In fact, they are crucial because of their own value for signal analysis but also because their terms are used to form correlation matrices.
Hence, a matrix equation can again be derived involving the AR coefﬁcients of the ARMA signal: Signals and Noise 2 33 rðN À 1Þ rðNÞ .. ÁÁÁ ÁÁÁ .. 32 3 2 3 1 e2 76 Àa1 7 6 0 7 76 7 6 7 76 .. 7 ¼ b0 bN 6 .. 7 54 . 5 4 . 5 0 rð2N À 1Þ Á Á Á rðNÞ ÀaN rðNÞ 6 rðN þ 1Þ 6 6 .. 4 . rð2NÞ rð0Þ rð1Þ .. ð2:79Þ For p > N, the sequence rðpÞ is again generated recursively from the N preceding terms. 76) gives xðnÞ ¼ N X bi eðn À iÞ þ i¼0 N X i¼1 ai 1 X hj eðn À i À jÞ j¼0 and xðnÞ ¼ N X bi eðn À iÞ þ i¼0 1 X eðn À kÞ N X ð2:81Þ ai hkÀi i¼1 k¼1 Clearly, the impulse response coefﬁcients can be computed recursively: h0 ¼ b0 ; hk ¼ 0 for k < 0 hk ¼ bk þ N X ai hkÀi ; ð2:82Þ k51 i¼1 In matrix form, for the N þ 1 ﬁrst terms we have 32 2 h0 0 0 1 0 0 ÁÁÁ 0 76 h1 6 Àa1 1 0 Á Á Á 0 h 0 0 76 6 7 6 Àa2 6 Àa 1 Á Á Á 0 h h h 1 1 0 76 2 6 6 ..
54 . 5 4 . 4 . 5 . . ÀaN ÀaN rðNÞ rðN À 1Þ Á Á Á rð0Þ ð2:145Þ This is an eigenvalue equation. The signal autocorrelation matrix is symmetric, and therefore all eigenvalues are greater than or equal to zero. For N sinusoids without noise, the ðN þ 1Þ Â ðN þ 1Þ autocorrelation matrix has one eigenvalue equal to zero; adding to the signal a white noise component of power e2 results in adding e2 to all eigenvalues of the autocorrelation matrix. Thus, the noise power e2 is the smallest eigenvalue of the signal, and the recursion coefﬁcients are the entries of the associated eigenvector.
Adaptive Digital Filters, 2nd Edition (Signal Processing and Communications) by Bellanger