Estimating the order of a FIR filter for noise cancellation

429 total variation classes, if the nominal rn; < < A, and for the band class, if m; < < A and mfi < < A. Then we have d h , = i T i 2 (t) A (t) dt. (22) To prove the first inequality in (22), we used the fact that the absolutely continuous part of a measure is no larger than the measure itself; to prove the second we used the fact that A is stochastically smallest under hi,, over all other elements in the capacity class. Case 2: Here we make use *of the discrete-set version of Lemma 1. The optimal filter is h, = s, /A,, where A, is the pmf corresponding to the measure A, singled out by Lemma 1 when applied to this case. The equivalent form to (21) is k k k k if. To prove the inequality in (23) we used the fact that A becomes stochastically smallest under h,, singled outAby Lemma 1. Cases 3 and 4: The optimal filter here is H (w) = S (w) / N (w) , where $ w) is the R-N derivative of the least favorable spectral measure hN, singled out by Lemma 1, with respect to A. Both the continuous-time and the discrete-time problems are represented here with common notation. The equivalent form to (21) is /Ifil2drn, =/lS\2hi-2dm, c / l S 1 2 ~-2 d h , = / l f i 1 2 d A N. To prove the inequality in (24), we used the fact that P? becomes stochastically smallest under h,,, singled out by Lemma 1. The integrals in (24) are over [-w o , w O ] or over [-T , a ] for the continuous-and discrete-time cases, respectively. The robust matched filter for uncertainty in the noise auto-correlation function or the noise spectral measure is derived for both continuous and discrete-time problems, when the uncertainty classes are generated by two-alternating capacities. In all cases the maximin robust matched filter depends on the inverse of the worst case noise statistic, which is obtained as the Huber-Strassen derivative of the capacity generating the uncertainty class with respect to the Lebesgue (or other equivalent measure) on a suitable interval. Abstruct-The problem of designing a finite impulse response (FIR) filter devoted to the joint process problem, and in particular to noise cancellation, is investigated. The goal is restricted to how …