Passive, active, and digital filters (3ed., CRC, 2009) - tiera.ru

Nonlinear Filtering Using Statistical Signal Models 26-25PROPERTY 26.5(No Undershoot=Overshoot)The output of the meridian estimator operating on samples fx(i) : i ¼ 1, 2, ..., Ng is always bounded byx[1] ^u(x) x[N]: (26:60)PROPERTY 26.6(Shift and Sign Invariance)Consider the observation set fx(i) : i ¼ 1, 2, ..., Ng and let z(i) ¼ x(i) þ b. Then,(1) ^u(z) ¼ ^u(x) þ b;(2) ^u(z) ¼ ^u( z).PROPERTY 26.7(Unbiasedness)Given a set of samples fx(i) : i ¼ 1, 2, ..., Ng that are independent and symmetrically distributed arounda symmetry center c, ^u(x) is also symmetrically distributed around c. In particular, if Ef^u(x)g exists, thenEf^u(x)g ¼c.The meridian characteristics can be broadened through the introduction of weights. The weightedmeridian possesses the same properties as the unweighted version and converges to the expected specialcases in the limit of the medianity parameter. The weighted case is formally defined and the limiting casesstated, but the properties are omitted due to their direct similarity to the previous formulations.THEOREM 26.10Consider a set of N independent samples fx(i) : i ¼ 1, 2, ..., Ng each obeying the Meridian distributionwith common location u and (possibly) varying scale parameters v(i) ¼ d=w(i). The ML estimate oflocation, or weighted meridian, is given by" #X^u N¼ arg min logfd þ w(i)jx(i) ujgbi¼1where ? denotes the weighting operation in the minimization problem.¼ meridianfw(i) ? x(i) : i ¼ 1, 2, ..., Ng, (26:61)COROLLARY 26.7Given a set of samples fx(i) : i ¼ 1, 2, ..., Ng and corresponding (positive) weights fw(i): i ¼ 1, 2, ..., Ng,the weighted meridian ^u converges to the weighted median as d !1. That is,lim ^u ¼ lim meridianfw(i) ? x(i) : i ¼ 1, 2, ..., N; dg¼ medianfw(i) x(i) : i ¼ 1, 2, ..., Ng: (26:62)d!1 d!1