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The main difference between the simpliﬁed PNLMS algorithm and PNLMS algorithm is the usage of the max function when calculating the time-varying update gain matrix by the PNLMS algorithm. 1]. info 48 PtNLMS Algorithms STEADY STATE MSE: SIMULATIONS VS. 13. 1. 1. 8], which are general to all PtNLMS algorithms. At this point, we need to ﬁnd the following expectations: E {gi zi }, E gi zi2 , E gi2 zi2 and E gi2 . Note that we assume, as before, E gi2 zj2 = E gi2 E zj2 if i = j. As we did when analyzing the simpliﬁed PNLMS, we assume that the expectation of the ratio is equal to the ratio of expectations, for example E {gi } = E ˆi |} max{γmin , |w 1 L L j=1 max{γmin , |w ˆj |} ≈ E {max{γmin , |w ˆi |}} E 1 L L j=1 max{γmin , |w ˆj |} After this assumption, the following terms will be calculated: E max{γmin , |w ˆi |} E max{γmin , |w ˆi |}zi E max{γmin , |w ˆi |}zi2 E max2 {γmin , |w ˆi |}zi2 E max2 {γmin , |w ˆi |} .

However, in order to ﬁnd the MSWD it will prove helpful to ﬁrst ﬁnd a more general recursion for E||z(k + 1)||2Σ , where Σ is an arbitrary positive deﬁnite diagonal matrix. We will refer to this recursion as the weighted variance recursion. We begin by writing the weighted variance at time k + 1 in terms of the weighted variance at time k. 3] into ||z(k + 1)||2Σ yields: ||z(k + 1)||2Σ = ||z(k)||2Σ − β||z(k)||2x(k)xT (k)Σ − β||z(k)||2Σx(k)xT (k) +β 2 ||z(k)||2x(k)xT (k)Σx(k)xT (k) +β 2 zT (k)x(k)v(k)||x(k)||2Σ+ β 2 ||x(k)||2Σ xT (k)z(k)v(k) +β 2 v 2 (k)||x(k)||2Σ −βv(k)zT (k)Σx(k) − βv(k)xT (k)Σz(k).

First, a general approach will be described and then the simpliﬁed PNLMS algorithm will be analyzed, followed by the analysis of the PNLMS algorithm. As we proceed, some results in each section can be applied to later algorithms. These results will be pointed out as needed. 1. Transient analysis of PtNLMS algorithm for white input In this section, the theory for analyzing the transient regime of the PtNLMS is presented. This analysis involves generating deterministic recursions for the MWD, MSWD, and MSE as a function time, so that these quantities can be evaluated at any given time.