By Makhmudov O. I.
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Extra resources for A Cauchy Problem for the System of Elasticity Equations
Indeed 2 ρ (cx,F , cy ) = = = 2 L T l=1 il Rcx ,l hl L l=1 hTl Rcx ,l hl · L T l=1 il Rcx ,l hl L l=1 2 λl + qTl Rv ql · L l=1 L hTl Rcx ,l hl · l=1 λl ρ2 (cx , cx,F ) · ρ2 (cx , cy ) . 34. 83) with equality when hl = il , ∀l. Proof. 82) since ρ2 (cx , cx,F ) ≤ 1. The SPCC between the two vectors cv and cy is ρ2 (cv , cy ) = L l=1 L l=1 qTl Rv ql qTl Ry ql 1 1 + iSNR = ρ2 (v, y) . 84) As expected, the SPCC between the two vectors v(k) and y(k) is identical to the SPCC between the two vectors cv (k) and cy (k).
1. 14) l=1 L oSNR(hl ) ≥ oSNR (h1:L ) . 15) 24 3 Performance Measures This means that the aggregation of the subband SNRs is greater than or equal to the fullband SNR. Proof. 16) where al and bl are positive reals. 2 Noise-Reduction Factor Another important measure in noise reduction is the noise-reduction factor, which quantiﬁes the amount of noise being attenuated by the ﬁlter. With the time-domain formulation, this factor is deﬁned as ,  ξnr (H) = tr (Rv ) tr HRv HT . 18) and ξnr (hl ) = qTl Rv ql , l = 1, 2, .
In the next chapter, we will present another criterion, called the Pearson correlation coeﬃcient, in which the output SNR appears naturally. 5 Pearson Correlation Coeﬃcient This chapter develops several forms of the Pearson correlation coeﬃcient in the diﬀerent domains. This coeﬃcient can be used as an optimization criterion to derive diﬀerent optimal noise reduction ﬁlters , but is even more useful for analyzing these optimal ﬁlters for their noise reduction performance. 1 Correlation Coeﬃcient Between Two Random Variables Let a and b be two zero-mean real-valued random variables.