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- where we have exploited ihe identity : (w-+2*) = 2Re{e}. Differentiation of €, with will give the corresponding relationship for the imaginary part; combining the two we get ( 1). Problem 2.3 Awelllknown result in estimation theory based on the minimum mean-squared-error criterion states that the minimum of €, is obtained when the error is orthogonal to cach of the functions in the series expansion. Hence : oo K / |“ - Yount fit =0, n= 1,2,.,K (1) 70 kel since the functions {f,(t)} are orthonormal, only the term with & == » will remain in the sum, so : f s(t) fa(tdt ~ s, = 0, Ne 1K 00 or: co Sr -/ a(t) FRE n= 1,2)... oo ‘The corresponding residual error & is : Ewin = [% oe safedt)) [9(0)~ DR sofa] at It Poa ae [2% TK a daltds ede ~ KL 98 J, [BC ~ DK a sn] Set oe ele de — £2, SH, sesielt)s” (ual sK io 42 = o> Vien el where we have exploited relationship (1) to go from the second to the third step in the above caleulation. Note : Relationship (1) can also be eblained by simple differentiation of the residual error with respect to the coefficients {s,}. Since s, is, in general, complex-valued s,, = ta + Jb, we have to differentiate with respect to both real and imaginary parts : Bb = ZI, [90 ~ TL, snsil] [500 - 8s saat] ae =0 so [an falt) wn 7 sn dalth]” + ah F200) [a(e) — TE suf] dé = 0 = Pan {%, Re { F2tt) [e(0) ~ OK sn fal! bare =f 8, Re { fate) [s) - ye, 1 Snfnlt)] } at = 0, NEL K respect to b, 
