A characterization of admissible linear estimators of fixed by Synowka-Bejenka E., Zontek S.

By Synowka-Bejenka E., Zontek S.

Within the paper the matter of simultaneous linear estimation of mounted and random results within the combined linear version is taken into account. an important and enough stipulations for a linear estimator of a linear functionality of mounted and random results in balanced nested and crossed type types to be admissible are given.

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M] + E(e[n])E(s[n]) = since s[n] and e[n] are independent and stationary. where we assumed e[n] has zero mean. ,(eJ"w). 83. ) (a) ••[m] = h[m]• h[-m]. Taking the Fourier transform, we get: + .. e '"' 1 . ). + 1-aeJ'*' Taking the Inverse Fourier transform, we get: alnl .. [m] = 1- 42. (b) Using pilrt (a), we get: IB(eJ""')I2 = H(eJ"W)B'(e'w) = B(e'w)B(e-i"') since h[n] is real = = •••(e'w) 1 1 (1- ae >w) (1- aei"') = _1-( 2 1- a 1 . + 1 . ). 84. The first-backward-difference system is given by: y[n] = z[n] - z[n- 1].

A) Since f[n] = e(n]- e(n- 1], +IJ(eiw) = HI(e'w)H,(e-iw)+,,(e'w) = (1- e-iw)(1 - = = cr~(2 e'w)cr~ _ e'w _ e-iw) ~(2- 2 cos(w)). ). -:(2o[m]- o{m + 1]- o[m- l]l. -. 1. -· = - L:(2zl" -1 00 L: n=-oo = 2z -1- 2z n=l 1 1 lzl < 2 1 = 1- lz2 (c) Z[G) n u[-nJ] = ioo (2z)" = 1! 2. :r[n] = { n :r[n] <* n u[n) <* n N, O 0 = n u[n]- (n- N)u[n- N] d I -z-X(z) =:> n u[n] <* -zdz lzl > 1 dz 1-z-1 z-1 (l _ z 1 )2. -... 3. 4. The pole-zero plot of X(z) appears below. X(z} 3 2 (a) For the Fourier transform of :z:(n] to exist, the z-transform of z[n] must have an ROC which includes the unit circle, therefore, Iii < lzl < 121.

7 4. =-OCl n:&[nje·jwn b:r,[n]e·iwn 2. 75. The output of an LTI system is obtained by the convolution sum, 00 y[n] = L 1=-oo z[kjh[n - kj. Taking the Fourier transform, Y(eiw) = = 1: 1: 00 = (1: ~f;oo z[k)h[n- k)) e-i~ 00 Cf;oo·h[n- kje-iwn) z[kje-i"> (_too h{n- k}e-iw(n->)) z[kj 00 Hence, 2. 76. _, (' d8 (' dw X (ei 1 )W(e'(w-B>)e->w• (21r) '-· L. 2... f' d8 X(ei1 )w[n]ei'• 21f )_ .. z[n]w[n] 2. 77. (a) The Fourier transform of y•[-n] is Y"(e'"), and X(ei")Y(e'") forms a transform pair with z[n] • y[n].

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