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Chapter_16

# Chapter_16 - CHAPTER 16 16.1(a The received signal of a...

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320 CHAPTER 16 16.1 (a) The received signal of a digital communication system in baseband form is given by where x k is the transmitted symbol, h ( t ) is the overall impulse response, T is the symbol period, and v ( t ) is the channel noise. Evaluating u ( t ) at times t 1 and t 2 : Hence, the autocorrelation function of u ( t ) is (1) where r x ( mT - lT ) is the autocorrelation function of the transmitted signal. From Eq. (1) we immediately see that which shows that u ( t ) is indeed cyclostationary in the wide sense. ut () x m ht mT vt + m =- = 1 x m ht 1 mT 1 + m =- = 2 x l 2 lT 2 + m =- = r u t 1 t 2 , Eut 1 u * t 2 [] = Ex m x l * 1 2 l =- m =- = Evt 1 v * t 2 + r x 1 h * t 2 l =- m =- = σ v 2 δ t 1 t 2 + r u t 1 Tt 2 T + , + r u t 1 t 2 , =

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321 (b) Applying the definitions to the result obtained in part (a), we may show that (2) As a check, we see that for k = 0, Eq. (2) reduces to the standard result: Let denote the phase response of and denote the phase response of the channel. Then recognizing that the power spectral density S x ( ω ) of the transmitted signal is real valued, we readily find from Eq. (2) that (3) (c) From the formula for the inverse Fourier transform, we have Applying these definitions to Eq. (3): r u α τ () 1 T --- r u t τ 2 -- t τ 2 , +   e j 2 πα t t d T 2 T 2 = S u α ω r u α τ e j –2 π f τ τ, d ω 2 π f == α kT , = k 012 , ± , ± , = S u 1 T He j ω jk π T + H * e j ω e π T S x ω k π T ----- + = σ v 2 δ k , k ± , ± , = + S u ω 1 T j ω 2 S x σ v 2 + = ψ k ω S u Φω Ψ k ω () Φω k π T + k π T , k –0 1 2 , ± , ± , ψ k τ 1 2 π ------ Ψ k ω e j ωτ ω d = φτ 1 2 π Φ k ω e j ω d = ψ k τ e j π k τ T e j π k τ T =
322 (4) On the basis of Eq. (4), we may make two important observations: (1) For k = 0 and k τ / T equal to an integer, is identically zero. For these values of k , φ ( τ ) cannot be determined. This means that for an arbitrary channel, the unknown phase response cannot be identified for k =0or k τ / T = integer by using cyclostationary second-order statistics of the channel output. (2) For k = +2 and higher in absolute value, the use of does not reveal any more information about the channel phase response than what can be obtained with k = +1. We may therefore just as well work with k = 1, for which has the largest support, as shown by That is, which shows that φ ( τ ) is identifiable from except for τ = mT , where m is an integer. 16.2 In the noise-free case, we have (1) where H is the LN -by-( M+N ) multichannel filtering matrix, x n is the ( M+N )-by-1 transmitted signal vector, and u n is the LN -by-1 multichannel received signal vector. Let u n be applied to a multichannel structure characterized by the ( M+N )-by- LN filtering matrix T such that we have (2) This zero-forcing condition ensures the perfect recovery of x n from u n . Substituting Eq.

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Chapter_16 - CHAPTER 16 16.1(a The received signal of a...

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