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Unformatted text preview: Kevin Buckley  2007 1 ECE 8770 Topics in Digital Communications  Sp. 2007 Lecture 7,8 4 Channel Equalization 4.2 Linear Equalization 4.2.2 Mean Squared Error (MSE) Criterion Consider the DT linear time invariant channel models established in Section 3 of the Course. This model has as its input the symbol sequence I k . It output could be, for example: 1) the sampled output of the receiver demodulator; or 2) the sampled output, denoted y k , of the receiver filter matched to the pulse shape h ( t ) at the channel output; or 3) the output, denoted v k , of the DT filter which noise whitens the sampled matched filter output y k . As an example we will consider processing v k . Here we consider a DT linear time invariant equalizer with input v k , transfer func tion C ( z ), impulse response c k , and output sequence ˆ I k which is to be considered an estimate of the symbol sequence I k . Note that ˆ I k is implicitly a function of the equalizer transfer function C ( z ), or equivalently the equalizer impulse response c k . Consider, as equalizer design objective, the MSE cost function J = E { I k ˆ I k  2 } . (1) The Minimum MMSE (MMSE) equalizer is the solution to the problem min C ( z ) J . (2) That is , the design problem is to select C ( z ) (or equivalently c k ) to minimize the cost J . FIR C(z) First consider an equalizer which has a structure which is constrained to be FIR. The formulation for this in the Course Text, shown here in Figure 1(a), is non causal. K is the FIR equalizer memory depth design parameter. The MMSE will decrease as K increases. However, increasing K increases computational require ments, and as we will see, for adaptive equalizers increasing K can actually lead to increased MSE. The formulation we will use is shown in Figure 1(b). It is a causal FIR equalizer of length K 1 + 1 and latency (delay) Δ. K 1 and Δ are design parameters. In terms of the noncausal equalizer design parameter K , reasonable values for K 1 and Δ are K 1 = 2 K and Δ. If the channel has some bulk propagation delay, say of B symbols, then Δ > B is desirable. For example, Δ = B + K 1 2 is reasonable. Kevin Buckley  2007 2 z1 z1 z1 v k v k1 v kK 1 c 1 c c K 1 K I k ∆ ..... z1 I k z1 z1 z1 z1 cK cK+1 c (b) (a) v v kK ..... v v k+K1 k+K k ..... c K Figure 1: Noncausal and causal linear time invariance channel equalizers. The output of the causal FIR equalizer is ˆ I k ∆ = c k * v k = K 1 summationdisplay j =0 c j v k j = c T v k (3) where v k = [ v k , v k 1 , ··· , v k K 1 ] T and c = [ c , c 1 , ··· , c K 1 ] T is the FIR equalizer coefficient vector. Define the error as e k ∆ = I k ∆ c T v k , (4) which is a linear function of the coefficient vector c . Consider the cost J = J ( c /K 1 , ∆) = E { e k ∆  2 } , (5) where the notation J ( c /K 1 , ∆) explicitly shows that the cost is a function of c and it depends on given values of K 1 and ∆. We have that J ( c /K 1 , ∆) = E { I k ∆...
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This document was uploaded on 10/12/2009.
 Spring '09

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