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U. Glaeser

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in<br />

FIGURE 34.53 Interpretation of PR equalization.<br />

F r(e j2πΩ ) = F(e j2πΩ )/H(e j2πΩ ) that accounts for the remaining distortion. The latter distortion has only a<br />

small amplitude component and can thus be undone without much noise enhancement by a linear<br />

equalizer with transfer function<br />

This is precisely the PR equalizer we sought for. It should be stressed that the subdivision in Fig. 34.53<br />

is only conceptual. The equalizer output is a noisy version of the “output” of the first filter in Fig. 34.53<br />

and is applied to the feedback detector of Fig. 34.52, to obtain decision variables ιˆ′ n and ιˆ n.<br />

The precoder<br />

and MM of Fig. 34.51 are, of course, also applicable and yield essentially the same performance.<br />

The choice of the coefficients of the PRE in Fig. 34.47 is the same as for full-response equalization<br />

and is explained in the subsection on “Adaptive Equalization and Timing Recovery.” Interestingly, zeroforcing<br />

here is not as bad as is the case with full-response signaling and yields approximately the same<br />

result as minimum mean-square equalization. To evaluate the performance of the PRE, let us assume<br />

that all past decisions that affect ιˆ n are correct and that the equalizer is zero forcing (see “Adaptive Equalization<br />

and Timing Recovery” for details). The only difference between ιˆ′ n and ιˆ n is now the filtered noise<br />

component (u ∗ c) n with variance<br />

Because |H(e j2πΩ )| was selected to be small wherever |F(e j2πΩ )| is small, the integrand never becomes very<br />

large, and the variance will be small. This is in marked contrast with full-response equalization. Here,<br />

H(e j2πΩ ) = 1 for all Ω, and the integrand in the above formula can become large at frequencies where<br />

|F(e j2πΩ )| is small. Obviously, the smallest possible noise enhancement occurs if H(e j2πΩ ) is selected so<br />

that the integrand is independent of frequency, implying that the noise at the output of the PRE is white.<br />

This is, in general, not possible if H(e j2πΩ ) is restricted to be PR (i.e., small memory-order, integer-valued).<br />

The generalized feedback detector of Fig. 34.52, on the other hand, allows a wide variety of causal<br />

responses to be used, and here |H(e j2πΩ )| can be chosen at liberty. Exploitation of this freedom leads to<br />

decision feedback equalization (DFE).<br />

Decision Feedback Equalization<br />

This subsection reviews the basics of decision feedback detection. It is again assumed that the channel<br />

characteristics are fixed and known, so that the structure of this detector need not be adaptive. Generalizing<br />

to variable channel characteristics and adaptive detector structure is tedious, but straightforward.<br />

A DFE detector shown in Fig. 34.54, utilizes the noiseless decision to help remove the ISI. There are<br />

two types of ISI: precursor ISI (ahead of the detection time) and postcursor (behind detection time). Feedforward<br />

equalization (FFE) is needed to eliminate the precursor ISI, pushing its energy into the postcursor<br />

domain. Supposing all the decisions made in the past are correct, DFE reproduces exactly the modified<br />

© 2002 by CRC Press LLC<br />

H ( e<br />

2<br />

σ ZFPRE<br />

j2π<br />

Ω<br />

hk<br />

∫<br />

)<br />

0.5<br />

F(<br />

e<br />

)<br />

xn + yn<br />

n xˆ<br />

j 2π Ω<br />

un<br />

j2π<br />

Ω<br />

Fr<br />

( e )<br />

Equalizer<br />

j2π<br />

Ω −1<br />

j2π<br />

Ω<br />

C(<br />

e ) = Fr<br />

( e )<br />

C e j2πΩ 1<br />

( )<br />

Fr e j2πΩ<br />

H e<br />

--------------------<br />

( )<br />

j2πΩ<br />

( )<br />

F e j2πΩ<br />

= = --------------------<br />

( )<br />

U e j2πΩ<br />

( ) C e j2πΩ<br />

( ) 2<br />

=<br />

dΩ<br />

=<br />

– 0.5<br />

∫<br />

0.5<br />

– 0.5<br />

U e j2πΩ<br />

( ) H e j2πΩ<br />

( ) 2<br />

F e j2πΩ<br />

( ) 2<br />

----------------------------------------------dΩ

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