U. Glaeser

FIGURE 34.52 Feedback detector.
Catastrophic error propagation can be avoided without precoder by forcing the output of the recursive
filter of Fig. 34.52 to be binary (Fig. 34.52(a)). An erroneous estimate î n−2= −i n−2 leads to a digit
whose polarity is obviously determined by in−2. Thus, the decision that is taken by the slicer in Fig. 34.52(a)
will be correct if in happens to be the opposite of in−2. If data is uncorrelated, this will happen with
probability 0.5, and error propagation will soon cease, since the average number of errors in a burst is
1 + 0.5 + (0.5) 2 ιˆ n
+… = 2. Error propagation is thus not a serious problem.
The feedback detector of Fig. 34.52 is easily generalized to arbitrary partial response H(D). For purposes
of normalization, H(D) is assumed to be causal and monic (i.e., hn = 0 for n < 0 and h0 = 1). The nontrivial
taps h1, h2,…together form the “tail” of H(D). This tail can be collected in P(D), with pn = 0 for n ≤ 0
and pn = hn for n ≥ 1. Hence, hn = δn + pn, where the Kronecker delta function δn represents the component
h0 = 1. Hence
The term (i ∗ p) n depends exclusively on past digits in−1, in−2,… that can be replaced by decisions ,
,…. Therefore, an estimate of the current digit in can be formed according to = xk( ∗ p) n as
in Fig. 34.52(b). As before, a slicer quantizes into binary decisions so as to avoid catastrophic error
propagation. The average length of bursts of errors, unfortunately, increases with the memory order of
H(D). Even so, error propagation is not normally a serious problem [21]. In essence, the feedback detector
avoids noise enhancement by exploiting past decisions. This viewpoint is also central to decision-feedback
equalization, to be explained later.
Naturally, all this can be generalized to nonbinary data; but in magnetic recording, so far, only binary
data are used (the so-called saturation recording). The reasons for this are elimination of hysteresis and
the stability of the recorded sequence in time.
Let us consider now the way the PR equalizer from Fig. 34.47 is constructed. In Fig. 34.53, a discretetime
channel with transfer function F(e j2πΩ ) transforms in into a sequence yn = (i ∗ f ) n + un, where un is
the additive noise with power spectral density U(e j2πΩ ), and yn represents the sampled output of a
whitened matched filter. We might interpret F(e j2πΩ ) as comprising two parts: a transfer function
H(e j2πΩ ιˆ n−2 ιˆ n ιˆ n ιˆ
ιˆ n
ιˆ n
) that captures most of the amplitude distortion of the channel (the PR target) and a function
© 2002 by CRC Press LLC
x n
in
in
hk
(a)
hn
(b)
+
+
iˆ
n−
2
Feedback detector
xn n
+
+
2
D
Feedback detector
xn n
P(D)
ιˆ n = xn + ιˆ n−2 = in – in−2 + ιˆ n−2 = in – 2in−2 = ( i ∗ h)n=
( i ∗ ( δ + p)
)n= in + i ∗ p
î
î
( )n
.
ιˆ n−1