U. Glaeser

The basic idea is to minimize the mean squared error with respect to some desired or ideal signal. Let
the desired or ideal signal be yˆ (n) in which case the error is e(n) = y(n) − yˆ (n). This minimization is
achieved by adjusting the tap value in the direction opposite to the derivative (with respect to the tap
values) of the expected value of the mean squared error. Dispensing with the expected value leads to the
LMS or stochastic gradient algorithm. The stochastic gradient for the kth tap weight is
© 2002 by CRC Press LLC
∂
∂ê( n)
∂yˆ ( n)
------------- 2 ------------- ∂y( n)
∆( k,n ) – ( e ( n)
) – 2e( n)
2e( n)
------------- yˆ
-------------
= = = –
– ( k)
∂h( k)
∂h( k)
∂h( k)
∂h( k)
∂y( n)
-------------
= – 2e( n)
∂h( k)
(34.4)
where the partial derivative of yˆ (n) with respect to h(k) is zero. We can now expand y(n) as in Eq. (34.3)
to further obtain
∆( k,n ) = – 2e( n)x(
n– k)
(34.5)
The gradient would actually be scaled by some tap weight update gain t ug to give the following tap update
equation:
h( k,n+ 1)
=
h( k,n)
– 2tuge( n)x(
n– k)
(34.6)
The choice of this update gain depends on several factors: (a) it should not be too large so as to cause
the tap adaptation loop to become unstable, (b) it should be large enough that the taps converge within
a reasonable amount of time, (c) it should be small enough that after convergence the adaptation noise
is small and does not degrade the BER performance. In practice, during drive optimization in the factory
the adaptation could take place in two steps, initially with higher update gain and then with lower update
gain. During the factory optimization different converged taps will be obtained for different radii on the
disk surface. Starting from factory optimized values means that the tap weights do not have to adapt
extremely fast and so allows the use of lower update gains during drive operation. Also, this means that
the tap weights need not all adapt every clock cycle—instead a round-robin approach can be taken, which
allows for sharing of the adaptation hardware across the various taps. A simpler implementation can also
be obtained by using the signed LMS algorithm whereby the tap update equation is based on using 2- or
3-level quantized version of x(n − k). For read channel applications, this can be done without hardly any
loss in performance.
A few other issues should be emphasized about the adaptive FIR. During a read event, the FIR filter
is usually adapted after the initial gain and timing recovery operations are performed over a preamble
field. Nevertheless, during the rest of the read event, the FIR filter equalizes the signal at the same time
that the gain and timing loops are operating. The adaptive gain loop uses an automatic gain control
(AGC) block to apply the correct gain to the signal to achieve the desired partial response target values.
Likewise the adaptive timing recovery loop works to adjust the sampling phase to achieve the desired PR
target values. It is necessary to minimize the interaction between these adaptive loops. The FIR filter will
typically have one tap as a “main” tap, which is fixed to minimize its interaction with the gain loop.
Another tap such as the one preceeding or following the main tap can be fixed (but allowed to be
programmable) to minimize interaction with the timing loop [14]. In some situations it may be advantageous
to have additional constraints to minimize the interaction with the timing loop [15].
Performance Characterization
The performance of various equalizer architectures based on bit error rate simulations can now be
characterized. The equalizer types (with reference to Fig. 34.11) actually simulated are of Types 2 (CTF +
analog FIR) and 3 (anti-aliasing CTF + analog FIR). One can consider the case where there are very few