Adaptive Equalization for Wireless Channels - Adaptive Signal Processing in Wireless Communications - page 260

equalizer away from the trivial all-zero solution. As with the DD-LMS and CMA vari-
ants of (9.33) and (9.34), the SWA variant of (9.35) only updates once per block.
To recap, traditional methods of trained and blind adaptive equalizer design can be
modified for use in PPM-based UWB systems. However, the cost functions and algo-
rithms must be modified to account for the block structure. The resulting algorithms
only update once per block, they generally require matrix-vector products in the update
term, as opposed to the more traditional scalar-vector products, and the decision func-
tion is a vector-to-vector mapping rather than a scalar-to-scalar mapping.

9.5 Interaction of Equalizer with Other

Adaptive Blocks

Many equalizer designs are produced in isolation: the equalizer explicitly or implicitly
assumes that all other adaptive blocks in the system are working perfectly. However, even
in a simple single-carrier receiver, the signal amplitude, timing frequency/phase, equal-
izer, and carrier frequency/phase are all interdependent [15, Table 9.2 ) . In newer modu-
lation formats with more or different blocks, there can be additional interdependence.
We conclude this chapter with a discussion of the merits of joint analysis and design of
the equalizer and other blocks in an adaptive receiver, though a full treatment of this
subject is beyond the current state of the art.
One of the classical examples of this dependence is the dependence of an adaptive
equalizer on accurate carrier frequency offset (CFO) estimation. For a trained, LMS
algorithm, the equalizer inputs and outputs will not match their model if there is a
residual CFO. At the same time, a trained CFO estimator needs ISI-free received data
in order to form a good estimate. This can be a chicken-and-egg problem. However, the
CMA equalizer uses a cost function that depends only on the magnitude of the equalizer
output, and not the phase. In the presence of a residual CFO, the received data will have
a linearly increasing additive phase, causing the signal constellation to spin about the
origin. Since CMA does not care about the phase, it can remove the ISI even in the pres-
ence of the CFO-induced spinning. Then the CFO estimator can operate on the ISI-free
equalizer output, and remove the CFO [15].
This method of delaying the need for CFO correction cannot be as easily incorporated
into multicarrier systems, since as discussed in section 9.3, blind multicarrier equaliz-
ers do make use of a constant modulus cost function in the time domain. An alternate
method of dealing with the coupling of an adaptive equalizer and another adaptive block
is to jointly adapt the two. In the case of multicarrier equalization and CFO correction,
the CNA, MERRY, and DD cost functions used for equalization can also be used to
adjust the CFO [53, 54, 63, 64]. By forming a single cost function at the output of the two
adaptive blocks in series, both algorithms may converge to their optimal setting [63].
As mentioned in section 9.3, in multicarrier systems, the CSE and FEQ must both
be adapted simultaneously. Algorithms such as DD-LMS and CMA cannot directly be
used for the CSE since the CSE output is not expected to be finite alphabet. However, the
final FEQ output is. By forming a decision-directed or constant modulus cost at the FEQ