Digital Signal Processing Reference
In-Depth Information
classic scenario of linear and supervised processing. As already commented,
many topics are devoted to this rich subject and present it in a more exhaus-
tive fashion. We opt for a brief and, to a certain extent, personal presentation
that facilitates the introduction of the central themes of the topic. First,
we discuss three emblematic problems in linear filter theory: identifica-
tion, deconvolution, and prediction. From there, the specific case of channel
equalization is introduced. Then, as usually done in the literature, we present
the Wiener filtering theory as the typical solution for supervised processing
and a paradigm for adaptive procedures. The sections on supervised adap-
tive filtering discuss the celebrated LMS and RLS algorithms, and also the
use of structures alternative to the linear FIR filter. Moreover, in Chapter 3
we introduce the notion of optimal and adaptive filtering without a refer-
ence signal, as a first step to consider blind techniques. In this context, we
discuss the problem of constrained filtering and revisit that of prediction,
indicating some relationships between linear prediction and unsupervised
equalization.
After establishing the necessary foundations in Chapters 2 and 3, the sub-
ject of unsupervised equalization itself is studied in Chapter 4, which deals
with single-input single-output (SISO) channels, and in Chapter 5, in which
the multichannel case is considered.
Chapter 4 starts with a general discussion on the problem of unsu-
pervised deconvolution, of which blind equalization may be viewed as a
particular case. After introducing the specific problem of equalization, we
state the two fundamental theorems: Benveniste-Goursat-Ruget and Shalvi-
Weinstein. Then we discuss the main adaptive techniques: the so-called
Bussgang algorithms that comprise different LMS-based blind techniques,
the Shalvi-Weinstein algorithm, and the super-exponential. Among Buss-
gang techniques, special attention is given to the decision-directed (DD)
and Godard/CMA approaches, due to their practical interest in communica-
tions schemes. We discuss important aspects about the equilibrium solutions
and convergence of these methods, having the Wiener MSE surface as a
benchmark for performance evaluation. Finally, based on a more recent
literature, we present some results concerning the relationships between
constant-modulus, Shalvi-Weinstein, and Wiener criteria.
The problem of blind equalization is extended to the context of systems
with multiple inputs and/or outputs in Chapter 5. First, we state some
theoretical properties concerning these systems. Then we discuss single-
input multiple-output (SIMO) channels, which may be engendered, for
instance, by two practical situations: temporal oversampling of the received
signal or the use of multiple antennas at the receiver. In the context of SIMO
equalization, we discuss equalization conditions in the light of Bezout's
identity and the second-order methods for blind equalization. Afterward,
we turn our attention to the most general scenario, that of multiple-input
multiple-output (MIMO) channels. In such case, special attention is given to
