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2. LINEAR OPTIMUM FILTERS
We may classify filters as linear or nonlinear. A filter is said to be linear if the filtered,
smoothed, or predicted quantity at the output of the filter is a linear function of the
observations applied to the filter input. Otherwise, the filter is nonlinear.
In the statistical approach to the solution of the linear filtering problem, we assume
the availability of certain statistical parameters (i.e., mean and correlation functions) of
the useful signal and unwanted additive noise, and the requirement is to design a linear
filter with the noisy data as input so as to minimize the effects of noise at the filter output
according to some statistical criterion. A useful approach to this filter-optimization prob-
lem is to minimize the mean-square value of the error signal defined as the difference
between some desired response and the actual filter output. For stationary inputs, the
resulting solution is commonly known as the Wiener filter, which is said to be optimum
in the mean-square-error sense. A plot of the mean-square value of the error signal ver-
sus the adjustable parameters of a linear filter is referred to as the error-performance
surface. The minimum point of this surface represents the Wiener solution.
The Wiener filter is inadequate for dealing with situations in which nonstationarity
of the signal and/or noise is intrinsic to the problem. In such situations, the optimum fil-
ter has to assume a time-varying form. A highly successful solution to this more difficult
problem is found in the Kalman filter, which is a powerful system with a wide variety of
engineering applications.
Linear filter theory, encompassing both Wiener and Kalman filters, is well devel-
oped in the literature for continuous-time as well as discrete-time signals. However, for
technical reasons influenced by the wide availability of computers and the ever increas-
ing use of digital signal-processing devices, we find in practice that the discrete-time
representation is often the preferred method. Accordingly, in subsequent chapters, we
only consider the discrete-time version of Wiener and Kalman filters. In this method of
representation, the input and output signals, as well as the characteristics of the filters
themselves, are all defined at discrete instants of time. In any case, a continuous-time
signal may always be represented by a sequence of samples that are derived by observing
the signal at uniformly spaced instants of time. No loss of information is incurred during
this conversion process provided, of course, we satisfy the well-known sampling theorem,
according to which the sampling rate has to be greater than twice the highest frequency
component of the continuous-time signal. We may thus represent a continuous-time
signal u(t) by the sequence u(n), n = 0, ;1, ;2, . . . , where for convenience we have nor-
malized the sampling period to unity, a practice that we follow throughout the book.
3. ADAPTIVE FILTERS
The design of a Wiener filter requires a priori information about the statistics of the
data to be processed. The filter is optimum only when the statistical characteristics of
the input data match the a priori information on which the design of the filter is based.
When this information is not known completely, however, it may not be possible to
design the Wiener filter or else the design may no longer be optimum. A straightforward
approach that we may use in such situations is the “estimate and plug” procedure. This
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