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1 Transversal Filters
Figure 2.1:
Transversal adaptive filter structure.

The most commonly used structure in implementing adaptive filters is the transversal
structure shown in Fig. 2.1. The transversal adaptive filter can be
split into two main parts, the filter part and the update part. The function of the
former is to calculate the filter output , while the function of the latter is to
adjust the set of N filter coefficients
(tap weights) so that the
output becomes as close as possible to a desired signal .
The filter part processes a single input sample and produces a single output sample
(assuming sample per sample implementation). The filter output is calculated as a
linear combination of the input sequence
composed of delayed
samples of ,

(1) 
Expressing the set of N filter coefficients at time index and the sequence of delayed
input samples in vector notations such that
and
, where is the vector transpose
operator, eq (2.1) can be written as

(2) 
The transversal filter structure is, therefore, a linear temporal filter that processes
the temporal samples of its input signal to produce the temporally and consequently
spectrally modified (filtered) output .
In fixed transversal filter applications, the set of filter coefficients are chosen at the
system design time to achieve the required spectral filtering and remain constant during the
filter operation. In adaptive filters applications, however, an adaptive
algorithm is used to continuously adjust the filter coefficients so that a certain performance
criterion is optimized in some sense. Regardless of the optimization method, it is usually
desired to adjust the filter coefficients such that the filter output resembles a desired
signal , or equivalently, the error signal must be minimized. The details of the
optimization process defines the adaptive algorithm and its behavior.
The adaptive signal processing toolbox contains several transversal adaptive algorithms such
as the Least Mean Squares (see Section 4.9), the Normalized Least Mean Squares (see
Section 4.11), the leaky Normalized Least Mean Squares (see Section 4.7),
the Variable Step Size Least Mean Squares (see Sections 4.10 and 4.20),
and the Recursive Least Squares (see Section 4.16). When
the number of filter coefficients , is large, it is much more efficient to perform filtering
and coefficient update in the frequency domain. This requires collecting a block of samples of
the input signal before the fast Fourier transform (FFT) can be calculated. For this reason, a
frequency domain transversal filter is usually a block processing filter that accepts a block of
input samples and produces a block of output samples. Several implementations of block frequency
domain adaptive filters are included in the adaptive signal processing toolbox, such as the Block
Frequency Domain Adaptive Filter (see Section 4.2), the Partitioned Block
Frequency Domain Adaptive Filter (see Section 4.12), and the Reduced Complexity
Partitioned Block Frequency Domain Adaptive Filter (see Section 4.13).
Next: 2 Linear Combiner Filters
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