Biomedical Signal Processing - Computational Intelligence in Biomedical Engineering - page 63

x k = s k − k

+

(Estimated signal)

Σ

s k = x k + k

−

(Corrupted signal)

Digital filter

r k

k

(Noise)

(Noise estimate)

Adaptive
algorithm

FIGURE 2.21
General adaptive filter structure. Filter coecients are modified by an adap-
tive algorithm based on the estimated signal recovered.

The filter receives two input signals, s k and r k simultaneously. The com-
posite signal, s k contains the original signal to be measured x k , corrupted by
noise η k . The reference noise signal, r k is correlated in some way with η k .In
this filtering technique, the noise signal η k is assumed to be independently
distributed to the required signal x k . The aim is to obtain an estimate of the
corrupting noise, remove it from the signal to obtain x k with less noise. The
estimated signal is

x k = s k − η k = x k + η k − η k (2.165)

The output signal x k is used as an estimate of the desired signal and also as
an error signal, which is fed back to the filter. The adaptive algorithms use
this error signal to adjust the filter coecients to adapt to the changing noise.
The FIR filter with N -points is given by the following equation:

N

η k = w k ( i ) r k−i (2.166)

i =1

where w k ( i ) for i =1 , 2 ,...,N are the filter coecients, r k−i the input noise
reference, and η k the optimum estimate of the noise. If multiple input signals
are present simultaneously, the equation can be extended to vector notation
to denote a system with multiple inputs and a single output.

2.7.2 The LMS Algorithm
Most adaptive algorithms are based on the discrete Wiener filter (Figure 2.22)
that we first review to facilitate description of the LMS algorithm. Two sig-
nals s k and r k are applied to the filter, where s k contains a component that