Adaptive signal processing can be considered to be a process in which the parameters used for the processing of signals changes according to some criterion. Usually the criterion is the estimated mean squared error or the correlation. Adaptive processing usually refers to adaptive filtering; that is, filtering in which the parameters of the filter can change with the independent variable (usually space or time). When data can be re-played (not real time), one can iterate the adaptation process to optimize the convergence process. It is pertinent at this stage to introduce a glossary of terms, which should be understood by the student:

Adaptive, adaptation - a process in which the parameters used for the processing of signals changes according to some criterion.

Adaptive filter - the adaptive process is applied to a filter; i.e., the values of filter parameters can change over the course of the independent variable (usually time or space) based on an error criterion.

Convergence - the weights for adaptation arrive at the optimized value.

Convergence Coefficients - the value of the parameter(s) that determine the speed of convergence of the adaptive weights toward the optimal weights.

Correlation - a measure of the similarity between two signals ranging from -1 to 1 when normalized. It can also be used as an error function (correlation coefficient closer to 0 means signals are more dissimilar).

Error - for signal processing, the difference between two signals. The less the error, the greater the similarity of the two signals.

Error criterion (function) - the error equation or equations used to optimize a set of weights. Often, the mean squared error estimate is used.

Locus - a range in weights centered around the optimal weight.

Incomplete convergence - the weights approach but do not reach the optimal values. This may occur because the convergence coefficient is too large, whereby the weights oscillate around the area of the locus. It may also happen because the convergence coefficient is too small, whereupon the weights stop converging before reaching the optimal value.

Optimal, optimality - implies that a set of weights associated with certain signal parameters are adjusted to minimize the error.

Optimal weights, weighting - the values of the weights when they have converged, i.e., when the error function is at a minimum.

Performance index - the error criterion or criteria used to adjust the values of the weights.

Performance surface - the relationship between the error function and the weighted parameter(s). Often this surface is concave and at the minimum value is the optimized weighting.

Weight - the value of a parameter used to adjust the signal shape. And example is the gain of the signal.

Weighting function - the set of parameter values used to adjust the signal shape.

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The following example illustrates how adaptive signal processing can be used to monitor biomedical events¹. In this case, we would like to track the morphologic changes to a cardiac signal that occur over a series of cardiac cycles by using a template. Each input (signal during one particular cardiac cycle) is matched to the template. The input is a signal taken from the surface of the heart, called an electrogram. The template can be an average signal or sometimes, the first in a series of electrogram signals. The template is adjusted for parameters of shape until it best matches the input signal in terms of the estimated mean squared error. At convergence, the weights (parameter values) are a measure of the difference between template and input signal. Tracked over a sequence of cardiac cycles, one can graph trends in the data based on the changes in signal shape that occur. In our paper¹, the amplitude, time duration, phase lag, and DC bias were weighted. If for example, the time duration of the input electrogram was found to gradually increase over a series of cardiac cycles, it would suggest that the speed of the activation wave-front was slowing at the location of the electrogram recording side, which would tend to broaden the wave-front shape.

Algorithm:

At a given value of the independent variable k (k is time, space, or another variable). For any parameter w, the gradient is the derivative of the MSE with respect to that parameter:

for the desired signal d_k, and estimated signals:

y_k = f (w_k, x_k) (5a)

The above equation states that the gradient is proportional to the product of the error and the derivative of the estimated signal with respect to any given parameter. If the parameter w_k is the signal gain, then the above equations reduce to the Widrow-Hoff LMS algorithm. Now let d_k be a fixed segment of the signal to be used as a template for matching. Equation 1 can then be computed via a time average of the MSE gradient³. Calculating the gradient as in Equation 6, the iterative parameter update i is:

1 - Ciaccio EJ, Dunn SM, Akay M et al. Characterization of Spontaneous Changes in Electrogram Morphology. Ciaccio et al. IEEE Computers in Cardiology 1994:701-704.

2 - Widrow B and McCool JM. A comparison of adaptive algorithms based on the method of steepest descent and random search. IEEE Trans Ant Prop 1976; AP-24:615-636

3 - Verhuecks NAM, van der Elzen HC, Snijders FAM et al. Distal echo cancellation for base band data transmission. IEEE Trans Acoust Speech Sig Proc 1979; ASSP-27:768-781.