Adaptive filter

An adaptive filter is a filter that "self-adjusts" its transfer function according to an optimizing algorithm. Because of the complexity of the optimizing algorithms, most adaptive filters are digital filters that perform digital signal processing and adapt their performance based on the input signal. By way of contrast, a non-adaptive filter has static filter coefficients (which collectively form the transfer function).

For some applications, adaptive coefficients are required since some parameters of the desired processing operation (for instance, the properties of some noise signal) are not known in advance. In these situations it is common to employ an adaptive filter, which uses feedback to refine the values of the filter coefficients and hence its frequency response.

Generally speaking, the adapting process involves the use of a cost function, which is a criterion for optimum performance of the filter (for example, minimizing the noise component of the input), to feed an algorithm, which determines how to modify of the filter coefficients to minimize the cost on the next iteration.

As the power of digital signal processors has increased, adaptive filters have become much more common and are now routinely used in devices such as mobile phones and other communication devices, camcorders and digital cameras, and medical monitoring equipment.

Example

Suppose a hospital is recording a heart beat (an ECG), which is being corrupted by a 50 Hz noise (the frequency coming from the power supply in many countries).

One way to remove the noise is to filter the signal with a notch filter at 50 Hz. However, due to slight variations in the power supply to the hospital, the exact frequency of the power supply might (hypothetically) wander between 47 Hz and 53 Hz. A static filter would need to remove all the frequencies between 47 and 53 Hz, which could excessively degrade the quality of the ECG since the heart beat would also likely have frequency components in the rejected range.

To circumvent this potential loss of information, an adaptive filter could be used. The adaptive filter would take input both from the patient and from the power supply directly and would thus be able to track the actual frequency of the noise as it fluctuates. Such an adaptive technique generally allows for a filter with a smaller rejection range, which means, in our case, that the quality of the output signal is more accurate for medical diagnoses.

Block diagram

The block diagram, shown in the following figure, serves as a foundation for particular adaptive filter realisations, such as Least Mean Squares (LMS) and Recursive Least Squares (RLS). The idea behind the block diagram is that a variable filter extracts an estimate of the desired signal.:To start the discussion of the block diagram we take the following assumptions:
*The input signal is the sum of a desired signal $d(n)$ and interfering noise $v(n)$:$x(n)\; =\; d(n)+v(n)$
*The variable filter has a Finite Impulse Response (FIR) structure. For such structures the impulse response is equal to the filter coefficients. The coefficients for a filter of order $p$ are defined as:$mathbf\{w\}\_\{n\}=left\; [w\_\{n\}(0),,w\_\{n\}(1),,\; ...,,w\_\{n\}(p)\; ight]\; ^\{T\}$.
*The error signal or cost function is the difference between the desired and the estimated signal:$e(n)\; =\; d(n)-hat\{d\}(n)$

The variable filter estimates the desired signal by convolving the input signal with the impulse response. In vector notation this is expressed as:$hat\{d\}(n)\; =\; mathbf\{w\}\_\{n\}^\{T\}mathbf\{x\}(n)$where :$mathbf\{x\}(n)=left\; [x(n),,x(n-1),,...,,x(n-p)\; ight]\; ^\{T\}$is an input signal vector. Moreover, the variable filter updates the filter coefficients at every time instant :$mathbf\{w\}\_\{n+1\}\; =\; mathbf\{w\}\_\{n\}+Deltamathbf\{w\}\_\{n\}$where $Deltamathbf\{w\}\_\{n\}$ is a correction factor for the filter coefficients. The adaptive algorithm generates this correction factor based on the input and error signals. LMS and RLS define two different coefficient update algorithms.

Applications of adaptive filters

* Noise cancellation
* Signal prediction
* Adaptive feedback cancellation
* Echo cancellation

Filter implementations

*Least mean squares filter
*Recursive least squares filter

References

* Monson H. Hayes "Statistical Digital Signal Processing and Modeling," Wiley, 1996, ISBN 0-471-59431-8
* Simon Haykin "Adaptive Filter Theory," Prentice Hall, 2002, ISBN 0-13-048434-2

ee also

*Kalman filter
*Wiener filter
*linear prediction
*filter (signal processing)