An adaptive filter is a digital filter that performs digital signal processing and can adapt its performance based on the input signal. By way of contrast, a non-adaptive filter has static filter coefficients (which collectively form the transfer function). An FIR filter In electronics, a digital filter is any electronic filter that works by performing digital math operations on an intermediate form of a signal. ... Digital signal processing (DSP) is the study of signals in a digital representation and the processing methods of these signals. ... A transfer function is a mathematical representation of the relation between the input and output of a linear time-invariant system. ...

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. Optimization is a branch of mathematics which is concerned with finding maxima and minima of real-valued functions. ...

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. A digital signal processor (DSP) is a specialized microprocessor designed specifically for digital signal processing, generally in real-time. ...

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. Least mean squares (LMS) algorithms are used in adaptive filters to find the filter coefficients that relate to producing the least mean squares of the error signal (difference between the desired and the actual signal). ... Recursive least squares (RLS) algorithm is used in adaptive filters to find the filter coefficients that relate to producing the recursively least squares of the error signal (difference between the desired and the actual signal) // Discussion The idea behind RLS filters is to minimize a weighted least squares error function. ...

To start the discussion of the block diagram we take the following assumptions: Image File history File links Download high resolution version (2275x884, 32 KB) Licensing I, the creator of this work, hereby grant the permission to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1. ...

• 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

.

• The error signal or cost function is the difference between the desired and the estimated signal

The variable filter estimates the desired signal by convolving the input signal with the impulse response. In vector notation this is expressed as A finite impulse response (FIR) filter is a type of a digital filter. ... Optimization is a branch of mathematics which is concerned with finding maxima and minima of real-valued functions. ...

where

is an input signal vector. Moreover, the variable filter updates the filter coefficients at every time instant

where 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.

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). A physician visiting the sick in a hospital. ... The heart and lungs, from an older edition of Grays Anatomy. ... ECG may also refer to the East Coast Greenway Lead II An Electrocardiogram (ECG or EKG, abbreviated from the German Elektrokardiogramm) is a graphic produced by an electrocardiograph, which records the electrical voltage in the heart in the form of a continuous strip graph. ... The hertz (symbol: Hz) is the SI unit of frequency. ... A power supply unit (sometimes abbreviated power supply or PSU) is a device or system that supplies electrical or other types of energy to an output load or group of loads. ...

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. A notch filter, also called a band-stop filter, sometimes a narrow band-pass filter, or T-notch filter, is an electronic filter typically used when the high frequency and the low frequency are less than 1 to 2 decades apart (that is, the high frequency is less than 10...

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 diagnosis.

Applications of adaptive filters

• Channel equalization
• Channel identification
• Noise cancellation
• Signal prediction
• Adaptive Feedback Cancellation

Active noise control (also known as noise cancellation or antinoise) is a method for preventing unwanted sound. ...

Filter implementations

• Least mean squares filter
• Recursive least squares filter

Least mean squares (LMS) algorithms are used in adaptive filters to find the filter coefficients that relate to producing the least mean squares of the error signal (difference between the desired and the actual signal). ... Recursive least squares (RLS) algorithm is used in adaptive filters to find the filter coefficients that relate to producing the recursively least squares of the error signal (difference between the desired and the actual signal) // Discussion The idea behind RLS filters is to minimize a weighted least squares error function. ...

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

See also

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