Adaptive filter

An adaptive filter in signal processing is a special analog filter or digital filter that can independently change its transfer function and frequency during operation.

Block diagram of an adaptive filter

For this purpose, an FIR filter is usually provided with a setting network that can change the filter coefficients of the transversal filter according to certain rules .

function

The sequence arriving at the input passes through the adjustable transversal filter and forms the output sequence . This sequence is compared with the sequence to be formed by the filter and supplies the error signal . Ideally, i. H. if the filter changes the input sequence exactly in such a way that it becomes the same , the error sequence is 0. If there is a discrepancy, the error signal has values ​​other than zero. The control algorithm then tries to minimize the error signal by changing the filter coefficients, that is to say to adapt the output sequence of the filter to the reference signal as well as possible. The current setting algorithm is based on a method for minimizing the error squares, the LMS algorithm . Better results are achieved with mathematically more complex, recursive methods such as the RLS algorithm . ${\displaystyle \mathbf {x} (n)}$ ${\displaystyle {\hat {d}}(n)}$ ${\displaystyle \mathbf {d} (n)}$ ${\displaystyle \mathbf {e} (n)}$ ${\displaystyle {\hat {d}}(n)}$ ${\displaystyle \mathbf {d} (n)}$ ${\displaystyle \mathbf {e} (n)}$ ${\displaystyle {\hat {d}}(n)}$ ${\displaystyle \mathbf {d} (n)}$

Application for channel equalization

Adaptive filters are used in signal processing for adaptive equalization of an input signal. Transmission channels such as an electrical line or a radio channel distort the signal to be transmitted. To minimize transmission errors, it may therefore be necessary on the receiving side to compensate for these distortions as well as possible. The input sequence represents the data sequence transmitted via a distorting channel. Since the receiver does not know how the channel changes the data sequence, a so-called training sequence that has been agreed and known between the transmitter and the receiver must be transmitted before user data transmission begins. This training process is sometimes incorrectly referred to as synchronization . With the help of the training sequence, which is denoted by in the above figure and which should also have certain properties such as a spectrally uniform distribution, the adaptive filter on the receiving side can be set so that the error signal is minimal. In this case the filter compensates for the distortion of the transmission channel. Under the mostly correct assumption that the channel properties regarding distortion do not change very quickly over time, the filter set in this way can then be used for channel equalization. ${\displaystyle \mathbf {x} (n)}$ ${\displaystyle \mathbf {d} (n)}$ ${\displaystyle \mathbf {e} (n)}$

If there is a slow increase in the error rate at the receiver during operation, as a result of changing channel transmission properties, you can repeatedly switch to training mode and set the adaptive filter to the changed conditions in the channel. There are also methods for channel equalization in which an ongoing adaptation to the slightly variable channel properties can take place during operation without switching.

There are further applications of adaptive filters in the area of echo cancellation and for the simulation of unknown transfer functions. As a practical application example, modems should be mentioned here, which use adaptive filters for channel equalization and echo compensation for more complex modulations.

See also

• Adaptive prediction

literature

• Simon Haykin, Adaptive Filter Theory 5th edition, international edition. Pearson Education 2013, ISBN 0-273-76408-X .
• Moschytz, G., Hofbauer, M., Adaptive Filters. Springer-Verlag Berlin Heidelberg, Berlin a. a. 2000. ISBN 978-3-540-67651-5 , 978-3-642-18250-1
• B. Widrow, S. Stearns, Adaptive Signal Processing , Prentice Hall, New Jersey 1985
• Bernhard Wirnitzer, Christian Schönig, Wolfram Seipp, in Spürnase , ELRAD 1994, issue 8, Heise Verlag, p. 28 ff. Plus the following issues