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When to use adaptive filters and where they have been used
The contamination of a signal of interest by other unwanted, often larger, signals or noise is a problem often encountered in many applications. Where the signal and noise occupy fixed and separate frequency bands, conventional linear filters with fixed coefficients are normally used to extract the signal. However, there are many instances when it is necessary for the filter characteristics to be variable, adapted to changing signal characteristics, or to be altered intelligently. In such cases, the coefficients of the filter must vary and cannot be specified in advance. Such is the case where there is a spectral overlap between the signal and noise (see figure bellow) or if the band occupied by the noise is unknown or varies with time.
Typical applications where fixed coefficient filters are inappropriate are the following.
- Electroencephalography (EEG), where artefacts or signal contamination produced by eye movements or blinks is much larger than the genuine electrical activity of the brain and shares the same frequency band with signals of clinical interest. It is not possible to use conventional linear filters to remove the artefacts while preserving the signals of clinical interest.
- Digital communication using a spread spectrum, where a large jamming signal, possibly intended to disrupt communication, could interfere with the desired signal. The interference often occupies a narrow but unknown band within the wideband spectrum, and can only be effectively dealt with adaptively.
- In digital data communication over the telephone channel at a high rate. Signal distortions caused by the poor amplitude and phase response characteristics of the channel lead to pulses representing different digital codes to interfere with each other (intersymbol interference), making it difficult to detect the codes reliably at the receiving end. To compensate for the channel distortions which may be varying with time or of unknown characteristics at the receiving end, adaptive equalization is used.
An adaptive filter has the property that its frequency response is adjustable or modifiable automatically to improve its performance in accordance with some criterion, allowing the filter to adapt to changes in the input signal characteristics. Because of their self-adjusting performance and in-built flexibility, adaptive filters have found use in many diverse applications such as telephone echo cancelling, radar signal processing, navigational systems, equalization of communication channels, and biomedical signal enhancement.
In summary we use adaptive filters
- when it is necessary for the filter characteristics to be variable, adapted to changing conditions,
- when there is spectral overlap between the signal and noise (see Figure 9.1), or
- if the band occupied by the noise is unknown or varies with time. The use of conventional filters in the above cases would lead to unacceptable distortion of the desired signal. There are many other situations, apart from noise reduction, when the use of adaptive filters is appropriate
Concepts of adaptive filtering
An adaptive filter consists of two distinct parts: a digital filter with adjustable coefficients, and an adaptive algorithm which is used to adjust or modify the coefficients of the filter (Figure bellow). Two input signals, y_k and x_b are applied simultaneously to the adaptive filter. The signal y_k is the contaminated signal containing both the desired signal, s_k and the noise, n_k assumed uncorrelated with each other. The signal, x_b is a measure of the contaminating signal which is correlated in some way with n_k . x_k is processed by the digital filter to produce an estimate, ῆ_k of n_k. An estimate of the desired signal is then obtained by subtracting the digital filter output, ῆ_k from the contaminated signal, y_k:
ŝ_k= y_k- ῆ_k= s_k+ n_k- ῆ_k(10.1)
The main objective in noise cancelling is to produce an optimum estimate of the noise in the contaminated signals and hence an optimum estimate of the desired signal. This is achieved by using ŝ_k in a feedback arrangement to adjust the digital filter coefficients, via a suitable adaptive algorithm, to minimize the noise in ŝ_k The output signal, ŝ_k serves two purposes: (i) as an estimate of the desired signal and (ii) as an error signal which is used to adjust the filter coefficients.
Main components of the adaptive filter
In most adaptive systems, the digital filter in Figure 10.2 is realized using a transversal or finite impulse response (FIR) structure (Figure 10.4). Other forms are sometimes used, for example the infinite impulse response (IIR) or the lattice structures, but the FIR structure is the most widely used because of its simplicity and guaranteed stability. For the N-point filter depicted in Figure 10.4, the output is given by
where w_k(i), i = 0, 1, ... , are the adjustable filter coefficients (or weights), and x_k(i) andῆ_k are the input and output of the filter. Figure 10.4 illustrates the single-input, single-output system. In a multiple-input single-output system, the x_k may be simultaneous inputs from N different signal sources.
