Nonlinear Filter - Frequency Domain

Frequency Domain

In signal processing one often deals with obtaining an input signal and processing it into an output signal. At times the signal may be transmitted through a channel which corrupts it, resulting in a noisy output. As a consequence, the user at the output end has to attempt to reconstruct the original signal given the noisy one. When the noise is additive, i.e. it is added to the signal (rather than multiplied for example) and the statistics of the noise process are known to follow the gaussian statistical law, then a linear filter is known to be optimal under a number of possible criteria (for example the mean square error criterion, aiming at minimizing the variance of the error). This optimality is one of the main reasons why linear filters are so important in the history of signal processing.

However, in several cases one cannot find an acceptable linear filter, either because the noise is non-additive or non-gaussian. For example, linear filters can remove additive high frequency noise if the signal and the noise do not overlap in the frequency domain. Still, in two-dimensional signal processing the signal may have important and structured high frequency components, like edges and small details in image processing. In this case a linear lowpass filter would blur sharp edges and yield bad results. Nonlinear filters should be used instead.

Nonlinear filters locate and remove data that is recognised as noise. The algorithm is 'nonlinear' because it looks at each data point and decides if that data is noise or valid signal. If the point is noise, it is simply removed and replaced by an estimate based on surrounding data points, and parts of the data that are not considered noise are not modified at all. Linear filters, such as those used in bandpass, highpass, and lowpass, lack such a decision capability and therefore modify all data. Nonlinear filters are sometimes used also for removing very short wavelength, but high amplitude features from data. Such a filter can be thought of as a noise spike-rejection filter, but it can also be effective for removing short wavelength geological features, such as signals from surficial features.

Examples of nonlinear filters include:

• phase-locked loops
• detectors
• mixers
• median filters
• ranklets

When moving to the time domain description of a system, in state space form, there is the possibility of describing more effectively the dynamics of a system, considering also the case where the time-evolution of the system is described by non-linear differential (or difference in discrete time) equations, or the case where the observations are a known nonlinear function of the signal then perturbed by additive noise. These are possible further sources of non-linearity, besides non-additive or non-gaussian noise, for which in most cases we have to move into the state space description of a system and abandon the frequency domain formulation.