A two-dimensional (2D) adaptive filter is very much like a one-dimensional
adaptive filter
An adaptive filter is a system with a linear filter that has a transfer function controlled by variable parameters and a means to adjust those parameters according to an optimization algorithm. Because of the complexity of the optimization algorit ...
in that it is a linear system whose parameters are adaptively updated throughout the process, according to some optimization approach. The main difference between 1D and 2D adaptive filters is that the former usually take as inputs signals with respect to time, what implies in
causality
Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the cau ...
constraints, while the latter handles signals with 2 dimensions, like x-y coordinates in the space domain, which are usually non-causal. Moreover, just like 1D filters, most 2D adaptive filters are
digital filter
In signal processing, a digital filter is a system that performs mathematical operations on a sampled, discrete-time signal to reduce or enhance certain aspects of that signal. This is in contrast to the other major type of electronic filter, t ...
s, because of the complex and iterative nature of the algorithms.
Motivation
The topic of 2D adaptive filters is very important in electrical engineering and
signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniq ...
since these filters have the ability to take into account the nonstationary statistical properties of 2D signals. Adaptive filters find applications in areas such as
Noise cancellation
Active noise control (ANC), also known as noise cancellation (NC), or active noise reduction (ANR), is a method for reducing unwanted sound by the addition of a second sound specifically designed to cancel the first. The concept was first develop ...
,
Signal prediction,
Equalization and
Echo cancellation
Echo suppression and echo cancellation are methods used in telephony to improve voice quality by preventing echo from being created or removing it after it is already present. In addition to improving subjective audio quality, echo suppression i ...
. Examples of applications of 2D adaptive filters include Image Denoising, Motion Tracking, OFDM channel estimation, magnetic recording equalization
Example Application
![Block diagram of 2d adaptive filters](https://upload.wikimedia.org/wikipedia/commons/1/1d/Block_diagram_of_2d_adaptive_filters.jpg)
2D Adaptive Filters can be used to identify systems. The system function of the unknown system is given by
, and
is the system function of the 2D adaptive filter when its output comes to steady. The error signal
between the unknown system output,
, and the adaptive filter output,
, is minimized if the unknown system and known 2D adaptive filter have the same input, and if the resulting outputs are similar. Then, it can be shown that
can be represented by
.
is known as the system identification model of the unknown system.
Problem Statement
![Adaptivefilter1](https://upload.wikimedia.org/wikipedia/commons/2/2e/Adaptivefilter1.png)
In digital signal processing, any
linear shift invariant
Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
system can be represented by the
convolution
In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is ...
of the signal with the filter's
impulse response
In signal processing and control theory, the impulse response, or impulse response function (IRF), of a dynamic system is its output when presented with a brief input signal, called an Dirac delta function, impulse (). More generally, an impulse ...
, given by the expression:
If this system is to model a desired response
, the adaptive system can be obtained by continuously adjusting the weight values according to some cost function