In
signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
, a matched filter is obtained by
correlating a known delayed
signal, or ''template'', with an unknown signal to detect the presence of the template in the unknown signal.
This is equivalent to
convolving the unknown signal with a
conjugated time-reversed version of the template. The matched filter is the optimal
linear filter
Linear filters process time-varying input signals to produce output signals, subject to the constraint of linearity. In most cases these linear filters are also time invariant (or shift invariant) in which case they can be analyzed exactly using ...
for maximizing the
signal-to-noise ratio (SNR) in the presence of additive
stochastic
Stochastic (, ) refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themselv ...
noise
Noise is unwanted sound considered unpleasant, loud or disruptive to hearing. From a physics standpoint, there is no distinction between noise and desired sound, as both are vibrations through a medium, such as air or water. The difference aris ...
.
Matched filters are commonly used in
radar
Radar is a detection system that uses radio waves to determine the distance (''ranging''), angle, and radial velocity of objects relative to the site. It can be used to detect aircraft, Marine radar, ships, spacecraft, guided missiles, motor v ...
, in which a known signal is sent out, and the reflected signal is examined for common elements of the out-going signal.
Pulse compression is an example of matched filtering. It is so called because the impulse response is matched to input pulse signals. Two-dimensional matched filters are commonly used in
image processing
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...
, e.g., to improve the SNR of X-ray observations.
Matched filtering is a demodulation technique with
LTI (linear time invariant) filters to maximize SNR.
It was originally also known as a ''North filter''.
Derivation
Derivation via matrix algebra
The following section derives the matched filter for a
discrete-time system
In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled.
Discrete time
Discrete time views values of variables as occurring at distinct, separate "po ...
. The derivation for a
continuous-time system is similar, with summations replaced with integrals.
The matched filter is the linear filter,
, that maximizes the output
signal-to-noise ratio.
:
where