Two-dimensional matched filters are commonly used in image processing, e.g., to improve the SNR of X-ray observations.
We can derive the linear filter that maximizes output signal-to-noise ratio by invoking a geometric argument.
, the inner product of our filter and the observed signal such that We now define the signal-to-noise ratio, which is our objective function, to be the ratio of the power of the output due to the desired signal to the power of the output due to the noise: We rewrite the above: We wish to maximize this quantity by choosing
This concept returns to the intuition behind the matched filter: this upper bound is achieved when the two vectors
, for which we can solve: yielding giving us our normalized filter, If we care to write the impulse response
of the filter for the convolution system, it is simply the complex conjugate time reversal of the input
Though we have derived the matched filter in discrete time, we can extend the concept to continuous-time systems if we replace
It can be shown that this eigenvalue equals yielding the following optimal matched filter This is the same result found in the previous subsection.
Matched filtering can also be interpreted as a least-squares estimator for the optimal location and scaling of a given model or template.
and the square within the sum yields The first term in brackets is a constant (since the observed signal is given) and has no influence on the optimal solution.
: The numerator can be upper-bounded by means of the Cauchy–Schwarz inequality: The optimization problem assumes its maximum when equality holds in this expression.
[5] If the transmitted signal possessed no unknown parameters (like time-of-arrival, amplitude,...), then the matched filter would, according to the Neyman–Pearson lemma, minimize the error probability.
What is commonly referred to as the Signal-to-noise ratio (SNR), which is supposed to be maximized by a matched filter, in this context corresponds to
For the case of an uncertain spectrum, the matched filter may be generalized to a more robust iterative procedure with favourable properties also in non-Gaussian noise.
[7] When viewed in the frequency domain, it is evident that the matched filter applies the greatest weighting to spectral components exhibiting the greatest signal-to-noise ratio (i.e., large weight where noise is relatively low, and vice versa).
In general this requires a non-flat frequency response, but the associated "distortion" is no cause for concern in situations such as radar and digital communications, where the original waveform is known and the objective is the detection of this signal against the background noise.
On the technical side, the matched filter is a weighted least-squares method based on the (heteroscedastic) frequency-domain data (where the "weights" are determined via the noise spectrum, see also previous section), or equivalently, a least-squares method applied to the whitened data.
To judge the distance of the object, we correlate the received signal with a matched filter, which, in the case of white (uncorrelated) noise, is another pure-tone 1-Hz sinusoid.
When the output of the matched filter system exceeds a certain threshold, we conclude with high probability that the received signal has been reflected off the object.
Using the speed of propagation and the time that we first observe the reflected signal, we can estimate the distance of the object.
To do so, we may correlate the received signal with several matched filters of sinusoids at varying frequencies.
The matched filter with the highest output will reveal, with high probability, the frequency of the reflected signal and help us determine the radial velocity of the object, i.e. the relative speed either directly towards or away from the observer.
complex-valued numbers corresponding to the relative amplitudes and phases of the sinusoidal components (see Moving target indication).
If the receiver were to sample this signal at the correct moments, the resulting binary message could be incorrect.
Precisely, the impulse response of the ideal matched filter, assuming white (uncorrelated) noise should be a time-reversed complex-conjugated scaled version of the signal that we are seeking.
[9] The first observation of gravitational waves was based on large-scale filtering of each detector's output for signals resembling the expected shape, followed by subsequent screening for coincident and coherent triggers between both instruments.
[11][12] Inference on the astrophysical source parameters was completed using Bayesian methods based on parameterized theoretical models for the signal waveform and (again) on the Whittle likelihood.
[13][14] Matched filters find use in seismology to detect similar earthquake or other seismic signals, often using multicomponent and/or multichannel empirically determined templates.
[15] Matched filtering applications in seismology include the generation of large event catalogues to study earthquake seismicity [16] and volcanic activity,[17][18] and in the global detection of nuclear explosions.
[20] Sensors that perceive the world "through such a 'matched filter' severely limits the amount of information the brain can pick up from the outside world, but it frees the brain from the need to perform more intricate computations to extract the information finally needed for fulfilling a particular task.