A matched filter is a linear optimal filter constructed on the basis of the known spectral characteristics of the useful signal and noise. Consistent filters are designed to distinguish signals of a known shape from noise. Optimality refers to the maximum signal-to-noise ratio at the filter output, while the waveform changes when passing through the filter.
Integrated Frequency Response
In order for the circuit to be used as a matched filter, the CFC of this circuit must be equal to the product of the expression complex conjugate with the expression describing the spectrum of the detected signal, exponentials of degree -j * ω * t0 and some coefficient, and t0 is the time to reach the maximum output filter signal (otherwise - signal observation time). In other words, the impulse response of such a filter is a copy of the input signal reflected relative to zero, shifted by the signal duration t0 and, in the general case, normalized to the coefficient indicated above.
Output Waveform
The output signal reaches its maximum at the end of the input signal, then during a time equal to the duration of the input signal, it decreases to zero.
Some special cases of input signals
For the input signal - a rectangular pulse, the output signal has the shape of a triangle with a vertex at the time the input pulse ends. For the input radio pulse (a section of a sinusoid with an envelope-rectangle), the output signal has the form of a sinusoid with an envelope, a form similar to the previous case.
Complex signals passing through a matched filter lose modulation and are compressed in time.
Let the shape of the processed signal be known in advance, and we need to determine only the fact of the presence of the signal against the background of noise . In this case, the filter should instead of preserving the waveform provide its maximum (compared with noise) output level. In this case, the signal-to-noise ratio , defined as:
SNR = where - standard deviation of the interference.
Consistent filtering has found widespread use in radar since the filter can be tuned to a pre-known type of received signal.
Consistent filtering when using a linear-frequency modulated LFM signal (used in radar ) allows to obtain a gain in SNR of up to 18 decibels .
A consistent filter is a device that maximizes the SNR at the output.