Matched filter

Under optimal filter ( engl. matched filter ) is meant optimized in communications technology, a filter, which is the signal -to-noise ratio ( engl. signal to noise ratio, SNR). In the literature, often finds the correlation filter names, signal - matched filter (SAF ) or only matched filter. The matched filter is used for optimal determination of the presence (detection ) of the amplitude or the position of a known waveform in the presence of disturbances ( parameter estimation ).

Problem and task

In signal transmission systems always arises a problem that the desired signal to be received (e.g., the single data bit of a sequence, the echo signal from a radar transmitter ) is overlapped by a more or less large interference signal. Characterized the detection of the useful signal is difficult in the receiver. In the "normal " ( so-called power) the receiver falls below or exceeds a threshold amplitude of the received signal -to-noise mixture as a " no signal " or " signal present" counted. If the signal is weak, there is always the danger that individual user signals not recognized or noise spikes being interpreted as useful signals.

This raises the fundamental question of the dimensions of an optimal structure of the receiver filter, which filters signals within the noise as well as possible, thus minimizing the probability of error.

The figure shows a message- technical system for transmitting a digital transmission data sequence which is to be transferred to the left in the picture on the AWGN channel. The AWGN channel is abstractly represents a disturbed with white noise transmission channel, such as a highly disturbed radio link. At the receiver, then comes the front of the matched filter shown strong with noise superimposed on the received signal. In it the original transmission signal sequence is no longer recognizable, it would apply in the direct evaluation of this signal to massive errors.

The strong interference received signal is thus supplied to the matched filter, which is optimally adapted to its impulse response of the transmit pulse shape shown on the left. By this adjustment, it is possible that the output of the filter, a signal can be obtained which corresponds to the original transmission signal sequence has a bit better. Can be reconstructed at the receiver through a filter right outside nachgeschaltene sampling stage and requantization it clearly and with minimum bit error probability, the original bit sequence of the transmitter.

Mathematical Foundations

The following considerations assume that the structure of the transmitted signal at the receiver is known. It is important that this assumption does not mean that the transmitted message is known - the knowledge of the time function of a data still says nothing about the information transmitted in a bit string.

The expected time-limited signal level ( in the sense mentioned about an individual bit or the echo signal of a radar system ) is. It was superimposed on a white noise signal having a power spectral density. The sought optimal filter structure is characterized by its response function to a Dirac pulse. The output signal of such a filter at the time is then

The response of the filter on the desired signal and the response of the filter to the interference signal representing the result from the convolution with the impulse response of the filter in each case:

The first term in (1) apparently describe the useful signal at the time, the second term is the noise signal at the time. The criterion for the safety of Nutzsignalerkennung the ratio of the instantaneous power of useful and interference signal is provided at a time; At this time, the filter output is sampled and the decision can be made about an existing payload. The larger the useful signal relative to the interference signal at the filter output, the larger the probability of detection will be apparent.

The power of the wanted signal at the time is. For the interference power applies to the Parseval theorem

The ratio is therefore

The energy of the time-limited useful signal is invariant to a time shift; it can therefore be written

(3) is expanded with (4 ), an expression evaluates to

The right part of the fraction can be interpreted as a square of the correlation factor between the response function of this filter and the signal () function:

Result

The ratio ( called signal -to-noise ratio or signal - to-noise ratio ) is at a maximum when, if so applies

( - Arbitrary constant ). It follows the main statement: In order to obtain maximum detection reliability of the useful signal in the noise, the desired impulse response of the optimal filter Nutzsignalfunktion must be equal to the time- mirrored ( " running backwards " ) be ( matched filter ).

In the noise-free case, in this filter would appear as a response to the information signal the duration of the autocorrelation function, and the time (that is, even when the total energy of the signal has entered into the filter ), which maximum value can be sampled.

In case of application of the optimal filter, ie (as opposed to the above- mentioned performance reception! ) Is not evaluated in the receiver, the waveform itself - which is unnecessary so, since it was assumed to be known - but its autocorrelation function (hence the designation as a correlation filter).

This fact is a further realization of the optimum reception to: In the receiver, and the full process of the correlation can be realized, that is, multiplying the incoming signal with the Störgemischs Nutzsignalfunktion at the location of the receiver so known and subsequent integration and sampling. This is recommended only if the expected timing of the desired signal is known.

Another significant finding from the matched filter condition is the first amazing fact that only the energy of the incoming (and thus also of the transmitted ) useful signal ( however just only when actually used a matched filter ) the value and thus the reliability of detection determined. Timing, frequency spectrum, signal bandwidth and other parameters can be chosen freely according to need of the transmission system without violating the optimum condition.

On the basis of this statement, it is for example possible to use a much wider (and therefore higher energy ) structured transmission pulse to be used instead of a more power-limited narrow single pulse in a radar system, if only the autocorrelation function has a single narrow peak and fast decaying values ​​beyond.

A first publication to analyze matched filter, this already applied to radar signals, derived from Dwight O. North in 1943.