Optimal filter

Among matched filter (engl. Matched filter ) is understood in the communications engineering a filter , which is the signal-to-noise ratio (engl. Signal to Noise Ratio , SNR) optimized. The terms correlation filter , signal-adapted filter (SAF) or just adapted filter are also often found in the literature . The optimal filter is used to optimally determine the presence ( detection ) of the amplitude or the position of a known signal shape in the presence of interference ( parameter estimation ).

Problem and task

In signal transmission systems, the problem always arises that the useful signal to be received (e.g. the individual data bit of a sequence, the echo signal from a radar transmitter) is superimposed by a more or less large interference signal . This makes it more difficult to recognize the useful signal in the receiver. In the “normal” (so-called power) receiver, falling below or exceeding an amplitude threshold of the received signal-to-noise mixture is assessed as “no signal” or “signal present”. If the signal is weak, there is always the risk that individual useful signals will not be recognized or that interference signal peaks will be incorrectly interpreted as useful signals.

The fundamental question therefore arises as to the dimensioning of an optimal filter structure of the receiver that filters a useful signal in the noise as well as possible and thus minimizes the probability of errors.

The figure shows a communications system for the transmission of a digital transmission data sequence which is to be transmitted on the left in the picture via the channel with additive white Gaussian noise ( English AWGN channel ). The AWGN channel abstractly represents a transmission channel disturbed by white noise , for example a radio link with severe disturbance. The received signal, which is shown in front of the matched filter and heavily superimposed with noise, then arrives at the receiver. The original transmission signal sequence is no longer recognizable there; direct evaluation of this signal would lead to massive errors.

The highly disturbed received signal is therefore fed to the matched filter, the impulse response of which is optimally adapted to the form of the transmitted pulse shown on the left. This adaptation makes it possible for a signal to be obtained at the output of the filter which already corresponds somewhat better to the original transmission signal sequence. By means of a sampling stage and requantization downstream of the filter on the far right, the original bit sequence of the transmitter at the receiver can be reconstructed from this unambiguously and with a minimal probability of bit errors.

Mathematical basics

The following considerations assume that the structure of the transmitted signal is known to the receiver. It is important that this assumption does not mean that the transmitted message is known - knowing the time function of a data bit does not say anything about the information transmitted in a bit sequence.

Let the expected time-limited useful signal (in the aforementioned sense, for example, a single bit or the echo signal of a radar system) be . It is superimposed by a white noise signal with a spectral power density . The optimal filter structure sought is characterized by its response function to a Dirac impulse . The output of such a filter at the time is then ${\displaystyle s}$ ${\displaystyle n}$ ${\displaystyle N_{0}}$ ${\displaystyle h}$ ${\displaystyle t}$

(1) , ${\displaystyle \!\,y(t)=([s+n]*h)(t)=(s*h)(t)+(n*h)(t)=g(t)+n_{e}(t)}$

where the response of the filter to the useful signal and the response of the filter to the interfering signal represent, each of which results from the convolution operation with the impulse response of the filter: ${\displaystyle g}$ ${\displaystyle s}$ ${\displaystyle n_{e}}$ ${\displaystyle n}$ ${\displaystyle h}$

${\displaystyle g(t)=(s*h)(t)=\int _{-\infty }^{+\infty }h(\tau )s(t-\tau )d\tau }$

${\displaystyle n_{e}(t)=(n*h)(t)=\int _{-\infty }^{+\infty }h(\tau )n(t-\tau )d\tau }$

The first term in (1) obviously describes the useful signal component at the point in time , the second term the interference signal component at the point in time . As a criterion for the security of the useful signal detection, the ratio of the instantaneous power of useful and interference signal components at a time is assumed; At this point in time, the filter output signal is to be sampled and the decision made about any useful signal that may be present. The larger the useful signal component is compared to the interference signal component at the filter output, the greater the detection probability will obviously be. ${\displaystyle g(t)}$ ${\displaystyle t}$ ${\displaystyle n_{e}(t)}$ ${\displaystyle t}$ ${\displaystyle T}$

The power of the useful signal component at the time is . Parseval's theorem applies to the disturbance power ${\displaystyle T}$ ${\displaystyle S=g^{2}(T)}$

(2) ${\displaystyle N=N_{0}\int _{-\infty }^{+\infty }h^{2}(t)dt}$

So the ratio will ${\displaystyle S/N}$

(3) ${\displaystyle {\frac {S}{N}}={\frac {[\int _{-\infty }^{+\infty }h(\tau )s(T-\tau )d\tau ]^{2}}{N_{0}\int _{-\infty }^{+\infty }h^{2}(t)dt}}}$

The energy of the time-limited useful signal is time-invariant ; so it can be written ${\displaystyle E}$

(4) ${\displaystyle E=\int _{-\infty }^{+\infty }s^{2}(t)dt=\int _{-\infty }^{+\infty }s^{2}(T-\tau )d\tau }$

If (3) is expanded with (4), an expression results

(5) ${\displaystyle {\frac {S}{N}}={\frac {E}{N_{0}}}{\frac {[\int _{-\infty }^{+\infty }h(\tau )s(T-\tau )d\tau ]^{2}}{\int _{-\infty }^{+\infty }h^{2}(\tau )d\tau \int _{-\infty }^{+\infty }s^{2}(T-\tau )d\tau }}}$

The right part of the fraction can be interpreted as the square of the correlation factor between the response function of the filter sought and the signal function ( ): ${\displaystyle \rho }$ ${\displaystyle h(t)}$ ${\displaystyle s(t)}$ ${\displaystyle -1\leq \rho \leq +1}$

(6) ${\displaystyle {\frac {S}{N}}={\frac {E}{N_{0}}}(\rho _{hs})^{2}}$

Result

The ratio (called the signal-to-noise ratio or signal-to-noise ratio) is then maximal if is, i.e. if applies ${\displaystyle {\frac {S}{N}}}$ ${\displaystyle \rho =\pm 1}$

${\displaystyle h(t)=k\cdot s(T-t)}$

( - any constant). From this follows the essential statement: In order to obtain maximum detection reliability of the useful signal in the noise, the impulse response of the optimal filter must be the same as the time-mirrored ("backwards running") useful signal function ( adapted filter ). ${\displaystyle k}$ ${\displaystyle h(t)}$ ${\displaystyle s(t)}$

In the noise-free case, this filter would appear as a response to the useful signal for the duration of its autocorrelation function, and its maximum value would be sampled at the point in time (i.e. precisely when the entire energy of the signal has entered the filter). ${\displaystyle T}$ ${\displaystyle T}$

In the case of using the optimal filter (in contrast to the above-mentioned power reception!), The signal form itself is not evaluated in the receiver - which is superfluous, since it was assumed to be known - but its autocorrelation function (hence the name as a correlation filter ) .

This fact allows a further realization of the optimal reception: The complete process of the correlation can also be realized in the receiver, i.e. a multiplication of the incoming signal-interference mixture with the useful signal function known at the location of the receiver and subsequent integration and sampling. However, this is only recommended if the expected time of the useful signal is known.

Another essential finding from the optimal filter condition is the initially astonishing fact that only the energy of the incoming (and thus also of the transmitted) useful signal determines the value and thus the detection reliability (but only if an optimal filter is actually used). Time course, frequency spectrum , signal bandwidth or other parameters can be freely selected without violating the optimal condition as required by the transmission system. ${\displaystyle S/N}$

On the basis of this statement, it is possible, for example, to use a much wider (and therefore more energetic) structured transmit pulse instead of an always power-limited, narrow individual pulse in a radar system , provided that its autocorrelation function has a single narrow maximum and rapidly decaying values ​​beyond . ${\displaystyle \tau =0}$

One of the first publications on the analysis of adapted filters, here already applied to radar signals , comes from Dwight O. North (1943).

Optimal filter as least squares method

The matched filter can be derived in various ways, but it is in particular also a special case of a least squares method . With this, the matched filter can be also as a maximum likelihood method ( English ML estimation ) in connection with Gaussian noise (and the corresponding Whittle likelihood ). If the transmitted signal had no unknown parameters (e.g. arrival time, amplitude, phase, ...), then according to the Neyman-Pearson lemma , the optimal filter (in the case of Gaussian noise) would minimize the error probability. Since the signal usually has unknown parameters to be estimated, the optimal filter, as ML detection statistic , represents a generalized likelihood quotient test statistic . From this it follows in particular that the error probability (in the sense of Neyman and Pearson) is not necessarily minimal. When designing an optimal filter, a known noise spectrum is also assumed. In fact, however, the spectrum is usually estimated from the relevant data and is actually only known with limited precision. The optimal filter can be generalized to an iterative method for the case of an only imprecisely known spectrum.

literature

• Jens-Rainer Ohm , Hans Dieter Lüke: Signal transmission: Basics of digital and analog communication systems . 10th edition. Springer, Berlin 2007, ISBN 3-540-69256-8 .
• PM Woodward: Probability and information theory with applications to radar . Pergamon Press, London 1953.

Individual evidence

1. ^ DO North: Analysis of the factors which determine signal / noise discrimination in radar . In: Report PPR-6C, RCA Laboratories, Princeton, NJ . 1943. Reprint: DO North: An Analysis of the factors which determine signal / noise discrimination in pulsed-carrier systems . In: Proceedings of the IEEE . tape
    51 , no. 7 , 1963, pp. 1016-1027 . ET Jaynes: Probability theory: The logic of science . Cambridge University Press, Cambridge 2003, Chapter 14.6.1 The classical matched filter .
   
2. ↑ GL Turin: An introduction to matched filters . In: IRE Transactions on Information Theory . tape 6 , no. 3 , June 1960, p. 311-329 , doi : 10.1109 / TIT.1960.1057571 .
3. ↑ N. Choudhuri, S. Ghosal, Roy, A .: Contiguity of the Whittle measure for a Gaussian time series . In: Biometrika . tape 91 , no. 4 , 2004, p. 211-218 , doi : 10.1093 / biomet / 91.1.211 .
4. ^ J. Neyman, ES Pearson: On the problem of the most efficient tests of statistical hypotheses . In: Philosophical Transactions of the Royal Society of London, Series A . tape 231 , 1933, pp. 289-337 , doi : 10.1098 / rsta.1933.0009 .
5. ^ AM Mood, FA Graybill, DC Boes: Introduction to the theory of statistics . 3. Edition. McGraw-Hill, New York.
6. ^ PD Welch: The use of Fast Fourier Transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms . In: IEEE Transactions on Audio and Electroacoustics . AU-15, no. 2 , June 1967, p. 70-73 , doi : 10.1109 / TAU.1967.1161901 .
7. ↑ C. Röver: Student-based filter for robust signal detection . In: Physical Review D . tape 84 , no. December 12 , 2011, p. 122004 , doi : 10.1103 / PhysRevD.84.122004 , arxiv : 1109.0442 .