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{{Short description|Filters used in signal processing that are optimal in some sense.}}In signal processing, the output of the matched filter is given by correlating a known delayed signal, or template, with an unknown signal to detect the presence of the template in the unknown signal.BOOK, Woodward, Philip Woodward, P. M., Probability and information theory with applications to radar, Pergamon Press, London, 1953, JOURNAL, Turin, G. L., An introduction to matched filters, IRE Transactions on Information Theory, 6, 3, 1960, 311–329, 10.1109/TIT.1960.1057571, 5128742,weblink 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 for maximizing the signal-to-noise ratio (SNR) in the presence of additive stochastic noise.Matched filters are commonly used in radar, 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, 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.WEB,weblink Demodulation, OpenStax CNX, en-US, 2017-04-18, It was originally also known as a North filter.After D.O. North who was among the first to introduce the concept: JOURNAL, An analysis of the factors which determine signal/noise discrimination in pulsed carrier systems, North, D. O., Report PPR-6C, RCA Laboratories, Princeton, NJ, 1943, Re-print: JOURNAL, North, D. O., An analysis of the factors which determine signal/noise discrimination in pulsed-carrier systems, Proceedings of the IEEE, 51, 7, 1963, 1016–1027, 10.1109/PROC.1963.2383, See also: BOOK, Jaynes, Edwin Thompson Jaynes, E. T., Probability theory: The logic of science, Cambridge University Press, Cambridge, 2003, 14.6.1 The classical matched filter,

Derivation

Derivation via matrix algebra

The following section derives the matched filter for a discrete-time system. The derivation for a continuous-time system is similar, with summations replaced with integrals.The matched filter is the linear filter, h, that maximizes the output signal-to-noise ratio. where x[k] is the input as a function of the independent variable k, and y[n] is the filtered output. Though we most often express filters as the impulse response of convolution systems, as above (see LTI system theory), it is easiest to think of the matched filter in the context of the inner product, which we will see shortly.We can derive the linear filter that maximizes output signal-to-noise ratio by invoking a geometric argument. The intuition behind the matched filter relies on correlating the received signal (a vector) with a filter (another vector) that is parallel with the signal, maximizing the inner product. This enhances the signal. When we consider the additive stochastic noise, we have the additional challenge of minimizing the output due to noise by choosing a filter that is orthogonal to the noise.Let us formally define the problem. We seek a filter, h, such that we maximize the output signal-to-noise ratio, where the output is the inner product of the filter and the observed signal x.Our observed signal consists of the desirable signal s and additive noise v: Let us define the auto-correlation matrix of the noise, reminding ourselves that this matrix has Hermitian symmetry, a property that will become useful in the derivation: where v^mathrm{H} denotes the conjugate transpose of v, and E denotes expectation (note that in case the noise v has zero-mean, its auto-correlation matrix R_v is equal to its covariance matrix).Let us call our output, y, 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 h. Expanding the denominator of our objective function, we have Now, our mathrm{SNR} becomes We will rewrite this expression with some matrix manipulation. The reason for this seemingly counterproductive measure will become evident shortly. Exploiting the Hermitian symmetry of the auto-correlation matrix R_v, we can write We would like to find an upper bound on this expression. To do so, we first recognize a form of the Cauchy–Schwarz inequality: which is to say that the square of the inner product of two vectors can only be as large as the product of the individual inner products of the vectors. This concept returns to the intuition behind the matched filter: this upper bound is achieved when the two vectors a and b are parallel. We resume our derivation by expressing the upper bound on our mathrm{SNR} in light of the geometric inequality above: Our valiant matrix manipulation has now paid off. We see that the expression for our upper bound can be greatly simplified: We can achieve this upper bound if we choose, where alpha is an arbitrary real number. To verify this, we plug into our expression for the output mathrm{SNR}: Thus, our optimal matched filter is We often choose to normalize the expected value of the power of the filter output due to the noise to unity. That is, we constrain This constraint implies a value of alpha, for which we can solve: yielding giving us our normalized filter, If we care to write the impulse response h of the filter for the convolution system, it is simply the complex conjugate time reversal of the input s.Though we have derived the matched filter in discrete time, we can extend the concept to continuous-time systems if we replace R_v with the continuous-time autocorrelation function of the noise, assuming a continuous signal s(t), continuous noise v(t), and a continuous filter h(t).

Derivation via Lagrangian

Alternatively, we may solve for the matched filter by solving our maximization problem with a Lagrangian. Again, the matched filter endeavors to maximize the output signal-to-noise ratio (mathrm{SNR}) of a filtered deterministic signal in stochastic additive noise. The observed sequence, again, is with the noise auto-correlation matrix, The signal-to-noise ratio is where y_s = h^mathrm{H} s and y_v = h^mathrm{H} v.Evaluating the expression in the numerator, we have and in the denominator, The signal-to-noise ratio becomes If we now constrain the denominator to be 1, the problem of maximizing mathrm{SNR} is reduced to maximizing the numerator. We can then formulate the problem using a Lagrange multiplier: which we recognize as a generalized eigenvalue problem Since s s^mathrm{H} is of unit rank, it has only one nonzero eigenvalue. It can be shown that this eigenvalue equals yielding the following optimal matched filter This is the same result found in the previous subsection.

Interpretation as a least-squares estimator

Derivation

Matched filtering can also be interpreted as a least-squares estimator for the optimal location and scaling of a given model or template. Once again, let the observed sequence be defined as where v_k is uncorrelated zero mean noise. The signal s_k is assumed to be a scaled and shifted version of a known model sequence f_k: We want to find optimal estimates j^* and mu^* for the unknown shift j_0 and scaling mu_0 by minimizing the least-squares residual between the observed sequence x_k and a "probing sequence" h_{j-k}: The appropriate h_{j-k} will later turn out to be the matched filter, but is as yet unspecified. Expanding x_k 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 last term has constant expected value because the noise is uncorrelated and has zero mean. We can therefore drop both terms from the optimization. After reversing the sign, we obtain the equivalent optimization problem Setting the derivative w.r.t. mu to zero gives an analytic solution for mu^*: Inserting this into our objective function yields a reduced maximization problem for just j^*: 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. According to the properties of the Cauchy–Schwarz inequality, this is only possible when for arbitrary non-zero constants nu or kappa, and the optimal solution is obtained at j^*=j_0 as desired. Thus, our "probing sequence" h_{j-k} must be proportional to the signal model f_{k-j_0}, and the convenient choice kappa=1 yields the matched filter Note that the filter is the mirrored signal model. This ensures that the operation sum_k x_k h_{j-k} to be applied in order to find the optimum is indeed the convolution between the observed sequence x_k and the matched filter h_k. The filtered sequence assumes its maximum at the position where the observed sequence x_k best matches (in a least-squares sense) the signal model f_k.

Implications

The matched filter may be derived in a variety of ways, but as a special case of a least-squares procedure it may also be interpreted as a maximum likelihood method in the context of a (coloured) Gaussian noise model and the associated Whittle likelihood.JOURNAL, Choudhuri, N., Ghosal, S., Roy, A., Contiguity of the Whittle measure for a Gaussian time series, Biometrika, 91, 4, 2004, 211–218, 10.1093/biomet/91.1.211, free, 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. However, since the exact signal generally is determined by unknown parameters that effectively are estimated (or fitted) in the filtering process, the matched filter constitutes a generalized maximum likelihood (test-) statistic.BOOK, Mood, A. M., Graybill, F. A., Boes, D. C., Introduction to the theory of statistics, 3rd, McGraw-Hill, New York, IX. Tests of hypotheses, 1974, The filtered time series may then be interpreted as (proportional to) the profile likelihood, the maximized conditional likelihood as a function of the time parameter.JOURNAL, Röver, C., Student-t based filter for robust signal detection, Physical Review D, 84, 12, 2011, 122004, 10.1103/PhysRevD.84.122004, 1109.0442, 2011PhRvD..84l2004R, This implies in particular that the error probability (in the sense of Neyman and Pearson, i.e., concerning maximization of the detection probability for a given false-alarm probabilityJOURNAL, Neyman, J., Pearson, E. S., On the problem of the most efficient tests of statistical hypotheses, Philosophical Transactions of the Royal Society of London A, 231, 694–706, 1933, 289–337, 10.1098/rsta.1933.0009, 1933RSPTA.231..289N, free, ) is not necessarily optimal.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 sqrt{2log(mathcal{L})}, where mathcal{L} is the (conditionally) maximized likelihood ratio. {{refn|group=nb|The common reference to SNR has in fact been criticized as somewhat misleading: "The interesting feature of this approach is that theoretical perfection is attained without aiming consciously at a maximum signal/noise ratio. As the matter of quite incidental interest, it happens that the operation [...] does maximize the peak signal/noise ratio, but this fact plays no part whatsoever in the present theory. Signal/noise ratio is not a measure of information [...]." (Woodward, 1953; Sec.5.1).}}The construction of the matched filter is based on a known noise spectrum. In reality, however, the noise spectrum is usually estimated from data and hence only known up to a limited precision. 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.

Frequency-domain interpretation

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.

Examples

Radar and sonar

Matched filters are often used in signal detection. As an example, suppose that we wish to judge the distance of an object by reflecting a signal off it. We may choose to transmit a pure-tone sinusoid at 1 Hz. We assume that our received signal is an attenuated and phase-shifted form of the transmitted signal with added noise.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. If we change the shape of the pulse in a specially-designed way, the signal-to-noise ratio and the distance resolution can be even improved after matched filtering: this is a technique known as pulse compression.Additionally, matched filters can be used in parameter estimation problems (see estimation theory). To return to our previous example, we may desire to estimate the speed of the object, in addition to its position. To exploit the Doppler effect, we would like to estimate the frequency of the received signal. 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. This method is, in fact, a simple version of the discrete Fourier transform (DFT). The DFT takes an N-valued complex input and correlates it with N matched filters, corresponding to complex exponentials at N different frequencies, to yield N complex-valued numbers corresponding to the relative amplitudes and phases of the sinusoidal components (see Moving target indication).

Digital communications

The matched filter is also used in communications. In the context of a communication system that sends binary messages from the transmitter to the receiver across a noisy channel, a matched filter can be used to detect the transmitted pulses in the noisy received signal.(File:Matched Filter Total System.jpg)Imagine we want to send the sequence "0101100100" coded in non polar non-return-to-zero (NRZ) through a certain channel.Mathematically, a sequence in NRZ code can be described as a sequence of unit pulses or shifted rect functions, each pulse being weighted by +1 if the bit is "1" and by -1 if the bit is "0". Formally, the scaling factor for the k^mathrm{th} bit is, begin{cases} +1, & text{if bit } k text{ is } 1, -1, & text{if bit } k text{ is } 0. end{cases}We can represent our message, M(t), as the sum of shifted unit pulses: where T is the time length of one bit and Pi(x) is the rectangular function.Thus, the signal to be sent by the transmitter is400pxIf we model our noisy channel as an AWGN channel, white Gaussian noise is added to the signal. At the receiver end, for a Signal-to-noise ratio of 3 dB, this may look like:400pxA first glance will not reveal the original transmitted sequence. There is a high power of noise relative to the power of the desired signal (i.e., there is a low signal-to-noise ratio). If the receiver were to sample this signal at the correct moments, the resulting binary message could be incorrect.To increase our signal-to-noise ratio, we pass the received signal through a matched filter. In this case, the filter should be matched to an NRZ pulse (equivalent to a "1" coded in NRZ code). 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. We choose In this case, due to symmetry, the time-reversed complex conjugate of h(t) is in fact h(t), allowing us to call h(t) the impulse response of our matched filter convolution system.After convolving with the correct matched filter, the resulting signal, M_mathrm{filtered}(t) is, where * denotes convolution.400pxWhich can now be safely sampled by the receiver at the correct sampling instants, and compared to an appropriate threshold, resulting in a correct interpretation of the binary message.400px

Gravitational-wave astronomy

Matched filters play a central role in gravitational-wave astronomy.JOURNAL, Schutz, B. F., Gravitational wave astronomy, Classical and Quantum Gravity, 16, 12A, 1999, A131–A156, 10.1088/0264-9381/16/12A/307, gr-qc/9911034, 1999CQGra..16A.131S, 19021009, 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.WEB,weblink LIGO: How We Searched For Merging Black Holes And Found GW150914, "A technique known as matched filtering is used to see if there are any signals contained within our data. The aim of matched filtering is to see if the data contains any signals similar to a template bank member. Since our templates should describe the gravitational waveforms for the range of different merging systems that we expect to be able to see, any sufficiently loud signal should be found by this method.", False-alarm rates, and with that, the statistical significance of the detection were then assessed using resampling methods.JOURNAL, Usman, Samantha A., The PyCBC search for gravitational waves from compact binary coalescence, 2016, Class. Quantum Grav., 33, 21, 215004, 10.1088/0264-9381/33/21/215004, 1508.02357, 2016CQGra..33u5004U, 53979477, JOURNAL, The LIGO Scientific Collaboration, the Virgo Collaboration, Abbott, B. P., GW150914: First results from the search for binary black hole coalescence with Advanced LIGO, 2016, Physical Review D, 93, 12, 122003, 10.1103/PhysRevD.93.122003, 32818163, 7430253, 1602.03839, 2016PhRvD..93l2003A, 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.JOURNAL, The LIGO Scientific Collaboration, the Virgo Collaboration, Abbott, B. P., Properties of the binary black hole merger GW150914, 2016, Physical Review Letters, 116, 24, 241102, 10.1103/PhysRevLett.116.241102, 1602.03840, 2016PhRvL.116x1102A, 27367378, 217406416, JOURNAL, Meyer, R., Christensen, N., Gravitational waves: A statistical autopsy of a black hole merger, 2016, Significance, 13, 2, 20–25, 10.1111/j.1740-9713.2016.00896.x, free,

Biology

Animals living in relatively static environments would have relatively fixed features of the environment to perceive. This allows the evolution of filters that match the expected signal with the highest signal-to-noise ratio, the matched filter.JOURNAL, Warrant, Eric J., October 2016, Sensory matched filters, Current Biology, 26, 20, R976–R980, 10.1016/j.cub.2016.05.042, 27780072, 0960-9822, free, 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."JOURNAL, Wehner, Rüdiger, 1987, 'Matched filters': neural models of the external world,weblink Journal of Comparative Physiology A, 161, 4, 511–531, 10.1007/bf00603659, 32779686, 0340-7594,

See also

• Periodogram
• Filtered backprojection (Radon transform)
• Digital filter
• Statistical signal processing
• Whittle likelihood
• Profile likelihood
• Detection theory
• Multiple comparisons problem
• Channel capacity
• Noisy-channel coding theorem
• Spectral density estimation
• Least mean squares (LMS) filter
• Wiener filter
• MUltiple SIgnal Classification (MUSIC), a popular parametric superresolution method
• SAMV

Notes

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References

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Further reading

• JOURNAL, Turin, G. L., An introduction to matched filters, IRE Transactions on Information Theory, 1960, 6, 3, 311–329, 10.1109/TIT.1960.1057571, 5128742,weblink
• BOOK, Wainstein, L. A., Zubakov, V. D., Extraction of signals from noise, Prentice-Hall, Englewood Cliffs, NJ, 1962,
• JOURNAL, Melvin, W. L., A STAP overview, IEEE Aerospace and Electronic Systems Magazine, 19, 1, 19–35, 2004, 10.1109/MAES.2004.1263229, 31133715,
• JOURNAL, Röver, C., Student-t based filter for robust signal detection, Physical Review D, 84, 12, 2011, 122004, 10.1103/PhysRevD.84.122004, 1109.0442, 2011PhRvD..84l2004R,
• ARXIV, Fish, A., Gurevich, S., Hadani, R., Sayeed, A., Schwartz, O., Computing the matched filter in linear time, December 2011, 1112.4883, cs.IT,