The Matched Filter

Computation of this function results in a three-dimensional plot for which one axis is time delay (or range), the second is Doppler frequency or radial velocity and the third is the output power of the matched filter (usually normalised to unity). The extent of the ambiguity function peak in the Tr and the fd dimensions determines the range and Doppler resolutions respectively. As we are using the directly received signal only we term this self-ambiguity as there is no inclusion of any system geometry dependence on the transmitter and receiver locations. 

The analysis is based on the matched filtering performed at the receiver. Before proceeding to the mathematical background, it is necessary to state the assumptions made when modeling the system. These are ... 

The mathematical representation of the ambiguity function for this second method is now different. The matched filtering is performed for each echo and the final output will be a summation of the bistatic ambiguity functions of the various bistatic pairs. The following equation outlines this ... 

The optimal (in the mean-square sense) detector is based on the matched filter concept. The output of the filter matched to the signal echo described by Equation (23) can be calculated as an integral in the form... 

To extend the integration time further, it is necessary to introduce acceleration into the target s motion model. The range to the target should now be expressed as r = ra + vat + a0t2/2. The matched filter concept now leads to the three-dimensional range-Doppler-velocity... 

The MFP process is illustrated in figure 8. The received signal, for all transmit/receive pairs, over a CPI forms a data cube. The matched filter, for a particular scene pixel and target velocity, also forms a data cube. For multiple operating frequencies, additional cubes would be formed. A single MFP output, a pixel and velocity, is the inner product... 

Note that the signal-to-noise ratio obtained is equal or even better than that obtained using the matched filter and that the spectral resolution is not sacrificed compared to the non-weighted spectrum. 

Test-tubes (75 x 10 mm and 100 x 12 mm) together with the matching filter tubes (55 x 7 mm and 80x9 mm) are standard products they are inexpensive and therefore eminently suitable for elementary and other students. The smaller size is satisfactory for most purposes. 

We now show how to kemelize the matched filter expression (29) where the resulting non-linear matched filter is called the kernel matched filter. The pseudoinverse (inverse) of the estimated background covariance matrix can be written in terms of its eigenvector decomposition as... 

We can see easily from this argument that if there is a range of linewidths in the spectrum we cannot find a value of the linebroadening which is the optimum for all the peaks. Also, the extra line broadening caused by the matched filter may not be acceptable on the grounds of the decrease in resolution it causes. Under these circumstances we may choose to use sufficient line broadening to cut off the excess noise in the tail of the FID, but still less than the matched filter. 

The matched filter theorem states that if we want to detect matches between / and g by cross-correlating a filter h with /, and the criterion for detection is the ratio of signal power to expected noise power, then the best filter to use is the template g itself Depending on the nature of / and the detection criterion, however, other filters may be better for example, if / is relatively smooth, better results are obtained by correlating the derivative of / with the derivative of g, or / with the second derivative of g. 

The simple cross-correlation estimator is used extensively in the form of a matched filter implementation to detect a finite number of known signals (in other words, simultaneous acquisition of multiple chaimels of known signals). When these deterministic signals are embedded in white Gaussian noise, the matched filter (obtained from cross-correlation estimate at zero lag, k = 0, between the known signal sequence and the observed noisy signal sequence) gives the optimum detection performance (in the Bayes sense ). 

In contrast to the principal component (PCA)-based approach for detection and discrimination, the cross-correlation or matched-filter approach is based on a priori knowledge of the shape of the deterministic signal to be detected. However, the PCA-based method also requires some initial data (although the data could be noisy, and the detector does not need to know a priori the label or the class of the diHerent signals) to evaluate the sample covariance matrix K and its eigenvectors. In this sense, the PCA-based detector operates in an unsupervised mode. Further, the matched-filter approach is optimal only when the interfering noise is white Gaussian. When the noise is colored, the PCA-based approach will he preferred. 

Correlation receiver A receiver structure that correlates received signal plus noise with rephcas of the possible received signal shapes. It is completely equivalent in terms of performance to the matched filter receiver. 

Integrate-and-dump detector A form of the matched filter or correlation detector that is specialized to rectangular signal shapes. Thus, the correlation operation reduces to an integration of the signal interval. 

Schematic diagrams of two receiver front ends that can be used to compute the coordinates of the data vector are shown in Fig. 12.60. The first is called the conelatorimplementation, and the second is called the matched filter implementation. Because the noise components are linear transformations of a Gaussian random process, they are also Gaussian, and can be shown to have zero means and covariances...

Pulse compression requires a separate matched filter be supported for each waveform used by the radar system. In practice, this filter is implemented via fast convolution as illustrated in Fig. 17.12. From Fourier transform theory, frequency-domain multiplication is equivalent to time-domain convolution. Hence, receiver digitized output data is input to a FFT, multiplied by the Fourier transform of the matched filter response, and then passed through an inverse FFT (I FFT) to output time-domain data. The matched filter reference functions transforms are generally computed off-line and stored in memory to support real-time processing. Fast convolution significantly reduces the number of operations required compared to time-domain direct convolution of returns and the appropriate matched filter function. 

An important concept in signal processing is that of the matched filter, with an apodization function that matches the envelope of the signal. 

For a Lorentzian line where the envelope decays as we use it as the match filter (the apodization function). 

This results in a Lorentzian line shape that is broadened by a factor of 2. Similarly, for Gaussian lines, applying the match filter... 

The matched filter is the most logical method of performing optical correlation this was first proposed by Vander Lugt in the 1960s [20]. The optical architecture is laid out in a linear fashion as a 4/ system. The lower part of Fig. 16 depicts the transmission of light from left to right, from planes 1 to 4. The upper part is performed off-line by electronic processing and is then stored as a matched filter. 

The FT of the reference is done off line on a computer and is defined as the matched filter R(u,v) for that particular reference r(x,y). In fact, the generation of the filter may be more complicated (to include invariances), and it is advantageous to use only the phase information of the reference FT rather than the full complex amplitude and phase as it gives a more detectable narrow peak [21]. The product of the input FT and the filter then undergoes a further FT to give the correlation in plane 4. The object in the reference r x,y) is centered in the process of generating the filter R(u,v), so that if a correlation peak occurs, its position is directly proportional to the object in the input... 

Great improvements can be made to the usefulness of the correlation peak by using a phase only matched filter (POMF). The matched filter R(m, v) is stripped of its phase information (i.e., the phase angle of the complex data at each pixel) and this is used as the filter in the correlator... 

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