Matched filter

An extension to two- resp. multidimensional filters is presented in Refs. 114, 167, 168. Matched filters are the most elegant solution, provided one knows what form the signal has. The choice between different filter types is discussed in Ref. 114. 

Matched filter The multiplication of the free induction decay with a sensitivity enhancement function that matches exactly the decay of the raw signal. This results in enhancement of resolution, but broadens the Lorentzian line by a factor of 2 and a Gaussian line by a factor of 2.5. 

The ambiguity function represents the output of a matched filter and may be written as... 

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 ... 

At this stage, the problem is to maximise SNIR at the instant of detection to over the input signal s(t) of finite energy and duration. Grouping the expressions for both whitening filter and matched filter ... 

Clearly the performance of (48) is bounded by the ideal matched filter output SINRideai obtained by letting R = R in (44). This gives... 

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... 

The single matched filter can be used only when the target s position and velocity are known. To detect a target at an unknown position, it is necessary to utilize a bank of filters matched to all possible target ranges and velocities. This approach leads directly to the range-Doppler correlation function, described by the formula... 

To detect the reflected signal in the presence of thermal white noise, the correlation process (matched filtering) is used according to the fundamentals given above. The received signal will be detected when its power is higher than the thermal noise power P/v multiplied by detectability factor D, e.g. 

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... 

In the complex domain, the demodulated signal is as described in (5). The signal is then filtered against another chosen signal v(t), often v is chosen to be the same as w (match-filtering)] that is, it is correlated with that signal, resulting in... 

This is the mutual information between the target variable (range and Doppler) X and the processed (with a matched filter) radar return Y, resulting from the use of the waveform identity matrix. We use this expected information as the MoE of the waveform more information we extract from the situation the better. 

Keywords automotive radar continuous wave radar CFAR processing rank-order filters FSK modulation matched filters unmatched filters. 

Rx Signals Data Cube - R Matched Filter - Fl(pixel, speed, Heading)... 

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... 

The design of a system for working with short pulses follows the same principles as other pulsed ultrasonic systems such as ultrasonic flaw detectors, but in this case it is necessary to achieve very much greater stability and higher bandwidth. A schematic circuit is shown in Fig. 5.4 A very short impulse is generated by a step recovery diode. The pulse has a width of half the period of the centre frequency of the lens if it is shorter than that the energy in the pulse is reduced without any improvement in the signal bandwidth. Thus the lens acts as a sort of matched filter with poor time resolution but optimal... 

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. 

Ju. For T2 <3C 1/(27//), maximal intensity is obtained for fj = 27 ./ and tf = 2T(. Of course, at these short time values, the sine build up of the interaction is small. Weighting functions of sin, sin2 or matched-filter-type are used to maximize the signal-to-noise ratio. In any case, since the T2 values in paramagnetic compounds are small, the cross peak intensity is drastically reduced. 

Figure 4.2 AR-based detection, P=50. (a) Prediction error filter (b) Matched filter.

Alles [Alles, 1977] further expands on the 4B oscillator module by describing the design of a matching filter and reverb module using the TRW multiplier. A separate switch module allows for the arbitrary interconnection of the functional units. The switch itself was a small processor with a simple instruction set. 

---