Radar Cross Section
3.4 RELATION OF SWERLING MODELS TO ACTUAL TARGETS
3.4.1 Simulating Swerling Targets
Analysts and radar testers often have a need to simulate the returns from fluctuating targets.
This might occur in simulation when attempting to reconcile the detection performance of radar simulations with predictions based on theory. It can also occur when evaluating the impact of target RCS fluctuations on target acquisition and tracking. In tower testing of actual radars (testing with signals generated from a test tower on a test range or through RF or IF injection in a laboratory environment), the use of fluctuating target returns provides more realistic estimates of detection performance than does the use of constant amplitude target returns.
Because of this perceived need, we present methods of simulating target returns with Swerling-like fluctuation characteristics. The methods make use of the fact that Swerling fluctuation statistics are governed by chi-squared probability density functions. As indicated earlier, the RCS (probability) density functions for SW1 and SW2 targets is a chi-squared density with two degrees of freedom. This means the density results from summing the square of two independent, zero-mean, equal variance, Gaussian random variables. In equation form, if x1 and x2 are random variables with the properties just described, then the random variable
will be governed by a chi-squared, two-degree-of-freedom density function. This further tells us that, if we want to generate random numbers that have statistics consistent with the SW1/SW2 RCS model, we can obtain them by generating two independent, zero-mean, equal variance, Gaussian random numbers, squaring them and taking the average of the squares.
The variance of the random numbers should be equal to the average RCS of the target, σAV. The resulting random variable will be governed by the density function of (3.9).
To simulate a SW2 target, we would create a new random number on every return pulse.
This stems from the fact that SW2 RCS values are, by definition, independent from pulse to pulse.
To simulate a SW1 target, we would generate a random number once every group of N pulses and maintain that as the RCS over the N pulses. Here N would be the number of pulses processed by the coherent and/or noncoherent processor (see Chapter 8). The idea of maintaining the RCS constant over the N pulses stems from the definition of SW1 RCS fluctuations, which states that the RCS remains constant during the time the radar beam scans by the target on a particular scan, but changes randomly from scan to scan.
As a note, the phase of the SW2 target also varies randomly from pulse to pulse and the phase of the SW1 target remains constant over the N pulses, but varies randomly from one group of N pulses to the next. We can achieve this phase behavior by defining the phase as
where the tan−1 is the four-quadrant arctangent. An alternate way of thinking about the above is to treat x1 and x2 as the real and imaginary parts of a complex number and defining the RCS as one-half times the magnitude squared and phase of the complex number, respectively.
While the above method of generating SW1 RCS fluctuations is accurate in terms of the SW1 fluctuation model, it can be cumbersome from an implementation perspective and is not representative of the fluctuation of RCS for actual targets. As illustrated in Figures 3.10 and 3.11, RCS tends to fluctuate continuously over time at rates that depend upon carrier frequency.
A method of achieving such a temporal characteristic and maintaining the SW1 statistics is to filter the Gaussian random numbers before squaring and adding them. Filtering the random numbers correlates them but does not change their Gaussian statistics.7 Thus, when the random numbers at the output of the filter are squared and added, the result will be a set of correlated, chi-squared, two degree-of-freedom, random variables that change fairly slowly over time.
A block diagram of the proposed method for generating SW1-like RCS values is shown in Figure 3.12. Sequences of independent, zero-mean, unit variance, Gaussian random numbers are generated and combined into a sequence of complex random numbers. The complex sequence is then filtered by a lowpass filter (LPF). The output of the LPF is then scaled so that the variance of the real and imaginary parts is equal to σAV. After scaling, the square of the magnitude is computed and divided by two [in compliance with (3.23)] to obtain the RCS. The angle of the complex number is formed to obtain the phase of the voltage that would result when the RCS is used to generate the complex return signal from the target.
In computer simulations, we prefer implementing the filter as an ideal “brick wall” LPF using the FFT.8 We prefer the FFT approach over a recursive filter approach because of the need to consider filter transients in the latter. We use the brick wall LFP because it is easy to implement. The length of the FFT is determined by the number of RCS samples needed in one execution of the simulation.
To set the filter bandwidth, we need the time between RCS samples. We normally choose this as the radar PRI for testing detection. For tracking studies we use the track update period or the PRI, depending upon whether or not we are modeling the signal processor.
As indicated by Figures 3.10 and 3.11, the bandwidth of the filter should be based on the operating frequency of the radar. If we assume the behavior in Figures 3.10 and 3.11 is representative, we would choose a bandwidth of about 0.5 Hz for radars operating in the S- to X-band and scale the bandwidth according to frequency from there.
In testing applications, it would be better to use recursive digital filters to generate the RCS values because the signals must persist over long time periods.
Fig ure 3.12 Block diagram of SW1 RCS generation algorithm.
Figures 3.13 and 3.14 contain plots that were generated by this technique. The filter bandwidth was set to 0.5 Hz for the top plot of the figures and 5 Hz for the bottom plot. As can be seen, the behavior is similar to the five-scatterer example of Figures 3.10 and 3.11.
Fig ure 3.13 RCS vs. time for SW1 RCS model.
Fig ure 3.14 Phase vs. time for SW1 RCS model.
The RCS generation technique for SW3 and SW4 targets is similar to the method used for SW1 and SW2 targets except that the RCS is based on the sum of four terms instead of two.
This is because SW3 and SW4 RCS fluctuations are governed by a chi-squared density with four degrees of freedom. In equation form,
To simulate a SW4 target, we would create a new random number on every return pulse. To simulate a SW3 target, we would generate a random number once every group of N pulses and maintain that as the RCS over the N pulses.
It is not clear how the phase should be modeled for this case. One approach would be to use (3.24). An alternative might be to use
That is, average the phase from two complex numbers represented by xa = x1 + jx2 and xb = x3 + jx4.
An alternative for the SW3 case would be to use an extension of the filter method suggested for SW1 targets. A block diagram of this method is shown in Figure 3.15. As can be seen, the method uses two of the SW1 filters and then averages the outputs of the magnitude square and angle computation blocks.
Fig ure 3.15 Block diagram of SW3 RCS generation algorithm.
Figures 3.16 and 3.17 contain plots of RCS and phase generated by the model of Figure 3.15. It is interesting that the RCS variations of Figure 3.16 appear to be smaller than those of Figure 3.13 and tend to be closer to the average RCS of 5 m2. This is consistent with the expected difference in RCS behavior between SW1 and SW3 targets.
Fig ure 3.16 RCS vs. time for SW3 RCS model.
Fig ure 3.17 Phase vs. time for SW3 RCS model.