# Probability of Detection and pulse-pulse decorrelation time

I'm trying to analyze the probability of detection ($$P_d$$) for a Swerling II target. I know that the Swerling II target assumes that the Radar Cross Section (RCS) is an exponential probability density function, and that the RCS fluctuates from pulse-to-pulse.

My question is: Do the Swerling models assume perfect decorrelation from pulse-pulse? As in, do they assume that the RCS on each pulse is purely independent of other pulses, or do they account for some correlation?

The target I'm trying to analyze does not perfectly decorrelate between pulses. So, the RCS on pulse #2 will be somewhat correlated to that on pulse #1.

• FYI - this isn't a homework question. I'm a graduate student working on a weather radar.
– Rob
Aug 24, 2015 at 13:06
• I would recommend asking the moderators to move this question to dsp.SE or electronics.SE. You can contact the moderators by clicking on the flag link below your question. Aug 24, 2015 at 13:17
• @DilipSarwate: I get a vibe that our sister site on statistics might be even better than DSP? You know this better, what do you think? Aug 27, 2015 at 7:16
• I've been monitoring the DSP site, it seems very similar. It is stats, but it's pretty radar-specific
– Rob
Aug 27, 2015 at 9:52
• @JyrkiLahtonen The frst question is about assumptions in Swering models. I think that it is more likely that someone with knowledge of the Swerling models will be found on dsp.SE than on stats.SE. The second question "What do I do about it?" might find a better answer on stats.SE but the question has likely been looked at already in the radar literature, and the person who knows about Swerling models might be able to tell the OP where to find the answer. That being said, if stats.SE wants the question, give it to them. Aug 27, 2015 at 12:51

The following is based on the report from Peter Swerling: Probability of Detection for Fluctuating Targets.

Case II is defined as fluctuations that are independent from pulse to pulse (page 2), that is, the RCS is pulse to pulse uncorrelated.

On page 8 of the report the expression for Case II probability of detection, $$P_{D}$$, is given as:

$$P_{D}=1 - I\left[\frac{Y_{b}}{(1+\bar{x})\sqrt{N}},N-1\right]$$

where $$I$$ is the incomplete gamma function, $$\bar{x}$$ is the expected value of the signal-to-noise ratio, $$N$$ is the number of pulses integrated, and $$Y_{b}$$ is the designed detection threshold.

For partial correlated pulses I will refer to Exact Detection Probability for Partially Correlated Rayleigh Targets. (My apologies for the pay wall.). While an exact formula for partial correlated pulses is presented in this paper it is too involved a computation to easily transcribe here. I am reproducing the abstract below.

The probability of detection of the sum of N square-law-detected pulses is derived for the case where the signal fluctuation obeys chi-square statistics with four degrees of freedom. P. Swerling's (1960) case III and IV represent the cases where the signal is completely correlated and completely decorrelated, respectively, from pulse to pulse. An exact expression for probability of detection is derived for the condition of partial signal correlation. The results given are compared with the approximate technique commonly used to handle partial signal correlation.