Verifying RHOH, RHOV, and RHOHV

These terms measure the normalized cross-channel covariance in a polarization radar. They all are computed having the form:

R H O A B = < S A n     S B n * > < S A 2     S B 2 >

Where the S A n and S B n are complex (I,Q) vectors from two receiver channels A and B, and "<>" denotes expected value. This suggests that some form of amplitude modulation (AM) of the input signal might be helpful.

Suppose that the S A n and S B n samples are coming from two signal generators installed on a dual-receiver system, and that only the B-Channel is AM modulated so that:

| S A n | = { S A , S A , S A , S A , S a ... } ,   | S B n | = { S B , 0 , S b , 0 , S B ... }

Then the above estimator reduces to:

R H O A B = ( 1 2 ) S A S B S A 2 × ( 1 2 )     S B 2 = 07.07

A simple way to create these data is to set the A-Channel siggen for 95% AM depth, and use a sinusoidal modulation source of, perhaps, 400 Hz. We do not choose 100% depth because we would lose the burst phase reference when the amplitude became smallest. The 26 dB reduction in SB is a close enough approximation to zero in the above formula.

If we now observe the two receive channels with the RVP10 at a PRF of 800Hz, we see the RHOAB terms varying with range; reaching a high value of 1.00, and a low value of 0.707. The plots are nearly stationary on the Ascope screen because the PRF is almost precisely twice the modulation rate (though they are free-running relative to each other).

Adjusting the amplitude of either signal generator is not affect the p terms, but it does have an interesting effect on SQI. If (T,Z,V,W) are computed from both channels combined, then the SQI is:

S Q I = S A 2 S A 2 + ( 1 2 ) S B 2

If we solve this equation for SQI=0.5 we find that the individual SA terms must have twice the power of the individual SB terms. This can be checked by adjusting either signal generator until the minimum plotted SQI is 0.5, and then verifying that the average H and V powers are identical; or, equivalently, that ZDR, LDRH and LDRV are 0.

The linear FM ramp (see Linear Ramp of Velocity with Range) can also be used as a test of RHOAB in a dual-receiver system. With one siggen modulated and the other fixed, one receive channel appears to rotate relative to the other. If the FM modulation is such that 1/N of a full revolution occurs per pulse at a given range, then if the sample size is N pulses we observe RHOAB = 0 at that range. The plot of RHOAB shows a characteristic sin (x)/x behavior as a function of range.