# Staggered PRT Processing

Presently ELDORA transmits a staggered PRT in the ratio T2/T1 = 5/4. For a staggered PRT the unambiguous velocity interval becomes: (8)

Data processing for each PRT of the staggered PRT is similar to uniform PRT processing with a notable complication in velocity unfolding.

#### Autocovariance Processing

The inphase and quadrature data is processed independently for each transmitted frequency. Covariance averaging is performed between pairs of pulses separated by T1 and T2 (denoted R1 and R2). This process gives zeroth lag power for both PRT's: (9) (10)

where n denotes the transmitted frequency number (1 £ n £ 5). It also yields two resultant vectors of the covariance averaging: (11) (12)

Vector addition is performed across the frequencies, yielding resultant vectors R1 and R2 for each PRT, as given by the following: (13) (14)

Scalar addition is performed across the frequencies yielding P1 and P2: (15) (16)

Where N is the total number of transmitted frequencies.

#### Spectral Moment Processing

The following spectral moments or associated products are computed for the staggered PRT: unfolded velocity, velocity for PRT#1, velocity for PRT#2, reflectivity, spectrum width, and NCP. The phase of each vector is determined as follows: (17) (18)

Velocity unfolding for ELDORA is done in vector space. This is advantageous because the signal processor can unfold the velocity without any knowledge of the PRTs or transmitted frequencies. Given T2/T1 = f 2/f 1 = K2/K1, where K1 and K2 are relatively prime integers and K2 > K1, there exists K1 + K2 regions where the difference, D = (K2/K1)f 2 - f 1 is unique. For K2/K1 = 5/4, nine such regions exist. Associated with each region is a constant phase offset which is added to the average phase, f = ((K2/K1)f 1 + f 2)/2 to place it in the correct Nyquist interval. These phase corrections are stored in a lookup table in the signal processor memory. Averaging the phases associated with each PRT improves the velocity estimate. The phase corrections for the first nine intervals are given in Table 1. Once the absolute phase is determined the unfolded velocity can be calculated as follows: (19)

where l is the average wavelength of the transmitted chips. As the frequency separation on ELDORA is around 10 MHz, this has a negligible effect on the accuracy of the velocity estimate.

 Region D =(K2-K1)f 1-f 2 Correction -5p < f 2£ -3(K2/K1)p p -2p (1+K2/K1) -3(K2/K1)p < f 2£ -3p -3p /2 -p (2+K2/K1) -3p < f 2£ -(K2/K1)p p /2 -p (1+K2/K1) -(K2/K1)p < f 2£ -p -2p -p -p < f 2£ p 0 0 p < f 2£ (K2/K1)p 2p p (K2/K1)p < f 2£ 3p -p /2 p (1+K2/K1) 3p < f 2£ 3(K2/K1)p 3p /2 p (2+K2/K1) 3(K2/K1)p < f 2£ 5p -p 2p (1+K2/K1)

Table 1 Phase Corrections for First 9 Intervals

Estimates for the mean velocities associated with T2 and T1 are given by the following: (20) (21)

Reflectivity is given by (22)

where RC is the radar constant and r is the range in km.

Spectrum width is computed as follows: (23)

where (24)

N is the noise power.

Normalized coherent power is defined as the ratio of the power calculated at lag one and the total received power: (25)

It should be noted that the value of l used is the average of all transmitted frequencies.