A frequency diversity pulse-pair algorithm for extending Doppler radar velocity Nyquist range

Millimeter wave radars have been widely used for atmospheric remote sensing and tracking hard-targets from airborne platforms. For these radars, the product of the unambiguous range and Doppler velocity is limited by the radar wavelength. This work focuses on a novel method to extend the Nyquist rate of millemeter radars, which uses frequency diversity pulse-pairs for Doppler phase estimation. Two short pulses with center-frequencies of f<sub>1</sub> followed by f<sub>2</sub> are transmitted during the first pulse repetition interval (PRI). During the next PRI, the pulses transmitted are in the order f<sub>2</sub> followed by f<sub>1</sub> respectively. There are two mechanisms for error reduction. First, the "beat" phases of the f<sub>1</sub>/f<sub>2</sub> and f<sub>2</sub>/f<sub>1</sub> pairs cancel out in the expected value sense. Second, since the f<sub>1</sub>/f<sub>2</sub> and f<sub>2</sub>/f<sub>1</sub> phase estimates are highly anti-correlated, the sum of the two phase estimates has a much smaller variance than the individual phase estimates. Based on Monte-Carlo simulations, the feasibility of this method is demonstrated herein. Ongoing data analysis is discussed.


I. INTRODUCTION
Millimeter wave radars have been widely used for atmospheric remote sensing and tracking hard-targets from airborne platforms [1][2][3][4][5]. In large part, the popularity of these millimeter wave radars is because fine antenna beamwidths can be realized while still allowing for physically small and lightweight designs, as well as higher backscattering efficiency for weather targets such as cloud particles. However, the product of the unambiguous range and Doppler velocity is limited by the radar wavelength (i.e. cλ 8 )). Airborne millimeter wave radars that are required to have long range coverage therefore may have Doppler Nyquist ranges that are much smaller than the relative speeds of competing aircrafts.
Different techniques to mitigate the Doppler-Range ambiguity dilemma have been explored. For example, the staggered PRT [6] and Polarization Diverse Pulse-Pair (PDPP) [7] techniques are methods that are already employed for these purposes. The "staggered" PRT is a multi-rate sampling method that unfolds the Doppler velocity based on the ratio of Doppler velocities measured by 2 PRTs. A weakness of this approach is that the resulting Doppler velocity estimates have increased sensitivity to noise, relative to the usage of a single PRT. The Polarization Diversty Pulse-Pair method utilizes two pulses with orthogoal polarizations. These two pulses can be placed with shorter time interval to extend the Doppler Nyquist range, while the PRT can be adjusted independently for range coverage. Notwithstanding, finite polarization isolation of the antenna and the receivers may overwhelm retrievals close to regions with high SNR [7].
This work focuses on a novel alternative -which uses frequency diversity (in lieu of polarization diversity) pulsepair for Doppler phase estimation. Along with the cutting-edge digital waveform generation and digital receiver/processor technologies, this frequency diversity pulse-pair technique can be easily implemented on a modern radar without additional microwave hardware. If proven practical, this frequency diverse pulse-pair method would potentially enable low-cost and light weight air-borne millimeter wave radars with wide Nyquist Doppler capability. The following section outlines the theoretical formulation of the Frequency Diversity Pulse-Pair algorithm (FDPP). Numerical simulations are then presented to validate the theory developed herein.

A. Conceptual description of FDPP algorithm
Two pulses at center frequencies of f 1 and f 2 are transmitted with a separation of lag ∆T . While retaining ∆T , the order of the pulses is reversed every alternate transmission. From the receive channels at f 1 and f 2 , the pulse-pair phase estimate of the two sequences are individually accumulated and stored as ∆φ order1 and ∆φ order2 . Finally Doppler velocity is estimated from the sum of an equal number of the two individual pulsepair phase estimates (denote as ∆φ). Note that the use of two closely spaced radar frequencies introduces two primary sources of error. First, a "beat" phase that scales as a function of range is introduced. Nonetheless, this term vanishes when the phases of the f 1 /f 2 pair and f 2 /f 1 pairs are added.Second, since there is little correlation between the f 1 and f 2 pulses, the variances of the f 1 /f 2 phase estimates is large. However, since the f 1 /f 2 and f 2 /f 1 phase estimates are highly anticorrelated, the sum of the two phase estimates has a much smaller variance.

B. Mathematical description of FDPP algorithm
Denote the transmitted waveform at frequency f 1 as E T x,f 1 (t). Let E 0,f 1 be the amplitude of the transmitted signal, the phase of the transmitted signal be Ψ T x,f 1 and t denote time. Two short pulses with center-frequencies of f 1 and f 2 are transmitted during the first pulse repetition interval (PRI). During the next PRI, the pulses are transmitted in the order of f 2 and f 1 respectively. There are two mechanisms for error cancellation. First, the "beat" phases of the f 1 -f 2 and f 2 -f 1 pairs cancel out in the expected value sense. Second, since the f 1 -f 2 and f 2 -f 1 phase estimates are highly anti-correlated, the sum of the two phase estimates has a much smaller variance than the individual phase estimates.
Denote c be the speed of light, f D1 be the Doppler shift and R the range to a point scatterer. Now, the received signal E Rx,f 1 at time t can be written as (2) Similarly, the T x and Rx signals at frequency f 2 and time t + ∆T can be written as follows. Note that the range to the point-scatterer is now R + v r ∆T . Here, v r denotes the radial velocity of the point-scatterer.
The implementation we consider herein typically has frequencies that are spead apart by a few MHz. Therefore, the term k 2 − k 1 can be neglected in the context of the rest of the equation. Further, we approximate k 1 + k 2 ≈ 2k 1 . The above equation simplifies to the following - Since all values other than ∆Φ are solely system dependent, the radial component of target mean radial velocity v r can be obtained from the ensemble-averaged ∆Φ. Two comments about the Nyquist velocity of the frequency diversity pulsepair equation are in order. First, for a given lag ∆T the Nyquist interval is diminished by a factor of 2 compared to a traditional pulse-pair or a polarization diversity pulse-pair. This is because the frequency diversity pulse-pair algorithm accumulates two phase estimates ∆Φ order1 and ∆Φ order2 and the resultant sum has a range of variation from [−π, π]. Second, the frequency diversity pulse-pair algorithm developed herein allows an extension of the Nyquist interval as compared to a traditional pulse-pair which necessarily needs long lags to accomodate sufficiently unambiguos ranges. For example, if the FDPP algorithm is denoted by lag ∆T and the pulse-pair algorithm denoted by a lag τ , then the Nyquist interval of the FDPP algorithm is improved by a factor τ 2∆T . A short analysis of the error of the FDPP phase estimate is in order. To this end, let σ denote the variance and ρ denote the correlation operators respectively.
Now, the covariance term can be conveniently decomposed as From the above relationships, Rearranging terms in (13), The underlying premise of the frequency-diversity pulsepair algorithm is that as ρ(∆Φ Order1 , ∆Φ Order2 ) → −1, the variance of the phase composite estimate σ(∆Φ) → 0.

C. Simulation methodology
Our approach to numerical simulations is to estimate the phases at lag ∆T using simulated backscattered electric fields at the antenna port using a Monte-Carlo based simulator. Specifically, the voltage due to backscattered electric fields at the receiver is given as Here, T the backscattering amplitude due to scattering from a Swerling III-type target, A denotes the antenna weighting function and R the range weighting function. The amplitude fluctations T of the target are samples from a fourth order Chi-squared distribution as illustrated in Figure 1. Here the black line shows power fluctuations due to a Swerling type III target. The red curve -with the x and y-axes indicated in red -shows the probability density function for the backscattering efficiency. In essence, this is the PDF for voltage at the receiver as the change in antenna and range weighting functions is small over the period of the dwell considered herein. The antenna weighting function is defined below where the antenna phase center is denoted by [X 0 , Y 0 ], the hard-target location is indicated by [X, Y ] and σ φ and σ θ are the second central moment of the effective two way antenna patterns in the azimuthal and elevational planes respectively. The range weighting function and its relationship to transmitted chirp bandwidth is given by where, Z 0 is the center of the resolution volution in range, Z indicates the scatterer location, c is the speed of light (m/s) and B is the transmitted chirp bandwidth. The constant 0.35 approximately accounts for losses due to amplitude modulation of the chirp and finite receiver bandwidth.
Scatterer locations are updated at every pulse repetition interval given mean and relatively small fluctating velocity components along each coordinate. Gaussian distributed pseudo-random values are used to generate the fluctuating velocity fields. Since the omission of temporal continuity of the fluctuating velocity causes the scatterer to have abrupt changes in position, we run a 5-point moving average filter to low-pass filter the fluctuating velocity in the temporal dimension. It is this low-pass filtering by the moving average filter that distinguishes velocity variations from random Brownian motion and from random changes in the phase of the composite signal introduced by thermal noise.
Once the time series data is generated , thermal noise is added . Phase estimates at the lag ∆T are then obtained for the sequences f 1 /f 2 and f 2 /f 1 . Typically, 1000 trials are performed in order to produce the statistics of the next section.

III. PRELIMINARY RESULTS
In this section, Monte-Carlo simulations are qualitatively compared with data-analysis results (all at W-band). Fig. 3 shows simulations of the FDPP Doppler velocity retrieval process. The basic idea is that a composite phase that is solely Doppler dependent is synthesized from noisy but highly anticorrelated frequency diverse pulse-pair lag-1 phase estimates. The phase estimate from the sequence in which f 1 leads f 2 is shown in Fig. 3a. After 1 PRT, the sequence of f 2 followed by f 1 is yields phase estimates shown in Fig. 3b. The sum of the two frequency diverse pulse-pair estimates is shown in Fig. 3c. Fig. 3d shows a scaled version of Fig.3c, where the composite phase is scaled to the Doppler Nyquist interval. Fig. 4 shows the FDPP algorithm Doppler retrieval accuracy for a W-band radar as a function of various design parameters. The simulation methodology employed herein is similar to that in [7]. Typically, 1000 Monte-Carlo tries were employed to generate the simulation statistics. The relevant parameters shown in the corresponding figures. In Fig. 4a, the increasing errors on the right side are a direct consequence of decreasing correlation between the f 1 /f 2 and f 1 /f 2 pair phase estimates. The increasing errors on the left hand side are due to sensitivity to thermal noise. As expected, this is exacerbated at shorter lags and decreasing SN R. Similar reasoning deems a 3 kHz pulse repetition frequency (PRF) optimal in Fig. 4b. In Fig. 4c,  the increasing errors on the right side are a direct consequence of decreasing correlation between the f 1 /f 2 and f 1 /f 2 pair phase estimates. The increasing errors on the left hand side are due to sensitivity to thermal noise. Fig. 5 shows preliminary data analysis results using NASA's Cloud Radar System (CRS) during the OLYMPEX mission. The CLoud Radar System was mounted on an ER-2 aircraft with the antenna pointed "nearly" at nadir. Aircraft attitude fluctuations cause the antenna to point slightly away from nadir, thereby inducing a surface velocity measurement contaminated with the motion of the aircraft. Traditional pulsepair measurements are used as "truth" and compared to the frequency diversity algorithm developed herein. Two comments are in order. First, high SNR cases from the surface were isolated for this analysis to be consistent with the simulation predictions that the algorithm requires high SNR to yield reasonable estimates. For most part, we also expect that they have lower spectrum width as compared to the layer of clouds evident in the Fig. 5a. Second, Fig. 5b shows excellent agreement of the frequency diverse pulse-pair algorithm with traditional pulse-pair measurements. In essence, this validates the approach developed herein to first order. More detailed analysis of the error space is ongoing at NASA Goddard Space Flight Center.

IV. SUMMARY
Herein, we have investigated a novel algorithm to overcome the Doppler-Range velocity dilemma using millimeter wave radars. The frequency diversity pulse-pair algorithm first transmits a sequence of two pulses modulated by frequencies f 1 and f 2 in that order. During the next sequence the order of f 2 and f 1 are reversed. In both cases, the pulses are seperated by a time lag ∆T . The sum of the phase estimates from the f 1 /f 2 and f 2 /f 1 pairs yields a resultant phase which is linearly related to Doppler velocity. There are two mechanisms for error reduction. First, the "beat" phases of the f 1 /f 2 and f 2 /f 1 pairs cancel out in the expected value sense. Second, since the f 1 /f 2 and f 2 /f 1 phase estimates are highly anticorrelated, the sum of the two phase estimates has a much smaller variance than the individual phase estimates. The error space of the algorithm was investigated with Monte-Carlo simulations.Based on numerical simulations, the algorithm was deemed to require high SNR. Data was collected using NASA's Cloud Radar System and the high SNR data points from the surface showed excellent agreement with traditional pulse pair-estimates of Doppler velocity. More detailed analysis is ongoing at Goddard.
With modern digital waveform generation, digital receiver and solid-state power amplifier technologies, FDPP can be implemented without additional microwave hardware like a second receiver channel. Compared to PDPP, FDPP provides better channel isolation, therefore better mitigates contamination from strong targets. Monte-Carlo simulations and preliminary data analysis indicate that the algorithm has strong potential for applications such as correcting space-borne radar with the antenna pointing "nearly" at nadir. Changes in aircraft attitude cause the antenna to point away from nadir thereby measuring a surface velocity contaminated with the aircraft motion. Note that the surface has high received power, and a layer of clouds above the surface has much lower received power. (b) Quantivative comparisons of the frequency diversity pulse-pair algorithm estimates with traditional pulse-pair algorithm measurements. Note that the FDPP estimates track the changes in pulse-pair velocity measurements very well. More quantitative analysis is underway at NASA Goddard Spaceflight Center. antenna pointing errors, air-traffic control. military aircraft and missile tracking.