A Proof for the Unbiased Nature of Range-Doppler Measurements in Coarse-Resolution Dechirp-on-Receive Feedback Synthetic Aperture Radar Navigation

2 COARSE-RESOLUTION SAR SIGNAL MODEL

This section derives a range-Doppler signal model for a coarse-resolution vertical SAR employing a dechirp-on-receive data collection architecture. The derivation will follow a similar derivation by Haydon and Humphreys (2023), but will specifically account for navigation errors and trace their path through the signal processing chain. The innovation of this publication lies not in the derivation of the signal model, but in the specific attention paid to the navigation errors. It will be shown that navigation errors enter radar imagery through the matched filtering process, but are canceled (to the first order) once the resulting range-Doppler cells are absolutely registered.

The following derivation will take advantage of a SAR image formation algorithm suitable only for coarse-resolution SAR imagery. Specifically, the rectangular format processing algorithm will be invoked, as this algorithm is simple to analyze and has already been established as sufficient for vertical SAR navigation purposes (Haydon & Humphreys, 2023). Rectangular format processing is perhaps the simplest SAR image formation algorithm, as it neglects all second-order and higher effects such as range migration and quadratic phase errors (Carrara et al., 1995). Neglecting these terms, however, limits the allowable scene size and resolution of the SAR system. For example, the scene size and range/Doppler resolutions are typically constrained by the following two relationships (Carrara et al., 1995):

$\begin{array}{cc}{X}_{CR}<\frac{2{\rho }_r{\rho }_a}{{\lambda }_c{K}_a},& Y_{DR}<\frac{2{\rho }_a^2}{{\lambda }_c{K}_a}\end{array}$1

Here, X_CR and Y_DR are the cross-range and down-range illuminated scene sizes, respectively, ρ_r is the range resolution, ρ_a is the ground-projected Doppler (azimuth) resolution, λ_c is the radar’s center frequency wavelength, and K_a is the azimuth mainlobe broadening factor (a near-unity constant introduced by aperture weighting and sidelobe suppression). For vertical SARs, the down-range scene size is related to the variation in local terrain elevation (the range extent between the closest-range and farthest-range ground points), and the cross-range scene size is determined by the radar’s beam width and altitude. In navigation applications, it is desirable to keep these scene sizes large so that the radar can detect geometrically distributed and diverse terrain features. Consequently, for a fixed wavelength λ_c and azimuth broadening factor K_a , the resolution products ρ_rρ_a and ${\rho }_a^2$ must also be large, and thus, the imagery must be coarse. Fortunately, fine image resolution is not the ultimate goal in navigation applications as it is in remote sensing applications, and coarse resolutions on the order of meters have been demonstrated to be sufficient for navigation (Haydon & Humphreys, 2023).

Figure 1(a) illustrates the closed-loop radar navigation architecture employing a dechirp-on-receive radar and the rectangular format image formation algorithm considered in this publication. The figure’s clockwise closed loop is this publication’s concern; specifically, the feedback loop’s effect on the range-Doppler navigation observables. In this architecture, the navigation system/Kalman filter (KF) position and velocity estimates condition replica radar pulses and then mix these pulses with received reflected pulses prior to low-pass filtering. The low-pass mixed signal is then quadrature-demodulated and sampled. The resulting in-phase and quadrature (I/Q) samples are stacked in a two-dimensional (2D) array of appropriate size, and a 2D discrete Fourier transform (DFT) is applied to the array, completing the rectangular format image formation algorithm. The result is a coarse-resolution range-Doppler image from which navigation observables are extracted. These navigation observables are consumed by the KF and correct the navigation system, completing the closed loop. The remainder of this publication is dedicated to tracing the navigation errors through the feedback loop identified in Figure 1(a).

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Figure 1

(a) Illustration of the architecture of a closed-loop, coarse-resolution feedback SAR navigation system; (b) coarse-resolution vertical SAR imaging geometry

Consider the patch radar sensing geometry shown in Figure 1(b). An airborne radar emits a series of linear frequency-modulated pulses (chirps) toward a patch center point and awaits their echoes off nearby point scatterers. The patch center point may be chosen a priori (e.g., a pre-planned point provided by a guidance system) or it may be determined immediately prior to a measurement (e.g., a ground point directly below the radar’s currently estimated position). This patch center serves as a linearization point relative to which the range and Doppler shift of nearby illuminated ground points are calculated. It is assumed that the radar system is linear and that the response of the radar to the nearby point scatterers is reasonably modeled as the superposition of the individual impulse responses of the radar to each individual point scatterer. This model assumes that the radar signal does not bounce between multiple point scatterers before returning to the radar. This single-bounce model may not be appropriate in, for example, urban situations; thus, for the purpose of this publication, it is also assumed that the radar is operating over natural terrain where this model is appropriate (such as the surface of the moon, as suggested by the lunar lander in Figure 1(b)).

Let us suppose that the radar emits N > 0 equally spaced chirps of duration T_p > 0 at chirp center times η₀,η₁,…,η_N-1∈ℝ where the center time of the midpoint chirp is defined as zero: ${\eta }_{\lfloor \frac{N}{2}\rfloor }:0$, where $\lfloor \cdot \rfloor$ is the floor operator. Each chirp is separated in time by the pulse repetition interval T_c > 0 such that the transmit synthetic aperture duration is T_a (N–1) T_c + T_p , and the center time of the n-th chirp is defined as ${\eta }_n:=(n-\lfloor \frac{N}{2}\rfloor )T_c$ The n-th transmitted chirp is modeled as follows:

$x_{\text{TX}}[t,{\eta }_n]=A\hspace{0.17em}\cos \left[2\pi f_0\right(t-{\eta }_n)+\pi k{(t-{\eta }_n)}^2]w\left[\right(t-{\eta }_n)/T_p]w[{\eta }_n/T_a]$2

Here, t ∈ ℝ is the continuous signal time, A > 0 is the transmitted signal amplitude, f₀ > 0 is the radar’s center frequency, k ∈ ℝ is the chirp rate, and w[τ] is the unit rectangle function defined as follows:

$w\left[\tau \right]=\{\begin{array}{cc}1& \left|\tau \right|<0.5\\ 0& \text{otherwise}\end{array}$3

The signal reflects off a point scatterer and returns to the radar as follows:

$\begin{array}{ccc}{x}_{\text{RX}}[t,{\eta }_n,s]& =& A\text{'}\left[s\right]\cos \left[2\pi f_0\right(t-{\eta }_n-2r*[{\eta }_n,s]/c)+\pi k{(t-{\eta }_n-2r*[{\eta }_n,s]/c)}^2+\varphi [s\left]\right]\\ & & w\left[\right(t-{\eta }_n-2r*[\eta ,s]/c)/T_p]w[{\eta }_n/T_a]\end{array}$4

where s ∈ ℝ³ is the vector from the patch center to the scatterer (see Figure 1(b)), A´[s] > 0 is the returned signal amplitude, r* [η_n,s] > 0 is the true range to the point scatterer at time η_n,ϕ[s] ∈ ℝ is the assumed-constant phase shift induced by the scatterer, and c is the speed of light. Here, it is assumed that the radar does not move significantly during the time of flight of a chirp so that the two-way range is reasonably modeled as 2r* [η_n,s]. This assumption is known as the commonly invoked “stop-and-go” or “stop-and-hop” assumption (Carrara et al., 1995) and is appropriate for low-speed systems. High- and orbital-speed systems may need to consider the motion of the radar during a signal’s time of flight (Carrara et al., 1995; Tsynkov, 2009), but this publication will assume a low-speed application such as a lunar lander on a ground approach.

Each received chirp is mixed with a replica chirp of the following form:

$\begin{array}{ccc}{x}_L[t,{\eta }_n]& =& \cos \left[2\pi f_0\right(t-{\eta }_n-2r\left[{\eta }_n\right]/c)+\pi k{(t-{\eta }_n-2r[{\eta }_n]/c)}^2]\\ & & w\left[\right(t-{\eta }_n-2r\left[{\eta }_n\right]/c)/T_p^{´}]w[{\eta }_n/T_a^{´}]\end{array}$5

where r[η_n] > 0 is the estimated range from the radar to the patch center at time η_n, $T_p^{´}>0$ is the replica pulse envelope, which is typically slightly longer than the transmitted pulse envelope, and $T_a^{´}=(N-1)T_c+T_p^{´}$ is the receive synthetic aperture duration. The estimated range r[η_n] is conditioned on the navigator’s position estimate at time η_n — this term is where the navigation errors enter into the radar signal model.

After mixing, low-pass filtering, quadrature demodulation, and sampling at M times per chirp, the discrete-time complex baseband signal is modeled as follows:

$x_{\text{BB}}[{\tau }_m,{\eta }_n,s]=A˝\left[s\right]\exp \left[\frac{-4\pi ik}{c}\right(r*[{\eta }_n,s]-r\left[{\eta }_n\right]\left){\tau }_m\right]\exp \left[\frac{-4\pi i{f}_0}{c}\right(r*[{\eta }_n,s]-r\left[{\eta }_n\right]\left)\right]\exp \left[\frac{4\pi ik}{c^2}{(r*[{\eta }_n,s]-r[{\eta }_n\left]\right)}^2\right]w[{\tau }_m/T_p]w[{\eta }_n/T_a]$6

where τ_m ∈ ℝ is the time of the m-th sample relative to the expected patch center return time of each chirp, A˝[s] ∈ ℂ is a complex value representing the combined effects of the transmitter’s power, antenna gain, path loss, receiver losses, and ground reflection coefficient of a scatterer ϕ[s], and $i:=\sqrt{-1}$ is the unit complex number. Note that a completion of the square is required to arrive at Equation (6). Also note the lack of noise in Equation (6), which is sometimes modeled on baseband samples; this publication is focused on the effects of navigation errors on radar imagery and not the computation of a signal-to-noise (SNR) ratio or other noise-dependent quantities, so noise is neglected.

Expanding the true range about the center pulse time η_n = 0 and patch center s = 0 produces the following:

$r*[{\eta }_n,s]\approx \underset{r_0^{*}}{\underbrace{r*[0,0]}}+\underset{{\dot{r}}_0^{*}}{\underbrace{\frac{\partial r*[0,0]}{\partial {\eta }_n}}}{\eta }_n+\underset{\Delta r_0^{*}\left[s\right]}{\underbrace{{\left(\frac{\partial r*[0,0]}{\partial s}\right)}^{{\rm T}}s}}\text{+}\underset{\Delta {\dot{r}}_0^{*}\left[s\right]}{\underbrace{\left({\left(\frac{{\partial }^2{r}^{*}[0,0]}{\partial {\eta }_n\partial s}\right)}^{{\rm T}}s\right)}}{\eta }_n$7

where $r_0^{*}>0$ and ${\dot{r}}_0^{*}\in ℝ$ are the true range and range rate, respectively, between the radar and the patch center at the center of the aperture, $\Delta r_0^{*}\left[s\right]>0$ and $\Delta {\dot{r}}_0^{*}\left[s\right]\in ℝ$ are the true offset range and offset range rate, respectively, of the point scatterer at the center of the aperture, and the subscript 0 is used to indicate quantities that occur at the center of the aperture (when η_n = 0). The offset range and range-rate are the first-order differences in range and range-rate between the radar/patch center and the radar/point scatterer at the center of the aperture, and it is with these offset values that the dechirp-on-receive range-Doppler image is initially formed. The first-order planar wavefront expansion invoked in Equation (7) is appropriate as long as $\Vert s\Vert /r_0^{*}\ll 1$ (the illuminated patch size is small relative to the standoff range).

Similarly, expanding the estimated range to the patch center around the center pulse time η_n = 0 produces the following:

$r\left[{\eta }_n\right]\approx \underset{r_0}{\underbrace{r\left[0\right]}}+\underset{{\dot{r}}_0}{\underbrace{\frac{\partial r\left[0\right]}{\partial {\eta }_n}}}{\eta }_n$8

where r₀ > 0 and ${\dot{r}}_0\in ℝ$ are the estimated range and range rate, respectively, from the radar to the patch center at the center of the aperture. The estimated range and range rate at the center of the aperture will be used at the end of this derivation; these terms are computed via standard geometric radar range and range-rate models with the estimated radar position p_r ∈ ℝ³ and velocity v_r ∈ ℝ³ at the center of the aperture and the location of the prescribed patch center point p_p ∈ ℝ³:

$r_0=\Vert p_p-p_r\Vert$9

${\dot{r}}_0=-v_r^T\frac{p_p-p_r}{\Vert p_p-p_r\Vert }$10

The quantity $r*[{\eta }_n,s]-r\left[{\eta }_n\right]$ in Equation (6) is then approximated as follows:

$r*[{\eta }_n,s]-r\left[{\eta }_n\right]\approx \Delta r_0^{*}\left[s\right]+\Delta {\dot{r}}_0^{*}\left[s\right]{\eta }_n+\underset{\delta r_0}{\underbrace{r_0^{*}-r_0}}+\underset{\delta {\dot{r}}_0}{\underbrace{({\dot{r}}_0^{*}-{\dot{r}}_0)}}{\eta }_n$11

where δr₀∈ ℝ and $\delta {\dot{r}}_0\in ℝ$ are the patch center range and range-rate estimation errors, respectively, at the center of the aperture due to position and velocity navigation errors. Note that δr₀ and $\delta {\dot{r}}_0$ could also be interpreted as offset range and range-rate errors in Equation (11) (indeed, that is how they will manifest in the following analysis).

Defining $\Delta {\tilde{r}}_0\left[s\right]:=\Delta r_0^{*}\left[s\right]+\delta r_0$ and $\Delta {\dot{\tilde{r}}}_0\left[s\right]:=\Delta {\dot{r}}_0^{*}\left[s\right]+\delta {\dot{r}}_0$ and substituting Equation (11) into Equation (6) produces the following:

$x_{\text{BB}}[{\tau }_m,{\eta }_n,s]\approx A"\left[s\right]\exp \left[\frac{-4\pi ik}{c}\right(\Delta {\tilde{r}}_0\left[s\right]+\Delta {\dot{\tilde{r}}}_0\left[s\right]{\eta }_n\left){\tau }_m\right]\exp \left[\frac{-4\pi i{f}_0}{c}\right(\Delta {\tilde{r}}_0\left[s\right]+\Delta {\dot{\tilde{r}}}_0\left[s\right]{\eta }_n\left)\right]\exp \left[\frac{4\pi ik}{c^2}{\left(\Delta {\tilde{r}}_0\right[s]+\Delta {\dot{\tilde{r}}}_0[s\left]{\eta }_n\right)}^2\right]w[{\tau }_m/T_p]w[{\eta }_n/T_a]$12

The goal is to represent the baseband signal in the following form:

$x_{\text{BB}}[{\tau }_m,{\eta }_n,s]=A"\text{'}\left[s\right]\exp \left[2\pi i{f}_r\right[s\left]{\tau }_m\right]\exp \left[2\pi i{f}_d\right[s\left]{\eta }_n\right]w[{\tau }_m/T_p]w[{\eta }_n/T_a]$13

where $A"\text{'}\left[s\right]\in ℂ$ is a modified complex value and f_r[s] ∈ ℝ and f_d[s] ∈ ℝ are frequency terms related to the offset range and range rate (analogously, offset Doppler). Once in this form, a 2D Fourier transform over the samples τ_m and η_n registers the energy reflected by a point scatterer at a patch center offset s to the coordinate pair [f_r[s], f_d[s]]:

$X_{\text{BB}}[\varsigma ,\xi ,s]=A"\text{'}\left[s\right]T_p{T}_a\sin c\left[T_p\right(\varsigma -f_r\left[s\right]\left)\right]\sin c\left[T_a\right(\xi -f_d\left[s\right]\left)\right]$14

Here, X_BB is the frequency-domain version of x_BB, ς ∈ ℝ and ξ ∈ ℝ are the frequency analogs of τ_m and η_n, and the normalized sinc function is defined as $\sin c\left[x\right]:=\frac{\sin \left[\pi x\right]}{\pi x}$.

Two approximations that are valid for coarse-resolution SARs are invoked to modify Equation (12) so that it takes the form of Equation (13). First, the term for which the product of η_n and τ_m is a factor in the first exponential expression is neglected. This term constitutes a range migration — a change in a scatterer’s offset range over the synthetic aperture duration. Provided that the range and Doppler resolutions are coarse and that the synthetic aperture duration T_a is small, this term can be safely neglected (equivalently, Equation (1) is satisfied). When this requirement is met, the change in the offset range of a point scatterer during aperture correlation does not exceed one range resolution cell, and the radar energy reflected by the point is focused within the cell. Note that this range migration requirement must also apply to the patch center range and range-rate estimation error terms δr₀ and $\delta {\dot{r}}_0$, which are components of $\Delta {\tilde{r}}_0\left[s\right]$ and $\Delta {\dot{\tilde{r}}}_0\left[s\right]$. That is, the navigation position and velocity errors should not induce a range migration greater than one resolution cell during aperture formation. Such a requirement can be demanding for fine-resolution SAR systems, but is not difficult to meet for coarse-resolution systems with range resolutions on the order of meters and aperture collection times shorter than 1 s. Second, the entire third exponential term is neglected. Division by the squared speed of light ensures that the phase contribution of this term is small, provided that the range extent (span of the offset range values) of the scene is small. Doerry (1994), for example, explored the limits of this second approximation.

By assuming that the requirements of these approximations are met and absorbing the constant phase terms into $A"\text{'}\left[s\right]$, one may approximate the baseband signal model by Equation (13), with the following relationships:

$\begin{array}{cc}{f}_r\left[s\right]=\frac{-2k}{c}\Delta {\tilde{r}}_0\left[s\right]& =\frac{-2k}{c}\left(\Delta r_0^{*}\right[s]+\delta r_0)\end{array}$15

$\begin{array}{cc}{f}_d\left[s\right]=\frac{-2f_0}{c}\Delta {\dot{\tilde{r}}}_0\left[s\right]& =\frac{-2f_0}{c}\left(\Delta {\dot{r}}_0^{*}\right[s]+\delta {\dot{r}}_0)\end{array}$16

From Equations (15) and (16), it is clear that the patch center range and range-rate estimation errors δr₀ and $\delta {\dot{r}}_0$ originating from the navigation errors bias the frequency values of the associated offset range-Doppler image. In other words, the energy of a point scatterer will be registered at a biased offset range-Doppler coordinate. A final step, however, eliminates this bias.

Once in the form of Equation (13), a 2D DFT over the M×N complex radar samples registers the energy from each point scatterer at an offset s to an offset frequency coordinate pair $\left[f_r\right[s],f_d[s\left]\right]$ To recover the absolute range and range rate between the radar and point scatterer, one must scale the offset frequencies and add the estimated range r₀ and range rate ${\dot{r}}_0$ to the patch center, completing the image formation block in Figure 1(a). The absolute measured range r_meas[s]>0 and range rate ${\dot{r}}_{\text{meas}}\left[s\right]\in ℝ$ are as follows:

$\begin{array}{cc}{r}_{\text{meas}}\left[s\right]=\frac{c}{-2k}{f}_r\left[s\right]+r_0& =r_0^{*}+\Delta r_0^{*}\left[s\right]\approx r^{*}[0,s]\end{array}$17

$\begin{array}{cc}{\dot{r}}_{\text{meas}}\left[s\right]=\frac{c}{-2f_0}{f}_d\left[s\right]+{\dot{r}}_0& ={\dot{r}}_0^{*}+\Delta {\dot{r}}_0^{*}\left[s\right]\approx {\dot{r}}^{*}[0,s]\end{array}$18

This final step of adding the estimated range and range rate to the patch center cancels the errors introduced by the same estimated values during the signal mixing, leaving an unbiased measurement of the range and range rate to the point scatterer. Consequently, it is appropriate, for example, to model the range r_meas > 0 from the radar’s true position at the center of the aperture $p_r^{*}\in {ℝ}^3$ to the true position of a feature point $p_s^{*}\in {ℝ}^3$ as the true (unbiased) geometric range:

$r_{\text{meas}}=\Vert p_s^{*}-p_r^{*}\Vert$19

Referring to Figure 1(a), this result indicates that the range-Doppler navigation observables produced by the block in the bottom-right corner (such as those that might be extracted by the trusted inertial terrain-aided navigation [TITAN] algorithm (Haydon & Humphreys, 2023)) can safely ignore the presence of the feedback loop and can be modeled as they normally might be: as standard measurements of the true navigation state. Stated differently and co-opting some feedback control language, the feedback response of the absolute range-Doppler measurements to navigation errors is nullified.

It is stressed that this result does not contradict the well-known fact that navigation errors induce blurring within a SAR image. Blurring occurs as a result of the range migration and quadratic phase error terms neglected in the baseband signal model, both of which are greatly mitigated by coarse range and Doppler resolutions. The key result of the foregoing analysis is that no absolute biases are introduced by the presence of navigation errors; consequently, standard navigation measurement models are appropriate for coarse-resolution dechirp-on-receive feedback SAR navigation systems. This result also makes intuitive sense: one would expect the time of flight and Doppler shift of a narrow-beam radio ping (a crude perspective on a single range-Doppler cell of a SAR image) to be unaffected by position or velocity errors.

3 SIMULATION

A brief simulation was performed to illustrate the key conclusions of this paper. A nadir-looking radar was simulated as flying flat and level at a ground clearance of 3 km with a speed of 80 m/s. The radar was configured to operate at a center frequency of f_c = 15 GHz with range and ground-projected Doppler resolutions of 3 m. To achieve these resolutions, the chirp duration, chirp rate, pulse repetition interval, and aperture duration were set as $T_p=2\mu s,k\approx 25\times {10}^{12}Hz/s,T_c\approx 600\mu s$, and T_a≈125 ms, respectively. The channel gain A′′ was set to unity. Five point scatterers were simulated, centered around the patch center (origin) with 10-m offsets in the along-track and vertical directions.

Two scenarios were simulated: (1) a scenario in which the radar experienced no navigation errors and (2) a scenario in which the radar experienced a position error of [20 10 10] m and a velocity error of [10 10 10] cm/s at the center of the aperture. Here, the coordinate axes are organized as [along-track, cross-track, vertical]. Baseband samples were simulated according to Equation (6), and range-Doppler images were formed according to the remainder of the paper (i.e., the samples were stacked in a 2D array, and a 2D DFT was applied to the data).

Figure 2 illustrates the resulting magnitude-squared range-Doppler images. The four range-Doppler images correspond to the the two scenarios and depict the images before and after the final range/Doppler correction. When the system experienced no navigation errors, the energy of the five point scatterers was correctly focused to individual range-Doppler cells corresponding to the scatterers’ true range and Doppler shift. When the system was subjected to moderate navigation errors, the energy of the five point scatterers was still sharply focused (a key benefit of the system’s coarse resolution), but the offset range-Doppler values were biased, as illustrated by the top-right subfigure. However, when the axes of the image with the estimated range and Doppler shift of the patch center were adjusted according to Equations (17)–(18), the errors in the offset range-Doppler coordinates were canceled, leaving a sharp and unbiased image of the five point scatterers.

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Figure 2

Simulated range-Doppler images without (left column) and with (right column) navigation errors

Offset range-Doppler images prior to Equations (17)–(18) are displayed in the top row, and absolute range-Doppler images after the correction are displayed in the bottom row.

The two images in the right column of Figure 2 capture the difference in conclusions between the geolocation work performed by, for example, Doerry and Bickel (2021) and Lindstrom et al. (2022b) and the direct measurement model derivation offered in this publication. The top-right subfigure illustrates the conclusion reached by the geolocation derivations: that navigation errors induce feature point geolocation errors (the geolocated positions of the point scatterers relative to the patch center are wrong). The bottom-right subfigure illustrates the conclusion reached by this paper: that these errors cancel out when one considers the absolute range/Doppler from the radar to ground feature points. Because the errors cancel, the absolute range-Doppler measurements from the radar to ground feature points can be safely modeled with the standard navigation measurement model as in Equation (19): z = h [x*] + w.

4 CONCLUSION

In dechirp-on-receive feedback SAR navigation, navigation estimates condition replica radar pulses that are used to form a SAR image, which then corrects the original navigation solution. Such a feedback mechanism could potentially bias radar range-Doppler measurements, which would require changes to the standard navigation model. This paper derived a signal model for a coarse-resolution dechirp-on-receive vertical SAR image and traced the effects of position and velocity navigation errors throughout the system. The analysis showed that the navigation errors do affect the offset range-Doppler measurements, although these effects cancel out when the image is absolutely registered; thus, standard navigation measurement models are appropriate to use with absolute range-Doppler measurements. This analysis was predicated on several simplifying assumptions, including constrained scene sizes, coarse range-Doppler resolutions, single bounce reflections, sub-orbital speeds, and an absence of radar motion during the signal time of flight; future work might extend this analysis to situations in which these assumptions are violated.

HOW TO CITE THIS ARTICLE:

Haydon, T., & Humphreys, T.E. (2026). FLP-Aided GNSS RTK positioning: A proof for the unbiased nature of range-doppler measurements in coarse-resolution dechirp-on-receive feedback synthetic aperture radar navigation. NAVIGATION, 73. https://doi.org/10.33012/navi.736