A possible way to accomplish our algorithm is given in Figure 5. We expect that we choose the correct interval in our preliminary processing and the final accuracy would be closely approximated by the Crammer-Rao low bound (CRLB). The Crammer-Rao low bound can be obtained by inverting the Fisher Information Matrix *F* whose elements are the expectation of the second derivative (with respect to *v* and *d*_{
T
} ) of the Log likelihood function [23]. The Fisher Information Matrix can be written as

F=\left[\begin{array}{cc}\hfill A\hfill & \hfill B\hfill \\ \hfill B\hfill & \hfill C\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill E\left[\frac{{\partial}^{2}ln\mathrm{\Lambda}\left(t,v,{d}_{T}\right)}{\partial {v}^{2}}\right]\hfill & \hfill E\left[\frac{{\partial}^{2}ln\mathrm{\Lambda}\left(t,v,{d}_{T}\right)}{\partial v\partial {d}_{T}}\right]\hfill \\ \hfill E\left[\frac{{\partial}^{2}ln\mathrm{\Lambda}\left(t,v,{d}_{T}\right)}{\partial v\partial {d}_{T}}\right]\hfill & \hfill E\left[\frac{{\partial}^{2}In\mathrm{\Lambda}\left(t,v,{d}_{T}\right)}{\partial {d}_{T}^{2}}\right]\hfill \end{array}\right]

(14)

Thus, it is not difficult to obtain the CRLB:

\mathsf{\text{CRLB}}={F}^{-1}=\frac{1}{AC-{B}^{2}}\left[\begin{array}{cc}\hfill C\hfill & \hfill -B\hfill \\ \hfill -B\hfill & \hfill A\hfill \end{array}\right]

(15)

The variations of *A, B*, and *C* in the case of *d*_{
T
} = 0.5*L* are shown in Figure 6a, and the variations of *A, B*, and *C* in the case of *d*_{
T
} = 0.2_{L} are shown Figure 6b. For a human target, the observation time is usually more than 25 s, and it can be known that *A* ≫ *C* > *B*, namely that *AC* ≫ *B*^{2} (shown in Figure 7), so we think the effect of *B* can be ignored when calculating the CRLB. That is to say, the CRLB can be simplified as

\mathsf{\text{CRLB}}={F}^{-1}\approx \frac{1}{AC}\left[\begin{array}{c}C\phantom{\rule{2.77695pt}{0ex}}-B\hfill \\ -B\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}A\hfill \end{array}\right]=\left[\begin{array}{c}{A}^{-1}\phantom{\rule{2.77695pt}{0ex}}-B/\left(AC\right)\hfill \\ -B/\left(AC\right)\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{C}^{-1}\hfill \end{array}\right]

(16)

For the non-random variable, we differentiate (10) and take the expectation, then we have

\begin{array}{c}E\left[\frac{{\partial}^{2}In\mathrm{\Lambda}\left[x\left(t\right),v,{d}_{T}\right]}{\partial {v}^{2}}\right]\\ =\frac{2}{{N}_{0}}\left\{E{\int}_{-\frac{{T}_{obs}}{2}}^{\frac{{T}_{obs}}{2}}\left[x\left(t\right)-{S}_{r}\left(t,v,{d}_{T}\right)\right]\frac{{\partial}^{2}{S}_{r}\left(t,v,{d}_{T}\right)}{\partial {v}^{2}}dt-E{\int}_{-\frac{{T}_{obs}}{2}}^{\frac{{T}_{obs}}{2}}{\left[\frac{\partial {S}_{r}\left(t,v,{d}_{T}\right)}{\partial v}\right]}^{2}dt\right\}\end{array}

(17)

\begin{array}{c}E\left[\frac{{\partial}^{2}In\mathrm{\Lambda}\left[x\left(t\right),v,{d}_{T}\right]}{\partial {{d}_{T}}^{2}}\right]\hfill \\ =\frac{2}{{N}_{0}}\left\{E{\int}_{-\frac{{T}_{obs}}{2}}^{\frac{{T}_{obs}}{2}}\left[x\left(t\right)-{S}_{r}\left(t,v,{d}_{T}\right)\right]\frac{{\partial}^{2}{S}_{r}\left(t,v,{d}_{T}\right)}{\partial {{d}_{T}}^{2}}dt-E{\int}_{-\frac{{T}_{obs}}{2}}^{\frac{{T}_{obs}}{2}}{\left[\frac{\partial {S}_{r}\left(t,v,{d}_{T}\right)}{\partial {d}_{T}}\right]}^{2}dt\right\}\hfill \end{array}

(18)

where *E*[*] means the expectation operator. In the first term of (17) and (18), we observe a factor that

E\left[x\left(t\right)-{S}_{r}\left(t,v,{d}_{T}\right)\right]=E\left[n\left(t\right)\right]=0

(19)

It will make the first integral term to be zero. In the second term of (17) and (18), there are no random quantities, and therefore the expectation operation gives the integral itself. In addition, the partial derivative of the signal can be rewritten as

\frac{\partial {S}_{r}\left(t,v,{d}_{T}\right)}{\partial v}\approx \frac{\partial \left({U}_{tg}sin\left(2\pi {f}_{d}t\right)\right)}{\partial v}\approx {U}_{tg}\cdot \frac{2\pi vL{t}^{2}}{\lambda {d}_{T}\cdot \left(L-{d}_{T}\right)}\cdot cos\left(\pi \frac{{v}^{2}L}{\lambda {d}_{T}\cdot \left(L-{d}_{T}\right)}{t}^{2}\right)

(20)

\frac{\partial {S}_{r}\left(t,v,{d}_{T}\right)}{\partial {d}_{T}}\approx \frac{\partial \left({U}_{tg}sin\left(2\pi {f}_{d}t\right)\right)}{\partial {d}_{T}}\approx {U}_{tg}\cdot \frac{\pi {v}^{2}{t}^{2}\left(2{d}_{T}-L\right)}{\lambda {{d}_{T}}^{2}\cdot {\left(L-{d}_{T}\right)}^{2}}\cdot cos\left(\pi \frac{{v}^{2}L}{\lambda {d}_{T}\cdot \left(L-{d}_{T}\right)}{t}^{2}\right)

(21)

Substituting (19)-(21) into (17) and (18), we have

{\gamma}_{v}^{2}=Var\left[{\widehat{v}}_{ml}-v\right]\ge \frac{{N}_{0}}{2{\int}_{-\frac{{T}_{\mathsf{\text{obs}}}}{2}}^{\frac{{T}_{\mathsf{\text{obs}}}}{2}}{\left[\frac{\partial {S}_{r}\left(t,v,{d}_{T}\right)}{\partial v}\right]}^{2}dt}\approx \frac{5}{4\cdot \frac{{E}_{r}}{{N}_{0}}\cdot {tan}^{2}\alpha \cdot \frac{{\pi}^{2}{T}_{\mathsf{\text{obs}}}^{2}}{{\lambda}^{2}}}

(22)

{\gamma}_{d}^{2}=Var\left[{\widehat{d}}_{T\phantom{\rule{2.77695pt}{0ex}}ml}-{d}_{T}\right]\ge \frac{{N}_{0}}{2{\int}_{-\frac{{T}_{\mathsf{\text{obs}}}}{2}}^{\frac{{T}_{\mathsf{\text{obs}}}}{2}}{\left[\frac{\partial {S}_{r}\left(t,v,{d}_{T}\right)}{\partial {d}_{T}}\right]}^{2}dt}\approx \frac{10}{\frac{{E}_{r}}{{N}_{0}}\cdot \frac{{\pi}^{2}}{{\lambda}^{2}}\cdot {\left({tan}^{2}\beta -{tan}^{2}\alpha \right)}^{2}}

(23)

Equations (22) and (23) give the analytical expression of the Crammer-Rao bounds for velocity estimation and baseline crossing point estimation. Both of these two estimation accuracies are in proportion to the SNR. The higher the SNR is, the better the estimation accuracies are. As for the velocity estimation accuracy, it is also in proportion to the azimuth angle *α*_{
h
} and the observation time *T*_{
obs
} . This is because the larger *α*_{
h
} and the longer *T*_{
obs
} will lead to a bigger Doppler shift. Meanwhile, a longer wavelength will reduce the sensitivity of the Doppler frequency, and therefore impact the estimation accuracy. For the crossing point estimation, a bigger difference between the *α*_{
h
} and *β*_{
h
} will lead to a bigger difference between the Doppler signals, and therefore results in a better estimation with higher accuracy. As for the wavelength, it has the same conclusion as the velocity estimation. Based on the analytical expression of (22) and (23), simulation results of the Crammer-Rao bounds for the velocity and crossing point estimation are presented in Figure 8, which is under the typical experimental parameter.

From Figure 8, we can see that, when the SNR is 15 dB, the square root of CRLB for the velocity estimation is less than 0.02 m/s, for the crossing point estimation it is less than 0.5 m. So, the estimated step should be set as 0.02 m/s for velocity estimation and 0.5 m for crossing point estimation (where the target model is human with a velocity of 1 m/s, the baseline length is 50 m). If so, the estimation accuracy of the relative velocity can reach 2%, which can well satisfy the requirement for target recognition.

Noting that in the ground-based FSR system, we do not estimate the velocity parameter and crossing position at the moment of crossing baseline. The velocity and crossing position are estimated via (11) with the whole received signal (double-side chirp signal shown in Figure 3). Take the velocity estimation for example, from (13), we can see that the target velocity estimation accuracy depends on the choice of velocity step Δ*v*= *v*_{
i
} - *v*_{i-1}. If we assume the target's true velocity is *v*, a velocity deviation Δ*v* introduces envelope and phase errors to the reference function (double-side chirp signal) used for signal compression. In general, the effect of envelope error is far less than that of phase error. Therefore, the required Δ*v* can be determined by the phase error size. As a rule of thumb, taking pi/4 as the maximum tolerable phase error, the choice of Δ*v* can be written as

\frac{\mathrm{\Delta}v}{v}\le \frac{\lambda}{8\cdot \left({d}_{T}+{d}_{R}\right)\cdot {tan}^{2}{\alpha}_{\mathsf{\text{Max}}}}

(24)

where *α*_{
Max
} is the maximal diffraction angle. We can see that velocity step depends on the carrier frequency, baseline distance, and target velocity, where wavelength and baseline distance are known in advance, the diffraction angle are also preset beforehand and approximately determined by the target FS patterns. Here, we assume, as an example, that baseline distance is equal to 50 m, *α*_{
Max
} is 30° (for a human target) and for typical velocity (for a human, 1 m/s) and carrier frequency (151 MHz), the velocity step should be chosen as 0.015 m/s at most.

Figure 9 gives the simulation and the experimental result of the signal processing method in this paper. For comparison, in Figure 10, we first present the processing result of the classical method in [22] under the same experimental condition. The target crosses the baseline perpendicularly at a distance of *d*_{
T
} = 10 m from the transmitter. The length of the baseline is 50 m. The true velocity of the target is 1.62 m/s. Figure 10a shows the received signal of the simulation and experimental data. In Figure 10b, it shows the estimated velocity by the classical method is 1.92 m/s which deviates from the true value a lot. The deviation percentage is \frac{1.92-1.62}{1.62}\times 100\%=18.5\%. While in Figure 9, we can see that both the velocity and the crossing point are estimated correctly. The estimated velocity is 1.62 m/s, the crossing point is estimated at 10 and 40 m.

The simulated and experimental results show that the classical processing algorithm cannot obtain the correct the velocity estimation because of the mismatched target crossing position. Our new algorithm cannot only estimate the velocity, but also estimate the target crossing position correctly, although there is no range resolution in the baseline direction. In addition, the new algorithm extend the operation area, that is to say no matter where is the target crossing position, we can always obtain good estimations of the *v* and *d*_{
T
} , contrary to the classical processing algorithm only works in the operation area around *d*_{
T
} /*L* = 0.5 (crossing near the midpoint).

We should point out that there is an ambiguous crossing position in Figure 8. This is because the reference signal is symmetrical when the constructed reference signal has the distance *d*_{
T
} to the transmitter or to the receiver. If we use the netted FSR, it is not difficult to solve the ambiguity problem of crossing position.