Range Bias Modeling for Hyperbolic-Frequency-Modulated Waveforms in Target Tracking

Abstract

Hyperbolic-frequency-modulated (HFM) waveforms offer detection performance that is "Doppler insensitive"-relatively unaffected by target range rate in matched filtering. They have been applied in radar; but more they are particularly useful in sonar where the wideband Doppler insensitivity is especially prized. However, Doppler insensitivity does come hand in hand with a range bias, which would significantly degrade the tracking performance. In this paper, we model the range bias as a function of range rate and system parameters, and then utilize that to calibrate the measurement equation in very precise target tracking. By doing so, the tracking performance can be improved, particularly for fast maneuvering targets.

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“…In this paper, the emitter signal is the HFM signal and it is an often-used waveform for target probing [21]. A HFM signal can be defined as follow [21]:$u\left(t\right)=A\mathrm{rect}\left(\frac{t}{T}\right)e^{-j2\pi T\frac{f_H{f}_L}{f_H-f_L}\ln \left(1-\frac{f_H-f_L}{T{f}_H}t\right)},t\in false[0,Tfalse],$where:$\mathrm{rect}\left(u\right)=\{\begin{array}{ll}1,& 0\le u\le 1\\ 0,& others\end{array},$$j=\sqrt{-1}$, and $A$ is the amplitude that represents the signal energy, $\mathrm{rect}\left(t/T\right)$ denotes a rectangular envelope, and $T$ is the pulse width. The time derivative of the signal phase is expressed as the instantaneous frequency, $f\left(t\right)$, which is given by [21]:$f\left(t\right)=\frac{T{f}_H{f}_L}{T{f}_H-\left(f_H-f_L\right)t}.$…”

“…The instantaneous frequency of $r\left(t\right)$ is $f_r\left(t\right)$, as given in Equation (12) [21]:$f_r\left(t\right)=\frac{T{f}_H{f}_L\kappa }{T{f}_H-\left(f_H-f_L\right)\kappa \left(t-\tau \right)}.$…”

“…The dashed lines depict the outputs when there are Doppler biases between the copied signals and the HFM signal, and the Doppler bias can be caused by the relative movement of the target. The HFM signal has a prized property of Doppler tolerance, that a waveform which satisfies the property also has a MF output peak for a moving target, even if the time bandwidth product is large [21]. As shown in Figure 3, the output amplitudes of the mismatched copied signals are slightly smaller than the exactly matched (no Doppler bias) copied signal, and they are distributed in sharp peak states as well.…”

“…Let $P\left(v\right)$ denote the amplitude of the WBAF ridge at velocity $v$ without the consideration of receiver noise, where $v=c\left(1-\kappa \right)/2$. The power loss ratio (namely the slope of the WBAF ridge) for an HFM signal with rectangular envelope can be loosely approximated as [12,21]:$\xi \left(v\right)=\frac{P\left(v\right)}{P\left(0\right)}\approx 1-\frac{2\left|v\right|}{c}\left|\frac{1}{2}-\frac{f_H}{f_H-f_L}\right|=1-\left|v\right|\frac{2f_0}{cB},$ where $0\le \xi \left(v\right)\le 1$, and it is in an approximate linear fashion. The power loss ratio is proportional to the center frequency $f_0$, and inversely proportional to the bandwidth propagation speed product $Bc$.…”

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“…Therefore, given the estimated ToAt p;q , the estimation of the ToFŝ p , the distancesr r;p and the radial velocitŷ v r;p requires some more calculations [18]:…”