In order to further improve tracking performance, we adaptively select the parameters of the MCPC CAZAC transmit waveform at each time step

*k* so that we can minimize the predicted tracking root mean-squared error (RMSE). The three MCPC CAZAC parameters we consider are

${\Theta}_{k}=\left({Q}_{k},{M}_{k},{\zeta}_{k}\right)$ in (4), where

*Q*_{
k
} is the number of cyclically permuted Björck CAZAC sequences at time

*k*,

*M*_{
k
} is the length of the sequences at time

*k*, and

*ζ*_{
k
} is a parameter that controls the cyclic frequency shift at time

*k*. The expected RMSE is given by the cost function

$\begin{array}{l}J\left({\Theta}_{k}\right)={E}_{{\widehat{\mathbf{X}}}_{k},{\mathbf{X}}_{k},{\mathbf{A}}_{k},{\mathbf{v}}_{k}|{\widehat{\mathbf{X}}}_{k-1},{\Theta}_{k}}\phantom{\rule{0.3em}{0ex}}\left[{\left({\mathbf{X}}_{k}-{\widehat{\mathbf{X}}}_{k}\right)}^{T}\mathbf{C}\left(\phantom{\rule{0.3em}{0ex}}{\mathbf{X}}_{k}-{\widehat{\mathbf{X}}}_{k}\phantom{\rule{0.3em}{0ex}}\right)\phantom{\rule{0.3em}{0ex}}\right]\end{array}$

(26)

where the weighting matrix **C** makes the units of the cost function consistent by compensating for the differing units of the state vector. The subscript in the expectation operator *E*_{·}[·] shows the dependance of the expected RMSE on the random target strength vector **A**_{
k
}, the random noise matrix **v**_{
k
}, the unknown true target state **X**_{
k
}, and the estimate${\widehat{\mathbf{X}}}_{k}$, given the multitarget state estimate${\widehat{\mathbf{X}}}_{k-1}$ at *k*−1 and the choice of Θ_{
k
}.

Next, we identify the set of values that the multitarget state estimate${\widehat{\mathbf{X}}}_{k}$ can take in terms of the delay-Doppler locations associated with it. As described in Section “IP-LPF algorithm”, in order to propose particles, we have considered a discrete finite set of delay-Doppler locations for each partition and each particle. This set corresponds to Cartesian coordinate locations that are most likely to occur according to the kinematic prior and the set of particles${\mathbf{X}}_{k-1}^{\left(i\right)}$, *i* = 1,…,*N* generated at the previous time step *k* − 1. The set is given in (13) and (14) as$\left({n}_{{j}_{n},\lambda ,u,k}^{\left(i\right)},{\nu}_{{j}_{\nu},\lambda ,u,k}^{\left(i\right)}\right)$,${j}_{\nu}=0,\dots ,{J}_{\nu}^{\left(i\right)}$ and${j}_{n}=0,\dots ,{J}_{n}^{\left(i\right)}$, for *λ* = 1,…,Λ, *u* = 1,…,2 and *i* = 1,…,*N*. We use index *ȷ* to denote a member of the set$\mathcal{G}$, of cardinality$\left|\mathcal{G}\right|$, consisting of the *N* particles that could be sampled by the IP-LPF proposal and subsequently weighted. Therefore,$\mathcal{G}$ is a large set including all combinations of possible delay-Doppler locations from two of the sensors for each target and particle. The process of forming partitions from sampled delay-Doppler locations is explained in Section “Stage 1: partitions sampling” and illustrated in Figure3. Subsequently, one possible outcome of the likelihood sampling process and particle weighting is *N* weight-particle pairs$\left\{{\Gamma}_{\u0237,k}^{\left(i\right)},{\mathbf{X}}_{\u0237,k}^{\left(i\right)}\right\},n=1,\dots ,N$ corresponding to delay-Doppler locations$\left({n}_{\u0237,\lambda ,u,k}^{\left(i\right)},{\nu}_{\u0237,\lambda ,u,k}^{\left(i\right)}\right)$, *λ* = 1,…,Λ, *u* = 1,…,*U*, *i* = 1,…,*N*. Similarly, based on the target motion model, we can identify a discrete finite set of possible true target states **X**_{
k
}. Each possible true target state${\mathbf{X}}_{{\u0237}^{\prime},k}$ with index${\u0237}^{\prime}$ is a member of the set${\mathcal{G}}^{\prime}$, of cardinality$\left|{\mathcal{G}}^{\prime}\right|$.${\mathbf{X}}_{{\u0237}^{\prime},k}$ is related to corresponding delay-Doppler locations$\left(\left.{n}_{{\u0237}^{\prime},l,u,k},{\nu}_{{\u0237}^{\prime},l,u,k}\right\}\right.$, *l* = 1,…,*L*, *u* = 1,…,*U*.

From the above, we may rewrite the cost function in (26) as

$\begin{array}{ccc}J\left({\Theta}_{k}\right)\hfill & =\hfill & {\int}_{{\mathbf{A}}_{k}}{\int}_{{\mathbf{v}}_{k}}\sum _{{\u0237}^{\prime}=1}^{\left|{\mathcal{G}}^{\prime}\right|}\sum _{\u0237=1}^{\left|\mathcal{G}\right|}{\left({\mathbf{X}}_{{\u0237}^{\prime},k}-\sum _{i=1}^{N}{\Gamma}_{\u0237,k}^{\left(i\right)}{\mathbf{X}}_{\u0237,k}^{\left(i\right)}\right)}^{T}\hfill \\ \times \mathbf{C}\left({\mathbf{X}}_{{\u0237}^{\prime},k}-\sum _{i=1}^{N}{\Gamma}_{\u0237,k}^{\left(i\right)}{\mathbf{X}}_{\u0237,k}^{\left(i\right)}\right)\hfill \\ \xb7\phantom{\rule{1em}{0ex}}p\left({\left\{{\Gamma}_{\u0237,k}^{\left(i\right)},{\mathbf{X}}_{\u0237,k}^{\left(i\right)}\right\}}_{i=1}^{N}|{\mathbf{X}}_{{\u0237}^{\prime},k},{\mathbf{A}}_{k},{\mathbf{v}}_{k},{\widehat{\mathbf{X}}}_{k-1},{\Theta}_{k}\right)\hfill \\ \times \phantom{\rule{1em}{0ex}}p\left({\mathbf{X}}_{{\u0237}^{\prime},k}|{\widehat{\mathbf{X}}}_{k-1}\right)\phantom{\rule{1em}{0ex}}p\left({\mathbf{A}}_{k}\right)\phantom{\rule{1em}{0ex}}p\left({\mathbf{v}}_{k}\right)\phantom{\rule{1em}{0ex}}d{\mathbf{A}}_{k}\phantom{\rule{1em}{0ex}}d{\mathbf{v}}_{k}\phantom{\rule{0.3em}{0ex}},\hfill \end{array}$

where the probability distributions$p\left({\mathbf{X}}_{{\u0237}^{\prime},k}|{\widehat{\mathbf{X}}}_{k-1}\right),$$p\left({\mathbf{A}}_{k}\right)$, and *p*(**v**_{
k
}) are defined in the context of the motion and measurement models in Sections “Tracking model” and “Matched filter statistic”.

In order to minimize the cost function, we need to minimize the probability$p\left({\left\{{\Gamma}_{\u0237,k}^{\left(i\right)},{\mathbf{X}}_{\u0237,k}^{\left(i\right)}\right\}}_{i=1}^{N}|{\mathbf{X}}_{{\u0237}^{\prime},k},{\mathbf{A}}_{k},{\mathbf{v}}_{k},{\widehat{\mathbf{X}}}_{k-1},{\Theta}_{k}\right)$ and the particle weights${\Gamma}_{\u0237,k}^{\left(i\right)}$ in (25) for particles (*i*) such that${\mathbf{X}}_{{\u0237}^{\prime},k}\ne {\mathbf{X}}_{\u0237,k}^{\left(i\right)}$. As the *ȷ* th set of particles${\left\{{\mathbf{X}}_{\u0237,k}^{\left(i\right)}\right\}}_{i=1}^{N}$ results from sampling by the IP-LPF, we will follow the sampling process of the IP-LPF and identify the selection probability for each partition of particles${\left\{{\mathbf{X}}_{\u0237,k}^{\left(i\right)}\right\}}_{i=1}^{N}$. According to Section “Stage 1: partitions sampling”, we obtain values${\stackrel{~}{\mathbf{x}}}_{\lambda ,k}^{\left(i\right)}$ for each partition *λ* = 1,…,Θ and each particle *i* = 1,…,*N* by sampling delay-Doppler bins from sensors *u* = 1,2 with probability$\prod _{u=1}^{2}{b}_{{j}_{n}^{\prime},{j}_{\nu}^{\prime},\lambda ,u,k}^{\left(i\right)}$ given by (19). The values${\stackrel{~}{\mathbf{x}}}_{\lambda ,k}^{\left(i\right)}$ allow us to evaluate the likelihoods for the *U* sensors in order to sample partitions. In Section “Stage 2: partitions sampling”, we obtain partitions${\mathbf{x}}_{\lambda ,k}^{\left(i\right)}$ with selection probability${b}_{\lambda ,k}^{\left(i\right)}$ given by (22). These sampled partitions are combined into particles into particles${\mathbf{X}}_{k}^{\left(i\right)}$ in Section “Particle weighting”. From the sampling process of each particle${\mathbf{X}}_{k}^{\left(i\right)}$, we conclude that the probability of each particle being selected is$\prod _{\lambda =1}^{\Theta}{b}_{\u0237,\lambda ,k}^{\left(i\right)}\prod _{u=1}^{2}{b}_{\u0237,{j}_{n}^{\prime},{j}_{\nu}^{\prime},\lambda ,u,k}^{\left(i\right)}$. Therefore, any set of particles${\left\{{\mathbf{X}}_{\u0237,k}^{\left(i\right)}\right\}}_{i=1}^{N}$ appears with probability$p\left({\left\{{\Gamma}_{\u0237,k}^{\left(i\right)},{\mathbf{X}}_{\u0237,k}^{\left(i\right)}\right\}}_{i=1}^{N}|{\mathbf{X}}_{k},{\mathbf{A}}_{k},{\mathbf{v}}_{k},{\left\{{\mathbf{X}}_{k-1}^{\left(i\right)}\right\}}_{i=1}^{N},{\Theta}_{k}\right)=\prod _{i=1}^{N}\prod _{\lambda =1}^{\Lambda}{b}_{\u0237,\lambda ,k}^{\left(i\right)}\prod _{u=1}^{2}{b}_{\u0237,{j}_{n}^{\prime},{j}_{\nu}^{\prime},\lambda ,u,k}^{\left(i\right)}$. Furthermore, from (21), (22), for${b}_{\u0237,\lambda ,k}^{\left(i\right)}$ and from (17), (19) for${b}_{\u0237,{j}_{n}^{\prime},{j}_{\nu}^{\prime},\lambda ,u,k}^{\left(i\right)}$ we observe that the above sampling probabilities depend on the single partition likelihood ratio which using (16) is proportional to$exp\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\left(\frac{\underset{\lambda ,1}{\overset{2}{\sigma}}-{\sigma}_{\lambda ,0}^{2}}{2\underset{\lambda ,1}{\overset{2}{\sigma}}{\sigma}_{\lambda ,0}^{2}}{\sum}_{i=1}^{N}{\sum}_{u=1}^{U}{\mathbf{y}}_{\u0237,\lambda ,u,k}^{\left(i\right)}\right)$. Since${\sigma}_{\lambda ,0}^{2}<{\sigma}_{\lambda ,1}^{2}$, the selection probability monotonically increases with the matched filter statistic${\mathbf{y}}_{\u0237,\lambda ,u,k}^{\left(i\right)}$. Therefore, in order to minimize the above selection probability the matched filter statistic needs to be minimized for the delay-Doppler values in (13) and (14) with the additional constraint that${n}_{\u0237,\lambda ,u,k}^{\left(i\right)}\ne {n}_{{\u0237}^{\prime},l,u,k},{\nu}_{{\u0237}^{\prime},l,u,k}\ne {\mathbf{\nu}}_{\u0237,\lambda ,u,k}^{\left(i\right)}$ for all partitions *λ*, particles *i* and sensors *u*. These sets of delay-Doppler locations correspond to the belief on target state as explained previously and only include delay-Doppler locations that imply erroneous target states${\mathbf{X}}_{{\u0237}^{\prime},k}\ne {\mathbf{X}}_{\u0237,k}^{\left(i\right)}$ (i.e., AF sidelobes). Since the matched filter statistic is a random variable it is minimized by minimizing its variance, given in (12) with Λ = 1, with respect to the waveform parameters.

Next, we observe that the particle weights${\Gamma}_{\u0237,k}^{\left(i\right)}$ in (25) contain the likelihood ratio both in the numerator and denominator. This, together with the fact that the prior has a wide spread compared to the likelihood, makes the particle weights nearly constant. Therefore, particle weights cannot be significantly reduced by adjusting the waveform parameters.

Therefore, the focus is on minimizing the matched filter statistic variance in (12) with respect to the waveform parameters specifically for the delay-Doppler values in (13) and (14) and such that${n}_{\u0237,\lambda ,u,k}^{\left(i\right)}\ne {n}_{{\u0237}^{\prime},l,u,k},{\nu}_{{\u0237}^{\prime},l,u,k}\ne {\mathbf{\nu}}_{\u0237,\lambda ,u,k}^{\left(i\right)}$. Since the matched filter statistic variance depends on the AF of the waveform, and since the above-mentioned set of delay-Doppler values refer to AF locations where the target is expected to be at time step *k*, excluding the AF mainlobe, the problem of minimizing the RMSE reduces to the problem of reducing AF sidelobes in the area where the target is expected to exist. This is a very well defined area in the delay-Doppler plane that is given by the sequential tracking process of the particle filter as explained in Section “IP-LPF algorithm”. Configuring the waveform so that zero sidelobes appear in selected areas of the AF surface was described in Section “AF surface of MCPC CAZAC waveforms”. Therefore, at each time step of the scenario, the parameters of the waveform to be transmitted at the next time step are selected such as to achieve low AF sidelobes in the areas where the weak target is expected to be found, resulting to a minimization of the expected RMSE. This is computationally efficient compared to iterative methods of waveform parameter selection[11, 12]. However, the entire multitarget particle filtering method proposed is still associated with a large computational load which is not expected to reach real time operation with state-of-the art hardware which also limits the number of targets that can be tracked.

It is noted that this method works well only if the number of weak targets is low. The AF surface valleys created by these waveforms are, as expected, of limited size since the uncertainty cannot be entirely eliminated. Therefore, if multiple weak targets happen to be relatively positioned such that AF surface valleys cannot be configured to contain them then these targets will be masked. The problem of unmasking a larger number of weak targets is, therefore, an open problem and a limitation of the proposed method. Moreover, there is a prediction error associated with each target location which is estimated based on the Bayesian methodology employed. In this study, the prediction error is minimized as targets are highly localized when using the high resolution, high AF surface peaked CAZAC-based waveforms. The AF surface valleys designed are then large enough to contain this uncertainty and guarantee the unmasking of a weak target.

Furthermore, it is noted that the weak targets need to have a signal strength that is well above the noise level so that they are observable. In this study, it is assumed that what keeps weak targets masked are in fact the sidelobes from stronger measurement returns and not the noise. In order to initially detect weak targets, when no prior tracking information on their state is available, a sequential selective positioning of the sidelobes over different regions of the field of view is necessary. Once a weak target is detected and the tracking process begins then the selective positioning of the sidelobes based on prior tracking information described in this study is possible to take place.