In the sequel, we will deal with the non-convex (the objective is non-convex and ‖*ξ*(*i*)‖ = ‖*η*(*i*)‖ = 1 defines a non-convex set) optimization problem \( \mathcal{P} \). By reformulating the problem using trigonometric parameters and the inspection of its computational complexity, we exploit the sequential minimum MSE estimation scheme to compute the optimal Tx/Rx polarizations instead of directly utilizing the minimum MSE estimation.

### 4.1 Problem reformulation

A first step toward the goal to tackle the optimization problem is represented by the following notations of transmission and reception polarizations. As to the transmission polarization, it can be parameterized into the following trigonometric function [2]:

$$ \boldsymbol{\xi} =\left\Vert \boldsymbol{\xi} \right\Vert {\mathrm{e}}^{j\varphi }Q\boldsymbol{w} $$

(13)

where \( Q=\left[\begin{array}{l} \cos \alpha \kern1.25em \sin \alpha \\ {}- \sin \alpha \kern0.5em \cos \alpha \end{array}\right] \), \( \boldsymbol{w}=\left[\begin{array}{l} \cos \beta \\ {}j \sin \beta \end{array}\right] \), ‖*ξ*‖e^{jφ} is the complex envelope of the source signal, *α* denotes the rotation angle between the system coordinates and the electric ellipse axes, and *β* determines the ellipse’s eccentricity, respectively, with the definition spaces of these trigonometric parameters being *φ* ∈ (−*π*, *π*], *α* ∈ [−*π*/2, *π*/2], and *β* ∈ [−*π*/4, *π*/4]. Since the transmitted signal is power limited, i.e., ‖*ξ*‖ = 1, by replacing *Q* and *w* with their explicit forms, we can rewrite (13) as

$$ \boldsymbol{\xi} \triangleq {\mathrm{e}}^{j\varphi}\left[\begin{array}{l}{\zeta}_{\mathrm{h}}\\ {}{\zeta}_v\end{array}\right]={\mathrm{e}}^{j\varphi}\left[\begin{array}{l} \cos \alpha \cos \beta +j \sin \alpha \sin \beta \\ - \sin \alpha \cos \beta +j \cos \alpha \sin \beta \end{array}\right] $$

(14)

Similarly, the trigonometric form of received polarization can be written as

$$ \boldsymbol{\eta} \triangleq {\mathrm{e}}^{j\phi}\left[\begin{array}{l}{\iota}_{\mathrm{h}}\\ {}{\iota}_v\end{array}\right]={\mathrm{e}}^{j\phi}\left[\begin{array}{l} \cos \theta \cos \vartheta +j \sin \theta \sin \vartheta \\ {}- \sin \theta \cos \vartheta +j \cos \theta \sin \vartheta \end{array}\right] $$

(15)

where *ϕ* ∈ (−*π*, *π*], *θ* ∈ [−*π*/2, *π*/2], and *ϑ* ∈ [−*π*/4, *π*/4].

We can observe that, once *ξ* and *η* are written into (14) and (15), respectively, Tr(*D*) can be uniquely determined by trigonometric parameters {*φ*, *α*, *β*, *ϕ*, *θ*, *ϑ*}. Before proceeding further, herein we introduce an interesting observation of Tr(*D*) with respect to such parameters, as summarized in following property.

**Property 1**
*The initial signal phases φ and ϕ do not affect the value of*
*D*.

*Proof* See Appendix 2.

It can be seen that with Property 1, Tr(*D*) can be uniquely determined by four trigonometric parameters {*α*, *β*, *θ*, *ϑ*} instead of six trigonometric parameters {*φ*, *α*, *β*, *ϕ*, *θ*, *ϑ*}. Thus, the problem \( \mathcal{P} \) can be equivalently recast as

$$ {\mathcal{P}}_1\left\{\begin{array}{c}\underset{\boldsymbol{D},{\left\{{\alpha}_i\right\}}_{i=1}^N,{\left\{{\beta}_i\right\}}_{i=1}^N,{\left\{{\theta}_i\right\}}_{i=1}^N,{\left\{{\vartheta}_i\right\}}_{i=1}^N}{min}\mathrm{T}\mathrm{r}\left(\boldsymbol{D}\right)\\ {}\mathrm{s}.\mathrm{t}.\kern0.5em \boldsymbol{D}={\left({\boldsymbol{C}}_{\mathrm{t}}^{-1}+{\mathbf{A}}^{\mathrm{H}}{\left(\mathbf{A}{\boldsymbol{C}}_{\mathrm{c}}{\mathbf{A}}^{\mathrm{H}}+{\sigma}_{\nu}^2{\mathbf{I}}_N\right)}^{-1}\mathbf{A}\right)}^{-1}\kern0.5em \mathrm{with}\\ {}\kern1.25em \mathbf{A}={\left[\boldsymbol{a}(1),\dots, \boldsymbol{a}(N)\right]}^{\mathrm{T}}\kern0.5em \mathrm{with}\\ {}\kern1.75em \boldsymbol{a}(i)={\left[{\xi}_{\mathrm{h}}^{(i)}{\eta}_{\mathrm{h}}^{(i)},{\xi}_{\mathrm{h}}^{(i)}{\eta}_v^{(i)}, {\xi}_v^{(i)}{\eta}_{\mathrm{h}}^{(i)},{\xi}_v^{(i)}{\eta}_v^{(i)}\right]}^{\mathrm{T}}\ \mathrm{with}\\ {}\kern1.75em \left[\begin{array}{l}{\xi}_{\mathrm{h}}^{(i)}\\ {}{\xi}_v^{(i)}\end{array}\right]=\left[\begin{array}{l} \cos {\alpha}_i \cos {\beta}_i+j \sin {\alpha}_i \sin {\beta}_i\\ {}- \sin {\alpha}_i \cos {\beta}_i+j \cos {\alpha}_i \sin {\beta}_i\end{array}\right]\kern0.5em \mathrm{and}\ \\ {}\kern1.75em \left[\begin{array}{l}{\eta}_{\mathrm{h}}^{(i)}\\ {}{\eta}_v^{(i)}\end{array}\right]=\left[\begin{array}{l} \cos {\theta}_i \cos {\vartheta}_i+j \sin {\theta}_i \sin {\vartheta}_i\\ {}- \sin {\theta}_i \cos {\vartheta}_i+j \cos {\theta}_i \sin {\vartheta}_i\end{array}\right]\kern0.5em \mathrm{with}\\ {}\kern1.75em {\alpha}_i\in \left[-\pi /2,\pi /2\right],\ {\beta}_i\in \left[-\pi /4,\pi /4\right],\ \\ {}\kern1.75em {\theta}_i\in \left[-\pi /2,\pi /2\right],\ {\vartheta}_i\in \left[-\pi /4,\pi /4\right],\ i=1,\dots, N.\ \end{array}\right. $$

(16)

We can see that \( {\mathcal{P}}_1 \) is still a non-convex optimization problem because the objective function is the same non-convex function as in \( \mathcal{P} \). Nevertheless, as opposed to \( \mathcal{P} \), the equivalent formulation provided by \( {\mathcal{P}}_1 \) shows that lattice search along the trigonometric parameters {*α*, *β*, *θ*, *ϑ*} can be employed to calculate the optimal polarization, as done in [2] and [5]. But since lattice search does not explore any optimization property, it requires a high computational burden. Precisely, the computational complexity is firstly exponential to the observation number *N*. Also, it highly depends on the maximum point search algorithm used for {*α*, *β*, *θ*, *ϑ*}. If we use lattice search with *l*
_{
α
}, *l*
_{
β
}, *l*
_{
θ
} and *l*
_{
ϑ
} points in each dimension of the domain space {*α*, *β*, *θ*, *ϑ*}, respectively, the relevant complexity burden is *O*((*l*
_{
α
}
*l*
_{
β
}
*l*
_{
θ
}
*l*
_{
ϑ
})^{N}).

### 4.2 Sequential minimum MSE estimation

Since to solve \( {\mathcal{P}}_1 \) straightforward requires a high computational burden, we exploit the sequential minimum MSE estimation to tackle it. By insight of \( {\mathcal{P}}_1 \), we can observe that it works at the case of block signal polarization design, namely designing the polarizations of *N* consecutive temporal signal samples at a time. However, in real radar signal processing applications, the returns are ongoing as time progresses. So it is reasonable to process the data sequentially in time. As to the problem at hand, the resulting solution belongs to the sequential minimum MSE estimation. Thus, by following the sequential minimum MSE estimation procedure in [Kay [17], Eq. (12.47)–(12.49)], we develop the sequential minimum MSE estimation to handle \( {\mathcal{P}}_1 \) as follows.

Let \( {\widehat{\boldsymbol{X}}}_{\mathrm{t},n} \) denote the minimum MSE estimator based on the observations [*y*(1), …, *y*(*n*)]^{T} and *D*
_{
n
} be the corresponding minimum MSE matrix (just the sequential version of (10) and (11)). Then, when the new sample *y*(*n* + 1) is available, the estimator is updated as

$$ {\widehat{\boldsymbol{X}}}_{\mathrm{t},n+1}={\widehat{\boldsymbol{X}}}_{\mathrm{t},n}+{\boldsymbol{K}}_{n+1}\left(y\left(n+1\right)-{\boldsymbol{a}}^{\mathrm{T}}\left(n+1\right){\widehat{\boldsymbol{X}}}_{\mathrm{t},n}\right), $$

(17)

where

$$ {\boldsymbol{K}}_{n+1}=\frac{{\boldsymbol{D}}_n\boldsymbol{a}\left(n+1\right)}{{\left[{\mathbf{A}}_{n+1}{\boldsymbol{C}}_{\mathrm{c}}{\mathbf{A}}_{n+1}^{\mathrm{H}}+{\sigma}_{\nu}^2{\mathbf{I}}_{n+1}\right]}_{n+1,n+1}+{\boldsymbol{a}}^{\mathrm{H}}\left(n+1\right){\boldsymbol{D}}_n\boldsymbol{a}\left(n+1\right)}, $$

(18)

is the gain factor weighting confidence in the new data with **A**
_{
n + 1} = [*a*(1), …, *a*(*n* + 1)]^{T} and *a*(*i*) having the same definition as before and [·]_{
n + 1,n + 1} denoting the *n* + 1th diagonal element of matrix [·]. In the meantime, the minimum MSE matrix is updated as

$$ {\boldsymbol{D}}_{n+1}=\left({\mathbf{I}}_{n+1}-{\boldsymbol{K}}_{n+1}{\boldsymbol{a}}^{\mathrm{T}}\left(n+1\right)\right){\boldsymbol{D}}_n. $$

(19)

As to *D*
_{
n + 1}, based on [Kay [17], p. 393], with *n* increasing, [*D*
_{
n + 1}]_{
ii
}, *i* = 1, 2, 3, 4, decreases and converges to a certain value. Hence, tr(*D*
_{
n + 1}) is also a monotonic decreasing sequence and converges to a certain value.

### 4.3 Optimal waveform parameter selection

By employing the sequential minimum MSE estimation procedure described earlier, our goal is to search for the optimal Tx/Rx polarizations for every iteration, under the criterion of minimizing the Bayesian MSE-based cost function tr(*D*
_{
n
}). As to each iteration, we still employ lattice search to compute the optimal Tx/Rx polarizations. Precisely, the sequential procedure to update the optimal polarizations is summarized as Algorithm 1. Also, Fig. 1 exhibits a pictorial representation of the sequential estimation processing.

Now let us examine the computational complexity of Algorithm 1. Let us still consider that there are *N* observation samples. Different from solving \( {\mathcal{P}}_1 \), Algorithm 1 is linear to the observation number *N*, with requiring the lattice search for the optimal polarizations in the domain space {*α*, *β*, *θ*, *ϑ*} in each iteration. Thus, the overall complexity is *O*(*l*
_{
α
}
*l*
_{
β
}
*l*
_{
θ
}
*l*
_{
ϑ
}
*N*). In comparison with the direct lattice search to solve \( {\mathcal{P}}_1 \), the computation burden is greatly reduced. It is not surprising since the proposed algorithm is sequential, namely it does not optimize for all observations and thus, it is clearly less expensive than the one proposed in (16).