In this section, we present the new joint estimator for the desired parameters. To this end, we consider the uplink transmission over a SIMO Rayleigh channel from a single source to *N*_{a} uniform linear array (ULA) at the receiver. The received signal in baseband at the *i* th antenna element is modeled as follows [15]:

\begin{array}{l}\phantom{\rule{-13.0pt}{0ex}}{x}_{i}\left(t\right)={\sigma}_{{x}_{i}}{\text{lim}}_{P\to +\infty}\frac{1}{\sqrt{P}}\sum _{p=1}^{P}{a}_{p}\text{exp}j\left[{\omega}_{\text{D}}\text{cos}\left({\theta}_{\text{p}}\right)t+{\varphi}_{\text{p}}\right]+{n}_{i}\left(t\right),\end{array}

(1)

where {\sigma}_{{s}_{i}}^{2} is the power of the received signal, *P* is the number of multipaths, *a*_{
p
} are random unknown complex constants normalized as follows:

\begin{array}{l}{\text{lim}}_{P\to +\infty}{p}^{-1}\sum _{p=1}^{P}{\left|{a}_{p}\right|}^{2}=1,\end{array}

(2)

so that {\sigma}_{{s}_{i}}^{2}=E\left[\phantom{\rule{0.3em}{0ex}}{\left|{x}_{i}\left(t\right)\right|}^{2}\right]-{\sigma}_{{n}_{i}}^{2} where {\sigma}_{{n}_{i}}^{2} is the power of the additive white Gaussian noise (AWGN), *n*_{
i
}(*t*), at the *i* th antenna. The AoAs *θ*_{p} of the received signals follow an angular distribution with a mean and a standard deviation defined by the mean AoA, *θ*_{m}, and the AS, *σ*_{
θ
}, respectively. The phases *ϕ*_{p} are uniformly distributed over (-*π* *π*]. *ω*_{D} denotes the normalized maximum DS and is given by *ω*_{D}= 2 *π* *F*_{D}*T*_{s} where *F*_{D} is the Doppler frequency [15] and *T*_{s} is the sampling interval.

As mentioned before, the estimation of the mean AoA, AS, and DS is useful in several applications by improving therein the potential performance gains from smart antennas. Most mean AoA and AS estimators as in [1, 6, 7] consider the following spatial correlation function:

\begin{array}{lll}\phantom{\rule{-6.0pt}{0ex}}{\mathbf{\text{R}}}_{\mathit{\text{kl}}}\phantom{\rule{1em}{0ex}}& =& \frac{E\left[{x}_{k}\left(t\right){x}_{l}^{\ast}\left(t\right)\right]}{\sqrt{E\left[\phantom{\rule{0.3em}{0ex}}{\left|{x}_{k}\left(t\right)\right|}^{2}\right]E\left[\phantom{\rule{0.3em}{0ex}}{\left|{x}_{l}\left(t\right)\right|}^{2}\right]}}\\ =& \underset{-\infty}{\overset{+\infty}{\int}}P\left({\theta}_{\text{p}}|{\theta}_{\text{m}},{\sigma}_{\theta}\right)\text{exp}\left(-j\frac{2\pi}{\lambda}{d}_{\mathit{\text{kl}}}\text{sin}\left({\theta}_{\text{m}}\right)\right)d{\theta}_{\text{p}},\end{array}

(3)

where *P*(*θ*_{p}|*θ*_{m},*σ*_{
θ
}) is the probability density function (PDF) of the incoming AoAs.

For the DS estimators, the temporal correlation function is exploited as in [3, 8, 13] and is defined by

\begin{array}{ll}\phantom{\rule{-15.0pt}{0ex}}\mathbf{\text{R}}(\tau )& =\frac{E\left[\phantom{\rule{0.3em}{0ex}}{x}_{k}\left(t\right){x}_{k}^{\ast}(t+\tau )\right]}{E\left[{\left|{x}_{k}\left(t\right)\right|}^{2}\right]}\\ =\underset{-\infty}{\overset{+\infty}{\int}}P\left({\theta}_{\text{D}}|{\theta}_{\text{m}},{\sigma}_{\theta}\right)\text{exp}\left(-j{\omega}_{\text{D}}\text{cos}\left({\theta}_{\text{D}}\right)\tau \right)d{\theta}_{\text{D}},\end{array}

(4)

where *P*(*θ*_{D}|*θ*_{m},*σ*_{
θ
}) is the PDF of the Doppler angles *θ*_{D}. The latter is given by (*θ*_{p}-*α*), where *α* is the direction of travel (DoT) defined as the angle between the direction of the mobile and the antenna axis as shown in Figure 1. In this work, instead of combining two methods from the ones developed in the literature, we propose a unique algorithm to jointly estimate the three parameters. To this end, we jointly exploit both the spatial and the temporal correlations. The cross-correlation matrix of the received signals is then given by

\begin{array}{lcr}{\mathbf{\text{R}}}_{\mathit{\text{kl}}}(\tau )& =& \frac{E\left[{x}_{k}\left(t\right){x}_{l}^{\ast}(t+\tau )\right]}{\sqrt{E\left[{\left|{x}_{k}\left(t\right)\right|}^{2}\right]E\left[{\left|{x}_{l}\left(t\right)\right|}^{2}\right]}}.\end{array}

After some algebraic manipulation and using (2), we obtain the following expression for the cross-corrrelation function:

\begin{array}{lcr}\phantom{\rule{-15.0pt}{0ex}}{\mathbf{\text{R}}}_{\mathit{\text{kl}}}(\tau )& =& \underset{-\pi}{\overset{\pi}{\int}}P\left({\theta}_{\text{p}};{\theta}_{\text{m}},{\sigma}_{\theta}\right)\text{exp}\left(-j\frac{2\pi}{\lambda}{d}_{\mathit{\text{kl}}}\text{sin}\left({\theta}_{\text{p}}\right)\right)\end{array}

(5)

\begin{array}{l}\text{exp}\left(-j{\omega}_{\text{D}}\tau \text{sin}\left({\theta}_{\text{p}}-\alpha \right)\right)d{\theta}_{\text{p}},\end{array}

(6)

with *d*_{
kl
} is the distance between the *k* th and the *l* th antenna elements.The estimated cross-correlation coefficients are given by

\begin{array}{lcr}{\widehat{\mathbf{\text{R}}}}_{\mathit{\text{kl}}}(\tau )& =& \frac{1}{{N}_{\text{s}}-\tau}\sum _{m=1}^{{N}_{\text{s}}-\tau}{x}_{k}\left(m\right)\phantom{\rule{1em}{0ex}}{x}_{l}(m+\tau ),\end{array}

(7)

where *N*_{s} is the number of the received signal samples.

In this paper, we consider both Gaussian and Laplacian angular distributions for the incoming AoAs [14]. Other angular distribution for the incoming AoAs can be applied like the uniform one, but this would yield to different closed-form expressions. The von Mises distribution approximate all these angular distributions over *κ* parameter value, but in our approach, it does not yield to closed-form expressions of the auto-correlation and cross-correlations functions.

### 2.1 Gaussian angular distribution

The PDF of the Gaussian angular distribution is given by

\begin{array}{l}P\left({\theta}_{\text{p}},{\theta}_{\text{m}},{\sigma}_{\theta}\right)=\frac{1}{{\sigma}_{\theta}\sqrt{2\pi}}\text{exp}\left[-\frac{{({\theta}_{\text{p}}-{\theta}_{\text{m}})}^{2}}{2\underset{\theta}{\overset{2}{\sigma}}}\right].\end{array}

(8)

The following entity is considered to solve the integral expression [16]:

\begin{array}{l}\underset{0}{\overset{+\infty}{\int}}\text{exp}\left(-a{x}^{2}\right)\text{cos}\mathit{bx}\phantom{\rule{1em}{0ex}}\mathit{dx}=\frac{1}{2}\sqrt{\frac{\pi}{a}}\text{exp}\frac{-{b}^{2}}{4\phantom{\rule{1em}{0ex}}a}.\end{array}

(9)

In this work, we assume small ASs, *σ*_{
θ
}. Indeed, in macrocell environments, the AS does not exceed 10° [15, 17, 18]. In this case, the following linearization is applied to ensure regular integrals:

\begin{array}{l}\text{sin}\left({\theta}_{\text{p}}\right)=\text{sin}\left({\theta}_{\text{m}}\right)+({\theta}_{\text{p}}-{\theta}_{\text{m}})\text{cos}\left({\theta}_{\text{m}}\right).\end{array}

(10)

We obtain the following closed-form expression, **R**_{
kl
}(*τ*), for the Gaussian angular distribution:

\begin{array}{l}\phantom{\rule{-15.0pt}{0ex}}\left|{\mathbf{\text{R}}}_{\mathit{\text{kl}}}\right(\tau \left)\right|\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\text{exp}\left[\phantom{\rule{0.3em}{0ex}}-\frac{\underset{\theta}{\overset{2}{\sigma}}}{2}{\left(\phantom{\rule{0.3em}{0ex}}-{\omega}_{\text{D}}\tau \text{cos}({\theta}_{\text{m}}\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}\alpha )\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}\frac{2\pi}{\lambda}{d}_{\mathit{\text{kl}}}\text{cos}\left({\theta}_{\text{m}}\right)\phantom{\rule{0.3em}{0ex}}\right)}^{2}\right],\end{array}

(11)

\begin{array}{l}\angle {\mathbf{\text{R}}}_{\mathit{\text{kl}}}(\tau )=-\frac{2\pi}{\lambda}{d}_{\mathit{\text{kl}}}\text{sin}\left({\theta}_{\text{m}}\right)-{\omega}_{\text{D}}\tau \text{sin}({\theta}_{\text{m}}-\alpha ).\end{array}

(12)

In our algorithm, we consider the modules and the phases of the estimated cross-correlation coefficients. The mean AoA is estimated using, respectively, the phase of the auto-correlation, {\widehat{\mathbf{\text{R}}}}_{\mathit{\text{kk}}}(\tau ), of the received signal at the *k* th antenna and the cross-correlation, {\widehat{\mathbf{\text{R}}}}_{\mathit{\text{kl}}}(\tau ), associated to the antenna pair (*k*,*l*) as follows:

\begin{array}{lcr}{\widehat{\theta}}_{\text{m}}(k,l)& =& \text{arcsin}\left(\frac{\angle {\widehat{\mathbf{\text{R}}}}_{\mathit{\text{kk}}}(\tau )-\angle {\widehat{\mathbf{\text{R}}}}_{\mathit{\text{kl}}}(\tau )}{\phantom{\rule{1em}{0ex}}\frac{2\pi}{\lambda}{d}_{\mathit{\text{kl}}}}\right),\end{array}

(13)

∀ *k* < *l* and (*k*,*l*) ∈ {1 … *N*_{a}} with *k* ≠ *l*.

The AS estimate is determined by exploiting the modules of the cross-correlations {\widehat{\mathbf{\text{R}}}}_{\mathit{\text{kj}}}(\tau ), {\widehat{\mathbf{\text{R}}}}_{\mathit{\text{kl}}}(\tau ), and the mean AoA estimates:

\begin{array}{lcr}\phantom{\rule{-15.0pt}{0ex}}{\widehat{\sigma}}_{\theta}(k,l,j)& =& \phantom{\rule{1em}{0ex}}\frac{\sqrt{-ln\left(\right|{\widehat{\mathbf{\text{R}}}}_{\mathit{\text{kj}}}(\tau )\left|\right)}-\sqrt{-ln\left(\right|{\widehat{\mathbf{\text{R}}}}_{\mathit{\text{kl}}}(\tau )\left|\right)}}{\frac{\sqrt{2}\pi}{\lambda}\text{cos}\left({\widehat{\theta}}_{\text{m}}\right)({d}_{\mathit{\text{kj}}}-{d}_{\mathit{\text{kl}}})},\end{array}

(14)

∀ *k* < *l* < *j* and (*k*,*l*,*j*) ∈ {1 … *N*_{a}} with *k* ≠ *l* ≠ *j*.

Finally, the maximum DS is deduced using both the module and phase of the cross-correlation, {\widehat{\mathbf{\text{R}}}}_{\mathit{\text{kl}}}(\tau ), the estimated values of the mean AoA and the AS and considering the trigonometric property (cos(*θ*_{m})^{2}+ sin(*θ*_{m})^{2}= 1). The maximum DS estimate is then expressed as follows:

\begin{array}{l}\widehat{\omega}(k,l)=1/\tau \left(\sqrt{{\left(\angle {\widehat{\mathbf{\text{R}}}}_{\mathit{\text{kl}}}(\tau )+\frac{2\pi}{\lambda}{d}_{\mathit{\text{kl}}}\text{sin}\left({\widehat{\theta}}_{\text{m}}\right)\right)}^{2}+{\left(\frac{\sqrt{-ln\left|{\widehat{\mathbf{\text{R}}}}_{\mathit{\text{kl}}}\right(\tau \left)\right|}}{{\widehat{\sigma}}_{\theta}}-\frac{2\pi}{\lambda}{d}_{\mathit{\text{kl}}}\text{cos}\left({\widehat{\theta}}_{\text{m}}\right)\right)}^{2}}\right),\end{array}

(15)

∀ *k* < *l* and (*k*,*l*) ∈ {1 … *N*_{a}} with *k* ≠ *l*.

### 2.2 Laplacian angular distribution

The PDF of the Laplacian angular distribution is defined by

\begin{array}{l}P({\theta}_{\text{p}},{\sigma}_{\theta},{\theta}_{\text{m}})=\frac{1}{{\sigma}_{\theta}\sqrt{2}}\text{exp}\left[-\frac{|{\theta}_{\text{p}}-{\theta}_{\text{m}}|}{\frac{{\sigma}_{\theta}}{\sqrt{2}}}\right]\end{array}

(16)

To overcome the integral expressions in (5), we can use the following identity [16]:

\begin{array}{l}\underset{0}{\overset{+\infty}{\int}}\text{exp}(-\mathit{ax})\text{cos}\mathit{bx}\phantom{\rule{1em}{0ex}}\mathit{dx}=\frac{a}{{a}^{2}+{b}^{2}}.\end{array}

(17)

Assuming a small AS, the following approximation can be considered:

\begin{array}{lcr}\text{cos}\left({\theta}_{\text{p}}\right)& =& \text{cos}\left({\theta}_{\text{m}}\right)-({\theta}_{\text{p}}-{\theta}_{\text{m}})\text{sin}\left({\theta}_{\text{m}}\right).\end{array}

(18)

After some algebraic manipulations, we obtain the cross-correlation coefficient for the Laplacian distribution **R**_{
kl
}(*τ*). As for the Gaussian version, we consider separately the magnitude and the phase of the cross-correlation coefficients as follows:

\begin{array}{l}\phantom{\rule{-15.0pt}{0ex}}\left|{\mathbf{\text{R}}}_{\mathit{\text{kl}}}\right(\tau \left)\right|\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\frac{1}{1+\frac{{\sigma}_{\theta}^{2}}{2}{\left[{\omega}_{\text{D}}\tau \text{cos}({\theta}_{\text{m}}-\alpha )\phantom{\rule{1em}{0ex}}+\phantom{\rule{1em}{0ex}}\frac{2\pi}{\lambda}{d}_{\mathit{\text{kl}}}\text{cos}\left({\theta}_{\text{m}}\right)\right]}^{2}},\end{array}

(19)

\begin{array}{lll}\phantom{\rule{-10.0pt}{0ex}}\angle {\mathbf{\text{R}}}_{\mathit{\text{kl}}}(\tau )& =& -\frac{2\pi}{\lambda}{d}_{\mathit{\text{kl}}}\text{sin}\left({\theta}_{\text{m}}\right)-{\omega}_{\text{D}}\tau \text{sin}({\theta}_{\text{m}}-\alpha ).\end{array}

(20)

The mean AoA is then obtained as for the Gaussian case (13). Using all the cross-correlation coefficients defined in (19) for two antenna couples (*k*,*l*) and (*k*,*j*), we obtain the following AS estimate:

\begin{array}{l}{\widehat{\sigma}}_{\theta}(k,l,j)=\phantom{\rule{1em}{0ex}}\frac{\sqrt{\frac{2}{\left|{\widehat{\mathbf{\text{R}}}}_{\mathit{\text{kl}}}\right(\tau \left)\right|}-2}-\sqrt{\frac{2}{\left|{\widehat{\mathbf{\text{R}}}}_{\mathit{\text{kj}}}\right(\tau \left)\right|}-2}}{\frac{2\pi}{\lambda}\text{cos}\left({\widehat{\theta}}_{\text{m}}\right(k,l\left)\right)({d}_{\mathit{\text{kl}}}-{d}_{\mathit{\text{kj}}})},\end{array}

(21)

where *k*,*l*,*j* ∈ {1 … *N*_{a}} and *k* ≠ *l* ≠ *j*.The maximum DS estimate is then deduced using the addition of the square \left|{\widehat{\mathbf{\text{R}}}}_{\mathit{\text{kl}}}\right(\tau \left)\right| in (19) and \angle {\widehat{\mathbf{\text{R}}}}_{\mathit{\text{kl}}}(\tau ) in (20):

\begin{array}{l}{\widehat{\omega}}_{\text{D}}(k,l)=1/\tau \left(\sqrt{{\left(\angle {\widehat{\mathbf{\text{R}}}}_{\mathit{\text{kl}}}(\tau )+2\pi {d}_{\mathit{\text{kl}}}\text{sin}\left({\widehat{\theta}}_{\text{m}}\right(k,l\left)\right)\right)}^{2}+{\left(\frac{\sqrt{\frac{2}{\left|{\widehat{\mathbf{\text{R}}}}_{\mathit{\text{kl}}}\right(\tau \left)\right|}-2}}{{\widehat{\sigma}}_{\theta}(k,l,j)}-2\pi {d}_{\mathit{\text{kl}}}\text{cos}\left({\widehat{\theta}}_{\text{m}}\right(k,l\left)\right)\right)}^{2}}\right),\end{array}

(22)

where *k*,*l* ∈ {1 … *N*_{a}} and *k* ≠ *l*.

The final estimates are then obtained by averaging over antenna branches separated by a half wavelength. As one can notice, only the cross-correlation matrix is used to jointly estimate the three parameters. Contrarily to the methods developed in [1, 13], the proposed algorithm does not require the additive noise power estimation nor the eigenvalue decomposition of the correlation matrix, which reduces considerably the overall computational complexity. In the next section, we study both performance and complexity of our joint estimator.