# DOA Estimation of GNSS Signals Based on Deconvolved Conventional Beamforming
## 3 The Proposed Method
### Deconvolved Conventional Beamforming  
Rewrite the array beam pattern, \(Y(\theta)\), in Equation (8) as a function of \(\sin(\theta)\). \[ \begin{array}{l}Y(\sin\theta)=A\sum\limits_{m=0}^{M}\alpha_{m}R_{c}(\Delta \tau_{m})e^{i\Delta\theta_{m}}\frac{\sin(\pi N(d/\lambda)(\sin\theta-\sin\theta _{m}))}{N\sin(\pi(d/\lambda)(\sin\theta-\sin\theta_{m}))}\\ =\sum\limits_{m=0}^{M}\widetilde{A}_{m}\frac{\text{sinc}(\pi N(d/\lambda)X_{m })}{\text{sinc}(\pi(d/\lambda)X_{m})}\end{array}  \tag{9}\]  
where \(\text{sinc}(x)=\sin(x)/x\) represents the sinc function. The complex form of the beam pattern can be represented in the following convolution form. \[ \begin{array}{l}Y(\sin\theta)=A\sum\limits_{m=0}^{M}\alpha_{m}R_{c}(\Delta \tau_{m})e^{i\Delta\phi_{m}}\int_{-1}^{1}Y_{p}(\sin\theta-\sin\theta)\delta( \sin\theta-\sin\theta_{m})d\sin\theta\\ =\int_{-1}^{1}Y_{p}(\sin\theta-\sin\theta)\sum\limits_{m=0}^{M}H_{m}(\sin \theta)d\sin\theta\end{array}  \tag{10}\]  
where \(Y_{p}(X)=\frac{\text{sinc}(\pi N(d/\lambda)X)}{\text{sinc}(\pi(d/\lambda)X)}\) represents the point spread function and \(H_{m}(\sin\theta)=\widetilde{A}_{m}\delta(\sin\theta-\sin\theta_{m})\) represents the magnitude distribution of the signal sources. \(\theta\) rep  
Figure 4: Normalized beam patterns using conventional beamforming method with different numbers of array elements. resents the angular variable in the convolution operation, which satisfies the condition that \(\sin\vartheta\) varies continuously within the range of \(-1\) to \(1\). The array pattern can be expressed as the convolution of the pattern and the signal distribution spectrum [22]. Therefore, estimation of the signal source distribution can be achieved using deconvolution methods. However, deconvolution using the beam pattern, \(Y(\sin\vartheta)\), can lead to an ill-posed problem. To solve this problem, deconvolution utilizes the beam energy spectrum instead of the complex beam. The energy spectrum of the beam pattern can also be represented in a convolution form as follows. \[B(\sin\theta)=\left|Y(\sin\theta)\right|^{2}=\int_{-1}^{1}B_{p}(\sin\theta- \sin\vartheta)\sum_{m=0}^{M}H^{\prime}{}_{m}(\sin\vartheta)d\sin\vartheta \tag{11}\]  
where \(B_{p}(X)=\left|\frac{\text{sinc}(\pi N(d/\lambda)X)}{\text{sinc}(\pi(d/\lambda) X)}\right|^{2}\) and the signal power distribution is represented by \(H^{\prime}{}_{m}(\sin\vartheta)=\left|\widetilde{A}_{m}\right|^{2}\delta(\sin \vartheta-\sin\theta_{m})\). The Richardson-Lucy method [23] is a commonly used deconvolution technique that has been widely applied in the field of image processing for image deblurring, producing good results. In this study, we applied this method for deconvolution processing. If the signal can be expressed in the following convolution form:  
\[r(y)=h(y|x)*f(x) \tag{12}\]  
where \(h(y|x)\) represents the channel impulse response (CIR) and \(f(x)\) represents the signal source, the R-L method can be utilized for deconvolution processing to estimate \(f(x)\). In this paper, \(f(x)\) denotes the spatial distribution of both the direct and multipath signals. The R-L algorithm is an iterative method and can be expressed as follows:  
\[ \begin{split} f^{(i+1)}(x)&=f^{(i)}(x)\int_{-\infty }^{\infty}\frac{h(y|x)}{\int_{-\infty}^{\infty}h(y|x^{\prime})f^{(i)}(x^{ \prime})dx^{\prime}}r(y)dy\\ &=f^{(i)}(x)\int_{-\infty}^{\infty}\frac{h(y|x)}{r^{(i)}(y)}r(y) dy\end{split}  \tag{13}\]  
where \(r^{(i)}(y)=\int_{-\infty}^{\infty}h(y\big{|}x^{\prime})f^{(i)}(x^{\prime})dx^ {\prime}\) and \(i\) represents the number of iterations. In order to achieve \(f(x)\) estimation using the R-L algorithm, it is essential to ensure that the beam pattern exhibits translational invariance in the angular domain, denoted as \(h(y|x)=h(y-x)\). We can observe from Equation (11) that the point spread function \(B_{p}(\sin\theta-\sin\vartheta)\) exhibits translational invariance characteristics when expressed as a function of \(\sin(\theta)\), while it does not exhibit translational invariance when expressed as a function of angle, \(\theta\).