# DOA Estimation of GNSS Signals Based on Deconvolved Conventional Beamforming
## 2 Signal Model  
### Multipath Signal Model  
The navigation signals transmitted by GNSS satellites are received by receivers after propagating through space. Due to potential signal reflections from obstacles around the receiver, as illustrated in Figure 1, a received navigation signal includes both direct and reflected signals. These reflected signals are commonly known as multipath signals, which will introduce pseudo-range errors. Typically, multipath errors may be about 5-6 m for BPSK signal modulation and are reduced to about 1 m for AltBOC modulation [19].  
When satellite signals propagate to the ground, the received signal at the receiver is a combination of the direct signal component and the multipath component. Assuming that the received navigation signal includes a direct signal and \(M-1\) multipath signals, the baseband signal after down-conversion can be represented as follows [20]:  
\[s(t)=A\sum_{m=0}^{M}\alpha_{m}c(t-\tau_{m})e^{j2\pi f_{d}t+j\phi_{m}} \tag{1}\]  
where \(A\) represents the signal amplitude; \(c(t)\) represents the pseudocode; \(f_{d}\) represents the Doppler frequency; \(m=0\) denotes the direct signal; and \(m=1,2,\cdots,M\) represent the multipath signal components. \(\alpha_{m}\), \(\tau_{m}\), and \(\phi_{m}\) represent the reflection coefficient, the time delay, and the carrier phase of the \(m\)th multipath signal component relative to the direct signal. \(m=0\) indicates the direct signal. Clearly, \(\alpha_{0}=1,\tau_{0}=0,\ \phi_{0}=0\). To simplify t the analysis, the amplitude, time delay, and phase parameters are considered as constants within the time period under consideration.  
Suppose the signal is in a tracking state, the Doppler frequency can be accurately estimated, and the local replica signal can be represented as follows [21]:  
\[s_{L}(t)=c(t-\widetilde{\tau})e^{-\beta\pi f_{d}t-i\widetilde{\psi}} \tag{2}\]  
where \(\widetilde{\phi}\) represents the carrier phase estimation value and \(\widetilde{\tau}\) represents the received signal's pseudocode delay estimation value.  
The accumulated correlation value between the input signal and the locally generated replica can be expressed as follows [21]:  
\[R(\varepsilon)=\frac{1}{T_{coh}}\int_{0}^{T_{coh}}s(t)\cdot s_{L}(t)dt\ =A\sum_{m=0}^{M}\alpha_{m}R_{c}(\Delta\tau_{m})e^{j\Delta\phi_{m}} \tag{3}\]  
where \(R_{c}(\cdot)\) represents the autocorrelation function of the GNSS signal and \(\Delta\tau_{m}=\tau_{m}-\widetilde{\tau}\) and \(\Delta\phi_{m}=\phi_{m}-\widetilde{\phi}\) indicate the code phase error and the carrier phase error of the \(m\)th component, respectively.  
Reflected multipath signals undergo delays, phase shifts, and fading, in contrast to direct signals. To begin with, let us consider the case of a direct signal and one multipath signal. When the direct and multipath signals are in phase (\(|\phi_{m}|\leq 90^{\circ}\)), the sum of their amplitudes is greater than that of any individual component. Conversely, when the direct and multipath signals are out of phase (\(|\phi_{m}|>90^{\circ}\)), the amplitudes of the two components will attenuate. A schematic diagram of the composite correlation peak is shown in Figure 2.