This section describes the two different beamforming solutions considered in this research, namely DAS and MPDR with and without spatial smoothing. The effect of correlation between LOS and multipath signals on beamformers is discussed and different numerical simulations are performed to evaluate the performance of these beamforming techniques to mitigate multipath signals for GNSS applications. The main difference between GNSS and other systems is that the measurement quality is of utmost importance beside signal strength improvement. Any type of filtering that distorts measurement quality affects GNSS receiver performance. Hence, special care is required for beamforming design and implementation.

### DAS beamformer

The DAS beamformer relies only on the spatial information of the LOS signal (Van Trees 2002). This beamformer does not guarantee a distortionless response as it just points the main beam in the direction of the LOS signal and does not consider any other constraints to preserve the desired correlation peak shape. From Eqs. (8) and (9), the steering vector of the LOS signal is given by \( {\text{\textbf{\textsf{a}}}}_{\mathsf{1}} \). The optimum weights for the DAS beamformer can be obtained as

$$ {\text{\textbf{\textsf{w}}}}_{\mathit{\mathsf{CONV}}}=\frac{\mathsf{1}}{\mathit{\mathsf{M}}\mathit{\mathsf{N}}}{\text{\textbf{\textsf{a}}}}_{\mathsf{1}} $$

(10)

where \( \mathit{\mathsf{M}}\mathit{\mathsf{N}} \) is the total number of antenna elements in the array.

### MPDR beamformer

The MPDR beamformer is a distortionless beamformer that minimizes total output power by constraining unity gain in the direction of the desired signal (Van Trees 2002). This beamformer relies on the covariance matrix of the received signal, which is normally computed by temporal averaging of the spatial samples. The covariance matrix of the received signal can be obtained as

$$ {\text{\textbf{\textsf{R}}}}_{\mathit{\mathsf{y}}\mathit{\mathsf{y}}}=\frac{\mathsf{1}}{{\mathit{\mathsf{K}}}_{\mathit{\mathsf{T}}}}{\displaystyle \sum_{\mathit{\mathsf{k}}=\mathsf{1}}^{{\mathit{\mathsf{K}}}_{\mathit{\mathsf{T}}}}\text{\textbf{\textsf{y}}}{\text{\textbf{\textsf{y}}}}^{\mathit{\mathsf{H}}}} $$

(11)

where \( {\mathit{\mathsf{K}}}_{\mathit{\mathsf{T}}} \) is the number of temporal snapshots.

The optimum weight vector for the MPDR beamformer is (Van Trees 2002)

$$ {\text{\textbf{\textsf{w}}}}_{\mathit{\mathsf{MPDR}}}=\frac{{\text{\textbf{\textsf{R}}}}_{\mathit{\mathsf{y}}\mathit{\mathsf{y}}}^{-\mathsf{1}}{\text{\textbf{\textsf{a}}}}_{\mathsf{1}}}{{\text{\textbf{\textsf{a}}}}_{\mathsf{1}}^{\mathit{\mathsf{H}}}{\text{\textbf{\textsf{R}}}}_{\mathit{\mathsf{y}}\mathit{\mathsf{y}}}^{-\mathsf{1}}{\text{\textbf{\textsf{a}}}}_{\mathsf{1}}} $$

(12)

### Effect of multipath signals on beamforming

The correlation between LOS and multipath signals has an adverse effect on the beamformer’s performance (Widrow et al. 1982; Reddy et al. 1987; Daneshmand et al. 2013b). As the covariance matrix is obtained by temporal averaging, the temporal cross correlation between the desired and the multipath signals is very high since their phase relation stays fairly constant during the averaging time. Therefore, the system regards the sum of the desired and multipath signals as one wave and computes weights to minimize the total output power. However, as desired and multipath signals are treated as one wave, the weights will have a destructive effect on the desired signal and in the process of mitigating multipath, the desired signal will also be cancelled (Widrow et al. 1982). In addition, the beamformer fails to form deep nulls in the direction of multipath (Chen et al. 2012). If the phase relation between the desired signal and multipath can be randomized, then the coherence between the signals will be reduced. This can be achieved by receiving the signals from different spatial locations by the antenna array; this can be performed either via moving the array (Daneshmand et al. 2013b) or through spatial smoothing techniques (Reddy et al. 1987). In the case of a static GNSS receiver, spatial smoothing can be applied to decorrelate the signals. In this method, antenna elements are grouped into a smaller number of overlapping subarrays (Van Trees 2002; Reddy et al. 1987). The basic requirement for spatial smoothing is that the steering vector should have a Vandermonde structure as in the case of linear and rectangular arrays (Van Trees 2002). The Vandermonde structure refers to the progressive linear phase shift of the signals across the array elements. The covariance matrices from all the subarrays are then averaged to form the spatially smoothed covariance matrix. The subarray concept emulates antenna array motion where signals received by different subarrays correspond to different spatial points. In this case, the phase relation between LOS and multipath is different for different subarrays and averaging the spatial covariance matrix over several subarrays reduces the correlation between the LOS and multipath signals. Along with forward smoothing, complex conjugated backward smoothing can be performed to improve the decorrelation as well as increase the antenna aperture (Reddy et al. 1987).

### MPDR beamformer with spatial smoothing (MPDRSS)

Consider an *M* × *N* array divided into overlapping subarrays of size {*J*,*L*}. Assume *P* subarrays in the x-direction and *Q* in the y-direction. Let \( {\text{\textbf{\textsf{R}}}}_{\mathit{\mathsf{f}}\mathit{\mathsf{p}}\mathit{\mathsf{q}}} \) be the covariance matrix of the [*p*, *q*]^{th} forward subarray. The forward spatially smoothed covariance matrix is the sample means of all the forward subarray covariance matrices and can be computed as

$$ {\text{\textbf{\textsf{R}}}}_{\mathit{\mathsf{f}}}=\frac{\mathsf{1}}{\mathit{\mathsf{P}}\mathit{\mathsf{Q}}}{\displaystyle \sum_{\mathit{\mathsf{p}}=\mathsf{1}}^{\mathit{\mathsf{P}}}{\displaystyle \sum_{\mathit{\mathsf{q}}=\mathsf{1}}^{\mathit{\mathsf{Q}}}{\text{\textbf{\textsf{R}}}}_{\mathit{\mathsf{f}}\mathit{\mathsf{p}}\mathit{\mathsf{q}}}}} $$

(13)

Similarly, if \( {\text{\textbf{\textsf{R}}}}_{\mathit{\mathsf{b}}} \) is the backward spatially smoothed covariance matrix, then the forward-backward spatially averaged covariance matrix is given by

$$ {\text{\textbf{\textsf{R}}}}_{\mathit{\mathsf{f}}\mathit{\mathsf{b}}}=\frac{{\text{\textbf{\textsf{R}}}}_{\mathit{\mathsf{f}}}+{\text{\textbf{\textsf{R}}}}_{\mathit{\mathsf{b}}}}{\mathsf{2}} $$

(14)

The optimum weight vector for the MPDR beamformer with spatial smoothing is (Van Trees 2002)

$$ {\text{\textbf{\textsf{w}}}}_{\mathit{\mathsf{MPDRSS}}}=\frac{{\text{\textbf{\textsf{R}}}}_{\mathit{\mathsf{f}}\mathit{\mathsf{b}}}^{-\mathsf{1}}{\text{\textbf{\textsf{a}}}}_{\mathsf{11}}}{{\text{\textbf{\textsf{a}}}}_{\mathsf{11}}^{\mathit{\mathsf{H}}}{\text{\textbf{\textsf{R}}}}_{\mathit{\mathsf{f}}\mathit{\mathsf{b}}}^{-\mathsf{1}}{\text{\textbf{\textsf{a}}}}_{\mathsf{11}}} $$

(15)

where \( {\text{\textbf{\textsf{a}}}}_{\mathsf{11}} \) is the steering vector of the LOS signal for the first subarray.

Beamformer’s performance depends on a number of factors such as the number of antenna elements, array configuration and incoming signal directions of arrival to name a few. The size and number of antenna elements are some of the limitations for practical applications in terms of cost and system complexity. Hence investigation of the performance of an antenna array based GNSS receiver with a limited number of antenna elements while still being able to perform spatial smoothing is important. In this research a Uniform Rectangular Array (URA) with six antenna elements is considered (M = 3, *N* = 2). The subarray formation for the spatial smoothing is shown in Fig. 2. Due to the limited number of elements in the array, only two subarrays (*P* = 2, Q = 1) are constructed with size {J = 2, L = 2}. The decorrelation obtained by spatial smoothing and in turn, the performance of the beamformer, is analyzed in the following sections.

### Numerical simulations

This section presents numerical simulation results for the array structure shown in Fig. 2 with inter-element spacing of 9.5 cm. The performance of the beamforming techniques in the presence of multipath signals is evaluated using the Signal-to-Multipath Ratio (SMR) (Egea et al. 2014) metric. SMR refers to the ratio between the LOS power and multipath power at the output of the beamformer and is expressed in dB. The pre-beamformer SMR is assumed to be 0 dB. Here, it is assumed that multipath is coming from (15°, 175°) and the LOS signal azimuth is (50°). Beamformer performance for different correlation coefficients of the LOS and multipath signals for different LOS signal elevations is assessed. For the two signals case, r_{1} (LOS) and r_{2} (multipath) the covariance matrix can be represented as

$$ {\text{\textbf{\textsf{R}}}}_{\mathit{\mathsf{y}}\mathit{\mathsf{y}}}=\text{\textbf{\textsf{A}}}{\text{\textbf{\textsf{R}}}}_{\mathit{\mathsf{s}}}{\text{\textbf{\textsf{A}}}}^{\mathit{\mathsf{H}}}+{\sigma}_{\eta}^{\mathsf{2}}\text{\textbf{\textsf{I}}} $$

(16)

where \( {\text{\textbf{\textsf{R}}}}_{\mathit{\mathsf{s}}} \) is the source covariance matrix and \( {\sigma}_{\eta}^{\mathsf{2}} \) is the noise variance. The source covariance can be defined as

$$ {\text{\textbf{\textsf{R}}}}_{\mathit{\mathsf{s}}}=\left[\begin{array}{l}{\sigma}_{\mathit{\mathsf{r}}\mathsf{1}}^{\mathsf{2}}\\ {}{\sigma}_{\mathit{\mathsf{r}}\mathsf{1}}{\sigma}_{\mathit{\mathsf{r}}\mathsf{2}}\rho \end{array}\right.\kern0.6em \left.\begin{array}{l}{\sigma}_{\mathit{\mathsf{r}}\mathsf{1}}{\sigma}_{\mathit{\mathsf{r}}\mathsf{2}}\rho \\ {}{\sigma}_{\mathit{\mathsf{r}}\mathsf{2}}^{\mathsf{2}}\end{array}\right] $$

(17)

where \( {\sigma}_{\mathit{\mathsf{r}}\mathsf{1}}^{\mathsf{2}} \) is the variance of the source signal, \( {\sigma}_{\mathit{\mathsf{r}}\mathsf{2}}^{\mathsf{2}} \) is the variance of the multipath signal and *ρ* is the correlation coefficient between the LOS and multipath, defined as

$$ \rho =\frac{E\left[{\mathit{\mathsf{r}}}_{\mathit{\mathsf{1}}}{\mathit{\mathsf{r}}}_{\mathsf{2}}^{\mathit{\mathsf{H}}}\right]}{\sqrt{E\left[{\mathit{\mathsf{r}}}_{\mathit{\mathsf{1}}}{\mathit{\mathsf{r}}}_{\mathsf{1}}^{\mathit{\mathsf{H}}}\right]\sqrt{E\left[{\mathit{\mathsf{r}}}_{\mathsf{2}}{\mathit{\mathsf{r}}}_{\mathsf{2}}^{\mathit{\mathsf{H}}}\right]}}} $$

(18)

The power of both LOS and multipath is set to 10 (\( {\sigma}_{\mathit{\mathsf{r}}\mathsf{1}}^{\mathsf{2}}={\sigma}_{\mathit{\mathsf{r}}\mathsf{2}}^{\mathsf{2}}=\mathsf{10} \)) and the noise variance is assumed to be 1. The elevation of the LOS signal varies from 0° to 90° for different magnitudes of the correlation coefficient between the signals and the SMR performance of both MPDR and MPDRSS is shown in Fig. 3. The MPDR performance is the same for different LOS signal elevations for a given correlation coefficient. For very low correlation coefficients, which is the case when LOS and multipath signals are uncorrelated to each other, the MPDR beamformer yields a SMR up to 40 dB. However, as correlation increases, beamformer performance decreases and results in low SMR. As seen in Fig. 3, when the correlation coefficient magnitude is above 0.6, the SMR is nearly 0 dB. The performance of MPDRSS is better for higher elevation satellites when signals are correlated to each other, as compared to MPDR. This is due to the fact that the angular separation of the LOS from multipath signals is higher and spatial smoothing is able to provide better decorrelation. As can be seen in Fig. 3, SMR up to 10 dB can be achieved using MPDRSS for higher elevation satellites even when signals are highly correlated. Since the decorrelation achieved by the spatial smoothing process is a function of the DOA of the incoming signals and the number of antenna elements, the MPDRSS beamformer performance will be different for different signals impinging on the array from different directions. However, it was observed that for the rectangular array considered, MPDRSS beamformer performance improves with an increase in the elevation angle of the LOS signal, considering the multipath signal is coming from a low elevation.

The beampatterns for the DAS, MPDR and MPDRSS beamformers for different correlation coefficients for a higher elevation satellite with multipath from low elevation are shown in Fig. 4. Here it is assumed that LOS is coming from (75°, 50°) and multipath from (15°, 175°). As the DAS beamformer does not rely on the statistics of the received signal, the performance will be same for any correlation between LOS and multipath signals. However, MPDR performance is improved only when the correlation between LOS and multipath is very low. However, MPDRSS provides better attenuation of the multipath signals. Even when signals are highly correlated, MPDRSS can attenuate multipath by up to 10 dB. Based on the LOS signal directions and correlation between LOS and multipath signals, the DAS beamformer performance could be similar to that of MPDR and MPDRSS. In some cases, it could be better than MPDR as correlation can degrade the performance of the latter.