This type of antennas attracted the attention in the past, mainly by the military branch, although it has a greater progress in recent years, as the result of technological developments, especially in digital processors and new signal-processing techniques. Nowadays, smart antennas are attractive for several areas including military applications, software-defined radio, satellites, mobile communications (especially in base stations), 4G MIMO and the emerging 5G MIMO.

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

An adaptive antenna consists, in a simple way, of an antenna array and a sophisticated signal processing. It continuously adjusts the feeding of its array elements, and therefore its radiation characteristics, to adapt to the changes in the environment around them, improving the communication.

The dynamic and flexible adjustment of the radiation pattern using a relevant signalprocessing capability allows to respond, in real time, to the communication constraints, first by detecting the signal directions of arrival (DOA), and then, according with them, applying beamforming techniques to shape and steer the radiation pattern of the array, by adjusting the locations of the maximums and nulls, to achieve a better reception or transmission.

In this sense, it is possible to recover a weak signal immersed in strong interference environment where signals come from various directions. Furthermore, these antennas enable the reduction of delay spread effects and a greater ability to reduce the impact of multipath fading.

Figure 1 shows the basic block diagram of an adaptive antenna, in the receiving mode, in which some of the different elements that comprise an adaptive antenna are noticeable.

This structure starts by the antenna array and the entire underlying radio frequency (RF) chain, which includes the possible frequency downconversion and analogue-to-digital conversion. Another important component is the DSP (digital signal processor) unit that handles all signal processing of the adaptive antenna.

Using samples of received signal from each array element, DOA algorithms are applied to estimate the arrival directions of the signal or signals that impinge on the antenna array. Then, using an auxiliary intelligence process, the detected signals are divided in terms of signals-ofinterest (SOI) and signals-not-of-interest/interferences. Finally, beamforming algorithms are employed using this information (DOA's and SOI's/SNOI's), computing the necessary weights (W1, W2, …, Wn) to apply to each array element, obtaining the desired radiation pattern, pointing the antenna to directions of interest signals, and reducing the impact of the interferences.

This chapter is divided into five sections and explores the main concepts presented in the review article [1], from which some tables and figures are reprinted with permission. It starts with an introduction to adaptive antennas, and an adaptive system applied to two-dimensional (2D)

Figure 1. Adaptive antenna block diagram.

arrays is presented in Section 2. In Section 3, two DOA algorithms are presented, which also are tested and compared in terms of important factors as runtime and estimation errors. In Section 4, a group of beamforming algorithms are reported, analysed and applied to a planar 4 · 4 array. Finally, the last section summarizes the main conclusions.

#### 2. Planar antenna array system

The system consists of planar antenna array, as illustrated in Figure 2, which is affected by a group of signals including SOI's and SNOI's, resulting in an input signal x(t) that is composed by contributions of all these arriving signals, and noise. A sample of this signal x(t) is then processed to estimate the angles of arrival of the various signals involved. Afterwards, the respective weights W to apply to each element of the array are calculated, to create a proper radiation pattern, pointing it to the direction of interest and reducing the influence of the signals from interfering directions.

Consider a planar antenna array with M · N elements, spaced by d1 in the rows and d2 in the columns, in which impinge J different signals s(t) of frequency with wavelength λ, coming from a direction (θ,φ), where θ is the elevation angle and φ the azimuth angle. The input signal received by each element (m,n) of the array has components from each of J arriving signals and noise n(t), and is given by [4, 5]

$$\mathbf{x}\_{mn} = \sum\_{i=1}^{J} \mathbf{s}\_i(t) e^{\frac{2\pi}{\lambda}[u\_i d\_1(m-1) + v\_i d\_2(n-1)]} + n(t) \tag{1}$$

with,

An adaptive antenna consists, in a simple way, of an antenna array and a sophisticated signal processing. It continuously adjusts the feeding of its array elements, and therefore its radiation characteristics, to adapt to the changes in the environment around them, improving the

The dynamic and flexible adjustment of the radiation pattern using a relevant signalprocessing capability allows to respond, in real time, to the communication constraints, first by detecting the signal directions of arrival (DOA), and then, according with them, applying beamforming techniques to shape and steer the radiation pattern of the array, by adjusting the

In this sense, it is possible to recover a weak signal immersed in strong interference environment where signals come from various directions. Furthermore, these antennas enable the reduction of delay spread effects and a greater ability to reduce the impact of multipath fading. Figure 1 shows the basic block diagram of an adaptive antenna, in the receiving mode, in which some of the different elements that comprise an adaptive antenna are noticeable.

This structure starts by the antenna array and the entire underlying radio frequency (RF) chain, which includes the possible frequency downconversion and analogue-to-digital conversion. Another important component is the DSP (digital signal processor) unit that handles all signal

Using samples of received signal from each array element, DOA algorithms are applied to estimate the arrival directions of the signal or signals that impinge on the antenna array. Then, using an auxiliary intelligence process, the detected signals are divided in terms of signals-ofinterest (SOI) and signals-not-of-interest/interferences. Finally, beamforming algorithms are employed using this information (DOA's and SOI's/SNOI's), computing the necessary weights (W1, W2, …, Wn) to apply to each array element, obtaining the desired radiation pattern, pointing the antenna to directions of interest signals, and reducing the impact of the interferences.

This chapter is divided into five sections and explores the main concepts presented in the review article [1], from which some tables and figures are reprinted with permission. It starts with an introduction to adaptive antennas, and an adaptive system applied to two-dimensional (2D)

locations of the maximums and nulls, to achieve a better reception or transmission.

communication.

54 Antenna Arrays and Beam-formation

processing of the adaptive antenna.

Figure 1. Adaptive antenna block diagram.

$$\mu\_i = \sin \theta\_i,\\ \upsilon\_i = \cos \theta\_i \sin \phi\_i,\\ \mathfrak{m} = 1...M,\\ \mathfrak{n} = 1...N, \; i = 1...J$$

The received data x(t) and the noise n(t) of the whole antenna array can be represented in a vector structure X(t) and N(t) as follows:

Figure 2. Planar adaptive array system.

$$X(t) = \begin{bmatrix} \mathbf{x}\_{11}(t)\mathbf{x}\_{21}(t) \ \dots \mathbf{x}\_{M1}(t)\mathbf{x}\_{12}(t) \ \dots \mathbf{x}\_{MN}(t) \end{bmatrix}^T \tag{2}$$

$$N(t) = \begin{bmatrix} n\_{11}(t)n\_{21}(t) \ \dots \ n\_{M1}(t)n\_{12}(t) \ \dots \ n\_{MN}(t) \end{bmatrix}^T \tag{3}$$

where T denotes the matrix transpose operation.

The total input signal X(t) can be also expressed by the following formula:

$$X(t) = \sum\_{i=1}^{J} s\_i(t) \ A\_i + N(t) \tag{4}$$

where A represents the steering matrix of the planar antenna array and is given by Eq. (5). The operation ⊗ denotes the Kronecker product and Cu and Cv are the steering vectors in each direction of the planar array.

The steering vector [6] contains the set of phase delays that a wave will take, relating to each element of the array, and for a planar array this can be represented by Eqs. (6) and (7) [4, 7].

$$A = \mathbb{C}\_{\boldsymbol{\iota}} \otimes \mathbb{C}\_{\boldsymbol{v}} \tag{5}$$

$$\mathcal{C}\_{u} = \begin{bmatrix} 1 \ e^{\hat{\rho}\frac{2\pi d\_1}{\lambda}(2-1)u\_i} \dots e^{\hat{\rho}\frac{2\pi d\_1}{\lambda}[(M-1)u\_i]} \end{bmatrix}^T \tag{6}$$

$$\mathbf{C}\_{v} = \begin{bmatrix} 1 \ e^{\frac{2\pi d\_{2}}{\lambda}[(2-1)v\_{i}]} \dots \ e^{\frac{2\pi d\_{2}}{\lambda}[(N-1)v\_{i}]} \end{bmatrix}^{T} \tag{7}$$

The output signal of the adaptive antenna is therefore the product of the received signal X(t) by the beamforming weights W, estimated through the beamforming algorithms, and applied to each array element, as the following:

$$y(t) = \mathbf{W}^H \mathbf{X}(t) \tag{8}$$

$$W(t) = \begin{bmatrix} w\_{11}(t)w\_{21}(t) \ \dots \ w\_{M1}(t)w\_{12}(t) \ \dots \ w\_{MN}(t) \end{bmatrix}^T \tag{9}$$

In the next sections, a system is implemented and studied using MATLAB [8]. The signal X(t), which consists of the sum of a number of generated signals from several directions (θ,φ), being some of these signals SOI and others SNOI, adding up some noise, is created. The locations of the generated incoming signals are totally unknown beyond this step. Then, X(t) is sent as a single input parameter for DOA algorithm, which provides as output parameters the estimated number of signals that reach at the array, and their locations (θ,φ).

After the locations of all the signals impinging at the antenna are identified, it is necessary to distinguish the signals between the SOI and interferences, and then a beamforming algorithm to determine the necessary weights to apply to each element of the array is applied.

By simulation, using high-frequency structural simulator (HFSS) [9], with the ability to change the amplitude and phase of feeding of a simulated array structure, these sets of estimated weights are applied to the array, and its radiation pattern is analysed.

### 3. DOA algorithms

XðtÞ¼½x11ðtÞx21ðtÞ … xM1ðtÞx12ðtÞ … xMNðtÞ�T ð2Þ

NðtÞ¼½n11ðtÞn21ðtÞ … nM1ðtÞn12ðtÞ … nMNðtÞ�T ð3Þ

siðtÞ Ai þ NðtÞ ð4Þ

A ¼ Cu ⊗ Cv ð5Þ

yðtÞ ¼ WHXðtÞ ð8Þ

WðtÞ¼½w11ðtÞw21ðtÞ … wM1ðtÞw12ðtÞ … wMNðtÞ�T ð9Þ

ð6Þ

ð7Þ

where T denotes the matrix transpose operation.

direction of the planar array.

56 Antenna Arrays and Beam-formation

each array element, as the following:

The total input signal X(t) can be also expressed by the following formula:

Cu ¼ 1 e

Cv ¼ 1 e

j 2πd1

j 2πd2

mated number of signals that reach at the array, and their locations (θ,φ).

weights are applied to the array, and its radiation pattern is analysed.

XðtÞ ¼ XJ

i¼1

where A represents the steering matrix of the planar antenna array and is given by Eq. (5). The operation ⊗ denotes the Kronecker product and Cu and Cv are the steering vectors in each

The steering vector [6] contains the set of phase delays that a wave will take, relating to each element of the array, and for a planar array this can be represented by Eqs. (6) and (7) [4, 7].

λ ½ð2�1Þui� … e

λ ½ð2�1Þvi� … e

The output signal of the adaptive antenna is therefore the product of the received signal X(t) by the beamforming weights W, estimated through the beamforming algorithms, and applied to

In the next sections, a system is implemented and studied using MATLAB [8]. The signal X(t), which consists of the sum of a number of generated signals from several directions (θ,φ), being some of these signals SOI and others SNOI, adding up some noise, is created. The locations of the generated incoming signals are totally unknown beyond this step. Then, X(t) is sent as a single input parameter for DOA algorithm, which provides as output parameters the esti-

After the locations of all the signals impinging at the antenna are identified, it is necessary to distinguish the signals between the SOI and interferences, and then a beamforming algorithm

By simulation, using high-frequency structural simulator (HFSS) [9], with the ability to change the amplitude and phase of feeding of a simulated array structure, these sets of estimated

to determine the necessary weights to apply to each element of the array is applied.

j 2πd1 λ ½ðM�1Þui� h iT

j 2πd2 λ ½ðN�1Þvi� h iT The processor that estimates the DOAs of arriving signals to the antenna array is a crucial part of an adaptive antenna, allowing to understand the environment in which the antenna is inserted. By processing the electromagnetic waves that reach an antenna array, it is possible to extract a number of information about them, particularly their arrival directions. This processing is done by using DOA algorithms.

The arriving signals could be divided into SOIs, which are important to steer the antenna towards them, and/or interfering signals (SNOIs), whose impact on the system should be reduced.

There are three main classes of DOA estimation methods referred in the literature, differing mainly in the performance and its computational requirements [10, 11]: the classical, the maximum likelihood and the subspace methods.

The classics are based in the beamforming, in which the central idea is to scan the antenna beam over the space, and the peaks of received power are the DOAs. These methods are theoretically simple but involve a high computational effort and provide a poor performance and a low resolution. The maximum likelihood methods present higher performance than the others (especially with low SNR conditions), because they can take advantage of using better signal and noise models to provide better DOA estimation. However, due to the needs to solve nonlinear multidimensional optimization problems, it increases the required computational load, which makes these methods less popular.

Finally, there are several subspace methods for DOA estimation, which have become popular and widely studied over the last decades due to their good trade-off between the computational complexity and good performance. These methods are based on the eigen decomposition of the estimated covariance matrix of the data received by antenna array, into a signal subspace and a noise subspace. The performance of these methods is essentially limited by the accuracy of distinguishing the signal and the noise subspaces in the presence of noise.

For a planar uniform antenna array, the most applied algorithms are 2D Multiple Signal Classification (MUSIC) and the 2D Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT), which are subspace-based methods, and will be subject of a more detailed description below.

#### 3.1. 2D MUSIC algorithm

The Multiple Signal Classification (MUSIC) is perhaps the most popular DOA estimation algorithm, which assumes that the steering vectors of the incoming signals lie in signal subspace, and are orthogonal to the noise subspace. The algorithm search in all possible steering vectors that are orthogonal to the noise subspace of the covariance matrix of the received data (Rxx) [11–14].

The MUSIC algorithm, as illustrated in Figure 3, uses the received information from each element of the array, and through eigenvalue or singular value decomposition of the Rxx matrix, it estimates the noise subspace (UN).

Figure 3. 2D MUSIC DOA estimator.

After the noise subspace UN is identified, the DOAs are the resulting peaks of the MUSIC spectrum PMUSIC(θ,φ), given by Eq. (10) [13]:

$$P\_{\rm MUSIC}(\theta,\phi) = \frac{1}{s^H(\theta,\phi)\mathcal{U}\_N\mathcal{U}\_N^H\mathbf{s}(\theta,\phi)}\tag{10}$$

where H represents the conjugate transpose matrix (Hermitian).

When a steering vector s(θ,φ) is related to one arriving signal, the product of sH(θ,φ)UN = 0, ideally, the function assumes a high value (peak), and therefore (θ,φ) is the DOA. There may be several signals from different angles of arrival, creating several peaks in the MUSIC spectrum.

The MUSIC algorithm is simpler to understand and can be applied in all antenna array geometries. However, computationally it requires a lot of resources, since it has to calculate the MUSIC spectrum, Eq. (10), for all the possible steering vectors to estimate the expected peaks.

The estimation error of the MUSIC algorithm is significantly influenced by the angle grid interval in which Eq. (10) is evaluated. In the presence of coherent signals, as in multipath environments, spatial smoothing schemes [15, 16] must be applied to suppress the correlations between the incoming signals.

#### 3.2. 2D ESPRIT algorithm

Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT) algorithm is a different subspace-based DOA estimator [17–21]. This algorithm solves some of drawbacks of the MUSIC, in terms of the high computational requirements, and the resulting effects of array calibration errors. The ESPRIT algorithm does not require a high level of calibration in the array since it employs the property of shift invariance of the antenna array. Also, the computational complexity of the ESPRIT is reduced in comparison to MUSIC because it imposes some constraints on array structure.

The ESPRIT algorithm assumes that the separation between equivalent elements in each subarray is fixed, as shown in Figure 4, and therefore the array presents a translational invariance. Detect and Pointing Algorithms Performance for a 2D Adaptive Antenna Array http://dx.doi.org/10.5772/67524 59

Figure 4. ESPRIT sub-array division with maximum overlap.

This translational invariance leads to a rotational invariance of the signal subspace that will enable to estimate the DOAs.

This algorithm performs three main stages: the signal subspace estimation, the solution of the invariance equation and the DOA estimation.

The algorithm procedure is [17–19] as follows:

Step 1. Signal subspace estimation

Computation of the Us

Step 2. Solve the invariance equation

$$\mathbb{K}\_{u1} \mathcal{U}\_s \mathcal{Y}\_u = \mathbb{K}\_{u2} \mathcal{U}\_s$$

After the noise subspace UN is identified, the DOAs are the resulting peaks of the MUSIC

When a steering vector s(θ,φ) is related to one arriving signal, the product of sH(θ,φ)UN = 0, ideally, the function assumes a high value (peak), and therefore (θ,φ) is the DOA. There may be several signals from different angles of arrival, creating several peaks in the MUSIC

The MUSIC algorithm is simpler to understand and can be applied in all antenna array geometries. However, computationally it requires a lot of resources, since it has to calculate the MUSIC spectrum, Eq. (10), for all the possible steering vectors to estimate the expected

The estimation error of the MUSIC algorithm is significantly influenced by the angle grid interval in which Eq. (10) is evaluated. In the presence of coherent signals, as in multipath environments, spatial smoothing schemes [15, 16] must be applied to suppress the correlations

Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT) algorithm is a different subspace-based DOA estimator [17–21]. This algorithm solves some of drawbacks of the MUSIC, in terms of the high computational requirements, and the resulting effects of array calibration errors. The ESPRIT algorithm does not require a high level of calibration in the array since it employs the property of shift invariance of the antenna array. Also, the computational complexity of the ESPRIT is reduced in comparison to MUSIC because it imposes

The ESPRIT algorithm assumes that the separation between equivalent elements in each subarray is fixed, as shown in Figure 4, and therefore the array presents a translational invariance.

sHðθ, φÞUNUH

Nsðθ, φÞ ð10Þ

PMUSICðθ, φÞ ¼ 1

where H represents the conjugate transpose matrix (Hermitian).

spectrum PMUSIC(θ,φ), given by Eq. (10) [13]:

Figure 3. 2D MUSIC DOA estimator.

58 Antenna Arrays and Beam-formation

spectrum.

peaks.

between the incoming signals.

some constraints on array structure.

3.2. 2D ESPRIT algorithm

Kv1UsYv = Kv2Us

where Ku1, Ku2, Kv1 and Kv2 represent the two pairs of transformed selection matrices, while Yu and Yv are the real-valued matrices.

Step 3. DOA estimation

λi i = 1…d! eigen values of Yu + j Yv

ui = 2 tan(Re{λi}),

vi = 2 tan(Im{λi}),

φi =arg(ui - jvi) θ i = sin-1(‖ui – j vi‖)

where θi and φi are the DOA angular information.

#### 3.3. Test and comparison of the DOA algorithms

A system composed by a signal generator (with signals of interest and interferences) followed by a DOA estimator and beamforming was implemented using the MATLAB [8]. The two DOA algorithms, 2D MUSIC and 2D ESPRIT, were accomplished and applied to a planar array configuration.

To test the beamforming results, a planar antenna array of 16 square microstrip elements, arranged in a 4 · 4 planar structure was designed (for a central frequency of 12 GHz), with spacing between elements in both directions of d = 0.5λ, as shown in Figure 5. The array was simulated with HFSS [9] electromagnetic simulator.

Several simulations were performed with good results. As an example, two signals with directions (θ,φ) = (45, 45) and (70, 0) were generated and 'received' by the 4 · 4 array. With the received signal from each element of the antenna array, given by Eq. (2), the DOAs were estimated using the 2D MUSIC algorithm.

The 2D MUSIC algorithm creates a two-dimensional grid, in the range which the angles vary θ∈[0, 90] and φ∈[0, 360], and then evaluates Eq. (10) for each point of the grid.

Figure 6 illustrates the spatial graph of the MUSIC spectrum, resulting in the algorithm, which has peaks in the position of incident signals.

Figure 5. Simulated planar array.

Figure 6. MUSIC spectrum.

According to Figure 6, the MUSIC spectrum contains two evidenced peaks. Note that there is another peak but is assumed to be repeated, since 0 and 360 are the same spatial locations.

In order to simplify the definition and extraction of the peaks of the 2D MUSIC spectrum, a function to correctly detect the number of maximum values was implemented. This function provides the points of zero gradient, and its result is shown in Figure 7, with the two well-defined peaks.

The output of the 2D MUSIC algorithm is that the incident signals arrive to antenna from (θ,φ) (45.3, 44.82) and (70.07, 0), which are quite close to the initially proposed angles.

Identical signal given to the 2D MUSIC algorithm was provided to the 2D ESPRIT algorithm, also implemented with MATLAB. The output of ESPRIT is just the pair (θ,φ) of estimated DOAs, since it does not have a grid to evaluate by a function. The ESPRIT algorithm was preformed and the output result estimates that the signals are arriving from (45.02, 45.11) and (69.82, 0.03).

The implemented DOA algorithms reveal estimated results very approximate to the original and expected values. These algorithms only receive the signal X(t), and provide the spatial position of each incoming source that compose it.

The analysis of the performance of the 2D DOA algorithms was performed for a large number of experiments. In Figure 8, the runtimes of both subspace-based DOA algorithms for n = 50 experiences are presented.

The upper part concerns the 2D MUSIC algorithm while the bottom is about the 2D ESPRIT algorithm. In both graphs, the line of the average time of all samples is identified. There is huge

Figure 7. 2D MUSIC spectrum peaks.

spacing between elements in both directions of d = 0.5λ, as shown in Figure 5. The array was

Several simulations were performed with good results. As an example, two signals with directions (θ,φ) = (45, 45) and (70, 0) were generated and 'received' by the 4 · 4 array. With the received signal from each element of the antenna array, given by Eq. (2), the DOAs were

The 2D MUSIC algorithm creates a two-dimensional grid, in the range which the angles vary

Figure 6 illustrates the spatial graph of the MUSIC spectrum, resulting in the algorithm, which

θ∈[0, 90] and φ∈[0, 360], and then evaluates Eq. (10) for each point of the grid.

simulated with HFSS [9] electromagnetic simulator.

estimated using the 2D MUSIC algorithm.

60 Antenna Arrays and Beam-formation

has peaks in the position of incident signals.

Figure 5. Simulated planar array.

Figure 6. MUSIC spectrum.

Figure 8. Runtime of DOA algorithms over n samples.

difference in the time that MUSIC algorithm takes to perform compared with the ESPRIT. This runtime of the MUSIC is much higher than that of ESPRIT essentially because the MUSIC algorithm needs to calculate the MUSIC function for each possible steering vector.

The 2D MUSIC algorithm takes between 5 and 6 s to estimate the DOAs with an average time of 5.4 s, while 2D ESPRIT, in the majority of samples, varies between 1 and 2 ms, with an average execution time of 1.57 ms.

Another property that can be analysed is the estimation error, between the real location of the incoming waves and the estimated DOAs from both algorithms. Figure 9 shows the progress of the estimating error over the set of n experiences, using the two DOA algorithms. The upper graph is related to the coordinate θ, while the bottom is about the φ, and the lines of the average error are further presented.

According to Figure 9, it is possible to notice that the 2D MUSIC algorithm has a constant estimation error, with an average value of 0.302 in θ and 0.3052 in φ coordinate. The 2D ESPRIT algorithm has average errors extremely lower than 2D MUSIC, about 0.037 in θ and 0.015 in φ coordinates.

Figure 9. Estimation error in θ and φ coordinates for MUSIC and ESPRIT algorithms.

Figure 10. Evaluation grid of 2D MUSIC algorithm.

difference in the time that MUSIC algorithm takes to perform compared with the ESPRIT. This runtime of the MUSIC is much higher than that of ESPRIT essentially because the MUSIC

The 2D MUSIC algorithm takes between 5 and 6 s to estimate the DOAs with an average time of 5.4 s, while 2D ESPRIT, in the majority of samples, varies between 1 and 2 ms, with an

Another property that can be analysed is the estimation error, between the real location of the incoming waves and the estimated DOAs from both algorithms. Figure 9 shows the progress of the estimating error over the set of n experiences, using the two DOA algorithms. The upper graph is related to the coordinate θ, while the bottom is about the φ, and the lines of the

According to Figure 9, it is possible to notice that the 2D MUSIC algorithm has a constant estimation error, with an average value of 0.302 in θ and 0.3052 in φ coordinate. The 2D ESPRIT algorithm has average errors extremely lower than 2D MUSIC, about 0.037 in θ and

Figure 9. Estimation error in θ and φ coordinates for MUSIC and ESPRIT algorithms.

algorithm needs to calculate the MUSIC function for each possible steering vector.

average execution time of 1.57 ms.

Figure 8. Runtime of DOA algorithms over n samples.

62 Antenna Arrays and Beam-formation

average error are further presented.

0.015 in φ coordinates.

The 2D ESPRIT errors are mainly due to the mathematical process and the noise of the signal; however, in the 2D MUSIC, the errors are strongly affected by the evaluation interval, as explained in Figure 10.

The accuracy of the 2D MUSIC algorithm depends on the number of points of its angle grid in which Eq. (10) is evaluated. More points lead to longer computational times. The number of points of this evaluation grid is then the main issue of MUSIC algorithm, and this value should be an agreement between the required accuracy and computational load allowed.

Figure 10 shows how the MUSIC algorithm of error is consistent with the error estimates of Figure 9, and its relationship with the number of grid points, that in these simulations were 150 in θ and 250 in φ.

#### 4. Beamforming algorithms

An antenna array, depending on the environment where it is inserted, is usually affected by several electromagnetic signals propagating around, where some are desired to be received, and others, called interferences, are undesirable. Signals of interest and intrusive signals occupy the same frequency range; however, they come from different spatial locations.

In order to improve system performance, the signals of interest must be received in the perfect way, while the impact of the unwanted interferences must be reduced or abolished.

In an adaptive antenna array, after the locations of arriving signals are identified, it is essential to use spatial-filtering techniques, also known as beamforming techniques. These techniques

Figure 11. Beamformer system.

deal with the radiation pattern (or beam pattern) of an antenna array, shaping it according to a group of constraints, improving the performance of the communication system.

The beamforming process consists of, as is illustrated in Figure 11, changing the relative amplitude and phase of feeding of each element of the array, estimated through beamforming algorithms, adjust the radiation pattern of the array, steering its main lobe and putting nulls in interference directions. There are a number of developed algorithms to estimate the complex weights (amplitude and phase) to apply to an antenna array.

These algorithms are classified as data independent or statistically optimum, according to how the weights are estimated [22, 23]. In the data-independent beamforming algorithms, the weights are estimated to provide a desired response independently of the data received through the antenna, while in the statistically optimum, the weights are estimated according to the statistics of the received data, to optimize its response.

Sometimes, the statistical information of the received data is not available or changes in time; therefore, adaptive algorithms are usually applied to estimate the weights. In this situation, the calculated weights tend to a solution statistically optimum.

#### 4.1. Data-independent algorithms

The data-independent beamformers are characterized by the estimation of its weights to be independent of the received data statistics from the array. One of the most used data-independent algorithms is the classical beamformer [22].

#### 4.1.1. Classical beamformer

The operating principle of this method is similar to the phased array, since it estimates the necessary weights w in order that the maximum of radiation is steered to a desired direction θSOI. The vector w adjusts the phases to feed each element, in a way that the signals of all the elements are added constructively in a certain direction. However, the main limitation is the impossibility of placing nulls, which at many times is required to eliminate the impact of unwanted signals.

#### 4.2. Statistically optimum algorithms

The statistically optimum beamformers are characterized by the calculation of its weights based on data statistics of received signals. The weights are estimated to steer the radiation pattern in the direction of the SOI while reducing the influence of interfering signals and noises. Some examples of statistically optimum algorithms mentioned in the literature [22] are the multiple side-lobe canceller (MSC), reference signal, SNR maximization and the linearly constrained minimum variance (LCMV).

#### 4.2.1. Multiple side-lobe canceller

The MSC beamformer [22] consists of a main channel and other auxiliary channels. The main idea of this algorithm is to estimate the appropriate weights to apply to the auxiliary channels, in order to cancel or reduce the impact of interference signals in the main channel. In the main channel, a data-independent beamformer can be used to steer the maximum of the array to the desired direction.

Although this method is simple and effective, when the signal of interest is weak relating to interferences, it presents some limitations since the weights are estimated with the absence of the desired signal.

#### 4.2.2. Reference signal

deal with the radiation pattern (or beam pattern) of an antenna array, shaping it according to a

The beamforming process consists of, as is illustrated in Figure 11, changing the relative amplitude and phase of feeding of each element of the array, estimated through beamforming algorithms, adjust the radiation pattern of the array, steering its main lobe and putting nulls in interference directions. There are a number of developed algorithms to estimate the complex

These algorithms are classified as data independent or statistically optimum, according to how the weights are estimated [22, 23]. In the data-independent beamforming algorithms, the weights are estimated to provide a desired response independently of the data received through the antenna, while in the statistically optimum, the weights are estimated according

Sometimes, the statistical information of the received data is not available or changes in time; therefore, adaptive algorithms are usually applied to estimate the weights. In this situation, the

The data-independent beamformers are characterized by the estimation of its weights to be independent of the received data statistics from the array. One of the most used data-indepen-

The operating principle of this method is similar to the phased array, since it estimates the necessary weights w in order that the maximum of radiation is steered to a desired direction θSOI. The vector w adjusts the phases to feed each element, in a way that the signals of all the elements are added constructively in a certain direction. However, the main limitation is the impossibility of placing nulls, which at many times is required to eliminate the impact of

group of constraints, improving the performance of the communication system.

weights (amplitude and phase) to apply to an antenna array.

to the statistics of the received data, to optimize its response.

calculated weights tend to a solution statistically optimum.

4.1. Data-independent algorithms

4.1.1. Classical beamformer

Figure 11. Beamformer system.

64 Antenna Arrays and Beam-formation

unwanted signals.

dent algorithms is the classical beamformer [22].

The use of reference signal beamformer [22] requires some knowledge about the desired signal to create a reference signal, and its objective is to minimize the mean square error between the output of the beamformer and the reference signal. The main difficulty of this method is to generate a proper reference signal.

#### 4.2.3. Maximum SNR

The maximum SNR [22] method requires the knowledge of the covariance matrix of the desired signal and of the noise, and the weights are estimated to maximize the SNR.

#### 4.2.4. Linear constrained minimum variance

One of the most important statistically optimum beamformers with higher applicability is the linear constrained minimum variance (LCMV) method [22], and a different approach of its formulation known as generalized side-lobe canceller (GSC).

Most of the times, the desired reference signal is unknown or we do not have enough information about it, being necessary to impose some linear constraints in the weight vector to minimize the variance of beamformer output. The LCMV technique can overcome the main drawbacks of most of the techniques presented before [24].

The principal idea of the LCMV beamformer is that its output is constrained in a way that signals of interest are preserved and the undesired (noise and interfering signals) are minimized.

The LCMV formulation problem is to select the complex weights that are suitable to the multiple linearly independent constraints:

$$\min\_{w} w^H \mathcal{R}\_x w \text{ subject to } \mathbb{C}^H w = f \tag{11}$$

where w is the vector of weights, Rx the covariance matrix, C is the constraint matrix and f is the response vector.

The solution of the constrained minimization of LCMV problem can be obtained using the method of Lagrange multipliers [22] resulting in

$$w\_{\rm opt} = R\_{\rm x}^{-1} \mathbb{C} (\mathbb{C}^H R\_{\rm x} \mathbb{C})^{-1} f \tag{12}$$

It is important to observe the dependence of the optimal weight vector of Eq. (12) with the data correlation matrix, and therefore with the statistics of the input signal.

#### 4.2.5. Generalized side-lobe canceller

The generalized side-lobe canceller (GSC) is a different approach to solve the LCMV problem, providing a simple implementation of the beamformer, changing the constrained minimization problem to an unconstrained arrangement [5, 25]. The GSC method, illustrated in Figure 12, splits the LCMV problem into two parts, one data independent and other data dependent.

This structure splits the optimum weight vector into two orthogonal components, which are in the range and in the null space of C, in such way that w=w0 -BwM. The beamformer output is y=w0 Hx-wMHBHx.

The vector w0 is designed to comply with the imposed constraints, consisting in the quiescent part of w. This weight vector is also independent of the input data and represents the nonadaptive component of the LCMV solution. Then, w0 must satisfy the linear constraints [5]:

$$\mathbb{C}^{H}\boldsymbol{w}\_{0} = f \Rightarrow \boldsymbol{w}\_{0} = \mathbb{C}(\mathbb{C}^{H}\mathbb{C})^{-1}f \tag{13}$$

The bottom branch of GSC is the data-dependent part, and consists of the blocking matrix B and wM that will block influence of interfering signals, while minimizing the variance of the output signal y. The blocking matrix B must be orthogonal to the constraint matrix C, so CHB ¼ 0.

Figure 12. Generalized side-lobe canceller.

The GSC unconstrained problem is

$$\min\_{w\_M} (w\_0 - Bw\_M)^H R\_\mathbf{x} (w\_0 - Bw\_M) \text{ subject to } \mathbb{C}^H \boldsymbol{w} = f \tag{14}$$

and the optimal solution is

The LCMV formulation problem is to select the complex weights that are suitable to the

where w is the vector of weights, Rx the covariance matrix, C is the constraint matrix and f is

The solution of the constrained minimization of LCMV problem can be obtained using the

It is important to observe the dependence of the optimal weight vector of Eq. (12) with the data

The generalized side-lobe canceller (GSC) is a different approach to solve the LCMV problem, providing a simple implementation of the beamformer, changing the constrained minimization problem to an unconstrained arrangement [5, 25]. The GSC method, illustrated in Figure 12, splits the LCMV problem into two parts, one data independent and other data

This structure splits the optimum weight vector into two orthogonal components, which are in the range and in the null space of C, in such way that w=w0 -BwM. The beamformer output is

The vector w0 is designed to comply with the imposed constraints, consisting in the quiescent part of w. This weight vector is also independent of the input data and represents the nonadaptive component of the LCMV solution. Then, w0 must satisfy the linear constraints [5]:

CHw0 ¼ f )w0 ¼ CðCHCÞ

The bottom branch of GSC is the data-dependent part, and consists of the blocking matrix B and wM that will block influence of interfering signals, while minimizing the variance of the output signal y. The blocking matrix B must be orthogonal to the constraint matrix C, so CHB ¼ 0.

�1

x CðCHRxCÞ

�1

wopt ¼ R�1

correlation matrix, and therefore with the statistics of the input signal.

minwwHRxw Subject to CHw ¼ f ð11Þ

f ð12Þ

f ð13Þ

multiple linearly independent constraints:

method of Lagrange multipliers [22] resulting in

4.2.5. Generalized side-lobe canceller

the response vector.

66 Antenna Arrays and Beam-formation

dependent.

Hx-wMHBHx.

Figure 12. Generalized side-lobe canceller.

y=w0

$$
\omega w\_M = \left(B^H R\_\mathbf{x} B\right)^{-1} B^H R\_\mathbf{x} w\_0 \tag{15}
$$

This implementation of GSC beamformer enables important benefits, since the w0 is dataindependent beamformer and wM is an unconstrained beamformer.

#### 4.3. Adaptive algorithms

Adaptive algorithms [22] solve the problem of the statistics of the received data, which changes over the time or may not be available, and affect the statistically optimum beamformers. Examples of adaptive algorithms are the least mean square (LMS), the recursive least squares (RLS) or Frost's algorithm.

#### 4.3.1. Frost's algorithm for LCMV beamforming

Frost's algorithm [23] belongs to the group of LCMV beamformers. Its weights are estimated based on the statistics of the received data, which sometimes are not available or are continually changing, the use of adaptive algorithms being necessary. Frost's algorithm minimizes the mean square error while maintains the specified response to the desired signal. The weight vector starts with an initial value:

$$w = \mathbb{C}(\mathbb{C}^H \mathbb{C})^{-1} f \tag{16}$$

and at each iteration, the vector is updated on negative-gradient direction with a factor μ, and the weights are defined by

$$w(n+1) = \mathbb{C}(\mathbb{C}^H \mathbb{C})^{-1} f + P(w(n) - \mu e^\*(n)\mathbf{x}(n)) \tag{17}$$

$$P = I - \mathbb{C}(\mathbb{C}^H \mathbb{C})^{-1} \mathbb{C}^H \tag{18}$$

#### 4.3.2. Least mean square

Least-mean-square adaptive algorithm estimates the gradient vector from the available data [22, 23, 26, 27]. This method follows an iterative procedure that successively adjusts the weight vector in the direction of the negative of the gradient vector at each iteration, eventually leading to the minimum mean square error:

$$
\pi w\_M(n) = \pi w\_M(n-1) + \mu \left. u(n-1) \right.\\
\left. y^\*(n-1) \right. \tag{19}
$$

$$y(n) = y\_d(n) - w\_M^H(n)u(n)\tag{20}$$

$$0 \le \mu \le \frac{1}{\lambda\_{\text{max}}}$$

where λmax is the largest eigenvalue of the correlation matrix.

The gain μ (0<μ<1) is the parameter that controls the convergence rate of the algorithm. Smaller values lead to slow convergence, but good approximation, while higher values result in a faster convergence and the stability around the minimum value is not ensured. This algorithm is simple since it requires no matrix inversion and by correctly choosing the μ-value, the weight vector tends to an optimum solution.

#### 4.3.3. Recursive least squares

The recursive least squares algorithm has a faster convergence rate than the LMS; however, it involves more complex mathematical operations and requires more computational resources [22, 23].

The RLS problem is

$$\min\_{\mathbf{w}\_{\mathcal{W}(k)}} \sum\_{n=0}^{N} \lambda^{N-n} |y\_d(n) - w\_M^H(n) \, u(n)|^2 \tag{21}$$

The RLS calculates the error signal αðnÞ and is given by

$$a(n) = y\_d(n) - w\_M^H(n-1)u(n) \tag{22}$$

and then updates the weight vector according to a gain vector kðnÞ

$$
\varpi\_M(n) = \varpi\_M(n-1) + k(n)\alpha^\*(n) \tag{23}
$$

$$k(n) = \frac{\lambda^{-1}v(n)}{1 + \lambda^{-1}u^H(n)v(n)}\tag{24}$$

The constant λ, 0<λ<1 is called forgetting factor.

$$
\sigma(n) = P(n-1)\mu(n) \tag{25}
$$

The matrix PðnÞ corresponds to the inverse correlation matrix of the input signal which has Pð0Þ as initial value.

$$P(n) = \lambda^{-1} P(n-1) - \lambda^{-1} k(n) v^H(n) \tag{26}$$

$$P(0) = \delta^{-1}I \tag{27}$$

#### 4.4. Test and comparison of the beamforming algorithms

The beamforming weights can be exhibited in the exponential form w = Aeφ , and consist of a set of amplitude and phases, that are applied to each element of an antenna array, to combine its signals in such a way that it produces constructive interference for the desired locals and destructive for undesired.

With the MATLAB, some beamforming algorithms were implemented, and the weights from each algorithm were computed. Then, their performance was analysed when applied to the aforementioned planar antenna array, as presented in Figure 5.

0 ≤ μ ≤

The gain μ (0<μ<1) is the parameter that controls the convergence rate of the algorithm. Smaller values lead to slow convergence, but good approximation, while higher values result in a faster convergence and the stability around the minimum value is not ensured. This algorithm is simple since it requires no matrix inversion and by correctly choosing the μ-value, the

The recursive least squares algorithm has a faster convergence rate than the LMS; however, it involves more complex mathematical operations and requires more computational resources

λN�njydðnÞ � wH

MðnÞ uðnÞj2 ð21Þ

ðnÞ ð23Þ

Mðn � 1ÞuðnÞ ð22Þ

uHðnÞvðnÞ ð24Þ

kðnÞvHðnÞ ð26Þ

, and consist of a

I ð27Þ

vðnÞ ¼ Pðn � 1ÞuðnÞ ð25Þ

where λmax is the largest eigenvalue of the correlation matrix.

minwMðkÞ

and then updates the weight vector according to a gain vector kðnÞ

PðnÞ ¼ λ�1

The beamforming weights can be exhibited in the exponential form w = Aeφ

4.4. Test and comparison of the beamforming algorithms

The RLS calculates the error signal αðnÞ and is given by

The constant λ, 0<λ<1 is called forgetting factor.

XN n¼0

αðnÞ ¼ ydðnÞ � wH

wMðnÞ ¼ wMðn � 1Þ þ kðnÞα�

1 þ λ�1

The matrix PðnÞ corresponds to the inverse correlation matrix of the input signal which has

Pðn � 1Þ � λ�1

Pð0Þ ¼ δ�1

set of amplitude and phases, that are applied to each element of an antenna array, to combine its signals in such a way that it produces constructive interference for the desired locals and

vðnÞ

kðnÞ ¼ λ�1

weight vector tends to an optimum solution.

4.3.3. Recursive least squares

68 Antenna Arrays and Beam-formation

The RLS problem is

Pð0Þ as initial value.

destructive for undesired.

[22, 23].

1 λmax

> The system (DOA and beamforming) simulation was tested using several groups of angles of arrival with excellent results, first by estimating the necessary weights, after, applying it to the simulated planar array to analyse its radiation pattern. In the simulations presented here, it was considered that the signals were coming from signal of interest: (θ,φ)=(45, 45) and interference signal: (θ,φ)=(70, 0).

> The LVCM algorithm was implemented, using as input parameters the two pairs of angles considered, and with a response vector f = [1 0]H in order to assume the first pair of angles the SOI direction, and the second the SNOI. The output weights, given in terms of amplitude and phase, to apply to the corresponding element of the 4 · 4 array, are presented in Table 1(a). The resultant radiation pattern when these weights are applied is shown in Figure 13(a).

Table 1. Estimated weights from beamforming algorithms.

Figure 14 shows the 2D gain distribution of the array as a function of θ and φ coordinates, for a better analysis of the 3D radiation patterns of Figure 13.

Through Figure 13(a) together with Figure 14(a), it is possible to observe the maximum of the radiation pattern pointed to the direction of the SOI (45, 45), while in the direction (70, 0) there is a low-gain value, which reduces significantly the influence of the interference signal in this direction.

The adaptive algorithms were also implemented because they are a more realistic approach, due to the environment changes.

The adaptive Frost's algorithm was tested applying the considered pair of angles. This algorithm was processed with n = 100 samples of the input signal, iteratively. The output resulting weights are shown in Table 1(b), and the produced radiation pattern is presented

Figure 13. 3D radiation pattern from the planar array with weights applied.

in Figures 13(b) and 14(b). The array points to the direction of interest and places a null in the interference zone, as desired.

Figure 14 shows the 2D gain distribution of the array as a function of θ and φ coordinates, for

Through Figure 13(a) together with Figure 14(a), it is possible to observe the maximum of the radiation pattern pointed to the direction of the SOI (45, 45), while in the direction (70, 0) there is a low-gain value, which reduces significantly the influence of the interference signal in

The adaptive algorithms were also implemented because they are a more realistic approach,

The adaptive Frost's algorithm was tested applying the considered pair of angles. This algorithm was processed with n = 100 samples of the input signal, iteratively. The output resulting weights are shown in Table 1(b), and the produced radiation pattern is presented

a) Optimum LCMV b) Adaptive Frost's LCMV

c) Adaptive LMS d) Adaptive RLS

Figure 13. 3D radiation pattern from the planar array with weights applied.

a better analysis of the 3D radiation patterns of Figure 13.

this direction.

due to the environment changes.

70 Antenna Arrays and Beam-formation

The least-mean-square algorithm was also performed in MATLAB using 100 samples of the received signal X(t). The output of this algorithm is shown in Table 1(c), and the respective radiation pattern is presented in Figure 13(c). Through Figures 13(c) and 14(c), it is possible to find out that using these weights, the antenna steers its main lobe in the direction of interest (45, 45) and places a null in the SNOI zone (70, 0).

Finally, the RLS algorithm was implemented and its weights for identical scenario were estimated. The output weights are shown in Table 1(d), leading to the radiation pattern of Figure 13(d). The array directs its maximum of radiation pattern to SOI direction (45, 45) as can be seen in Figure 14(d).

The four beamforming algorithms being implemented and tested, a set of 50 consecutive experiments were done, performing a comparative statistical analysis, evaluating the runtime for each iteration. Figure 15 shows the progress of the execution time of each beamforming algorithm along the n experiences. In each sub-figure, a straight line (red) corresponding to the average runtime of all the 50 samples is also presented.

According to Figure 15, there are a couple of samples with a more accentuated discrepancy, observed by a couple of peaks in the runtime compared to the average value, property that is common in all beamforming algorithms.

Figure 14. 2D gain distribution of the radiation pattern from the planar array with weights applied.

Figure 15. 3D radiation pattern from the planar array with weights applied.

The LCMV algorithm, Figure 15(a), has an average runtime of about 0.62 ms. Concerning the adaptive algorithms, Frost's shown in Figure 15(b) present a mean value of 0.5 ms, that even being an adaptive presents a better result than the LCMV (but similar).

The LMS and RLS algorithms, presented in Figure 15(c) and, have an average runtime over all the 50 experiences, of about 2.8 and 7.6 ms, with the RLS, as expected, requiring more computational resources than LMS.

After the evaluation of the DOA and beamforming algorithms independently, a comparative analysis about their performance in the system was made, operating together as an adaptive antenna array. The system consists of DOA estimation followed by beamforming weights computation. It was evaluated in terms of total runtime (for all algorithms), the changeability of its results when the noise level modifies, and in terms of estimation errors. The results are presented in Tables 2–5, for each beamforming algorithm.

According to Table 2, the runtime of LCMV algorithm is similar using both DOA algorithms (MUSIC and ESPRIT), and its variation with SNR is not significantly noted when SNR changes from 10 to 15 dB. However, it almost doubles when SNR changes from 10 to 5 dB. Another important and already referred characteristic is that the MUSIC algorithm is heavier than the ESPRIT in terms of runtime, and while the τMUSIC does not show a significant variation, the τESPRIT tends to reduce with the increase of SNR. In terms of DOA estimation errors, they tend to reduce with the increase of the SNR.

Table 2. Variation of the runtime and estimation error, using LCMV beamforming algorithm.

Table 3. Variation of the runtime and estimation error, using Frost's beamforming algorithm.

The LCMV algorithm, Figure 15(a), has an average runtime of about 0.62 ms. Concerning the adaptive algorithms, Frost's shown in Figure 15(b) present a mean value of 0.5 ms, that even

c) Adaptive LMS d) Adaptive RLS

a) Optimum LCMV b) Adaptive Frost's LCMV

The LMS and RLS algorithms, presented in Figure 15(c) and, have an average runtime over all the 50 experiences, of about 2.8 and 7.6 ms, with the RLS, as expected, requiring more

After the evaluation of the DOA and beamforming algorithms independently, a comparative analysis about their performance in the system was made, operating together as an adaptive antenna array. The system consists of DOA estimation followed by beamforming weights computation. It was evaluated in terms of total runtime (for all algorithms), the changeability of its results when the noise level modifies, and in terms of estimation errors. The results are

According to Table 2, the runtime of LCMV algorithm is similar using both DOA algorithms (MUSIC and ESPRIT), and its variation with SNR is not significantly noted when SNR changes from 10 to 15 dB. However, it almost doubles when SNR changes from 10 to 5 dB. Another important and already referred characteristic is that the MUSIC algorithm is heavier than the ESPRIT in terms of runtime, and while the τMUSIC does not show a significant variation, the τESPRIT tends to reduce with the increase of SNR. In terms of DOA estimation errors, they tend

being an adaptive presents a better result than the LCMV (but similar).

Figure 15. 3D radiation pattern from the planar array with weights applied.

presented in Tables 2–5, for each beamforming algorithm.

computational resources than LMS.

72 Antenna Arrays and Beam-formation

to reduce with the increase of the SNR.

Regarding ESPRIT algorithm, a reduction of the estimation errors is observed; however, using the MUSIC algorithm this error maintains constant. This fact is due to the choice of the angle grid for the evaluation of music function PMUSIC, Eq. (10), as was previously referred. Its interval must be an agreement between runtime and estimation error.

Using Frost's algorithm, Table 3, its runtime reduces with the SNR. The performance of DOA algorithms maintains with the already described characteristics, in terms of execution time and estimation errors.

The performance using LMS algorithm is presented in Table 4. The runtime decreases when the SNR increases, whereas the MUSIC and ESPRIT algorithms maintain with similar characteristics to the preceding cases.

Finally, using the RLS algorithm, Table 5, it is possible to observe a high reduction of the runtime when SNR increases from 5 to 10 dB. The error follows the expected behaviour, reducing when SNR increases, using the DOA ESPRIT algorithm, and with the DOA MUSIC algorithm the estimation error holds up constant.

Table 4. Variation of the runtime and estimation error, using LMS beamforming algorithm.

Table 5. Variation of the runtime and estimation error, using RLS beamforming algorithm.

#### 5. Conclusions

In this chapter, a set of DOA and beamforming algorithms was studied and implemented. Furthermore, an analysis of each algorithm was carried out, allowing to understand and consolidate the differences between the various algorithms. Regarding the DOA algorithms, both have good estimates in all the simulations performed. It was possible to conclude the reason why the MUSIC algorithm has much larger runtimes and estimation errors; this is due to the grid in which the algorithm is applied. However, it was also concluded that it is a simpler algorithm to implement compared to the ESPRIT. In terms of beamforming algorithms, four algorithms have been implemented and tested and the main notes retained are that all of them present good estimations of weights, and the radiation patterns obtained point to the desired direction. It is also possible to remark the reduction of the runtime of the algorithms with an increase of SNR.

#### Acknowledgements

reducing when SNR increases, using the DOA ESPRIT algorithm, and with the DOA MUSIC

DOA Performance 5 10 15 MUSIC τMUSIC (s) 5.35 5.37 5.46

ESPRIT τESPRIT (s) 0.0095 0.0012 0.0011

In this chapter, a set of DOA and beamforming algorithms was studied and implemented. Furthermore, an analysis of each algorithm was carried out, allowing to understand and consolidate the differences between the various algorithms. Regarding the DOA algorithms, both have good estimates in all the simulations performed. It was possible to conclude the reason why the MUSIC algorithm has much larger runtimes and estimation errors; this is due to the grid in which the algorithm is applied. However, it was also concluded that it is a simpler algorithm to implement compared to the ESPRIT. In terms of beamforming algorithms, four algorithms have

Table 5. Variation of the runtime and estimation error, using RLS beamforming algorithm.

Table 4. Variation of the runtime and estimation error, using LMS beamforming algorithm.

DOA Performance 5 10 15 MUSIC τMUSIC (s) 5.82 5.75 5.71

ESPRIT τESPRIT (s) 0.00181 0.00180 0.0015

τLCMV (s) 0.0163 0.0070 0.0161 εθ 0.302 0.302 0.302 εφ 0.305 0.305 0.305

τLCMV (s) 0.020 0.007 0.002 εθ 0.302 0.302 0.302 εφ 0.305 0.305 0.305

τLCMV (s) 0.0054 0.0050 0.0046 εθ 0.203 0.042 0.02 εφ 0.090 0.045 0.024

τLCMV (s) 0.0101 0.01 0.01 εθ 0.2294 0.0378 0.0972 εφ 0.156 0.031 0.01

algorithm the estimation error holds up constant.

74 Antenna Arrays and Beam-formation

Algorithms SNR (dB)

Algorithms SNR (dB)

5. Conclusions

The authors are grateful to Fundação para a Ciência e a Tecnologia in the scope of R&D Unit 50008, financed by the applicable financial framework (FCT/MEC through national funds and when applicable co-funded by FEDER-PT2020 partnership agreement under the project UID/ EEA/50008/2013), to European Structural and Investment Funds (FEEI) through the Competitiveness and Internationalization Operational Program - COMPETE 2020 and by National Funds through FCT - Foundation for Science and Technology under the project POCI-01- 0145- FEDER- 016432, for the partial funding support of this work.

### Author details

Tiago Varum1,2\*, João N. Matos1,2 and Pedro Pinho1,3

#### References

[21] J. Hui and Y. Gang. An improved algorithm of ESPRIT for signal DOA estimation. 2012 International Conference on Industrial Control and Electronics Engineering (ICICEE). 2012;(8):317–320.

[6] L. C. Godara. Applications of antenna arrays to mobile communications. I. Performance improvement, feasibility, and system considerations. Proceedings of the IEEE. 1997;85

[7] R. Li, C. Rao, L. Dai and S. Zhao. Adaptive-adaptive beamforming algorithm of planar array based on one-dimensional auxiliary beam. 3rd International Congress on Image

[9] Ansoft Corporation. User's Guide: High Frequency Structure Simulator, Edition: REV 1.0

[10] A. S. Foutz and M. Banavar. Narrowband direction of arrival estimation for antenna arrays. Synthesis Lectures on Antennas: Morgan & Claypool Publishers, USA; 2008. [11] L. C. Godara. Application of antenna arrays to mobile communications. II. Beamforming.

[12] R. Schmidt. Multiple emitter location and signal parameter estimation. IEEE Transactions

[13] S. Sekizawa. Estimation of arrival directions using music algorithm with a planar array. ICUPC '98. IEEE 1998 International Conference on Universal Personal Communications.

[14] K. V. Rangarao and S. Venkatanarasimhan. Gold-music: a variation on music to accurately determine peaks of the spectrum. IEEE Transactions on Antennas and Propagation.

[15] C. C. Yeh, J. H. Lee and Y. M. Chen. Estimating two-dimensional angles of arrival in coherent source environment. IEEE Transactions on Acoustics, Speech, and Signal

[16] Y. M. Chen. On spatial smoothing for two-dimensional direction-of-arrival estimation of coherent signals. IEEE Transactions on Signal Processing. 1997;45(7):1689–1696.

[17] Z. Chen, G. Gokeda and Y. Yu. Introduction to Direction-of-Arrival Estimation. Artech

[18] M. Haardt, M. D. Zoltowski, C. P. Mathews and J. Nossek. 2d Unitary esprit for efficient 2d parameter estimation. ICASSP-95, International Conference on Acoustics, Speech, and

[19] M. D. Zoltowski, M. Haardt and C. P. Mathews. Closed-form 2-d angle estimation with rectangular arrays in element space or beamspace via unitary esprit. IEEE Transactions

[20] Y. Y. Wang and W. W. Chen. A low complexity 2d-doa estimation algorithm using signal decomposition. 4th International in High Speed Intelligent Communication Forum

House Signal Processing Library: Artech House, USA; 2010.

[8] MATLAB. Version 7.10.0 (R2010a). Natick, MA: The MathWorks Inc.; 2010.

(7):1031–1060.

76 Antenna Arrays and Beam-formation

Software Version 10.0; 2005.

1998;1(10):555–559.

2013;61(4):2263–2268.

(HSIC). 2012;(5):1–4.

Processing. 1989;37(1):153–155.

Signal Processing. 1995;3(3):2096–2099.

on Signal Processing. 1996;44(2):316–328.

and Signal Processing (CISP). 2010;7:3300–3303.

Proceedings of the IEEE. 1997;85(8):1195–1245.

on Antennas and Propagation. 1986;34(3):276–280.

### **Beamformer Based on Quaternion Processes**

Jian-wu Tao and Wen-xiu Chang

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/67859

#### Abstract

In this chapter, the problem of quaternion beamformer based on linear and widely linear hypercomplex processing is investigated in scenarios, where there exist one signal and one interference that are uncorrelated. First, we introduce brief information about the quaternion algebra and a quaternion model of linear symmetric array with twocomponent electromagnetic (EM) vector-sensors is presented. Based on array's quaternion model, a quaternion MVDR (QMVDR) beamformer is derived and its performance is analysed. Second, we propose the general expression of a quaternion semi-widely linear (QSWL) beamformer and derive its useful implementation and the array's gain expression. Finally, we give the main results of Monte Carlo simulation.

Keywords: quaternion beamforming, hypercomplex processes, widely linear hypercomplex processes, polarization signal processes, EM vector-sensor array

#### 1. Introduction

As an important tool of multidimensional signal processing, the quaternion algebra has been applied to spatio-temporal-polarization beamformer based on an electromagnetic (EM) vectorsensor array [1–5]. The potential advantages of multidimensional signal processing are: (1) the correlation and coupling between each dimension are naturally considered, leading to improved accuracies of signal processing; (2) signals of different geometric nature in different dimensions are being represented as a single signal, leading to reduced complexity of processing approaches. For example, we consider an array consisting of M two-component vector-sensors (If an EM vector-sensor consists of only two components, such as two magnetic loops [6], one electric dipole plus one magnetic loop [7] and two electric dipoles [3], it is referred to a two-component vector-sensor in this chapter.), the output of complex 'long vector' beamformer [8, 9] is yc ¼ wH c xc where xc ¼ ½x11,x12,…xM1,xM2� T is the observed vector of array; wc ¼ ½w11,w12,…wM1,wM2� T is a complex weighted vector; the symbol (.)H denotes the complex conjugation transposition operator. Whereas the output of quaternion-based beamformer [3–5] is yh ¼ wΔ h xh ¼ yc þ j ye where

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

xh ¼ ½xh1,xh2,…xhM� T is the quaternion-valued, observed vector of array and xhm ¼ xm1 þ jxm2 ðm ¼ 1, …, MÞ. wh ¼ ½wh1,wh2,…whM� T is a quaternion-valued, weighted vector and whm ¼ wm1þ jwm2 ðm ¼ 1, …, MÞ. The symbol ð:Þ Δ denotes the quaternion conjugation transposition operator and j denotes an imaginary unit of quaternions. Comparing yh with yc, we can see that the output of quaternion-based beamformer has one more extra information ye than the output of complex 'long vector' beamformer. By employing this extra information ye, we can further improve the performance of beamformer.

In this chapter, our aim is to investigate the beamformer of EM vector-sensor array, based on quaternion processes. First, a QMVDR beamformer and its interference and noise canceller (INC) algorithm are proposed. The output signal to interference-plus-noise ratio (SINR) expression of INC algorithm is derived in a scenario where there exist one signal and one interference that are uncorrelated. By analysing the effect of sources parameters on the output SINR, the fact is explicitly revealed that even though no separation between the DOA's of the desired signal and interference, the maximum value of output SINR can be obtained using the orthogonality between the polarizations of the desired signal and interference. Second, we propose a quaternion semi-widely linear beamformer and its useful implementation, i.e., quaternion semi-widely linear (QSWL) Generalized sidelobe canceller (GSC). Since the QSWL GSC consists of two-stage beamformers, it has more information than the complex 'long vector' beamformer. The increase in information results in the improvement of the beamformer's performance. By designing the weight vectors of two-stage beamformers, the interference is completely cancelled in the output of QSWL GSC and the desired signal is not distorted.

#### 2. Quaternion algebra and vector-sensor array model

#### 2.1. Quaternion algebra

Quaternion came up in the investigations of constructing multidimensional analogues of the field of complex numbers C. The field of quaternion numbers Q is also algebra over the field of real numbers R. The dimension of this algebra is four, and four basis elements are 1, i, j and k. In field of quaternion numbers Q, following multiplication is satisfied

$$\mathbf{i}^2 = \mathbf{j}^2 = k^2 = -1; \text{ iĭ} = k = -j \mathbf{i}; \mathbf{j}k = \mathbf{i} = -k \mathbf{j}; \mathbf{k}i = \mathbf{j} = -\mathbf{k} \tag{1}$$

A quaternion variable x∈ Q has two forms of representation. One form is x = x1 + ix2 + jx3 + kx4, where x1, x2, x3 and x4 are real coefficients. It is referred to as the R-expansion of quaternions. We call x1 the real/scalar part of x, and it is denoted by RðxÞ. ix2 + jx3 + kx4 is called the imaginary/vector part of x, and it is denoted by IðxÞ. We refer to x\* = x1 � ix2 � jx3 � kx4 as the conjugate of x and jxj¼ðx2 1 þ x2 2 þ x2 3 þ x2 4Þ 1=2 as the modulus of x. The other form is x = z1 + jz2, where z1 and z2 are complex coefficients. It is called the C-expansion of quaternions or Cayley-Dickson. x� ¼ z� 1 � jz2 is the conjugate of x and |x|=(|z1|2 +|z2|2 ) 1/2 is the modulus of x. Several properties of quaternions are discussed in Table 1 [10–12].

x and y denotes two quaternions

xh ¼ ½xh1,xh2,…xhM�

80 Antenna Arrays and Beam-formation

ðm ¼ 1, …, MÞ. wh ¼ ½wh1,wh2,…whM�

of QSWL GSC and the desired signal is not distorted.

2. Quaternion algebra and vector-sensor array model

In field of quaternion numbers Q, following multiplication is satisfied

1 þ x2

properties of quaternions are discussed in Table 1 [10–12].

2 þ x2

1 � jz2 is the conjugate of x and |x|=(|z1|2

3 þ x2 4Þ

jwm2 ðm ¼ 1, …, MÞ. The symbol ð:Þ

performance of beamformer.

2.1. Quaternion algebra

conjugate of x and jxj¼ðx2

Dickson. x� ¼ z�

i 2 ¼ j 2 ¼ k

T is the quaternion-valued, observed vector of array and xhm ¼ xm1 þ jxm2

and j denotes an imaginary unit of quaternions. Comparing yh with yc, we can see that the output of quaternion-based beamformer has one more extra information ye than the output of complex 'long vector' beamformer. By employing this extra information ye, we can further improve the

In this chapter, our aim is to investigate the beamformer of EM vector-sensor array, based on quaternion processes. First, a QMVDR beamformer and its interference and noise canceller (INC) algorithm are proposed. The output signal to interference-plus-noise ratio (SINR) expression of INC algorithm is derived in a scenario where there exist one signal and one interference that are uncorrelated. By analysing the effect of sources parameters on the output SINR, the fact is explicitly revealed that even though no separation between the DOA's of the desired signal and interference, the maximum value of output SINR can be obtained using the orthogonality between the polarizations of the desired signal and interference. Second, we propose a quaternion semi-widely linear beamformer and its useful implementation, i.e., quaternion semi-widely linear (QSWL) Generalized sidelobe canceller (GSC). Since the QSWL GSC consists of two-stage beamformers, it has more information than the complex 'long vector' beamformer. The increase in information results in the improvement of the beamformer's performance. By designing the weight vectors of two-stage beamformers, the interference is completely cancelled in the output

Quaternion came up in the investigations of constructing multidimensional analogues of the field of complex numbers C. The field of quaternion numbers Q is also algebra over the field of real numbers R. The dimension of this algebra is four, and four basis elements are 1, i, j and k.

A quaternion variable x∈ Q has two forms of representation. One form is x = x1 + ix2 + jx3 + kx4, where x1, x2, x3 and x4 are real coefficients. It is referred to as the R-expansion of quaternions. We call x1 the real/scalar part of x, and it is denoted by RðxÞ. ix2 + jx3 + kx4 is called the imaginary/vector part of x, and it is denoted by IðxÞ. We refer to x\* = x1 � ix2 � jx3 � kx4 as the

where z1 and z2 are complex coefficients. It is called the C-expansion of quaternions or Cayley-

2 ¼ �1; ij ¼ k ¼ �ji; jk ¼ i ¼ �kj; ki ¼ j ¼ �ik ð1Þ

+|z2|2 )

1=2 as the modulus of x. The other form is x = z1 + jz2,

1/2 is the modulus of x. Several

T is a quaternion-valued, weighted vector and whm ¼ wm1þ

Δ denotes the quaternion conjugation transposition operator

Table 1. Several properties of quaternions.

#### 2.2. Vector-sensor array model

Consider a scenario with one narrowband, completely polarized source, which is travelling in an isotropic and homogeneous medium, impinges on a uniform linear symmetric array from direction (θ, ϕ). This array consists of 2M two-component vector-sensors, which is depicted in Figure 1, and the spacing between the adjacent two vector-sensors is assumed to be half wavelength. All the vector-sensors are indexed by �M,…, �1,1,…,M from left to right.

Let the array centre be the phase reference point, two highly complex series xm1(n) and xm2(n) are recorded on first and second components of the mth two-component vector-sensor, respectively. xm1(n) and xm2(n) are given by Ward [10]

$$
\begin{bmatrix} \mathfrak{x}\_{\mathfrak{m}1}(n) \\ \mathfrak{x}\_{\mathfrak{m}2}(n) \end{bmatrix} = \begin{bmatrix} a\_1(\theta, \varphi, \gamma, \eta) \\ a\_2(\theta, \varphi, \gamma, \eta) \end{bmatrix} q\_{\mathfrak{m}}(\theta, \varphi) \mathbf{s}(n) \tag{2}
$$

where 0 ≤ θ ≤ π and 0 ≤ ϕ < π denote the incidence source's elevation angle measured from the positive z-axis and the azimuth angle measured from the positive x-axis, respectively. 0 ≤ γ < π=2 represents the auxiliary polarization angle, and �π ≤ η < π signifies the polarization phase difference. a1ðθ,ϕ, γ, ηÞ and a2ðθ,ϕ, γ, ηÞ are the responses on first and second components of two-component vector-sensor, respectively. The two-component vector-sensor consists of one electric dipole plus one magnetic loop co-aligned along the x-axis, where a1ðθ,ϕ, γ, ηÞ ¼ eiηcosϕcosθsinγ � sinϕcosγ and a2ðθ,ϕ, γ, ηÞ¼ � eiηsinϕsinγ � cosϕcosθcosγ [7] or two magnetic loops co-aligned along the x-axis, where a1ðθ,ϕ, γ, η޼�eiηsinϕsinγ� cosϕcosθcosγ and a2ðθ,ϕ, γ, ηÞ ¼ eiηcosϕsinγ � sinϕcosθcosγ [6]. qmðθ,ϕÞ is the spatial phase factor describing wave-field propagation along the array, and qmðθ,ϕÞ ¼ q� �mðθ,ϕÞ due to the

Figure 1. A uniform linear symmetric array.

symmetric structure of array. s(n) is the complex envelope of the waveform, assumed to be a zero-mean, stationary stochastic process.

Using xm1(n) and xm2(n) (m ¼ �M,…, M), a quaternion-valued series xm(n) can be constructed in the C-expansion of quaternions, as the output of the mth two-component vector-sensor:

$$\begin{split} \mathbf{x}\_{m}(n) &= \mathbf{x}\_{m1}(n) + j\mathbf{x}\_{-m2}(n) = q\_{m}(\boldsymbol{\Theta}, \boldsymbol{\uprho}) \Big( \mathbf{a}\_{1}(\boldsymbol{\Theta}, \boldsymbol{\uprho}, \boldsymbol{\uprho}, \boldsymbol{\upeta}) + j\mathbf{a}\_{2}(\boldsymbol{\Theta}, \boldsymbol{\uprho}, \boldsymbol{\uprho}, \boldsymbol{\upeta}) \Big) \mathbf{s}(n) \\ &= q\_{m}(\boldsymbol{\Theta}, \boldsymbol{\uprho}) \mathbf{P}(\boldsymbol{\uptheta}, \boldsymbol{\uprho}, \boldsymbol{\uprho}, \boldsymbol{\upeta}) \mathbf{s}(n) \end{split} \tag{3}$$

where j denotes an imaginary unit of quaternions. Pðθ,ϕ, γ, ηÞ is the quaternion-valued response on two-component vector-sensor. This transformation maps the complex series xm1(n) on scalar and i imaginary fields of a quaternion, and the complex series xm2(n) is simultaneously mapped to the j and k imaginary fields. When the quaternion-valued additive noise is considered, the quaternion-valued output of the mth two-component vector-sensor is given by

$$\mathbf{x}\_m(n) = q\_m(\theta, \varphi) P(\theta, \varphi, \mathbf{y}, \eta) \mathbf{s}(n) + n\_m(n) \tag{4}$$

where nmðnÞ ¼ nm1ðnÞ þ jn�m2ðnÞ: nm1ðnÞ is the complex-valued additive noises recorded on first component of the mth vector-sensor and n�m2ðnÞ is the complex-valued additive noises recorded on second component of the mth vector-sensor, which are assumed to be zero mean, Gaussian noise with identical covariance σ2 n. And it is assumed that nmðnÞ and nnðnÞ, where m 6¼ n, are uncorrelated.

#### 3. Quaternion MVDR (QMVDR) beamformer

It is assumed that two uncorrelated, completely polarized plane-waves impinge on an array with 2M two-component vector-sensor. One is the desired signal characterized by its arrival angles (θs, ϕs) and polarization parameters (γs, ηs); the other is the interference characterized by its arrival angles (θi, ϕi) and polarization parameters (γi, ηi). Assumed interference's DOA and polarization are unknown but signal's DOA and polarization are known or may be priorly estimated from techniques. Thus, the quaternion-valued measurement vector of array can be written as

$$\mathbf{x}(n) = \begin{bmatrix} \mathbf{x}\_{-M}(n), \ \cdots, \mathbf{x}\_{M}(n) \end{bmatrix}^{T} = \mathbf{v}\_{s}\mathbf{s}\_{s}(n) + \mathbf{v}\_{i}\mathbf{s}\_{i}(n) + \mathbf{n}(n) \tag{5}$$

where nðnÞ¼½n�MðnÞ, ⋯, nMðnÞ�T denotes the quaternion-valued additive noise vector. vs ¼ qðθs, ϕsÞPðθs, ϕs, γs, ηsÞ;vi ¼ qðθi, ϕi ÞPðθi, ϕi , γi , ηi Þ are the quaternion-valued steering vector associated with the desired signal and the interference, respectively, where qðθτ, ϕτÞ ¼ ½q�Mðθτ, ϕτÞ, ⋯, qMðθτ, ϕτÞ�T (τ ¼ s, i) denotes the spatial phase factor vector of array.

Using the quaternion-valued measurement vector of an array x(n), the output of a beamformer is

$$\mathbf{y}(n) = \mathbf{w}^{\Lambda} \mathbf{x}(n) \tag{6}$$

where w is the quaternion-valued weight vector and the symbol ð:Þ Δ denotes the quaternion transposition-conjugation operator. Then, the QMVDR beamformer can be derived by solving the following constrained optimization problem [2]:

$$J(\mathbf{w}) = \min \{ \mathbf{w}^{\Lambda} \mathbf{R}\_{\mathbf{x}} \mathbf{w} \}; \\ \text{subject to } \mathbf{w}^{\Lambda} \mathbf{v}\_{\mathbf{s}} = 1 \tag{7}$$

where Rx ¼ EfxðnÞxΔðnÞg is the covariance matrix of the measurement vector. By using Lagrange multipliers, the solution of Eq. (7) is obtained, i.e.,

$$J = \mathbf{w}^{\Delta} \mathbf{R}\_x \mathbf{w} + \lambda \mathbf{w}^{\Delta} \mathbf{v}\_s \tag{8}$$

where λ is a real number. Based on the quaternion-valued gradient operator defined in Ref. [11], the following gradients need to be calculated:

$$\frac{\partial \mathbf{J}}{\partial \mathbf{w}^{\Delta}} = \mathbf{R}\_x \mathbf{w} + \lambda \mathbf{v}\_s \tag{9}$$

Let Eq. (9) is equal to zero, then

symmetric structure of array. s(n) is the complex envelope of the waveform, assumed to be a

Using xm1(n) and xm2(n) (m ¼ �M,…, M), a quaternion-valued series xm(n) can be constructed in the C-expansion of quaternions, as the output of the mth two-component vector-sensor:

where j denotes an imaginary unit of quaternions. Pðθ,ϕ, γ, ηÞ is the quaternion-valued response on two-component vector-sensor. This transformation maps the complex series xm1(n) on scalar and i imaginary fields of a quaternion, and the complex series xm2(n) is simultaneously mapped to the j and k imaginary fields. When the quaternion-valued additive noise is considered, the

where nmðnÞ ¼ nm1ðnÞ þ jn�m2ðnÞ: nm1ðnÞ is the complex-valued additive noises recorded on first component of the mth vector-sensor and n�m2ðnÞ is the complex-valued additive noises recorded on second component of the mth vector-sensor, which are assumed to be zero mean,

It is assumed that two uncorrelated, completely polarized plane-waves impinge on an array with 2M two-component vector-sensor. One is the desired signal characterized by its arrival angles (θs, ϕs) and polarization parameters (γs, ηs); the other is the interference characterized by its arrival angles (θi, ϕi) and polarization parameters (γi, ηi). Assumed interference's DOA and polarization are unknown but signal's DOA and polarization are known or may be priorly estimated from techniques. Thus, the quaternion-valued measurement vector of array can be

where nðnÞ¼½n�MðnÞ, ⋯, nMðnÞ�T denotes the quaternion-valued additive noise vector. vs ¼

, γi , ηi

tor associated with the desired signal and the interference, respectively, where qðθτ, ϕτÞ ¼

Using the quaternion-valued measurement vector of an array x(n), the output of a beamformer is

ÞPðθi, ϕi

½q�Mðθτ, ϕτÞ, ⋯, qMðθτ, ϕτÞ�T (τ ¼ s, i) denotes the spatial phase factor vector of array.

quaternion-valued output of the mth two-component vector-sensor is given by

¼ qmðθ,ϕÞPðθ,ϕ, γ, ηÞsðnÞ ð3Þ

a1ðθ,ϕ, γ, ηÞ þ ja2ðθ,ϕ, γ, ηÞ

xmðnÞ ¼ qmðθ,ϕÞPðθ,ϕ, γ, ηÞsðnÞ þ nmðnÞ ð4Þ

xðnÞ¼½x�MðnÞ, ⋯,xMðnÞ�T¼ vsssðnÞ þ visiðnÞ þ nðnÞ ð5Þ

n. And it is assumed that nmðnÞ and nnðnÞ, where

Þ are the quaternion-valued steering vec-

yðnÞ ¼ wΔxðnÞ ð6Þ

 sðnÞ

zero-mean, stationary stochastic process.

82 Antenna Arrays and Beam-formation

Gaussian noise with identical covariance σ2

qðθs, ϕsÞPðθs, ϕs, γs, ηsÞ;vi ¼ qðθi, ϕi

3. Quaternion MVDR (QMVDR) beamformer

m 6¼ n, are uncorrelated.

written as

xmðnÞ ¼ xm1ðnÞ þ jx�m2ðnÞ ¼ qmðθ,ϕÞ

$$\mathbf{w} = -\lambda \mathbf{R}\_x^{-1} \mathbf{v}\_s \tag{10}$$

Since wΔvs ¼ �λvΔ s R�1 x vs ¼ 1, we have,

$$
\lambda = \frac{-1}{\nu\_s^\Lambda \mathbf{R}\_x^{-1} \mathbf{v}\_s} \tag{11}
$$

Substituting Eq. (11) into Eq. (10), the weight vector of the QMVDR beamformer can be written as

$$\mathbf{w} = \frac{\mathbf{R}\_x^{-1} \mathbf{v}\_s}{\mathbf{v}\_s^\Lambda \mathbf{R}\_x^{-1} \mathbf{v}\_s} \tag{12}$$

Substituting Eq. (12) into Eq. (6), the quaternion-valued output of the QMVDR beamformer is given by

$$\mathbf{y}(n) = \mathbf{s}\_s(n) + \mathbf{w}^\Lambda \mathbf{v}\_{i\bar{\imath}i}(n) + \mathbf{w}^\Lambda \mathbf{n}(n) \tag{13}$$

where wΔvs ¼ 1. To a linear symmetric array with 2M two-component vector-sensors, the signal to interference-plus-noise ratio (SINR) of quaternion-valued output y(n) can be written in the simple form (the proof is in Appendix 1 of Ref. [4])

$$SINR\_y = \xi\_s |P\_s|^2 (M - \frac{|P\_i|^2 |q\_s^H q\_i|^2}{4\xi\_i^{-1} + 4M|P\_i|^2})\tag{14}$$

where the input signal-to-noise ratio (SNR) ξs ¼ σ2 s σ2 n and the input interference-to-noise ratio (INR) ξi ¼ σ2 i σ2 n .

#### 4. Interference and noise canceller (INC) based on QMVDR

Using the C-expansion of quaternions, y(n) can be written as

$$y(n) = y\_1(n) + j \, y\_2(n) \tag{15}$$

where y1(n) and y2(n) are two complex-valued components of y(n), i.e.,

$$\mathbf{y}\_1(n) = \mathbf{s}\_i(n) + (\mathbf{w}^\Delta \mathbf{v}\_i)\_1 \mathbf{s}\_i(n) + (\mathbf{w}^\Delta \mathbf{n}(n))\_1; \ y\_2(n) = (\mathbf{w}^\Delta \mathbf{v}\_i)\_2 \mathbf{s}\_i(n) + (\mathbf{w}^\Delta \mathbf{n}(n))\_2 \tag{16}$$

ð:Þ1 and ð:Þ2 denote, respectively, first and second complex-valued components of a quaternion. The expansion (16) highlights the fact that y2(n) does not include the desired signal, but include only the interference and noise. By employing y2(n), we can partly cancel the interference and noise component in y1(n). Thus, an INC based on the QMVDR is shown in Figure 2.

The INC is a spatio-temporal processing, i.e., first part is a spatio filter and second part is a temporal filter. Then, the output of INC may be written as

$$\begin{array}{l} y\_s(n) = y\_1(n) - w\_\varepsilon^\* \ y\_2(n) \\ = s\_\circ(n) + ((\mathbf{w}^\Delta \mathbf{v}\_i)\_1 - w\_\varepsilon^\*(\mathbf{w}^\Delta \mathbf{v}\_i)\_2) s\_i(n) + ((\mathbf{w}^\Delta \mathbf{n}(n))\_1 - w\_\varepsilon^\*(\mathbf{w}^\Delta \mathbf{n}(n))\_2) \\ = s\_\circ(n) + w\_\varepsilon s\_i(n) + \varepsilon(n) \end{array} \tag{17}$$

where wc is a complex weight, which can be given by the Wiener-Hoft equation

$$w\_{\mathcal{C}} = \frac{r\_{y\_2 y\_1}}{R\_{y\_2}} \tag{18}$$

where ry2y1 ¼ Efy2ðnÞy� 1ðnÞg and Ry2 ¼ Efy2ðnÞy� 2ðnÞg. Following the proof given in Appendix 2 of Ref. [4], we have

$$w\_c = \frac{(\mathbf{w}^{\Delta} \mathbf{v}\_i)\_2 (\mathbf{w}^{\Delta} \mathbf{v}\_i)\_1^\* \xi\_i}{\left| (\mathbf{w}^{\Delta} \mathbf{v}\_i)\_2 \right|^2 \xi\_i + \left\| \mathbf{w} \right\|^2} \tag{19}$$

If ξi is very small, wc is approximately equal to 0. Whereas, if ξi is very large, wc is approximately equal to ðwΔviÞ � 1 ðwΔviÞ � 2 . In this case, the interference can be cancelled. From Eq. (17), the SINR in the complex output ys(n) is given by

Figure 2. INC based on the QMVDR.

Beamformer Based on Quaternion Processes http://dx.doi.org/10.5772/67859 85

$$\text{SINR}\_{\mathcal{Y}\_s} = \frac{\xi\_s}{\|\mathbf{w}\|^2} \left( 1 - \frac{\sigma\_i^2 |(\mathbf{w}^\Lambda \mathbf{v}\_i)\_1|^2}{\sigma\_n^2 \|\mathbf{w}\|^2 + \sigma\_i^2 |\mathbf{w}^\Lambda \mathbf{v}\_i|^2} \right) = \mathbf{x}(1 - \kappa\_i) \tag{20}$$

where we define κ ¼ ξs kwk2 and κi ¼ σ2 i jðwΔviÞ1j 2 σ2 nkwk2 þσ2 i jwΔvij 2 . The proof is in Appendix 3 of Ref. [4]. Clearly, SINRys increases with an increase in κ but decreases with an increase in κi.

Next, we show the effect of sources parameters on κ and κi. Following the proof given in Appendix 4 of Ref. [4], we have

$$\kappa = \xi\_s |P\_s|^2 \left( 2M - \frac{|P\_i|^2 |\mathbf{q}\_s^H \mathbf{q}\_i|^2}{2\xi\_i^{-1} + 2M|P\_i|^2} \right) \frac{2M - \mu}{2M - \mu(1 + \varepsilon)} = 2SINR\_y \beta \tag{21}$$

where

4. Interference and noise canceller (INC) based on QMVDR

where y1(n) and y2(n) are two complex-valued components of y(n), i.e.,

yðnÞ ¼ y1ðnÞ þ j y2ðnÞ ð15Þ

c ðwΔnðnÞÞ2Þ

2ðnÞg. Following the proof given in Appendix

ξi þ kwk2 ð19Þ

ð17Þ

ð18Þ

y1ðnÞ ¼ ssðnÞþðwΔviÞ1siðnÞþðwΔnðnÞÞ1; y2ðnÞ¼ðwΔviÞ2siðnÞþðwΔnðnÞÞ2 ð16Þ

c ðwΔviÞ2ÞsiðnÞ þ ððwΔnðnÞÞ1 � w�

� 1ξi

. In this case, the interference can be cancelled. From Eq. (17), the SINR

ð:Þ1 and ð:Þ2 denote, respectively, first and second complex-valued components of a quaternion. The expansion (16) highlights the fact that y2(n) does not include the desired signal, but include only the interference and noise. By employing y2(n), we can partly cancel the interference and

The INC is a spatio-temporal processing, i.e., first part is a spatio filter and second part is a

wc ¼ ry2y1 Ry2

wc ¼ ðwΔviÞ2ðwΔviÞ

2

If ξi is very small, wc is approximately equal to 0. Whereas, if ξi is very large, wc is approxi-

jðwΔviÞ2j

noise component in y1(n). Thus, an INC based on the QMVDR is shown in Figure 2.

where wc is a complex weight, which can be given by the Wiener-Hoft equation

1ðnÞg and Ry2 ¼ Efy2ðnÞy�

Using the C-expansion of quaternions, y(n) can be written as

temporal filter. Then, the output of INC may be written as

¼ ssðnÞ þ wisiðnÞ þ εðnÞ

c y2ðnÞ ¼ ssðnÞ þ ððwΔviÞ1 � w�

ysðnÞ ¼ y1ðnÞ � w�

� 1 ðwΔviÞ � 2

in the complex output ys(n) is given by

Figure 2. INC based on the QMVDR.

where ry2y1 ¼ Efy2ðnÞy�

84 Antenna Arrays and Beam-formation

2 of Ref. [4], we have

mately equal to ðwΔviÞ

$$\beta = \frac{2M - \mu}{2M - \mu(1 + \varepsilon)}; \ \mu = \frac{|P\_i|^2 |q\_s^H q\_i|^2}{2\xi\_i^{-1} + 2M|P\_i|^2}; \ \varepsilon = \frac{\xi\_i^{-1}}{\xi\_i^{-1} + M|P\_i|^2} \tag{22}$$

Obviously, the gain β ≥ 1 because of 0 ≤ ε < 1. Then, κ ≥ 2SINRy.

From Eq. (21), κ depends mainly on separation between the DOAs of the desired signal and interference (i.e., jqH s qij 2 ). The dependencies of κ on jqH s qij 2 are shown in the following consequences:

In addition, κ depends also on the input INR ξi, the array's element number 2M and the interference response jPij 2 and the desired signal response jPsj 2 .

In order to illustrate the previous discussions, Figure 3(a) and (b) displays, respectively, the variations of κ as a function of the desired signal's arrival angles θs and ϕs for several values of ξi, where θi ¼ 90� ϕi ¼ 60� ; ϕs ¼ 60� in Figure 3(a) and θs ¼ 90� in Figure 3(b), where a linear symmetric antenna array is used with four (i.e., M = 2) two-component vector-sensor.

Figure 3. Variations of κ as a function of the desired signal's arrival angles θs and ϕs.

Each two-component vector-sensor has one electric dipole and one magnetic loop co-aligned along the x-axis, and four vector-sensors are spaced half a wavelength apart. It is assumed that the desired signal with ξs = 1 and the interference have the same polarization parameters (γs ¼ γi ¼ 30� , ηs ¼ ηi ¼ 30� ). Simulation results in Figure 3 are in agreement with the previous discussions. From Figure 3(a), it is seen that jqH s qij ≈ 4 and κ ≈ 3 as θs ≈ 90� . The cause of this phenomenon is that jPsj 2 ¼ sin2 ϕs þ cos2ϕs cos2θs for the two-component vector-sensor used in this example. When θs ≈ 90� , jPsj 2 ≈ 0:75. In cases of jqH s qij ¼ 4, κ ≈ 3. Along θs is away from 90� , jqH s qij decreases. This results in the reduction of κ. From Figure 3(b), it is seen that jqH s qij ≈ 4 and κ ≈ 3 as ϕs ≈ 60� . This is as same as Figure 3(a). In addition, it is noted that jqH s qij ≈ 0 and κ ≈ 4 as ϕs ≈ 100� . The cause of this phenomenon is that jPsj 2 ≈ 1 in cases of ϕs ≈ 100� and κ ≈ 4 in cases of jqH s qij ¼ 0 . When ϕs ¼ 0� or ϕs ¼ 180� , jPsj 2 ¼ 0, then, κ ¼ 0.

Since the output interference-plus-noise power wΔRinw ¼ σ2 nkwk2 þ σ2 i jwΔvij 2 , we have

$$\kappa\_i = \frac{\sigma\_i^2 |(\mathbf{w}^\Lambda \mathbf{v}\_i)\_1|^2}{\sigma\_n^2 \left|\mathbf{w}\right|^2 + \sigma\_i^2 \left|\mathbf{w}^\Lambda \mathbf{v}\_i\right|^2} = \frac{\sigma\_i^2 |(\mathbf{w}^\Lambda \mathbf{v}\_i)\_1|^2}{\mathbf{w}^\Lambda \mathbf{R}\_{\text{in}} \mathbf{w}} \tag{23}$$

where 0 ≤ κi ≤ 1 because of 0 ≤ jðwΔviÞ1j 2 ≤ jwΔvij 2 . Following the proof given in Appendix 1 of Ref. [4], κi is further written as

$$\kappa\_i = \xi\_i |P\_s|^2 \left( M - \frac{|P\_i|^2 |\mathbf{q}\_s^H \mathbf{q}\_i|^2}{4\xi\_i^{-1} + 4M|P\_i|^2} \right) |(\mathbf{w}^\Delta \mathbf{v}\_i)\_1|^2 \tag{24}$$

Obviously, κi = 0 at jðwΔviÞ1j ¼ 0 or ξi ¼ 0. And κi increases with an increase in jðwΔviÞ1j 2 . In addition, κi depends also on the array's element number 2M and the interference response jPij 2 .

Following the proof given in Appendix 5 of Ref. [4], we have

$$(\mathbf{w}^{\Lambda}\mathbf{v}\_i)\_1 = \alpha (\mathbf{v}\_s^{\Lambda}\mathbf{v}\_i)\_1 = \alpha |\mathbf{q}\_s^H\mathbf{q}\_i|(P\_s^{\ast}P\_i)\_1\tag{25}$$

where

Each two-component vector-sensor has one electric dipole and one magnetic loop co-aligned along the x-axis, and four vector-sensors are spaced half a wavelength apart. It is assumed that the desired signal with ξs = 1 and the interference have the same polarization parameters

0.5 1 1.5 2 2.5 3 3.5 4

κ or |qH

s qi

2 ≈ 0:75. In cases of jqH

i jðwΔviÞ1j

2 ≤ jwΔvij

2 M � jPij

4ξ�1

Obviously, κi = 0 at jðwΔviÞ1j ¼ 0 or ξi ¼ 0. And κi increases with an increase in jðwΔviÞ1j

addition, κi depends also on the array's element number 2M and the interference response

nkwk2 þ σ2

s qij decreases. This results in the reduction of κ. From Figure 3(b), it is seen that

s qij ¼ 0 . When ϕs ¼ 0� or ϕs ¼ 180�

2

i jwΔvij

2

2 jqH s qij 2

!

i þ 4MjPij

2

jðwΔviÞ1j

. The cause of this phenomenon is that jPsj

2 ¼ σ2

). Simulation results in Figure 3 are in agreement with the previ-

s qij ≈ 4 and κ ≈ 3 as θs ≈ 90�

ϕs þ cos2ϕs cos2θs for the two-component vector-sensor used

. This is as same as Figure 3(a). In addition, it is noted that

i jðwΔviÞ1j

nkwk2 þ σ2

2

. The cause of this

ξ i =1 ξ i =3 ξ i =5 ξ i =7 |qs Hqi |

2 ≈ 1 in cases of

2 ¼ 0, then, κ ¼ 0.

, we have

2 ð24Þ

2 . In

s qij ¼ 4, κ ≈ 3. Along θs is away from

0 20 40 60 80 100 120 140 160 180 0

φs [deg]

, jPsj

i jwΔvij 2

. Following the proof given in Appendix 1 of

wΔRinw ð23Þ

(γs ¼ γi ¼ 30�

1.5

2

2.5

κ or |qH

s qi 3

3.5

4

86 Antenna Arrays and Beam-formation

90� , jqH

jqH

jqH

jPij 2 .

phenomenon is that jPsj

in this example. When θs ≈ 90�

s qij ≈ 4 and κ ≈ 3 as ϕs ≈ 60�

s qij ≈ 0 and κ ≈ 4 as ϕs ≈ 100�

ϕs ≈ 100� and κ ≈ 4 in cases of jqH

where 0 ≤ κi ≤ 1 because of 0 ≤ jðwΔviÞ1j

Ref. [4], κi is further written as

, ηs ¼ ηi ¼ 30�

ous discussions. From Figure 3(a), it is seen that jqH

0 20 40 60 80 100 120 140 160 180 1

ξ i =1 ξ i =3 ξ i =5 ξ i =7 |qs Hqi |

(a) (b)

θs [deg]

2 ¼ sin2

, jPsj

Figure 3. Variations of κ as a function of the desired signal's arrival angles θs and ϕs.

Since the output interference-plus-noise power wΔRinw ¼ σ2

κi ¼ ξijPsj

κi ¼ σ2

σ2

$$\alpha = \frac{\xi\_i^{-1}}{|P\_s|^2} \left(\frac{2}{4M\xi\_i^{-1} + |P\_i|^2(4M^2 - |\mathbf{q}\_s^H \mathbf{q}\_i|^2)}\right) \tag{26}$$

From Eqs. (25) and (26), ðwΔviÞ1 depends not only on the separation between the DOAs of the desired signal and interference (i.e., jqH s qij), but also on the difference between the polarizations of the desired signal and interference (i.e., ðPs �PiÞ1). The dependencies of ðwΔviÞ1 and κi on jqH s qij and ðPs �PiÞ1) are shown in the following consequences:

Finally, we analyse the performance of the INC. From Eq. (20) and above analysis, we can obtain the following consequences:

Further, jðPs �PiÞ1j increases with a decrease in the difference in polarizations. Thus, SINRys will reduce due to the increase in κi.

3. When ξi ¼ 0 (i.e., in the absence of interference), κ ¼ 2MξsjPsj 2 and κi ¼ 0. In this case, we can obtain the maximum value of SINRys , i.e., SINRysmax ¼ 2MξsjPsj 2 . Further, SINRys decreases with an increase in ξi. In the presence of a strong interference (i.e., ξ�1 i ≈ 0), SINRys can be approximated as

$$\text{SINR}\_{\mathcal{Y}\_s} \approx \xi\_s |P\_s|^2 \left( 2M - \frac{|\mathbf{q}\_s^H \mathbf{q}\_i|^2}{2M} \right) \left( 1 - \frac{|\left(\mathbf{v}\_s^\Lambda \mathbf{v}\_i\right)\_1|^2}{|\mathbf{v}\_s^\Lambda \mathbf{v}\_i|^2} \right) \tag{27}$$

where κ ≈ ξsjPsj 2 2M � jqH s qij 2 2M � � and κi ¼ jðvs ΔviÞ1j 2 jvs Δvij 2 . Expression (27) highlights the fact that in the presence of a strong interference, SINRys ≈ 0 in the case of no separation between the DOAs of the desired signal and interference (i.e., qs ¼ qi) or no difference between the polarizations of the desired signal and interference (i.e., Ps ¼ Pi). This implies that the INC fails.

4. When the vector-sensor number 2M in array increases, κ increases and κi decreases. Thus, SINRys is an increasing function of 2M. Since κ increases with an increase in jPsj 2 , but κi decreases with an increase in jPsj 2 , SINRys is an increasing function of jPsj 2 . Further, SINRys is a decreasing function of jPij 2 because κ decreases with an increase in jPij 2 .

#### 5. The quaternion semi-widely linear (QSWL) beamformer

According to the definition in Ref. [12], the involution of a quaternion x over a pure unit quaternion i is xðiÞ ¼ ixi�1 ¼ ixi� ¼ �ixi and it represents the reflection of x over the plane spanned by {1, i}. A quaternion vector x is Ci -proper iff it can be represented by means of two jointly proper complex vectors in the plane spanned by {1, i}. The augmented covariance matrix of a Ci -proper quaternion vector x can be written as

$$\mathbf{R}\_{\overline{\mathbf{x}}',\overline{\mathbf{x}}} = \mathbb{E}\{\overline{\mathbf{x}}'\overline{\mathbf{x}}^{\Lambda}\} = \begin{bmatrix} \mathbf{R}\_{\widetilde{\mathbf{x}},\widetilde{\mathbf{x}}} & \mathbf{0} \\ \mathbf{0} & \mathbf{R}\_{\widetilde{\mathbf{x}},\widetilde{\mathbf{x}}} \mathbf{y}^{j} \end{bmatrix} \tag{28}$$

where x ¼ ½xT,xðiÞT,xðjÞT,xðkÞT� T is the augmented quaternion vector; ~x ¼ ½xT,xðiÞT� T is the semiaugmented quaternion vector and R~x,~x ¼ Ef~x ~xΔg is the semi-augmented covariance matrix of quaternion vector x. In comparison with the semi-augmented covariance matrix R~x,~x, the augmented covariance matrix Rx, x has not more extra information. In other words, the fullwidely linear processing is equivalent to the semi-widely linear processing in handling the Ci -proper quaternion vector. We should not expect that the performance is improved by replacing semi-widely linear processing with full-widely linear processing.

The most general linear processing is full-widely linear processing, which consists in the simultaneous operation on the quaternion vector and its three involutions. Then, a quaternion widely linear beamformer can be written as

$$\mathbf{y}(n) = \mathbf{w}^{\Lambda}\mathbf{x}(n) + \mathbf{G}^{\Lambda}\mathbf{x}^{(i)}(n) + \mathbf{H}^{\Lambda}\mathbf{x}^{(j)}(n) + \mathbf{F}^{\Lambda}\mathbf{x}^{(k)}(n) \tag{29}$$

where W, G, H and F denote the quaternion-valued weight vectors. xðiÞ ðnÞ, xðjÞ ðnÞ and xðkÞ ðnÞ denote the quaternion involution of x(n) over a pure unit imaginary i, j and k, respectively.

The full-widely linear processing is optimal processing for the Q-improper quaternion vector. Since the quaternion-valued vector x(n) is Ci -proper vector, however, the optimal processing reduces to semi-widely linear processing. Because the semi-widely linear processing consists only in the simultaneous operation on the quaternion vector and its involution over i, the general expression of a quaternion semi-widely linear (QSWL) beamformer can be written as Ref. [5]

$$\mathbf{y}(n) = \mathbf{W}^{\Lambda}\mathbf{x}(n) + \mathbf{G}^{\Lambda}\mathbf{x}^{(i)}(n) \tag{30}$$

where xðiÞ ðnÞ is given by

Further, jðPs

88 Antenna Arrays and Beam-formation

where κ ≈ ξsjPsj

fails.

matrix of a Ci

Ci

where x ¼ ½xT,xðiÞT,xðjÞT,xðkÞT�

augmented covariance matrix Rx,

will reduce due to the increase in κi.

SINRys can be approximated as

decreases with an increase in jPsj

spanned by {1, i}. A quaternion vector x is Ci

is a decreasing function of jPij

can obtain the maximum value of SINRys

2 2M � jqH

3. When ξi ¼ 0 (i.e., in the absence of interference), κ ¼ 2MξsjPsj

SINRys ≈ ξsjPsj

s qij 2 2M � �

�PiÞ1j increases with a decrease in the difference in polarizations. Thus, SINRys

decreases with an increase in ξi. In the presence of a strong interference (i.e., ξ�1

!

s qij 2

2M

ΔviÞ1j 2

in the presence of a strong interference, SINRys ≈ 0 in the case of no separation between the DOAs of the desired signal and interference (i.e., qs ¼ qi) or no difference between the polarizations of the desired signal and interference (i.e., Ps ¼ Pi). This implies that the INC

jvs Δvij

4. When the vector-sensor number 2M in array increases, κ increases and κi decreases. Thus, SINRys is an increasing function of 2M. Since κ increases with an increase in jPsj

According to the definition in Ref. [12], the involution of a quaternion x over a pure unit quaternion i is xðiÞ ¼ ixi�1 ¼ ixi� ¼ �ixi and it represents the reflection of x over the plane

jointly proper complex vectors in the plane spanned by {1, i}. The augmented covariance

augmented quaternion vector and R~x,~x ¼ Ef~x ~xΔg is the semi-augmented covariance matrix of quaternion vector x. In comparison with the semi-augmented covariance matrix R~x,~x, the

widely linear processing is equivalent to the semi-widely linear processing in handling the

2 2M � jqH

and κi ¼ jðvs

2

5. The quaternion semi-widely linear (QSWL) beamformer

replacing semi-widely linear processing with full-widely linear processing.

Rx, x ¼ Efx x Δg ¼

, i.e., SINRysmax ¼ 2MξsjPsj

1 � jðvs

, SINRys is an increasing function of jPsj

2 because κ decreases with an increase in jPij

R~x,~x 0 0 R~x,~x

T is the augmented quaternion vector; ~x ¼ ½xT,xðiÞT�

" #

j

x has not more extra information. In other words, the full-

jvs Δvij 2

!

ΔviÞ1j 2

2 . Expression (27) highlights the fact that

2

2 .

2 and κi ¼ 0. In this case, we

. Further, SINRys

2 , but κi

. Further, SINRys

ð28Þ

T is the semi-

i ≈ 0),

ð27Þ

2

$$\mathbf{x}^{(i)}(n) = -i\mathbf{x}(n)i = \mathbf{v}\_{\mathbf{s}}^{(i)}\mathbf{s}\_{\mathbf{s}}(n) + \mathbf{v}\_{\mathbf{i}}^{(i)}\mathbf{s}\_{i}(n) + \mathbf{n}^{(i)}(n) \tag{31}$$

Moreover, we can write the quaternion-valued output series yðnÞ in the following Cayley-Dickson representation

$$y(n) = ((\mathbf{W}^\Lambda \mathbf{x}(n))\_1 + (\mathbf{G}^\Lambda \mathbf{x}^{(i)}(n))\_1) + j((\mathbf{W}^\Lambda \mathbf{x}(n))\_2 + (\mathbf{G}^\Lambda \mathbf{x}^{(i)}(n))\_2) = y\_1(n) + jy\_2(n) \tag{32}$$

where y1 and y2 denote, respectively, first and second complex-valued components of a quaternion y. Thus, the QSWL beamformer has two complex-valued output series y1ðnÞ and y2ðnÞ in the planes spanned by {1, i}, where y1ðnÞ¼ðWΔxðnÞÞ1 þ ðGΔxðiÞ ðnÞÞ1Þ and y2ðnÞ¼ðWΔxðnÞÞ2þ ðGΔxðiÞ ðnÞÞ2Þ. Since the complex 'long vector' beamformers have only one complex-valued output series y1ðnÞ, the QSWL beamformer can obtain more information than the conventional 'long vector' beamformer. The increase of information results in the improvement of QSWL beamformer's performance. In addition, we incorporate both the information on x(n) and xðiÞ ðnÞ, so that the QSWL beamformer with different characteristics may be obtained by designing two weight vectors W and G under some different criterions.

#### 6. The QSWL generalized sidelobe canceller

In this section, a useful implementation of the QSWL beamformer, i.e., QSWL generalized sidelobe sanceller (GSC), is proposed. The QSWL GSC, which is depicted in Figure 4, consists of two-stage beamformers. In first-stage beamformer (weight vector is W), we attempt to extract a desired signal without any distortion from observed data. To cancel interferences, we attempt to estimate interferences in second-stage beamformer (weight vector is G). By

Figure 4. The structure of QSWL GSC.

employing the output of second-stage beamformer to cancel interferences in the output of firststage beamformer, there is no interference in the output of the QSWL GSC. Compared with the complex 'long vector' beamformers, the advantages of two-stage beamformers are that the main beam can always point to desired signal's direction, even if the separation between the DOAs of the desired signal and interference is less, and the robustness to DOA mismatch is improved.

In the following, we derive the expressions of quaternion-valued weight vectors W in firststage beamformer and G in second-stage beamformer. Because the quaternion-valued output y(n) has two complex-valued components in the planes spanned by {1, i}, i.e., y1(n) and y2(n), we define the first complex-valued output component as the output of the QSWL GSC. Thus, the complex-valued output of the QSWL GSC is written as

$$y\_{\rm GSC}(n) = (y(n))\_1 = y\_w(n) - \ y\_{\mathcal{g}}(n) \tag{33}$$

where ywðnÞ is the complex-valued output of the first-stage beamformer, i.e., ywðnÞ ¼ ðWΔxðnÞÞ1; ygðnÞ is the complex-valued output of the second-stage beamformer, i.e., ygðnÞ ¼ ðGΔxðiÞ ðnÞÞ1

#### 6.1. The first-stage beamformer

From Eq. (5), we have

$$y\_w(n) = (\mathbf{W}^\Lambda \mathbf{v}\_s)\_1 \mathbf{s}\_i(n) + (\mathbf{W}^\Lambda \mathbf{v}\_i)\_1 \ s\_i(n) + (\mathbf{W}^\Lambda \mathbf{n}(n))\_1 \tag{34}$$

In the first-stage beamformer, we attempt to minimize the interference-plus-noise energy in ywðnÞ, subject to the constraint ðwΔvsÞ1 ¼ 1.

Since the Cayley-Dickson representations of W, vs, vi and n(n) are, respectively, W = W1+j W2, vs ¼ vs1 þ j vs2, vi ¼ vi1 þ j vi2 and nðnÞ ¼ n1ðnÞ þ j n2ðnÞ, we have

$$(\mathbf{W}^{\Lambda}\mathbf{v}\_{s})\_{1} = \mathbf{W}\_{1}^{H}\mathbf{v}\_{s1} + \mathbf{W}\_{2}^{H}\mathbf{v}\_{s2} = \overline{\mathbf{W}}^{H}\overline{\mathbf{V}}\_{s} \tag{35}$$

$$(\mathbf{W}^{\Delta}\mathbf{v}\_i)\_1 = \mathbf{W}\_1^H \mathbf{v}\_{i1} + \mathbf{W}\_2^H \mathbf{v}\_{i2} = \overline{\mathbf{W}}^H \overline{\mathbf{V}}\_i \tag{36}$$

$$(\mathbf{W}^{\Delta}\mathbf{n}(n))\_1 = \mathbf{W}\_1^H \mathbf{n}\_1(n) + \mathbf{W}\_2^H \mathbf{n}\_2(n) = \overline{\mathbf{W}}^H \overline{\mathbf{N}}(n) \tag{37}$$

where W ¼ W1 W2 � �, Vs ¼ vs1 vs2 � �, Vi ¼ vi1 vi2 � �, NðnÞ ¼ n1ðnÞ n2ðnÞ � � . Superscript (.)H denotes the complex conjugate and transpose operator. Thus, Eq. (34) can be rewritten as

$$\mathbf{y}\_w(n) = \mathbf{W}^H \mathbf{V}\_s \mathbf{s}\_s(n) + \mathbf{W}^H \nabla\_l \mathbf{s}\_i(n) + \mathbf{W}^H \mathbf{N}(n) \tag{38}$$

Then, W can be derived by solving the following constrained optimization problem:

$$f(\overline{\mathbf{W}}) = \min \{ \overline{\mathbf{W}}^H \mathbf{R}\_{\text{in}} \overline{\mathbf{W}} \}; \text{ subject to } \overline{\mathbf{W}}^H \overline{\mathbf{V}}\_s = 1 \tag{39}$$

where

employing the output of second-stage beamformer to cancel interferences in the output of firststage beamformer, there is no interference in the output of the QSWL GSC. Compared with the complex 'long vector' beamformers, the advantages of two-stage beamformers are that the main beam can always point to desired signal's direction, even if the separation between the DOAs of the desired signal and interference is less, and the robustness to DOA mismatch is

In the following, we derive the expressions of quaternion-valued weight vectors W in firststage beamformer and G in second-stage beamformer. Because the quaternion-valued output y(n) has two complex-valued components in the planes spanned by {1, i}, i.e., y1(n) and y2(n), we define the first complex-valued output component as the output of the QSWL GSC. Thus,

where ywðnÞ is the complex-valued output of the first-stage beamformer, i.e., ywðnÞ ¼ ðWΔxðnÞÞ1; ygðnÞ is the complex-valued output of the second-stage beamformer, i.e., ygðnÞ ¼

In the first-stage beamformer, we attempt to minimize the interference-plus-noise energy in

Since the Cayley-Dickson representations of W, vs, vi and n(n) are, respectively, W = W1+j W2,

1 vs1 þ WH

1 vi1 þ WH

1 n1ðnÞ þ WH

yGSCðnÞ¼ðyðnÞÞ1 ¼ ywðnÞ � ygðnÞ ð33Þ

ywðnÞ¼ðWΔvsÞ1ssðnÞþðWΔviÞ1 siðnÞþðWΔnðnÞÞ1 ð34Þ

2 vs2 ¼ WHVs ð35Þ

2 vi2 ¼ WHVi ð36Þ

2 n2ðnÞ ¼ WHNðnÞ ð37Þ

the complex-valued output of the QSWL GSC is written as

improved.

ðGΔxðiÞ

ðnÞÞ1

From Eq. (5), we have

6.1. The first-stage beamformer

Figure 4. The structure of QSWL GSC.

90 Antenna Arrays and Beam-formation

ywðnÞ, subject to the constraint ðwΔvsÞ1 ¼ 1.

vs ¼ vs1 þ j vs2, vi ¼ vi1 þ j vi2 and nðnÞ ¼ n1ðnÞ þ j n2ðnÞ, we have

ðWΔnðnÞÞ1 ¼ WH

ðWΔvsÞ1 ¼ WH

ðWΔviÞ1 ¼ WH

$$\mathbf{R\_{\dot{m}}} = \begin{bmatrix} \operatorname{E} \{ (\mathbf{x\_{\dot{m}}(n)})\_1 \ (\mathbf{x\_{\dot{m}}(n)})\_1^H \end{bmatrix} \quad \operatorname{E} \{ (\mathbf{x\_{\dot{m}}(n)})\_1 \ (\mathbf{x\_{\dot{m}}(n)})\_2^H \} \\ \operatorname{E} \{ (\mathbf{x\_{\dot{m}}(n)})\_2 \ (\mathbf{x\_{\dot{m}}(n)})\_1^H \} \quad \operatorname{E} \{ (\mathbf{x\_{\dot{m}}(n)})\_2 \ (\mathbf{x\_{\dot{m}}(n)})\_2^H \} \end{bmatrix} \tag{40}$$

is the covariance matrix and xinðnÞ ¼ visiðnÞ þ nðnÞ is the measurement vector of array in the absence of the desired signal. The solution of this constrained optimization problem is obtained by using Lagrange multipliers, that is

$$\overline{\mathbf{W}} = \frac{\mathbf{R}\_{\text{in}}^{-1} \overline{\mathbf{V}}\_{s}}{\nabla\_{\text{s}}^{H} \mathbf{R}\_{\text{in}}^{-1} \overline{\mathbf{V}}\_{s}} \tag{41}$$

If the interferences are uncorrelated with the additive noise, W can be written in the simple form (the proof is in Appendix A of Ref. [5])

$$\overline{\mathbf{W}} = \frac{\varepsilon \overline{\mathbf{V}}\_s - (P\_i^{\rm A} P\_s)\_1 \mathbf{q}\_i^H \mathbf{q}\_s \overline{\mathbf{V}}\_i}{\mu} \tag{42}$$

where

$$\mu = 2M|P\_s|^2 \varepsilon - \left| (P\_i^{\Lambda} P\_s)\_1 \right|^2 |\mathbf{q}\_i^H \mathbf{q}\_s|^2 \text{ : } \varepsilon = \xi\_i^{-1} + \ 2M|P\_i|^2 \tag{43}$$

where ξi denotes the input interference-to-noise ratio (INR). Moreover, the quaternion-valued optimal weight vector Wo may be given by

$$\mathbf{W}\_{\rm o} = \mathbf{J}\_1 \,\overline{\mathbf{W}} + j \,\mathbf{J}\_2 \,\overline{\mathbf{W}} \tag{44}$$

where J1 ¼ ½I2M�2M, 02M�2M� and J2 ¼ ½02M�2M, I2M�2M� are two selection matrices. It is noted that in some applications, such as Radar, Rin may be estimated in intervals of no transmitted signal. But, Rin is not obtained in other applications, such as Communications. In these applications, we may replace Rin by Rx, where

$$\mathbf{R\_{x}} = \begin{bmatrix} \operatorname{E} \{ (\mathbf{x}(n))\_1 \ (\mathbf{x}(n))\_1^H \} & \operatorname{E} \{ (\mathbf{x}(n))\_1 \ (\mathbf{x}(n))\_2^H \} \\ \operatorname{E} \{ (\mathbf{x}(n))\_2 \ (\mathbf{x}(n))\_1^H \} & \operatorname{E} \{ (\mathbf{x}(n))\_2 \ (\mathbf{x}(n))\_2^H \} \end{bmatrix} \tag{45}$$

is the covariance matrix and x(n). When the distortionless constraint is perfectly matched with the desired signal, the weight vector Wo is identical in both Rin and Rx.

By using the optimal weight vector Wo, the complex output of first-stage beamformer can be given by

$$\mathbf{y}\_w(n) = \mathbf{s}\_s(n) + (\mathbf{W}\_\mathbf{o}^\Lambda \mathbf{v}\_i)\_1 \mathbf{s}\_i(n) + (\mathbf{W}\_\mathbf{o}^\Lambda \mathbf{n}(n))\_1 \tag{46}$$

#### 6.2. The second-stage beamformer

From Eq. (31), we have

$$\boldsymbol{y}\_{\mathcal{S}}(n) = (\mathbf{G}^{\Lambda}\boldsymbol{\nu}\_{\mathrm{s}}^{(\hat{t})})\_1 \mathbf{S}\_{\boldsymbol{s}}(n) + (\mathbf{G}^{\Lambda}\boldsymbol{\nu}\_{\mathrm{i}}^{(\hat{t})})\_1 \mathbf{S}\_{\boldsymbol{i}}(n) + (\mathbf{G}^{\Lambda}\boldsymbol{\pi}^{(\hat{t})}(n))\_1 \tag{47}$$

In the second-stage beamformer, we attempt to minimize the noise energy in ygðnÞ, subject to the constraints ðGΔv ðiÞ S Þ1 ¼ 0 and ðGΔv ðiÞ i Þ1 ¼ ðWΔ o viÞ1 . In the following, two schemes are presented to implement this aim.

#### 6.2.1 The Scheme 1, i.e., combined QPMC and MVDR

Let ¼ wqswMV, where wqs is a quaternion-valued diagonal weight matrix and wMV is a complex weight vector. In this scheme, the first is to achieve the constraint ðGΔv ðiÞ s Þ1 ¼ 0 by designing wqs, which is referred to quaternion polarization matched cancellation (QPMC); the second is to minimize the noise energy in ygðnÞ subject to the constraint ðGΔv ðiÞ i Þ1 ¼ ðWΔ o viÞ1 by designing wMV, which is referred to MVDR.

Let wqs ¼ diag fwqsð�MÞ, ⋯, wqsðMÞg; then we have

$$\mathbf{G}^{\Delta} \mathbf{v}\_{s}^{(i)} = \mathbf{w}\_{MV}{}^{H} \mathbf{w}\_{qs}^{\Delta} \mathbf{v}\_{s}^{(i)} = \mathbf{w}\_{MV}{}^{H} \begin{bmatrix} \mathbf{w}\_{qs}{}^{\*} (-M) \boldsymbol{q}\_{-M} (\boldsymbol{\Theta}\_{s} \cdot \boldsymbol{q}\_{s}) P\_{s}^{(i)} \\ \vdots \\ \mathbf{w}\_{qs}{}^{\*} (M) \boldsymbol{q}\_{M} (\boldsymbol{\Theta}\_{s} \cdot \boldsymbol{q}\_{s}) P\_{s}^{(i)} \end{bmatrix} \tag{48}$$

where superscript (.)\* denotes the quaternion conjugate operator. From Eq. (48) and the constraint ðGΔv ðiÞ s Þ1 ¼ 0, we have the constraint ðwqs�ðmÞqmðθs, ϕsÞPðiÞ s Þ1 ¼ 0, where m ¼ f�M, ⋯, Mg. When wqsðmÞ ¼ qmðθs, ϕsÞ ða� s2 þ j a� s1Þ, this constraint is satisfied. Thus, we can obtain

$$\mathbf{w}\_{\mathfrak{g}\*} = \text{diag}\left\{ \mathbf{q}\_{\mathfrak{s}} \right\} \left( a\_{\mathfrak{s}2}^{\*} + j \, a\_{\mathfrak{s}1}^{\*} \right) \tag{49}$$

where diag fqsg ¼ diag fq�Mðθs, ϕsÞ, ⋯, qMðθs, ϕsÞg . In the constraint ðGΔv ðiÞ s Þ1 ¼ 0, we insert G ¼ wqswMV into (47). Thus, ygðnÞ can be rewritten as

$$\mathbf{y}\_{\mathcal{g}}(n) = \mathbf{w}\_{MV}{}^{H}(\mathbf{w}\_{\mathcal{g}s}^{\Lambda}\mathbf{v}\_{i}^{(i)})\_{1} \ s\_{i}(n) + \mathbf{w}\_{MV}{}^{H}(\mathbf{w}\_{\mathcal{g}s}^{\Lambda}\mathbf{n}^{(i)}(n))\_{1} \tag{50}$$

Then, wMV can be derived by solving the following constrained optimization problem:

Beamformer Based on Quaternion Processes http://dx.doi.org/10.5772/67859 93

$$J(\mathbf{w}\_{MV}) = \min \{ \mathbf{w}\_{MV}^H \mathbf{R}\_{\mathbf{q}s} \mathbf{w}\_{MV} \}; \text{ subject to } \mathbf{w}\_{MV}^H \tilde{\mathbf{V}}\_i = \mathbf{W}^H \boldsymbol{\nabla}\_i \tag{51}$$

where Rqs ¼ EfðwΔ qsxðiÞ ðnÞÞ1 wΔ qsxðiÞ ðnÞ H 1 g is the covariance matrix and V~ i ¼ ðwΔ qsv ðiÞ i Þ1 . The solution of this constrained optimization problem is obtained by using Lagrange multipliers, i.e.,

$$\mathbf{w}\_{MV} = \frac{\mathbf{R}\_{qs}^{-1}}{\tilde{V}\_i{}^H \mathbf{R}\_{qs}^{-1} \tilde{V}\_i} \overline{\mathbf{V}}\_i^H \overline{\mathbf{W}} \tag{52}$$

If the desired signal and interference are uncorrelated with the additive noise, wMV can be written in the simple form (the proof is in Appendix B of Ref. [5])

$$\mathbf{w}\_{\rm MV} = \frac{\mathbf{g}\_1}{\kappa} \tilde{V}\_i \tag{53}$$

where

is the covariance matrix and x(n). When the distortionless constraint is perfectly matched with

By using the optimal weight vector Wo, the complex output of first-stage beamformer can be

ΔviÞ1 siðnÞþðWo

ðiÞ

In the second-stage beamformer, we attempt to minimize the noise energy in ygðnÞ, subject to

Let ¼ wqswMV, where wqs is a quaternion-valued diagonal weight matrix and wMV is a com-

designing wqs, which is referred to quaternion polarization matched cancellation (QPMC); the

H

where superscript (.)\* denotes the quaternion conjugate operator. From Eq. (48) and the constraint

2 6 4 i Þ1 SiðnÞþðGΔnðiÞ

wqs�ð�MÞq�Mðθs, ϕsÞPðiÞ

⋮ wqs�ðMÞqMðθs, ϕsÞPðiÞ

s1Þ, this constraint is satisfied. Thus, we can obtain

HðwΔ qsnðiÞ

s2 þ j a�

i Þ1 siðnÞ þ wMV

ΔnðnÞÞ1 ð46Þ

o viÞ1 . In the following, two schemes are

ðnÞÞ1 ð47Þ

ðiÞ

5 ð48Þ

ðiÞ i Þ1 ¼ ðWΔ

s

3 7

s Þ1 ¼ 0, where m ¼ f�M, ⋯, Mg.

s1Þ ð49Þ

ðiÞ

ðnÞÞ1 ð50Þ

s Þ1 ¼ 0, we

s

s Þ1 ¼ 0 by

o viÞ1 by

the desired signal, the weight vector Wo is identical in both Rin and Rx.

ywðnÞ ¼ ssðnÞþðWo

s Þ1SsðnÞþðGΔv

ðiÞ i Þ1 ¼ ðWΔ

plex weight vector. In this scheme, the first is to achieve the constraint ðGΔv

s ¼ wMV

wqs ¼ diag fqsg ða�

Then, wMV can be derived by solving the following constrained optimization problem:

where diag fqsg ¼ diag fq�Mðθs, ϕsÞ, ⋯, qMðθs, ϕsÞg . In the constraint ðGΔv

HðwΔ qsv ðiÞ

second is to minimize the noise energy in ygðnÞ subject to the constraint ðGΔv

ygðnÞ¼ðGΔvðiÞ

S Þ1 ¼ 0 and ðGΔv

ðiÞ

designing wMV, which is referred to MVDR.

GΔvðiÞ

When wqsðmÞ ¼ qmðθs, ϕsÞ ða�

Let wqs ¼ diag fwqsð�MÞ, ⋯, wqsðMÞg; then we have

s ¼ wMV

HwΔ qsvðiÞ

s Þ1 ¼ 0, we have the constraint ðwqs�ðmÞqmðθs, ϕsÞPðiÞ

s2 þ j a�

insert G ¼ wqswMV into (47). Thus, ygðnÞ can be rewritten as

ygðnÞ ¼ wMV

6.2.1 The Scheme 1, i.e., combined QPMC and MVDR

given by

6.2. The second-stage beamformer

presented to implement this aim.

From Eq. (31), we have

92 Antenna Arrays and Beam-formation

the constraints ðGΔv

ðGΔv ðiÞ

$$\kappa = \tilde{\mathbf{V}}\_i^H \tilde{\mathbf{V}}\_i = 2M(|a\_{s2}|^2 |a\_{i1}|^2 + |a\_{s1}|^2 |a\_{i2}|^2) - 2\mathcal{R}(a\_{s1} a\_{s2}^\* a\_{i2} a\_{i1}^\* (\mathbf{q}\_i^2)^H \mathbf{q}\_s^{-2}) \tag{54}$$

$$\mathcal{g}\_1 = (\mathbf{W}^H \mathbf{V}\_i)^H = \frac{\xi\_i^{-1} (P\_i^\Lambda P\_s)\_1 \mathbf{q}\_i^H \mathbf{q}\_s}{\mu} \tag{55}$$

where R (.) denotes the real part of a complex number. μ is given by Eq. (43).

#### 6.2.2. The scheme 2, i.e. Linearly constrained minimum variance (LCMV) beamformer

In this scheme, we employ the LCMV beamformer as the second-stage beamformer. Since the Cayley-Dickson representations of G, v ðiÞ s , v ðiÞ i and nðiÞ ðnÞ are, respectively, G ¼ G1 þ j G2, v ðiÞ s ¼ vs1 � jvs2, v ðiÞ i ¼ vi1 � j vi2 and nðiÞ ðnÞ ¼ n1ðnÞ � j n2ðnÞ, we have

$$(\mathbf{G}^{\Lambda} \mathbf{v}\_s^{(i)})\_1 = \mathbf{G}\_1^H \mathbf{v}\_{s1} - \mathbf{G}\_2^H \mathbf{v}\_{s2} = \overline{\mathbf{G}}^H \overline{\mathbf{V}}\_s^{(i)} \tag{56}$$

$$(\mathbf{G}^{\Lambda}\mathbf{v}\_{i}^{(i)})\_{1} = \mathbf{G}\_{1}^{H}\mathbf{v}\_{i1} - \mathbf{G}\_{2}^{H}\mathbf{v}\_{i2} = \overline{\mathbf{G}}^{H}\overline{\mathbf{V}}\_{i}^{(i)} \tag{57}$$

$$(\mathbf{G}^{\Delta}\mathbf{n}^{(i)}(n))\_1 = \mathbf{G}\_1^H\mathbf{n}\_1(n) - \mathbf{G}\_2^H\mathbf{n}\_2(n) = \overline{\mathbf{G}}^H\overline{\mathbf{N}}^{(i)}(n) \tag{58}$$

where G ¼ G1 G2 ; VðiÞ s ¼ vs1 �vs2 ; VðiÞ i ¼ vi1 �vi2 ; NðiÞ ðnÞ ¼ n1ðnÞ �n2ðnÞ

Thus, Eq. (47) can be rewritten as

$$\boldsymbol{y}\_{\mathcal{S}}(n) = \overline{\mathbf{G}}^H \overline{\mathbf{V}}\_{\mathbf{s}}^{(i)} \boldsymbol{s}\_{\boldsymbol{s}}(n) + \overline{\mathbf{G}}^H \overline{\mathbf{V}}\_{i}^{(i)} \boldsymbol{s}\_{i}(n) + \overline{\mathbf{G}}^H \overline{\mathbf{N}}^{(i)}(n) \tag{59}$$

Then, G can be derived by solving the following constrained optimization problem:

$$J(\overline{\mathbf{G}}) = \min \{ \overline{\mathbf{G}}^H \mathbf{R}\_{\text{in}G} \overline{\mathbf{G}} \}; \text{subject to } \overline{\mathbf{G}}^H \mathbf{C} = \mathbf{g}^H \tag{60}$$

where

$$\mathbf{R}\_{\dot{\mathbf{m}}G} = \begin{bmatrix} \operatorname{E} \{ (\mathbf{x}\_{\dot{\mathbf{m}}}^{(\dot{t})}(n))\_1 \ (\mathbf{x}\_{\dot{\mathbf{m}}}^{(\dot{t})}(n))\_1^H \end{bmatrix} \quad \operatorname{E} \{ (\mathbf{x}\_{\dot{\mathbf{m}}}^{(\dot{t})}(n))\_1 \ (\mathbf{x}\_{\dot{\mathbf{m}}}^{(\dot{t})}(n))\_2^H \end{bmatrix} \tag{61}$$

$$\operatorname{E} \{ (\mathbf{x}\_{\dot{\mathbf{m}}}^{(\dot{t})}(n))\_2 \ (\mathbf{x}\_{\dot{\mathbf{m}}}^{(\dot{t})}(n))\_1^H \ \operatorname{E} \{ (\mathbf{x}\_{\dot{\mathbf{m}}}^{(\dot{t})}(n))\_2 \ (\mathbf{x}\_{\dot{\mathbf{m}}}^{(\dot{t})}(n))\_2^H \} \tag{62}$$

is the covariance matrix and xðiÞ in ðnÞ ¼ v ðiÞ i siðnÞ þ nðiÞ ðnÞ is the quaternion involution of xinðnÞ. C ¼ ½VðiÞ i , VðiÞ s � and gH ¼ ½gH 1 , 0�, where g1 is given by Eq. (55). The solution of Eq. (60) is given by Van Trees [13]