Beamformer Based on Quaternion Processes

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 two-component electromagnetic (EM) vector-sensors is presented. Based on array ’ s quater- nion 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. The increase of information results in the improvement of the beamformer ’ s performance. Analyses in theory and simulation results verify the advantages of quaternion beamformers.


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][2][3][4][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 y c ¼ w H c x c where x c ¼ ½x 11 , x 12 , …x M1 , x M2 T is the observed vector of array; w c ¼ ½w 11 , w 12 , …w M1 , w M2 T is a complex weighted vector; the symbol (.) H denotes the complex conjugation transposition operator. Whereas the output of quaternion-based beamformer [3][4][5] is y h ¼ w Δ h x h ¼ y c þ j y e where x h ¼ ½x h1 , x h2 , …x hM T is the quaternion-valued, observed vector of array and x hm ¼ x m1 þ jx m2 ðm ¼ 1, …, MÞ. w h ¼ ½w h1 , w h2 , …w hM T is a quaternion-valued, weighted vector and w hm ¼ w m1 þ jw m2 ðm ¼ 1, …, MÞ. The symbol ð:Þ Δ denotes the quaternion conjugation transposition operator and j denotes an imaginary unit of quaternions. Comparing y h with y c , we can see that the output of quaternion-based beamformer has one more extra information y e than the output of complex 'long vector' beamformer. By employing this extra information y e , 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.

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 A quaternion variable x ∈ Q has two forms of representation. One form is x = x 1 + ix 2 + jx 3 + kx 4 , where x 1 , x 2 , x 3 and x 4 are real coefficients. It is referred to as the R-expansion of quaternions.
We call x 1 the real/scalar part of x, and it is denoted by RðxÞ. ix 2 + jx 3 + kx 4 is called the imaginary/vector part of x, and it is denoted by IðxÞ. We refer to x* = x 1 À ix 2 À jx 3 À kx 4 as the conjugate of x and jxj ¼ ðx 2 1 þ x 2 2 þ x 2 3 þ x 2 4 Þ 1=2 as the modulus of x. The other form is x = z 1 + jz 2 , where z 1 and z 2 are complex coefficients. It is called the C-expansion of quaternions or Cayley-Dickson. x Ã ¼ z Ã 1 À jz 2 is the conjugate of x and |x|=(|z 1 | 2 +|z 2 | 2 ) 1/2 is the modulus of x. Several properties of quaternions are discussed in Table 1 [10-12].

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 ÀM,…, À1,1,…,M from left to right.
Using x m1 (n) and x m2 (n) (m ¼ ÀM, …, M), a quaternion-valued series x m (n) can be constructed in the C-expansion of quaternions, as the output of the mth two-component vector-sensor: x m ðnÞ ¼ x m1 ðnÞ þ jx Àm2 ðnÞ ¼ q m ðθ, ϕÞ a 1 ðθ, ϕ, γ, ηÞ þ ja 2 ðθ, ϕ, γ, ηÞ sðnÞ ¼ q m ðθ, ϕÞPðθ, ϕ, γ, ηÞsðnÞ ð3Þ 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 x m1 (n) on scalar and i imaginary fields of a quaternion, and the complex series x m2 (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 where n m ðnÞ ¼ n m1 ðnÞ þ jn Àm2 ðnÞ: n m1 ðnÞ is the complex-valued additive noises recorded on first component of the mth vector-sensor and n Àm2 ðnÞ 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 σ 2 n . And it is assumed that n m ðnÞ and n n ðnÞ, where m 6 ¼ n, are uncorrelated.

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 (θ s , ϕ s ) and polarization parameters (γ s , η s ); the other is the interference characterized by its arrival angles (θ i , ϕ i ) and polarization parameters (γ i , η i ). 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 where nðnÞ ¼ ½n ÀM ðnÞ, ⋯, n M ðnÞ T denotes the quaternion-valued additive noise vector. v s ¼ qðθ s , ϕ s ÞPðθ s , ϕ s , γ s , η s Þ;v i ¼ qðθ i , ϕ i ÞPðθ i , ϕ i , γ i , η i Þ are the quaternion-valued steering vector associated with the desired signal and the interference, respectively, where qðθ τ , ϕ τ Þ ¼ ½q ÀM ðθ τ , ϕ τ Þ, ⋯, q M ðθ τ , ϕ τ Þ 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 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]: where R x ¼ EfxðnÞx Δ ðnÞg is the covariance matrix of the measurement vector. By using Lagrange multipliers, the solution of Eq. (7) is obtained, i.e., where λ is a real number. Based on the quaternion-valued gradient operator defined in Ref. [11], the following gradients need to be calculated: Let Eq. (9) is equal to zero, then Substituting Eq. (11) into Eq. (10), the weight vector of the QMVDR beamformer can be written as Substituting Eq. (12) into Eq. (6), the quaternion-valued output of the QMVDR beamformer is given by 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]) where the input signal-to-noise ratio (SNR) ξ s ¼ σ 2 s σ 2 n and the input interference-to-noise ratio

Interference and noise canceller (INC) based on QMVDR
Using the C-expansion of quaternions, y(n) can be written as where y 1 (n) and y 2 (n) are two complex-valued components of y(n), i.e., ð:Þ 1 and ð:Þ 2 denote, respectively, first and second complex-valued components of a quaternion. The expansion (16) highlights the fact that y 2 (n) does not include the desired signal, but include only the interference and noise. By employing y 2 (n), we can partly cancel the interference and noise component in y 1 (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 where w c is a complex weight, which can be given by the Wiener-Hoft equation where r y 2 y 1 ¼ Efy 2 ðnÞy Ã 1 ðnÞg and R y 2 ¼ Efy 2 ðnÞy Ã 2 ðnÞg. Following the proof given in Appendix 2 of Ref. [4], we have If ξ i is very small, w c is approximately equal to 0. Whereas, if ξ i is very large, w c is approximately equal to In this case, the interference can be cancelled. From Eq. (17), the SINR in the complex output y s (n) is given by where we define κ ¼ ξ s kwk 2 and κ i ¼ The proof is in Appendix 3 of Ref. [4]. Clearly, SINR y s increases with an increase in κ but decreases with an increase in κ i .
Next, we show the effect of sources parameters on κ and κ i . Following the proof given in Appendix 4 of Ref. [4], we have Obviously, the gain β ≥ 1 because of 0 ≤ ε < 1. Then, κ ≥ 2SINR y .
From Eq. (21), κ depends mainly on separation between the DOAs of the desired signal and interference (i.e., jq H s q i j 2 ). The dependencies of κ on jq H s q i j 2 are shown in the following consequences: In the case that M is constant, β increases with an increase in ξ i and jP i j 2 , but κ increases with an increase in ξ s and jP s j 2 .

When the separation between the DOAs of the desired signal and interference increases,
jq H s q i j decreases. This results in the reduction of μ. Then, the both β and κ also reduce.
Along with an increase in ξ i , the value of jq H s q i j, which results in a minimum value of κ, tends to 2M. Thus, the minimum value of κ tends to 0. Afterwards, κ will increase with a decrease in jq H s q i j.

When jq
In addition, κ depends also on the input INR ξ i , the array's element number 2M and the interference response jP i j 2 and the desired signal response jP s j 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 θ s and ϕ s for several values of Figure 3(a) and θ s ¼ 90 in Figure 3(b), where a linear symmetric antenna array is used with four (i.e., M = 2) two-component vector-sensor.
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 ξ s = 1 and the interference have the same polarization parameters Simulation results in Figure 3 are in agreement with the previous discussions. From Figure 3(a), it is seen that jq H s q i j ≈ 4 and κ ≈ 3 as θ s ≈ 90 . The cause of this phenomenon is that jP s j 2 ¼ sin 2 ϕ s þ cos 2 ϕ s cos 2 θ s for the two-component vector-sensor used in this example. When θ s ≈ 90 , jP s j 2 ≈ 0:75. In cases of jq H s q i j ¼ 4, κ ≈ 3. Along θ s is away from 90 , jq H s q i j decreases. This results in the reduction of κ. From Figure 3(b), it is seen that jq H s q i j ≈ 4 and κ ≈ 3 as ϕ s ≈ 60 . This is as same as Figure 3(a). In addition, it is noted that jq H s q i j ≈ 0 and κ ≈ 4 as ϕ s ≈ 100 . The cause of this phenomenon is that jP s j 2 ≈ 1 in cases of ϕ s ≈ 100 and κ ≈ 4 in cases of jq H s q i j ¼ 0 . When ϕ s ¼ 0 or ϕ s ¼ 180 , jP s j 2 ¼ 0, then, κ ¼ 0.
Since the output interference-plus-noise power Following the proof given in Appendix 1 of Ref. [4], κ i is further written as Obviously, In addition, κ i depends also on the array's element number 2M and the interference response jP i j 2 .  Following the proof given in Appendix 5 of Ref. [4], we have From Eqs. (25) and (26), ðw Δ v i Þ 1 depends not only on the separation between the DOAs of the desired signal and interference (i.e., jq H s q i j), but also on the difference between the polarizations of the desired signal and interference (i.e., ðP s Ã P i Þ 1 ). The dependencies of ðw Δ v i Þ 1 and κ i on jq H s q i j and ðP s Ã P i Þ 1 ) are shown in the following consequences: Whereas if the polarizations of the desired signal are an orthogonal with that of interfer- 2. When the separation between the DOAs of the desired signal and interference increases, jq H s q i j decreases. This results in the reduction of ðw Δ v i Þ 1 . In addition, the increase in the difference between the polarizations of the desired signal and interference also results in the reduction of ðw Δ v i Þ 1 . Thus, κ i reduces.
Finally, we analyse the performance of the INC. From Eq. (20) and above analysis, we can obtain the following consequences: 1. When jq H s q i j ¼ 0, κ ¼ 2Mξ s jP s j 2 and κ i ¼ 0. This implies that the separation between the DOA's of the desired signal and interference reaches to maximum. In this case, we can obtain the maximum value of SINR y s , i.e., SINR y s max ¼ 2Mξ s jP s j 2 . Further, jq H s q i j increases with a decrease in the separation in DOA. Thus, SINR y s will reduce due to the decrease in κ and the increase in κ i . When jq H s q i j ¼ 2M , κ reaches to a minimum value. In this case, we can obtain the minimum value of SINR y s if κ i reaches to a maximum value.
2. When ðP s Ã P i Þ 1 ¼ 0, κ i ¼ 0. In this case, we can obtain the maximum value of SINR y s , i.e., SINR y s max ¼ 2Mξ s jP s j 2 , if jq H s q i j ¼ 2M. This implies that even though no separation between the DOAs of the desired signal and interference, SINR y s can reach to maximum by using the orthogonally between the polarizations of the desired signal and interference. Further, jðP s Ã P i Þ 1 j increases with a decrease in the difference in polarizations. Thus, SINR y s will reduce due to the increase in κ i .

3.
When ξ i ¼ 0 (i.e., in the absence of interference), κ ¼ 2Mξ s jP s j 2 and κ i ¼ 0. In this case, we can obtain the maximum value of SINR y s , i.e., SINR y s max ¼ 2Mξ s jP s j 2 . Further, SINR y s decreases with an increase in ξ i . In the presence of a strong interference (i.e., ξ À1 i ≈ 0), SINR y s can be approximated as where κ ≈ ξ s jP s j 2 2M À jq H s q i j 2 2M and κ i ¼ jðvs Δ viÞ 1 j 2 jvs Δ vij 2 . Expression (27) highlights the fact that in the presence of a strong interference, SINR y s ≈ 0 in the case of no separation between the DOAs of the desired signal and interference (i.e., q s ¼ q i ) or no difference between the polarizations of the desired signal and interference (i.e., P s ¼ P i ). This implies that the INC fails.

4.
When the vector-sensor number 2M in array increases, κ increases and κ i decreases. Thus, SINR y s is an increasing function of 2M. Since κ increases with an increase in jP s j 2 , but κ i decreases with an increase in jP s j 2 , SINR y s is an increasing function of jP s j 2 . Further, SINR y s is a decreasing function of jP i j 2 because κ decreases with an increase in jP i j 2 .

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 x ðiÞ ¼ ixi À1 ¼ ixi Ã ¼ Àixi and it represents the reflection of x over the plane spanned by {1, i}. A quaternion vector x is C i -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 C i -proper quaternion vector x can be written as augmented quaternion vector and Rx ,x ¼ Efxx Δ g is the semi-augmented covariance matrix of quaternion vector x. In comparison with the semi-augmented covariance matrix Rx ,x , the augmented covariance matrix R x , x has not more extra information. In other words, the fullwidely linear processing is equivalent to the semi-widely linear processing in handling the C i -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 where W, G, H and F denote the quaternion-valued weight vectors. x ðiÞ ðnÞ, x ðjÞ ðnÞ and x ðkÞ 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 C i -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] yðnÞ ¼ W Δ xðnÞ þ G Δ x ðiÞ ðnÞ ð 30Þ where x ðiÞ ðnÞ is given by where y 1 and y 2 denote, respectively, first and second complex-valued components of a quaternion y. Thus, the QSWL beamformer has two complex-valued output series y 1 ðnÞ and y 2 ðnÞ in the planes spanned by {1, i}, where y 1 ðnÞ ¼ ðW Δ xðnÞÞ 1 þ ðG Δ x ðiÞ ðnÞÞ 1 Þ and y 2 ðnÞ ¼ ðW Δ xðnÞÞ 2 þ ðG Δ x ðiÞ ðnÞÞ 2 Þ. Since the complex 'long vector' beamformers have only one complex-valued output series y 1 ðnÞ, 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 x ðiÞ ðnÞ, so that the QSWL beamformer with different characteristics may be obtained by designing two weight vectors W and G under some different criterions.

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 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., y 1 (n) and y 2 (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 GSC ðnÞ ¼ ðyðnÞÞ 1 ¼ y w ðnÞ À y g ðnÞ ð 33Þ where y w ðnÞ is the complex-valued output of the first-stage beamformer, i.e., y w ðnÞ ¼ ðW Δ xðnÞÞ 1 ; y g ðnÞ is the complex-valued output of the second-stage beamformer, i.e., y g ðnÞ ¼ ðG Δ x ðiÞ ðnÞÞ 1

The first-stage beamformer
From Eq. (5), we have In the first-stage beamformer, we attempt to minimize the interference-plus-noise energy in y w ðnÞ, subject to the constraint ðw Δ v s Þ 1 ¼ 1.
Since the Cayley-Dickson representations of W, v s , v i and n(n) are, respectively,  where W ¼ Then, W can be derived by solving the following constrained optimization problem: where is the covariance matrix and x in ðnÞ ¼ v i s i ðnÞ 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 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]) where ξ i denotes the input interference-to-noise ratio (INR). Moreover, the quaternion-valued optimal weight vector W o may be given by where J 1 ¼ ½I 2MÂ2M , 0 2MÂ2M and J 2 ¼ ½0 2MÂ2M , I 2MÂ2M are two selection matrices. It is noted that in some applications, such as Radar, R in may be estimated in intervals of no transmitted signal. But, R in is not obtained in other applications, such as Communications. In these applications, we may replace R in by R x , where is the covariance matrix and x(n). When the distortionless constraint is perfectly matched with the desired signal, the weight vector W o is identical in both R in and R x .
By using the optimal weight vector W o , the complex output of first-stage beamformer can be given by 6.2. The second-stage beamformer From Eq. (31), we have In the second-stage beamformer, we attempt to minimize the noise energy in y g ðnÞ, subject to In the following, two schemes are presented to implement this aim.

The Scheme 1, i.e., combined QPMC and MVDR
Let ¼ w qs w MV , where w qs is a quaternion-valued diagonal weight matrix and w MV is a complex weight vector. In this scheme, the first is to achieve the constraint ðG Δ v ðiÞ s Þ 1 ¼ 0 by designing w qs , which is referred to quaternion polarization matched cancellation (QPMC); the second is to minimize the noise energy in y g ðnÞ subject to the constraint ðG Δ v ðiÞ i Þ 1 ¼ ðW Δ o v i Þ 1 by designing w MV , which is referred to MVDR.
Let w qs ¼ diag fw qs ðÀMÞ, ⋯, w qs ðMÞg; then we have where superscript (.)* denotes the quaternion conjugate operator. From Eq. (48) and the constraint Þ, this constraint is satisfied. Thus, we can obtain where diag fq s g ¼ diag fq ÀM ðθ s , ϕ s Þ, ⋯, q M ðθ s , ϕ s Þg . In the constraint ðG Δ v ðiÞ s Þ 1 ¼ 0, we insert G ¼ w qs w MV into (47). Thus, y g ðnÞ can be rewritten as Then, w MV can be derived by solving the following constrained optimization problem: where R qs ¼ Efðw Δ qs x ðiÞ ðnÞÞ 1 w Δ qs x ðiÞ ðnÞ H 1 g is the covariance matrix andṼ i ¼ ðw Δ qs v ðiÞ i Þ 1 . The solution of this constrained optimization problem is obtained by using Lagrange multipliers, i.e., If the desired signal and interference are uncorrelated with the additive noise, w MV can be written in the simple form (the proof is in Appendix B of Ref. [5]) where R (.) denotes the real part of a complex number. μ is given by Eq. (43).
and n ðiÞ ðnÞ ¼ n 1 ðnÞ À j n 2 ðnÞ, we have where If the desired signal and interference are uncorrelated with the additive noise, G can be written in the simple form (the proof is in Appendix C of Ref. [5]) μ is given by (43). Moreover, the quaternion-valued optimal weight vector G o may be given by where J 1 ¼ ½I 2MÂ2M , 0 2MÂ2M and J 2 ¼ ½0 2MÂ2M , I 2MÂ2M are two selection matrices.
By using the optimal weight vector G o , the complex output of second-stage beamformer can be given by Thus, the complex output of QSWL GSC may be rewritten as From above equation, we see that the interference component is completely cancelled in the output y GSC ðnÞ.

The performance analysis
Since the QSWL GSC can totally remove the interference, its output signal-to-interference ratio (SIR) tends to infinite. Thus, we focus our attention on the output signal-to-noise ratio (SNR) and array's gain. Then, ρ n can be written as When the combined QPMC and MVDR are adopted in the second-stage beamformer, ρ n can be written in the simple form (the proof is in Appendix D of Ref. [5]) À 2MjP i j 2 , κ is given by Eq. (54). From Eq. (67), the expression of output SNR and array's gain A q may be written as where ξ s denotes the input signal-to-noise ratio (SNR) and ε is given by Eq. (43).
When the LCMV is adopted in the second-stage beamformer, ρ n can be written in the simple form (the proof is in Appendix E of Ref. [5]) where λ l ¼ 2Mξ À2 i jPsj 2 ν À 2MjP i j 2 , ν is given by Eq. (64). Then, the expression of output SNR and array's gain A l may be written as From Eqs. (71), (72), (74) and (75), we can see that the output SNR and array's gain depend on not only separation between the DOA's of the desired signal and interference (i.e., jq H i q s j), but also difference between the polarizations of the desired signal and interference (i.e., jðP Δ i P s Þ 1 j). The dependencies of them on jq H i q s j and jðP i Δ P s Þ 1 j are shown in following consequences:

1.
When jq H i q s j ¼ 0, the separation between the DOAs of the desired signal and interference reaches to maximum. In this case, A q ¼ A l ¼ 2MjP s j 2 . Further, jq H i q s j increases with a decrease of the DOA's separation. Thus, the array's gain A q and A l will reduce if jP s j 2 is a constant. When q i ¼ q s , jq H i q s j ¼ 2M. This implies that there is no separation between the DOAs of the desired signal and interference. In this case, the array's gain is given by Further, P i ¼ P s if γ s ¼ γ i and η s ¼ η i . Thus, the array's gain A q ¼ A l ¼ 0 due to λ ¼ ∞. This implies that the QSWL GSC fails.

When jðP
In the cases that θ s ¼ θ i 6 ¼ 0 and ϕ s ¼ ϕ i 6 ¼ 0 (i.e., This implies that even though there is no separation between the DOAs of the desired signal and interference, the array's gain can also reach to 2MjP s j 2 by using the orthogonality between the polarizations of the desired signal and interference. Further, the array's gain decreases with an increase of jðP i Δ P s Þ 1 j if jP s j 2 is a constant. When P i ¼ P s , jðP i Δ P s Þ 1 j ¼ jP s j 2 . This implies that there is no difference between the polarizations of the desired signal and interference. But, the array's gain is not equal to zero if q i 6 ¼ q s .
In addition, the output SNR and array's gain depend also on the input INR ξ i , the array's element number 2M, the interference response's power jP i j 2 and the desired signal response's power jP s j 2 .

Monte Carlo simulations
In this section, we investigate the performance of the proposed beamformers by two experiments. More results of simulations were shown in Refs. [2,4,5].

Experiment 1: the performance of QMVDR beamformer
In practice, if there is a misalignment between the desired signal's DOA and the look direction, the SINR of the complex MVDR beamformer degrades in the case of a scalar vector array [13].
In this experiment, we investigate the robustness of the beamformer against the DOA mismatch.
We consider each two-component vector-sensor has two orthogonal magnetic loop co-aligned along the x-axis, and assume that M = 1. The result is the average of the output SINR obtained by 1000 Monte Carlo runs. To compare the performance, the complex 'long vector' MVDR (CLVMVDR) [13], QMVDR and its INC beamformers are included in simulation results. For the CLVMVDR and QMVDR beamformers, the SINR is in general expressed in the form of logarithm. In order to be identical with the SINR of CLVMVDR and QMVDR beamformers, we define SINR ¼ 10log in this experiment and assume that input SINR ¼ 0 dB. Figure 5 displays the output SINR as the function of the DOA error of the desired signal where the DOA error is between À8 and 8 . In the case of θ s ¼ 0 , , the polarization of the desired signal and interference is identical, but the DOA is not identical), simulation results for two different sample sizes N = 20 and N = 500 are given in Figure 5(a) and (b), respectively. From Figure 5, it is seen that the output SINR behaviour is different with different values of sample size N. In case of small N, as shown in Figure 5(a), the INC and the QMVDR have a better robustness than the CLVMVDR. But the output SINR of the QMVDR and CLVMVDR is more than that of the INC. In case of large N, as shown in Figure 5(b), the robustness against the DOA mismatch is almost identical for three beamformers. But the INC has the largest output SINR in three beamformers.

Experiment 2: the performance of QSWL beamformer
In the second experiment, we illustrate the performance of the proposed QSWL GSC in the presence of a single interference. We assume M = 6, ϕ s ¼ ϕ i ¼ 60 ; γ s ¼ γ i ¼ 30 , η s ¼ η i ¼ 30 , and that the covariance matrix R x , instead of R in , is available. Figure 6 displays the power patterns for three values of jΔθj : 60 , 20 and 10 , where θ s ¼ jΔθj, θ i ¼ 0 . From Figure 6, it is seen that three beamformers steer almost a zero towards the interference's DOA (located at 0 ) in all cases. When jΔθj decreases, the main-lobe of the QSWL GSC points almost to the source location, but the main-lobe of the complex 'long vector' LCMV (CLCMV) is away from  the source location. This implies that the QSWL GSC outperforms obviously CLCMV as the desired signal moves towards the interference. In addition, the side-lobes are amplified with a decrease of jΔθj. These side-lobes lead the beamformer to capture the white noise, which spans the whole space, so that the performance of beamformer degrades.

Conclusion
The problem of beamformer based on quaternion processes is considered in this chapter. The quaternion beamformers has more information than the complex 'long vector' beamformer. The increase of information results in the improvement of the beamformer's performance. Analyses in theory and simulation results verify the advantages of quaternion beamformers.