Open access peer-reviewed chapter

# Beamformer Based on Quaternion Processes

Written By

Jian-wu Tao and Wen-xiu Chang

Submitted: September 8th, 2016 Reviewed: February 13th, 2017 Published: May 10th, 2017

DOI: 10.5772/67859

From the Edited Volume

## Antenna Arrays and Beam-formation

Edited by Modar Shbat

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## 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 two-component 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) vector-sensor array [15]. 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=wcHxc  where xc=[x11,x12,xM1,xM2]T is the observed vector of array; wc=[w11,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 [35] is yh=whΔxh=yc+j ye where xh=[xh1,xh2,xhM]T  is the quaternion-valued, observed vector of array and xhm=xm1+jxm2 (m=1,,M). wh=[wh1,wh2,whM]T  is a quaternion-valued, weighted vector and whm=wm1+jwm2 (m=1,,M). The symbol (.)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

i2=j2=k2=1;ij=k=ji;jk=i=kj;ki=j=ikE1

A quaternion variable xQ 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 R(x). ix2 + jx3 + kx4 is called the imaginary/vector part of x, and it is denoted by I(x). We refer to x* = x1 ix2 jx3kx4 as the conjugate of x and |x|=(x12+x22+x32+x42)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*=z1*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 [1012].

x and y denotes two quaternions
ConjugationNorm, noted .Inverse, noted x−1Multiplication
xx*=x*x=|x|2x=0 if and only if x=0If x0, x−1= x*/|x|2a:Rax = xa
|x|=|x*|xy=yx=xycC jc = c*j, jcj*=c*
(xy)*=y*x*x+yx+yq Qxqqx

### 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]

[xm1(n)xm2(n)]=[a1(θ,φ,γ,η)a2(θ,φ,γ,η)]qm(θ,φ)s(n)E2

where 0θπ and 0φ<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. 0γ<π/2 represents the auxiliary polarization angle, and πη<π signifies the polarization phase difference. a1(θ,φ,γ,η) and a2(θ,φ,γ,η) 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 a1(θ,φ,γ,η)= eiηcosφcosθsinγsinφcosγ and a2(θ,φ,γ,η)= eiηsinφsinγcosφcosθcosγ [7] or two magnetic loops co-aligned along the x-axis, where a1(θ,φ,γ,η)=eiηsinφsinγcosφcosθcosγ and a2(θ,φ,γ,η)= eiηcosφsinγsinφcosθcosγ [6]. qm(θ,φ) is the spatial phase factor describing wave-field propagation along the array, and qm(θ,φ)=qm*(θ,φ) due to the 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=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:

xm(n)=xm1(n)+jxm2(n)=qm(θ,φ)(a1(θ,φ,γ,η)+ja2(θ,φ,γ,η))s(n)=qm(θ,φ)P(θ,φ,γ,η)s(n)E3

where j denotes an imaginary unit of quaternions. P(θ,φ,γ,η) 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

xm(n)=qm(θ,φ)P(θ,φ,γ,η)s(n)+nm(n)E4

where nm(n)=nm1(n)+jnm2(n). nm1(n) is the complex-valued additive noises recorded on first component of the mth vector-sensor and nm2(n)  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 σn2. And it is assumed that nm(n) and nn(n), where mn, 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

x(n)=[xM(n), ,xM(n)]Tvsss(n)+visi(n)+n(n)E5

where n(n)=[nM(n), , nM(n)]T denotes the quaternion-valued additive noise vector. vs=q(θs, φs)P(θs, φs,γs, ηs),vi=q(θi, φi)P(θi, φi,γi, ηi are the quaternion-valued steering vector associated with the desired signal and the interference, respectively, where q(θτ, φτ)=[qM(θτ, φτ), , qM(θτ, φτ)]T (τ=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

y(n)=wΔx(n) E6

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

J(w)=min{wΔRxw};subject towΔvs=1E7

where Rx=E{x(n)xΔ(n)} is the covariance matrix of the measurement vector. By using Lagrange multipliers, the solution of Eq. (7) is obtained, i.e.,

J=wΔRxw+λwΔvsE8

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

JwΔ=Rxw+ λvsE9

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

w=λRx1vsE10

Since wΔvs=λvsΔRx1vs=1, we have,

λ= 1vsΔRx1vsE11

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

w= Rx1vsvsΔRx1vsE12

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

y(n)=ss(n) + wΔvisi(n) + wΔn(n)E13

where wΔvs=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])

SINRy=ξs|Ps|2(M|Pi|2|qsHqi|24ξi1+4M|Pi|2)E14

where the input signal-to-noise ratio (SNR) ξs=σs2σn2  and the input interference-to-noise ratio (INR) ξi=σi2σn2.

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

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

y(n)=y1(n)+j y2(n)E15

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

y1(n)=ss(n) + (wΔvi)1si(n) + (wΔn(n))1; y2(n)=(wΔvi)2si(n) + (wΔn(n))2E16

(.)1 and (.)2 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

ys(n)=y1(n)wc* y2(n)=ss(n)+((wΔvi)1wc*(wΔvi)2)si(n)+((wΔn(n))1wc*(wΔn(n))2)=ss(n) + wisi(n) + ε(n)E17

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

wc=ry2y1Ry2E18

where ry2y1=E{y2(n)y1*(n)} and Ry2=E{y2(n)y2*(n)}. Following the proof given in Appendix 2 of Ref. [4], we have

wc=(wΔvi)2(wΔvi)1*ξi|(wΔvi)2|2ξi+w2E19

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

SINRys=ξsw2(1σi2|(wΔvi)1|2σn2w2+σi2|wΔvi|2)=κ(1κi)E20

where we define κ=ξsw2 and κi=σi2|(wΔvi)1|2σn2w2+σi2|wΔvi|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

κ=ξs|Ps|2(2M|Pi|2|qsHqi|22ξi1+2M|Pi|2)2Mμ2Mμ(1+ε)=2SINRyβE21

where

β=2Mμ2Mμ(1+ε); μ=|Pi|2|qsHqi|22ξi1+2M|Pi|2; ε=ξi1ξi1+M|Pi|2E22

Obviously, the gain β ≥ 1 because of 0 ≤ ε < 1. Then, κ2SINRy.

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

1. When qs = qi (no separation in DOA), μ=2|Pi|2M2ξi1+M|Pi|2 because of qsHqi=2M. Then, β=1+Mξi|Pi|2 and κ=2Mξi|Pi|2. In the case that M is constant, β increases with an increase in ξi and |Pi|2, but κ increases with an increase in ξs and |Ps|2.

2. When the separation between the DOAs of the desired signal and interference increases, |qsHqi| decreases. This results in the reduction of μ. Then, the both β and κ also reduce. When μ=2|Pi|2M22ξi1+M|Pi|2 (i.e., |qsHqi|=2M1+Mξi|Pi|22+Mξi|Pi|2), κ=2Mξs|Ps|2(4(1+Mξi|Pi|2)(2+Mξi|Pi|2)2)  reaches to a minimum value. In this case, if |qsHqi|=2M and κ=2Mξs|Ps|2. Along with an increase in ξi, the value of |qsHqi|, 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 |qsHqi|.

3. When |qsHqi|=0 (i.e., μ = 0), β = 1 and κ=2Mξs|Ps|2. In this case, κ=2SINRy.

In addition, κ depends also on the input INR ξi, the array’s element number 2M and the interference response |Pi|2 and the desired signal response |Ps|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=90° φi=60°; φs=60° in Figure 3(a) and θs=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 ξs = 1 and the interference have the same polarization parameters (γs=γi=30°,ηs=ηi=30°). Simulation results in Figure 3 are in agreement with the previous discussions. From Figure 3(a), it is seen that |qsHqi|4 and κ ≈ 3 as θs90°. The cause of this phenomenon is that |Ps|2=sin2φs+ cos2φs cos2θs for the two-component vector-sensor used in this example. When θs90°, |Ps|20.75. In cases of |qsHqi|=4, κ3. Along θs is away from 90°, |qsHqi|  decreases. This results in the reduction of κ. From Figure 3(b), it is seen that |qsHqi|4 and κ3 as φs60°. This is as same as Figure 3(a). In addition, it is noted that |qsHqi|0 and κ4 as φs100°. The cause of this phenomenon is that |Ps|21 in cases of φs100° and κ4 in cases of |qsHqi|=0 . When φs=0° or φs=180°, |Ps|2=0, then, κ=0.

Since the output interference-plus-noise power wΔRinw=σn2w2+σi2|wΔvi|2, we have

κi=σi2|(wΔvi)1|2σn2w2+σi2|wΔvi|2=σi2|(wΔvi)1|2wΔRinwE23

where 0κi1 because of 0|(wΔvi)1|2|wΔvi|2. Following the proof given in Appendix 1 of Ref. [4], κi is further written as

κi=ξi|Ps|2(M|Pi|2|qsHqi|24ξi1+4M|Pi|2)|(wΔvi)1|2E24

Obviously, κi = 0 at |(wΔvi)1|=0 or ξi=0. And κi increases with an increase in |(wΔvi)1|2. In addition, κi depends also on the array’s element number 2M and the interference response |Pi|2.

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

(wΔvi)1=α(vsΔvi)1=α|qsHqi|(Ps*Pi)1E25

where

α=ξi1|Ps|2(24Mξi1+|Pi|2(4M2|qsHqi|2))E26

From Eqs. (25) and (26), (wΔvi)1 depends not only on the separation between the DOAs of the desired signal and interference (i.e., |qsHqi|), but also on the difference between the polarizations of the desired signal and interference (i.e., (Ps*Pi)1). The dependencies of (wΔvi)1 and κi on |qsHqi| and (Ps*Pi)1) are shown in the following consequences:

1. When qs=qi (i.e., no separation in DOA), (wΔvi)1=Ps1*Pi1+Ps2*Pi2|Ps|2 because of |qsHqi|=2M. At the same time, if Ps=Pi (i.e., no difference in polarization), (wΔvi)1=1  because of Ps1*Pi1+Ps2*Pi2=|Ps|2 So, κi=Mξi|Ps|2(1M|Pi|2ξi1+M|Pi|2)  reaches to a maximum value. Whereas if the polarizations of the desired signal are an orthogonal with that of interference (i.e., γs+γi=π2,ηsηi=π), (wΔvi)1=0 because of (Ps*Pi)1=0. So, κi reaches to a minimum value.

2. When the separation between the DOAs of the desired signal and interference increases, |qsHqi| decreases. This results in the reduction of (wΔvi)1. In addition, the increase in the difference between the polarizations of the desired signal and interference also results in the reduction of (wΔvi)1. Thus, κi reduces.

3. When |qsHqi|=0 or (Ps*Pi)1=0, (wΔvi)1=0 . Thus, κi=0. In the absence of the interference (i.e., ξi=0), κi=0.

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

1. When |qsHqi|=0, κ=2Mξs|Ps|2 and κi=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 SINRys, i.e., SINRysmax=2Mξs|Ps|2. Further, |qsHqi| increases with a decrease in the separation in DOA. Thus, SINRys will reduce due to the decrease in κ and the increase in κi. When |qsHqi|=2M1+Mξi|Pi|22+Mξi|Pi|2), κ reaches to a minimum value. In this case, we can obtain the minimum value of SINRys if κi reaches to a maximum value.

2. When (Ps*Pi)1=0, κi=0. In this case, we can obtain the maximum value of SINRys, i.e., SINRysmax=2Mξs|Ps|2, if |qsHqi|=2M. This implies that even though no separation between the DOAs of the desired signal and interference, SINRys  can reach to maximum by using the orthogonally between the polarizations of the desired signal and interference. Further, |(Ps*Pi)1| increases with a decrease in the difference in polarizations. Thus, SINRys will reduce due to the increase in κi.

3. When ξi=0 (i.e., in the absence of interference), κ=2Mξs|Ps|2 and κi=0. In this case, we can obtain the maximum value of SINRys, i.e., SINRysmax=2Mξs|Ps|2. Further, SINRys decreases with an increase in ξi. In the presence of a strong interference (i.e., ξi10), SINRys can be approximated as

SINRysξs|Ps|2(2M|qsHqi|22M)(1|(vsΔvi)1|2|vsΔvi|2)E27

where κξs|Ps|2(2M|qsHqi|22M) and κi=|(vsΔvi)1|2|vsΔvi|2. Expression (27) highlights the fact that in the presence of a strong interference, SINRys0 in the case of no separation between the DOAs of the desired signal and interference (i.e., qs=qi) or no difference between the polarizations of the desired signal and interference (i.e., Ps=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 |Ps|2, but κi decreases with an increase in |Ps|2, SINRys is an increasing function of |Ps|2. Further, SINRys is a decreasing function of |Pi|2 because κ decreases with an increase in |Pi|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 x(i)=ixi1=ixi*=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

Rx¯,x¯=E{x¯x¯Δ}=[Rx˜,x˜00Rx˜,x˜j]E28

where x¯=[xT,x(i)T,x(j)T,x(k)T]T is the augmented quaternion vector; x˜=[xT,x(i)T]T is the semi-augmented quaternion vector and Rx˜,x˜=E{x˜x˜Δ} is the semi-augmented covariance matrix of quaternion vector x. In comparison with the semi-augmented covariance matrix Rx˜,x˜, the augmented covariance matrix Rx¯,x¯ has not more extra information. In other words, the full-widely 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

y(n)=wΔx(n)+ GΔx(i)(n)+HΔx(j)(n)+FΔx(k)(n)E29

where W, G, H and F denote the quaternion-valued weight vectors. x(i)(n), x(j)(n) and x(k)(n) 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]

y(n)=WΔx(n)+ GΔx(i)(n)E30

where x(i)(n) is given by

x(i)(n)=ix(n)i=vs(i)ss(n)+vi(i)si(n)+n(i)(n)E31

Moreover, we can write the quaternion-valued output series y(n) in the following Cayley-Dickson representation

y(n)=((WΔx(n))1+(GΔx(i)(n))1)+j((WΔx(n))2+(GΔx(i)(n))2)=y1(n)+jy2(n)E32

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 y1(n) and y2(n) in the planes spanned by {1, i}, where y1(n)=(WΔx(n))1+(GΔx(i)(n))1)  and y2(n)=(WΔx(n))2+(GΔx(i)(n))2). Since the complex ‘long vector’ beamformers have only one complex-valued output series y1(n), 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(i)(n), 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 employing the output of second-stage beamformer to cancel interferences in the output of first-stage 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 first-stage 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

yGSC(n)=(y(n))1=yw(n) yg(n) E33

where yw(n) is the complex-valued output of the first-stage beamformer, i.e., yw(n)=(WΔx(n))1; yg(n) is the complex-valued output of the second-stage beamformer, i.e., yg(n)=(GΔx(i)(n))1

### 6.1. The first-stage beamformer

From Eq. (5), we have

yw(n)=(WΔvs)1ss(n) + (WΔvi)1 si(n)+(WΔn(n))1E34

In the first-stage beamformer, we attempt to minimize the interference-plus-noise energy in yw(n), subject to the constraint (wΔvs)1=1.

Since the Cayley-Dickson representations of W, vs, vi and n(n) are, respectively, W = W1+j W2, vs=vs1+j vs2, vi=vi1+j vi2 and n(n)=n1(n)+j n2(n), we have

(WΔvs)1=W1Hvs1+W2Hvs2=W¯HV¯sE35
(WΔvi)1=W1Hvi1+W2Hvi2=W¯HV¯iE36
(WΔn(n))1=W1Hn1(n)+W2Hn2(n)=W¯HN¯(n)E37

where W¯=[W1W2],V¯s=[vs1vs2], V¯i=[vi1vi2], N¯(n)=[n1(n)n2(n)] . Superscript (.)H denotes the complex conjugate and transpose operator. Thus, Eq. (34) can be rewritten as

yw(n)=W¯HV¯sss(n)+W¯HV¯i si(n)+W¯HN¯(n)E38

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

J(W¯)=min{W¯HRinW¯};subject toW¯HV¯s=1E39

where

Rin=[E{(xin(n))1 (xin(n))1H}E{(xin(n))1 (xin(n))2H}E{(xin(n))2 (xin(n))1H}E{(xin(n))2 (xin(n))2H}]E40

is the covariance matrix and xin(n)=visi(n)+n(n)  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

W¯= Rin1 V¯sV¯sHRin1 V¯sE41

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])

W¯=εV¯s(PiΔPs)1qiHqsV¯iμE42

where

μ=2M|Ps|2ε|(PiΔPs)1|2|qiHqs|2 ; ε=ξi1+ 2M|Pi|2 E43

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

Wo=J1 W¯+j J2 W¯E44

where J1=[I2M×2M, 02M×2M] and J2=[02M×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

Rx=[E{(x(n))1 (x(n))1H}E{(x(n))1 (x(n))2H}E{(x(n))2 (x(n))1H}E{(x(n))2 (x(n))2H}]E45

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

yw(n)=ss(n)+(WoΔvi)1 si(n)+(WoΔn(n))1E46

### 6.2. The second-stage beamformer

From Eq. (31), we have

yg(n)=(GΔvs(i))1Ss(n)+(GΔvi(i))1 Si(n)+(GΔn(i)(n))1E47

In the second-stage beamformer, we attempt to minimize the noise energy in yg(n), subject to the constraints (GΔvS(i))1=0 and (GΔvi(i))1=(WoΔvi)1 . In the following, two schemes are presented to implement this aim.

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

Let =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 (GΔvs(i))1=0 by designing wqs, which is referred to quaternion polarization matched cancellation (QPMC); the second is to minimize the noise energy in yg(n) subject to the constraint (GΔvi(i))1=(WoΔvi)1  by designing wMV, which is referred to MVDR.

Let wqs=diag {wqs(M), ,wqs(M)}; then we have

GΔvs(i)=wMVHwqsΔvs(i)=wMVH[wqs*(M)qM(θs, φs)Ps(i)wqs*(M)qM(θs, φs)Ps(i)]E48

where superscript (.)* denotes the quaternion conjugate operator. From Eq. (48) and the constraint (GΔvs(i))1=0, we have the constraint (wqs*(m)qm(θs, φs)Ps(i))1=0, where m={M, ,M}. When wqs(m)=qm(θs, φs) (as2*+j as1*), this constraint is satisfied. Thus, we can obtain

wqs=diag {qs} (as2*+j as1*)E49

where diag {qs}=diag {qM(θs, φs),,qM(θs, φs)} . In the constraint (GΔvs(i))1=0, we insert G=wqswMV into (47). Thus, yg(n) can be rewritten as

yg(n)=wMVH(wqsΔvi(i))1 si(n)+wMVH(wqsΔn(i)(n))1E50

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

J(wMV)=min{wMVHRqswMV};subject towMVH V˜i=W¯HV¯iE51

where Rqs=E{(wqsΔx(i)(n))1 (wqsΔx(i)(n))1H}  is the covariance matrix and V˜i=(wqsΔvi(i))1 . The solution of this constrained optimization problem is obtained by using Lagrange multipliers, i.e.,

wMV=Rqs1 V˜iV˜iHRqs1V˜iV¯iHW¯E52

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])

wMV= g1κV˜iE53

where

κ=V˜iHV˜i=2M(|as2|2|ai1|2+|as1|2|ai2|2)2R(as1as2*ai2ai1*(qi2)Hqs2) E54
g1=(W¯HV¯i)H=ξi1(PiΔPs)1qiHqsμ E55

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, vs(i), vi(i) and n(i)(n) are, respectively, G=G1+j G2, vs(i)=vs1jvs2, vi(i)=vi1j vi2 and n(i)(n)=n1(n)j n2(n), we have

(GΔvs(i))1=G1Hvs1G2Hvs2=G ¯HV¯s(i)E56
(GΔvi(i))1=G1Hvi1 G2Hvi2=G ¯HV¯i(i)E57
(GΔn(i)(n))1=G1Hn1(n) G2Hn2(n)=G ¯HN¯(i)(n)E58

where G ¯=[G1G2];V¯s(i)=[vs1vs2]; V¯i(i)=[vi1vi2]; N¯(i)(n)=[n1(n)n2(n)]

Thus, Eq. (47) can be rewritten as

yg(n)=G ¯HV¯s(i)ss(n)+G ¯HV¯i(i) si(n) +G ¯HN¯(i)(n)E59

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

J(G¯)=min{G¯HRinGG¯};subject toG¯HC=gHE60

where

RinG=[E{(xin(i)(n))1 (xin(i)(n))1H}E{(xin(i)(n))1 (xin(i)(n))2H}E{(xin(i)(n))2 (xin(i)(n))1H}E{(xin(i)(n))2 (xin(i)(n))2H}]E61

is the covariance matrix and xin(i)(n)=vi(i)si(n)+n(i)(n)  is the quaternion involution of xin(n). C=[V¯i(i),V¯s(i)] and gH=[g1H, 0], where g1 is given by Eq. (55). The solution of Eq. (60) is given by Van Trees [13]

G¯=RinG1C(CHRinG1C)1gE62

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])

G ¯=g1ν(2M|Ps|2V¯i(i)(PsΔPi)1qsHqiV¯s(i))E63

where

ν=(2M)2|Ps|2|Pi|2|(PiΔPs)1|2|qiHqs|2 =μ2Mξi1 |Ps|2E64

μ is given by (43). Moreover, the quaternion-valued optimal weight vector Go may be given by

Go=J1 G¯+j J2 G¯E65

where J1=[I2M×2M, 02M×2M] and J2=[02M×2M, I2M×2M] are two selection matrices.

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

yg(n)=(WoΔvi)1 si(n)+(GoΔn(i)(n))1E66

Thus, the complex output of QSWL GSC may be rewritten as

yGSC(n)=yw(n)yg(n)=ss(n)+(WoΔn(n))1 (GoΔn(i)(n))1E67

From above equation, we see that the interference component is completely cancelled in the output yGSC(n).

### 6.3. 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. Let ρn=E{|(WoΔn(n))1(GoΔn(i)(n))1|2} is the power of output noise. From Eqs. (37) and (58), we have

(WoΔn(n))1(GoΔn(i)(n))1=(W1H G1H)n1(n)+ (W2H + G2H)n2(n)E68

Then, ρn can be written as

ρn=σn2(W1HG1H)(W1G1) + σn2(W2H+G2H)(W2+G2) E69

When the combined QPMC and MVDR are adopted in the second-stage beamformer, ρn can be written in the simple form (the proof is in Appendix D of Ref. [5])

ρn=σn2μ2(2M|Ps|2ε2+|(PiΔPs)1|2|qiHqs|2λq)E70

where λq=ξi2|Ps|2κ2M|Pi|2, κ is given by Eq. (54). From Eq. (67), the expression of output SNR and array’s gain Aq may be written as

SNRo=ξs(2M|Ps|2ε|(PiΔPs)1|2|qiHqs|2)2(2M|Ps|2ε2+|(PiΔPs)1|2|qiHqs|2λq)E71
Aq=(2M|Ps|2ε|(PiΔPs)1|2|qiHqs|2)2(2M|Ps|2ε2+|(PiΔPs)1|2|qiHqs|2λq) E72

where ξs 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, ρn can be written in the simple form (the proof is in Appendix E of Ref. [5])

ρn=σn2μ2(2M|Ps|2ε2+|(PiΔPs)1|2|qiHqs|2λl)E73

where λl=2Mξi2|Ps|2ν2M|Pi|2, ν is given by Eq. (64). Then, the expression of output SNR and array’s gain Al may be written as

SNRo=ξs(2M|Ps|2ε|(PiΔPs)1|2|qiHqs|2)2(2M|Ps|2ε2+|(PiΔPs)1|2|qiHqs|2λl)E74
Al=(2M|Ps|2ε|(PiΔPs)1|2|qiHqs|2)2(2M|Ps|2ε2+|(PiΔPs)1|2|qiHqs|2λl) E75

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., |qiHqs|), but also difference between the polarizations of the desired signal and interference (i.e., |(PiΔPs)1|). The dependencies of them on |qiHqs| and |(PiΔPs)1| are shown in following consequences:

1. When |qiHqs|=0, the separation between the DOAs of the desired signal and interference reaches to maximum. In this case, Aq=Al=2M|Ps|2. Further, |qiHqs|  increases with a decrease of the DOA’s separation. Thus, the array’s gain Aq and Al will reduce if |Ps|2 is a constant. When qi=qs, |qiHqs|=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

Aq=Al=2M(|Ps|2ε2M|(PiΔPs)1|2)2|Ps|2ε2+2M|(PiΔPs)1|2λ E76

where

λ=ξi2|Ps|22M(|Ps|2|Pi|2|(PiΔPs)1|2)2M|Pi|2E77

Further, Pi=Ps if γs=γi and ηs=ηi. Thus, the array’s gain Aq=Al=0 due to λ=. This implies that the QSWL GSC fails.

2. When |(PiΔPs)1|=0, Aq=Al=2M|Ps|2. In the cases that θs=θi 0 and φs=φi0 (i.e., qi=qs), we have |(PiΔPs)1|=(sin2θscos2φs+sin2φs)cos(γiγs)cos(ηiηs). If γiγs=±π/2 or ηiηs=±π/2, then |(PiΔPs)1|=0. 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 2M|Ps|2 by using the orthogonality between the polarizations of the desired signal and interference. Further, the array’s gain decreases with an increase of |(PiΔPs)1| if |Ps|2 is a constant. When Pi=Ps, |(PiΔPs)1|=|Ps|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 qiqs.

In addition, the output SNR and array’s gain depend also on the input INR ξi, the array’s element number 2M, the interference response’s power |Pi|2 and the desired signal response’s power |Ps|2.

## 7. 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].

### 7.1. 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=10logσs2σi2+σn2 in this experiment and assume that input SINR=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 8° and 8°. In the case of θs=0°,θi=80°, φs=φi=0°;γs=γi=60°,ηs=ηi=30°, (i.e., 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.

### 7.2. 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, φs=φi=60°; γs=γi=30°,ηs=ηi=30°, and that the covariance matrix Rx, instead of Rin, is available. Figure 6 displays the power patterns for three values of |Δθ|:60°, 20°  and 10°, where θs=|Δθ|,θi=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 |Δθ| 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 |Δθ|. These side-lobes lead the beamformer to capture the white noise, which spans the whole space, so that the performance of beamformer degrades.

## 8. 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.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grants 61571462 and 60872088.

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Written By

Jian-wu Tao and Wen-xiu Chang

Submitted: September 8th, 2016 Reviewed: February 13th, 2017 Published: May 10th, 2017