# MIMO Antenna Array Design with Polynomial Factorization

Authors: Wen-Qin Wang; Huaizong Shao; Jingye Cai
Journal: International Journal of Antennas and Propagation
 (2013)

---

## Abstract

One of the main advantages of multiple-input multiple-output (MIMO) antenna is that the degrees-of-freedom can be significantly increased by the concept of virtual antenna array, and thus the MIMO antenna array should be carefully designed to fully utilize the virtual antenna array. In this paper, we design the MIMO antenna array with the polynomial factorization method. For a desired virtual antenna array, the polynomial factorization method can optimally design the specified MIMO transmitter and receiver. The array performance is examined by analyzing the degrees-of-freedom and statistical output signal-to-interference-plus-noise ratio (SINR) performance. Design examples and simulation results are provided.


---

## Body

## 1. Introduction

Multiple-input multiple-output (MIMO) antenna array has received much attention in communication, radar, and navigation systems [1–4]. MIMO idea is not new; its origin in control systems can be traced back to 1970s [5]. The early 1990s saw an emergence of MIMO ideas into the field of wireless communication systems [6–8]. More recently, the ideas of MIMO appear in sensor and radar systems [9–12]. Different from conventional phased-array systems [13], in MIMO antenna systems each antenna transmits a unique waveform, orthogonal to the waveforms transmitted by other antennas, and a matched filter bank is used to extract the orthogonal waveform components. When the orthogonal signals are transmitted from different antennas, the return of each orthogonal signal will carry independent information. The phase difference caused by different transmitting antennas along with the phase differences caused by different receiving antennas can form a new virtual antenna array steering vector. With optimally designed antenna array geometry, we can create a very long array steering vector with a small number of antennas [14, 15]. More importantly, this provides good flexibility and reconfigurability in array configuration, thus enabling a flexible and reconfigurable system.Although MIMO antenna arrays are widely used in various applications, radar-related application is assumed in this paper. Even so, the rich literature of research in this area appears in threefold: (a) the comparison between conventional phased-array and MIMO array, and the development of MIMO architecture [16–18]; (b) MIMO waveform diversity and transmitter design [19, 20]; and (c) the system performance and the receiver design for enhanced performance [21, 22]. In fact, one of the main advantages of MIMO antenna array is that the degrees-of-freedom can be significantly increased by the concept of virtual antenna array, and thus the MIMO antenna array should be carefully designed to fully utilize the virtual antenna array. This paper focuses on the MIMO sparse antenna array design to obtain a uniform virtual array for the system.Many array design methods have been proposed for conventional phased arrays [23–26]. Specially, linear programming-based algorithms were proposed by [27, 28] for reduced sidelobe thinned array design. Genetic algorithm was proposed by [29] to optimally design thin large phased arrays. Simulated annealing-based technique was proposed by [30] to synthesize the position and weight coefficients of a linear array that minimizes peak sidelobes. A hybrid approach combining particle swarm optimization with combinational techniques was proposed by [31] to synthesize planar thinned arrays. Recently a general polynomial factorization-based design of sparse periodic linear arrays was proposed by [32] to design sparse periodic arrays. In the method, transmit and receive aperture polynomials are selected such that their product results in a polynomial representing the desired combined transmit/receive effective aperture function.In this paper, we use the polynomial factorization method proposed in [32] to design MIMO antenna array and comparatively analyze the designed array performance. To obtain the desired virtual MIMO antenna array, we use the polynomial factorization-based method to design the transmit and receive antenna arrays. Since there may be multiple solutions for the transmit and receive antenna arrays and different solutions may have different array performance, we use adaptive beamforming algorithm to comparatively analyze the statistical output signal-to-interference-plus-noise ratio (SINR) performance of the designed arrays. Extensive design examples and simulation results are also provided.The remaining sections are organized as follows: after introducing the polynomial factorization in Section2, we describe the polynomial factorization-based MIMO antenna array design in Section 3. Next, the output SINR performance of the designed MIMO antenna arrays is analyzed by adaptive beamforming on the receiver in Section 4, and design examples and simulation results are provided in Section 5. Finally, this paper is concluded in Section 6 with a short discussion of future work.


## 2. General Polynomial Factorization Method

The polynomial factorization method is based on factorizing a polynomialPM(x) of degree M in positive powers of x and with unity coefficients [33]
(1)PL(x)=∑l=0L‍xl.
Let the positive integer of the number of coefficients L+1 be expressible as a product of K+1 irreducible positive integers Lk+1, 0≤k≤K:
(2)L+1=∏k=0K‍(Lk+1).Considering the productPL0(x)PL1(xS0), we have
(3)PL0(x)PL1(xS0)=(∑l=0L0‍xl)(∑l=0L1‍xlS0)=∑l=0L0‍xl+xS0(∑l=0L0‍xl)+x2S0(∑l=0L0‍xl)+⋯+xL1S0(∑l=0L0‍xl)=∑l=0L1S0+L0‍xl.Therefore,PL(x) can be expressed as [32, 33]
(4)PL(x)=∏k=0K‍PLk(xSk),
with
(5)Sk=∏i=0k‍(Li-1+1),
where L-1=0.The factorization of a specifiedL+1 into a product of irreducible integers can be carried out using Euclid’s algorithm [34]. When L+1 can be expressed as a power of two, that is, L+1=3ν, PL(x) can be factorized as
(6)PL(x)=(1+x)(1+x2)⋯(1+x2ν-1).
Note that the solutions to the cases when L+1 cannot be expressed as a power of two can be found in [32]. The decomposition in (6) has been used in designing a sparse antenna array with a uniform aperture function [35] and a linearly tapered aperture function [36].As an example, we supposeL+1=10; using Euclid’s algorithm we then have
(7)L+1=10=2×5=5×2.Then, according to the above polynomial factorization method, we can get(8)PL(x)=(x0+x1)[(x2)0+(x2)1+(x2)2+(x2)3+(x2)4]=(x0+x1)(x0+x2+x4+x6+x8)=(1+x)(1+x2+x4+x6+x8)
or
(9)PL(x)=(x0+x1+x2+x3+x4)[(x5)0+(x5)1]=(x0+x1+x2+x3+x4)(x0+x5)=(1+x+x2+x3+x4)(1+x5).
That is to say, there may be multiple possible factorizations for a given polynomial.


## 3. Polynomial Factorization-Based MIMO Array Design

Consider a MIMO radar system with a transmitter equipped withM colocated antennas and a receiver equipped with N colocated antennas. Suppose that both the transmit and receive arrays are close to each other in space (possibly the same array) so that they see targets at the same directions. The signal received by each antenna is the weighted summation of all the transmitted waveform
(10)xn(t)=∑m=1M‍αn,mϕm(t),m=1,2,…,M;n=1,2,…,N,
where xn(t) is the received signal at the nth antenna, ϕm(t) is the transmitted waveform at the mth antenna, and αn,m is the channel coefficient with the mth antenna as input and the nth antenna as output. Suppose x(t)=[x1(t),x2(t),…,xN(t)] and ϕ(t)=[ϕ1(t),ϕ2(t),…,ϕN(t)], and the above equation can be rewritten in a matrix
(11)x(t)=σs[aT(θ)ϕ(t)]b(θ),
where θ is the target direction, σs is the complex-valued reflection coefficient of the focal point θs, (·)T is the transpose operator, and a(θ) and b(θ) are the actual transmit and receive steering vectors associated with the direction θ.When the transmitted waveforms are orthogonal,(12)∫‍ϕm(t)ϕn*(t)dt={δ(t),m=m*0,m≠m*,
where ()* denotes a conjugate operator. The radar return due to the mth transmitted waveform can be separated by matched-filtering the received signal with ϕm(t), m=1,2,…,M; that is,
(13)sm≜∫T0‍x(t)ϕm*(t)dt,m=1,2,…,M,
where T0 is the radar pulse width. Therefore, there are a total of M×N independent signals, which can be written as a MN×1 virtual data vector
(14)y≜[s1T,s2T,…,sMT]T=σsa(θ)⊗b(θ),
where ⊗ is the Kronecker product; that is, the virtual antenna array is
(15)v(θ)=a(θ)⊗b(θ).Suppose that themth transmitting antenna is located at xT,m=(λ/2)um and the nth receiving antenna is located at xR,n=(λ/2)vn, where λ is the wavelength. Considering a far-field point target, the transmitter and receiver steering vectors can be represented, respectively, by
(16)a(θ)=[eju1πsinθ,eju2πsinθ,…,ejuMπsinθ]T,b(θ)=[ejv1πsinθ,ejv2πsinθ,…,ejvNπsinθ]T.Note that the first antenna ofa(θ) and b(θ), respectively, is the reference antenna. According to (15), we can get
(17)v(θ)=[ej(v1+u1)πsinθej(v1+u2)πsinθ⋯ej(v1+uM)πsinθej(v2+u1)πsinθej(v2+u2)πsinθ⋯ej(v2+uM)πsinθ⋮⋮⋯⋮ej(vN+u1)πsinθej(vN+u2)πsinθ⋯ej(vN+uM)πsinθ].
The array response in the nth receiving antenna for the mth transmitted waveform can be expressed as
(18)vm,n(θ)=ej(vn+um)πsinθ.
It can be noticed that the MIMO antenna array response is the same as the target response received by a receiving array with MN antennas located at
(19){xT,m+xR,n},m=1,2,…,M,n=1,2,…,N.
This MN-element array is just the virtual MIMO array. An utmost of MN-element virtual array can be obtained by using only M+N physical antenna elements. It is as if we have a receiving array of MN elements.According to (18), the far-field radiation pattern at an angle θ away from the broadside of the virtual MIMO array with uniform weighting is expressed as
(20)PV(θ)=∑m=1M‍∑n=1N‍ej(vn+um)πsinθ=∑m=1M‍ejumπsinθ∑n=1N‍ejvnπsinθ=PT(θ)PR(θ),
where PT(θ)=∑m=1M‍ejumπsinθ and PR(θ)=∑n=1N‍ejvnπsinθ denote, respectively, the far-field radiation pattern for the transmitter and receiver. For a more general format, by substituting x=ej2πsinθd/λ with d being the interelement spacing, the virtual array, transmit array, and receive array radiation patterns can be expressed, respectively, as (21a)PV(x)=∑i=0M+N-1‍wV(i)xi,(21b)PT(x)=∑i=0M-1‍wT(i)xi,(21c)PR(x)=∑i=0N-1‍wR(i)xi,where wV(i), wT(i), and wV(i) denote the element weighting functions for the virtual array, transmit array, and receive array, respectively. It follows also that [32]
(22)PV(x)=PT(x)PR(x).Therefore, the far-field radiation pattern of the virtual array is equal to the product of the transmitter radiation pattern and receiver radiation pattern. This means that we can use different transmit and receive arrays with a combined virtual MIMO array that is a close equivalent to that of a single full array with no missing elements. To reach this aim, like the general polynomial factorization-based design of sparse periodic linear array discussed in [32], we can design the transmit and receive arrays based on factorizing the desired MIMO virtual array polynomial PV(x) as a product of two lower order polynomials with some zero-valued coefficients. In this paper, we aim to optimally design the transmit and receive arrays for a given virtual MIMO antenna array. The design steps are illustrated in Figure 1.Figure 1
Illustration of the MIMO antenna array design steps.


## 4. Array Performance Analysis by Adaptive Beamforming

The output of the matched filter of the MIMO array system can be written as(23)y=α0v(θ)+n,
where n denotes the noise plus interference. The output SINR of the beamformer directly impacts the MIMO array performance. The minimum variance distortionless response (MVDR) [37] is a widely employed beamformer choice. The MDVR beamformer for a look angle θ is given by
(24)w(θ)=R-1v(θ)vH(θ)R-1v(θ),
where (·)H is the conjugate transpose and R is the covariance matrix of the noise and interference signal. The beamformer output can then be expressed as
(25)z=wH(θ)y=α0+wH(θ)n.Correspondingly, the output SINR can be expressed as(26)η≜|α0|2wH(θ)Rw(θ).Since the output SINR is random due to the random covariance matrixR, we can examine the SINR distribution under the standard assumption that n has a complex Gaussian distribution with zero mean and covariance matrix R. In this case, the probability density function (PDF) of SINR can be represented by [38, 39]
(27)fη(x)=K!(Ndof-2)!(K+1-Ndof)!ηmax×(1-xηmax)Ndof-2(xηmax)K+1-Ndof,0≤x≤ηmax,
where K is the number of the training signals for obtaining the covariance matrix, Ndof is the number of the system degrees-of-freedom, and ηmax is the maximum output SINR which is achieved when the covariance matrix is known exactly
(28)ηmax=|α0|2vH(θ)R-1v(θ).
Figure 2 shows the PDF fη(x) as a function of x/ηmax.Figure 2
The PDFfη(x) as a function of  x/ηmax.The detailed derivation of (27) can be found in the appendix of [38]. The corresponding cumulative distribution function is
(29)Fη(x)=K!(Ndof-2)!(K+1-Ndof)!×Γ(xηmax;K+2-Ndof,Ndof-1),
where Γ(x;a,b) is the incomplete Beta function
(30)Γ(x;a,b)≜∫0x‍ta-1(1-t)b-1dt.The expectation and variance of the output SINR can be derived, respectively, as(31)E(η)=ηmaxK+2-NdofK+1,(32)var(η)=ηmax2(Ndof-1)(K+2-Ndof)(K+2)(K+1)2.
It can be noticed from (31) that the expectation of the output SINR depends mainly on the ratio Ndof/K, not the number of the training data, K.Equations (31) and (32) can be used to evaluate the designed MIMO array performance. Figure 3 gives the comparative standard deviation (STD) of the output SINR performance, where K=20 is assumed. It can be noticed that, for a given number of the training signals, more degrees-of-freedom mean a better STD of the SINR performance.Figure 3
Comparative STD of the output SINR performance.Define(33)E(η)=αηmax.
The corresponding K is
(34)K=11-α(Ndof-2+α)≈Ndof1-α.
To ensure that the mean output SINR of the adaptive beamformer is within 3dB from its maximum achievable value, the training signals should be satisfactory with [38]
(35)K≥2Ndof-3.
If we want to achieve α=0.95, the K should be
(36)K≥3Ndof.


## 5. Design Examples and Performance Analysis

We consider two design examples with given virtual MIMO antenna arrays.Example 1.
L
=
9 and PV(x)=∑i=09‍xi.According to (2), we factorize L+1=10 into two prime integers 2 and 5, that is, 10=2×5. Then, according to (4), the virtual array radiation pattern PV(x) can then be factorized as
(37)PV(x)=(1+x)(1+x2+x4+x6+x8).
We can use the first factor for the transmit array and the second factor for the receive array or the first factor for the receive array and the second factor for the transmit array; that is, (38a)PT(x)=1+x,(38b)PR(x)=1+x2+x4+x6+x8 or (39a)PT(x)=1+x2+x4+x6+x8,(39b)PR(x)=1+x.Suppose that the number of the training signals isK=25; Figure 4 shows the possible two transmit-receive arrays. It can be noticed that 7 antennas are required for the MIMO antenna arrays and the number of the system degrees-of-freedom is 10. Figure 5 gives the statistical mean and STD of the output SINR performance. Note that, although the two arrays have the same statistical SINR performance (because they have the same number of degrees-of-freedom), they should be seen as two different MIMO antenna arrays in the design because they have different transmit array gain and receive array gain (MIMO antenna array has no transmit array gain whereas phased array has both transmit array gain, and receive array gain).Illustration of the designed MIMO antenna array positions. Thex-axis stands for the MIMO antenna position. y=1 denotes an antenna and y=0 means no antenna.
(a)
virtual antenna array positions
(b)
transmit antenna array positions
(c)
receive antenna array positionsFigure 5
Statistical output SINR performance.Example 2.
L
=
11 and PV(x)=∑i=011‍xi.According to (2), we can factorize L+1=12 into the following three ways.Factorization 1. We can factorize 12 into three prime integers 2, 2, and 3, that is, 12=2×2×3. Then, according to (4), the virtual array radiation pattern PV(x) can then be factorized as
(40)PV(x)=(1+x)(1+x2)(1+x4+x8).Factorization 2. We can factorize 12 into three prime integers 2, 3, and 2, that is, 12=2×3×2. Then, according to (4), the virtual array radiation pattern PV(x) can then be factorized as
(41)PV(x)=(1+x)(1+x2+x4)(1+x6).Factorization 3. We can factorize 12 into three prime integers 3, 2, and 2, that is, 12=3×2×2. Then, according to (4), the virtual array radiation pattern PV(x) can then be factorized as
(42)PV(x)=(1+x+x2)(1+x3)(1+x6).Then there are nine possible transmit-receive array designs:(43a)PT(x)=(1+x),PR(x)=(1+x2)(1+x4+x8),(43b)PT(x)=(1+x2),PR(x)=(1+x)(1+x4+x8),(43c)PT(x)=(1+x4+x8),PR(x)=(1+x2)(1+x),(43d)PT(x)=(1+x),PR(x)=(1+x2+x4)(1+x6),(43e)PT(x)=(1+x2+x4),PR(x)=(1+x)(1+x6),(43f)PT(x)=(1+x6),PR(x)=(1+x)(1+x2+x4),(43g)PT(x)=(1+x+x2),PR(x)=(1+x3)(1+x6),(43h)PT(x)=(1+x3),PR(x)=(1+x+x2)(1+x6),(43i)PT(x)=(1+x6),PR(x)=(1+x+x2)(1+x3).Suppose that the number of the training signals isK=64; Figure 6 shows the possible nine transmitter-receiver arrays. It can be noticed that 8 antennas are required for the transmit-receive pairs (a), (b), (d), (f), (h), and (i), and 7 antennas are required for the transmit-receive pairs (c), (e), and (g). All of the nine transmitter-receiver arrays have 10 degrees-of-freedom. Figure 7 gives the statistical mean and STD of the output SINR performance. Since they have the same number of degrees-of-freedom, the nine transmit-receive arrays have the same statistical SINR performance.Illustration of the nine possible transmit-receive array positions for the MIMO antenna array. Thex axis stands for the MIMO antenna position. y=1 denotes an antenna and y=0 means no antenna.Figure 7
Statistical output SINR performance.


## 6. Conclusion

In MIMO antenna array systems, the transmitter and receiver should be carefully designed to fully utilize the virtual MIMO antenna array. In this paper, we designed the MIMO antenna array with the polynomial factorization method and comparatively analyzed the designed array performance. For a desired virtual antenna array, the polynomial factorization method can optimally design the specified transmitter and receiver antenna arrays. Design examples are also provided, which are verified by simulation results. Note that two MIMO antenna arrays may have the same statistical SINR performance; they often should be seen as two different MIMO antenna arrays in the design because they have different transmit array gain and receive array gain. In this paper, uniform virtual MIMO antenna array elements are assumed. We are aware that nonuniform virtual MIMO antenna array elements may be desired in actual MIMO antenna arrays. This topic will be further investigated in our subsequent work.



---

*Source: 358413-2013-06-27.xml*