Effect of Mutual Coupling on the Performance of STCM-MIMO Systems
Fatemeh Asghari Azhiri, Reza Abdolee, Behzad Mozaffari Tazehkand
aa r X i v : . [ ee ss . SP ] M a r c (cid:13) ffect of Mutual Coupling on the Performance ofSTCM-MIMO Systems Fatemeh Asghari Azhiri, Reza Abdolee, and Behzad Mozaffari Tazehkand
Abstract —Space-time coded massive (STCM) multiple-inputmultiple-output (MIMO) system provides superior bit error rate(BER) performance compared with the conventional space-timecoding and massive MIMO techniques. The transmitter of theSTCM-MIMO system consists of a large antenna array. In apractical system, the self-interference created by the signalstransmitted by the elements of this antenna array, known asmutual coupling (MC), degrades the performance of the system.The MC effect is pronounced in communication systems with alarge antenna array. On the other hand, increasing the numberof transmitting antennas results in improved BER performance.Hence, there is a trade off in selecting the optimum numberof transmitting antennas in an STCM-MIMO system. In orderto take the impact of MC into account, we have derived ananalytical expression for the received signal to accurately modelthe STCM-MIMO system under the existence of the MC effect.We present an algorithm to select the optimal number of antennasto minimize mutual coupling and the system bit error rate(BER). Through computer simulations, we investigate the BERperformance of the STCM-MIMO system for different numbersof array elements.
Index Terms —massive MIMO, 5G, mutual coupling effect,STCM-MIMO, space-time coding.
I. I
NTRODUCTION T HE massive multiple-input multiple-output (MIMO)technology has been introduced to meet the growingdemands of bandwidth hungry applications and multiusercommunications in next generation wireless systems such asfifth generation (5G) cellular networks [1]. The base station(BS) of a massive MIMO system is able to encode andsend data to multiple users simultaneously and in the samefrequency band, using a large number of antennas. Therefore,incorporating massive MIMO technology results in robustand highly reliable communications for upcoming ultra densewireless communication networks [2].In order to further improve the performance of massiveMIMO systems, Ice et al. [5] introduced ingenious space-time coded massive MIMO (STCM-MIMO) technique thatbenefits from the advantages of both massive MIMO andspace-time coding technologies. The data symbols in a STCM-MIMO system are coded by a space-time code technique inthe transmitter. Then, each of the coded symbols is transmittedby a subset of the antenna array in the BS. Using antennaarrays in STCM-MIMO for transmitting the space-time codedsymbols significantly improves the overall system performance
F. Asghari Azhiri and B. Mozaffari Tazehkand are with the Departmentof Electrical and Computer Engineering, University of Tabriz, Tabriz, Irane-mail: (f.asghari, mozaffary)@tabrizu.ac.irR. Abdolee is with Department of Electrical and Computer Engineering,California State University, Bakersfield, USA e-mail: [email protected] compared to the space-time coded MIMO and conventionalmassive MIMO system [6].The transmitter of the STCM-MIMO system contains alarge number of antennas that must be located close to eachother in order to achieve an appropriate size of the base station.This compactness leads to an increase in the mutual coupling(MC) effect between elements of the antenna arrays.The effect of the MC should be considered in the designand implementation of the communication systems specificallymassive MIMO based systems due to the deployment of largeantenna arrays in their structure. The impact of MC and thespatial correlation have been studied on various types of sys-tems with array antennas in their structure. The effect of MCin BER performance of Alamouti space-time coded systemshas been investigated in [7] and a performance degradation inlow correlated channels has been indicated. In [8] an antennaselection algorithm is used in the receiver side in the presenceof mutual coupling to achieve best BER performance foran Alamouti coding scheme. The authors in [9] proposed arigorous network-theory framework for the analysis of mutualcoupling in MIMO wireless communications. This methodattains an upper bound for the capacity expression in thepresence of mutual coupling in the studied system.The growing application of massive MIMO technology inthe new generations of communication networks such as 5Gsystems has attracted increasing attention to study its hardwareperformance. In [10], signal to noise and interference ratioof a massive MIMO system for various types of antennaarrays are calculated. The mutual coupling model of antennaarrays is applied to the 3GPP 3D channel model in [11]and by a matching network, the coupling effect has beenpartly compensated. Most articles in the literature assume thatthe antenna arrays have a regular structure such as uniformlinear or uniform planar arrays. The antenna arrays may haveirregular structures as shown in [12]. It is indicated that insome specific cases, the irregular arrays outperform the regularantenna arrays in the achievable rate.Since STCM-MIMO systems are introduced recently, theirhardware implementation has not been investigated thor-oughly. It has been shown that increasing the number of an-tenna elements in the transmitter of an STCM-MIMO systemresults in better BER performance, while the destructive effectof MC of the antenna array is not considered [6]. However,the effect of MC is not negligible in practice, especially forlarge antenna arrays. In this paper, we study the performanceof STCM-MIMO system in the presence of the MC effect. Wederive the analytical expression of the received signal vectorby considering the MC effect of the antenna elements in theransmitter. The BER performance of an STCM-MIMO systemwhich consists of a uniform linear array (ULA) antenna isinvestigated. The simulation results determined the amount ofthe performance degradation of the STCM-MIMO system withdifferent number of antenna elements and element distancesin the presence of the coupling effect. In order to designa proper array structure for the transmitter of an STCM-MIMO system, we compare the performance of the systemwith various number of antennas within identically sized arraysand different styles of sub-array selection in the transmitter.The simulation results confirm that we face a trade off inspecifying the optimum number of elements of the antennaarray to achieve appropriate system performance. Even thoughincreasing the number of antennas in STCM-MIMO transmit-ter improves the BER in ideal systems, it ends in increasingthe destructive effect of mutual coupling when the MC effect istaken into account. We investigate this trade off and determinethe optimum number of elements for the inspected antennaarrays.The remainder of the paper is organized as follows. Insection II the system model is discussed and the received sig-nal vector is formulated. The simulation results are discussedin section III followed by concluding remarks in section IV.II. S
YSTEM M ODEL
We consider an STCM-MIMO system with a base stationequipped with M antennas and K single antenna mobileusers. We divide the base station array into two sub-arrayseach having N = M/ elements. Two information symbolswith a certain modulation scheme such as M-ary QAM orPSK modulation are transmitted by two transmit antenna sub-arrays in each time slot. This system applies the Alamoutispace-time codes and uses Hermitian pre-coding scheme [5].At time t , the first sub-array sends the symbol s and thesecond sub-array sends s . At time t + T , the symbols − s ∗ and s ∗ are sent by first and second sub-arrays, respectively.The Hermitian pre-coding is applied on each symbol beforetransmitting (see Figure 1). The weight vector w ( w ) is theHermitian pre-coding vector according to the channel vectorof the first (second) sub-array and the receiver.The received signals can be expressed as [5] ˜ r nc =˜ r nc ( t ) = w H h s + w H h s + K − X j =0 ( w H j h s j + w H (2 j +1) h s (2 j +1) ) + ˜ n ˜ r nc =˜ r nc ( t + T ) = − w H h s ∗ + w H h s ∗ + K − X j =0 ( − w H j h s ∗ j +1 + w H (2 j +1) h s ∗ (2 j ) ) + ˜ n (1)where ˜ r nc is the received signal at time slot t without MCeffect, ˜ r nc is the received signal at time slot t + T withoutMC effect, K is the number of users, ( . ) H and ( . ) ∗ indicateHermitian transform and complex conjugate respectively. Thecoefficient vector of the channel between the antenna array atthe transmitter and the single antenna receiver is expressed as Space-
Time
Code M d User kUser KUser 1 h h Pre- coding ) w ( Pre- coding ) w ( s, s ŝ , ŝ N+1 C * s- , 0 s *0 s , 1 s Fig. 1. N × STCM-MIMO h = [ h ; h ] , where h and h are N × vectors which exhibitthe channel coefficients through first and second sub-arrays tothe receiver, respectively. The massive MIMO Hermitian pre-coding vector, w j , is defined as w j = 1 N h j (2)The receiver combining scheme for two branches of STCM-MIMO system can be expressed as ˜ s = k h k ˜ r nc + k h k ˜ r nc ∗ ˜ s = k h k ˜ r nc − k h k ˜ r nc ∗ (3)where k v k is the L -norm of the vector v . Using theequation (3) the maximum likelihood detector can estimatethe transmitted symbols.Equation (1) assumes that the antenna elements of the arraydo not interfere with each other. When antenna elementsare close, the electromagnetic field produced by one antennainfluences the output of its neighbor antennas. The interactionbetween two or more antennas, that affects coefficients of theantenna array is called mutual coupling.Assuming that a communication system consists of anantenna array with M elements as a transmitter and a singleantenna receiver, the received signal at the receiver can bewritten as y nc = gx (4)where y nc is the received signal without MC effect, x is thetransmitting signal vector and g is the wireless channel vector.Mutual coupling effect can be incorporated into (4) asfollows y c = Cgx = ˆ gx (5)where C represents the M × M coupling matrix and ˆ g can bereplaced as the channel vector in order to consider MC effect.herefore, in the case of STCM-MIMO, the receiving signalincluding mutual coupling can be written as ˜ r c =˜ r c ( t ) = w H ˆ h s + w H ˆ h s + K − X j =0 ( w H j ˆ h s j + w H (2 j +1) ˆ h s (2 j +1) ) + ˜ n ˜ r c =˜ r c ( t + T ) = − w H ˆ h s ∗ + w H ˆ h s ∗ + K − X j =0 ( − w H j ˆ h s ∗ j +1 + w H (2 j +1) ˆ h s ∗ (2 j ) ) + ˜ n (6)where the wireless channel between the transmitter array andthe receiver of the system including the MC effect is definedas ˆ h = Ch = C " h h = " ˆ h ˆ h (7)The mutual coupling matrix, C can be acquired from elec-tromagnetics analysis and measurement. The antennas havethe reciprocity property which means that the receive andtransmit properties of an antenna are identical [13]. Therefore,in the case that the receiver uses an antenna array, the mutualcoupling effect will be the same.For a uniform linear array consisting of M dipole elements, C can be written as [14] C = ( Z A + Z L )( Z + Z L I ) − (8)where I is the identity matrix of size M × M and Z A and Z L are the antenna impedance and load impedance, respectively.The elements of matrix Z can be calculated as follows [15] Z mn = η π [0 .
577 + ln (2 π ) − Ci (2 π )+ jSi (2 π )] m = nη π { [2 Ci ( βd ) − Ci ( βu ) − Ci ( βu )] − j [2 Si ( βd ) − Si ( βu ) − Si ( βu )] } m = n (9)where η = p µ /ǫ ≈ π is the intrinsic impedance and β = 2 π/λ is the wave number, λ is the wavelength and ( u = √ d + L + Lu = √ d + L − L (10)where d is the distance between array elements and L is thelength of the dipole antenna. Ci ( x ) and Si ( x ) are the cosineand sine integrals defined as Ci ( x ) = Z x −∞ cos( x ) x dxSi ( x ) = Z x −∞ sin( x ) x dx (11)III. C OMPUTER E XPERIMENT R ESULTS
In order to investigate the effect of MC on the performanceof STCM-MIMO systems, it is essential to calculate thecoupling matrix which depends on the array structure. In thissection, we analyze an STCM-MIMO communication system
SNR(dB) -4 -3 -2 -1 Fig. 2. BER performance of STCM-MIMO with MC effect, M = 100 with a ULA as its transmitter and following assumptions andparameters. A N × STCM-MIMO system with one basestation is assumed which contains a transmitting antenna arraywith M elements. The antenna array is a ULA consisting ofdipole antennas with length L and element spacing d . Thecarrier frequency is assumed to be 2 GHz. Data symbols arechosen from QPSK modulation constellation. The full channelstate information (CSI) is available at the transmitter forcalculating pre-coding weights and at the receiver for decodingthe data. The STCM-MIMO system utilizes × Alamouticode, so two consecutive time slots are required for detectingtwo transmitted symbols. The transmission environment isassumed to be dispersive such that the channel coefficientsfollow i.i.d. complex Gaussian distribution; also, there existsadditive white complex Gaussian noise at the receiver. All theresults are achieved by 20000 times Monte-Carlo simulations.Figure 2 represents the MC effect on the BER performanceof STCM-MIMO system. In this simulation, the transmittingarray consists of 100 antenna elements with various distances.The antenna impedance is Z A = 73 + 42 j and the loadimpedance is assumed to be Z L = Z ∗ A in order to providefull matching. The simulation results show that, decreasingthe distances between array elements increases the MC effectbetween array antennas. Hence, the bit error rate increases forthe same SNR value.In a STCM-MIMO system as long as the MC effect of theantenna elements is negligible, it’s straightforward to increasethe number of elements of the antenna array in the transmitterto improve the BER performance. However, by increasing thenumber of elements in the antenna array of the transmitter ordecreasing the distances between them, the MC effect becomesignificant. Figure 3 demonstrates the BER performances ofthe STCM-MIMO systems with different numbers of antennaelements in the presence of MC effect.To achieve the optimum performance in a system withcompact transmitter, it is important to utilize the appropriatenumber of antenna elements with specific distances in its basestation. In a STCM-MIMO system without considering theMC effect, increasing the number of transmitting antennasimproves the BER performance [6]. Figure 4 compares the SNR(dB) -5 -4 -3 -2 -1 Fig. 3. BER performance of STCM-MIMO with MC effect for differentnumber of array elements, d = 0 . λ, d = 0 . λ BER performance of a STCM-MIMO system with and withoutconsidering the MC effect at an SNR of 10dB. Our simulationresults demonstrate a trade-off to select the optimum numberof antenna elements. Increasing the number of elements mayimprove the performance of STCM-MIMO, however to keepthe size of the transmitter constant, the distances betweenelements have to be decreased, which increases MC effect anddegrades BER performance. The antenna array of the trans-mitter is considered to be a uniform linear array consisting of M elements with uniform spacing. The analytical formulationof mutual coupling effect presented in (6) facilitates deter-mining the optimal number of antennas in a STCM MIMOsystem. We propose a heuristic search algorithm inspired byNewton-Raphson method to achieve the optimum number ofantennas (Algorithm 1). We first divide the searching intervalby the arbitrarily chosen ’step’ size which gives us samplenumbers of antennas, n i . The sample numbers must be evennumbers in a × STCM-MIMO system. Then we calculatethe MC matrix for n i s using (8). In the next step, theBER of the STCM-MIMO system is calculated utilizing the(6) and by simulation for the intended SNR values. Then n j = arg min( average ( BER )) , n ≤ n j ≤ n m is se-lected. The searching is continued among n j , ⌈ n j − + n j ⌉ and ⌈ n j + n j +1 ⌉ where ⌈ . ⌉ denotes the ceil function. The numberwith min( average ( BER )) is selected to be the middle sampleand the searching continues in the adjacency of this number.The rest of searching process continues until reaching theoptimal number of antennas with minimum average ( BER ) .Figures 5 and 6 show the BER performance of STCM-MIMO system for total length = 30 λ and λ at variousSNR values. In Table I we provide the optimum number oftransmitting antennas in a × STCM-MIMO system forthe antenna arrays with specified total lengths. The requiredinformation for the search algorithm have been considered as search interval = (50 , and step = 50 .In the previous simulations, the antenna sub-arrays offirst and second branches have been selected as shown inFigure 1 (style 1). Simulation results don’t show significantperformance differences with the case that the sub-arrays werechosen one in between (style 2). Figure 7 demonstrates the
50 100 150 200 250Number of antennas10 -4 -3 -2 -1 Fig. 4. BER performance of 2 ×
50 100 150 200 250
Total number of antennas -3 -2 -1 Fig. 5. BER performance of STCM-MIMO with MC effect versus totalnumber of antennas determined for array size of λ at different SNRconditions
50 100 150 200 250
Total number of antennas -4 -3 -2 -1 Fig. 6. BER performance of STCM-MIMO with MC effect versus totalnumber of antennas determined for array size of λ at different SNRconditions results of this comparison for an antenna array with 200elements. SNR(dB) -4 -3 -2 -1 Fig. 7. Antenna selection effect on BER performance, M=200TABLE IO
PTIMAL NUMBER OF ANTENNAS IN THE TRANSMITTER OF
STCM-MIMO
SYSTEMS
Total length of array Optimum number of antennas λ λ λ λ λ IV. C
ONCLUSION
In this paper, the STCM-MIMO system has been modeledconsidering the MC effect using an analytical expressionderived for the received signal of the system. The simulationresults indicated a performance degradation in the presenceof MC. The existence of a trade-off has been demonstratedfor determining the appropriate number of antennas in thetransmitters with predefined physical sizes. Increasing thenumber of array elements improves the BER performance dueto the increment of the number of antennas in STCM-MIMOtransmitter, on the other hand it increases the destructive effectof mutual coupling and causes degradation in system perfor-mance. We proposed an algorithm to calculate the optimalnumber of antennas to achieve the minimum bit error rate ofSTCM-MIMO systems. R
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Algorithm 1:
Search Algorithm to find the optimal numberof antennas in the transmitter of a 2 × Input:
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