Frequency domain adaptive antenna array (FDAAA) is an effective method to suppress interference caused by frequency selective fading and multiple-access interference (MAI) in single-carrier (SC) transmission. However, the performance of FDAAA receiver will be affected by the antenna placement parameters such as antenna separation and spread of angle of arrival (AOA). On the other hand, hybrid frequency reuse can be adopted in cellular system to improve the cellular capacity. However, optimal frequency reuse factor (FRF) depends on the channel propagation and transceiver scheme as well. In this paper, we analyze the impact of antenna separation and AOA spread on FDAAA receiver and optimize the cellular capacity by using hybrid FRF.
1. Introduction
Single-carrier (SC) transmission has been adopted for uplink transmission in LTE [1] and LTE-A [2] systems due to its wide coverage and lower peak-to-average power ratio (PAPR) than multicarrier (MC) transmission. However, the wireless channel becomes severely frequency selective as the data rate increases due to the multiple paths with different time delays [3]. In such a frequency-selective channel, interblock interference (IBI) and intersymbol interference (ISI) are produced and degrade the transmission significantly. To deal with this problem, cyclic prefix (CP) will be inserted at the transmitter side, and then be removed at the receiver side to avoid IBI and frequency domain equalization (FDE) technique has been proposed to suppress ISI in SC transmission [3]. FDE equalizes the frequency domain receive signal by applying equalization weight to combat the channel fluctuation in each frequency. Several methods to calculate FDE weight have been introduced, such as zero-forcing (ZF), minimum-mean-square error (MMSE), and so forth.
In cellular system, neighboring cells will use the same carrier frequency/frequencies to save the bandwidth and cochannel interference (CCI) exists and CCI power will be determined by the distance between cochannel cells. In addition, the existence of multiple users within the same cell will cause multiuser interference (MUI). As a result, cellular capacity will be limited by multiple access interference (MAI, which includes both CCI and MUI) instead of additive noise. Adaptive antenna array (AAA) is a powerful method to combat MAI. It has been proved in [4] that AAA receiver can effectively deal with up to Nr-1 interferences when flat fading channel is assumed where Nr represents the number of antenna elements in the array.
Therefore, in order to combat the MAI in frequency selective fading channel, it is natural for us to resort to both FDE and AAA. In our previous study [5], frequency domain adaptive antenna array (FDAAA) has been proposed and proved to be more effective to suppress MUI in severe frequency selective fading channel than the other frequency domain algorithms such as diversity combing algorithm. However, the performance of antenna array will depend on antenna placement, which determines the coupling and radiation pattern between antennas. In addition, the angle of arrival (AOA) of the receive signals’ waveform will also affect the performance [6, 7]. These considerations have not been addressed in our previous study and, to the best of our knowledge, have not been addressed in the literature.
On the other hand, cellular capacity is determined by the bandwidth efficiency and the available bandwidth. By increasing the frequency reuse factor (FRF), the distance between cochannel cells will increase, and the CCI power will be reduced. However, the available bandwidth is reduced at the same time. As a result, cellular capacity becomes a tradeoff between bandwidth efficiency and available bandwidth, which is determined by FRF. In our previous study [8], hybrid FRF has been proposed to optimize the cellular capacity. However, the optimal FRF depends on the channel propagation model and transceiver structure. And the cellular capacity optimization problem has to be re-formulated when FDAAA receiver is considered.
In this paper, we optimize the cellular capacity in SC uplink transmission using FDAAA receiver by taking into consideration the impact of antenna placement. The remaining of the paper is organized as follows: uplink FDAAA receiver for cellular system is introduced in Section 2; Hybrid FRF algorithm and analysis on cellular capacity is given in Section 3; In Section 4, the impact of antenna placement, that is, antenna separation and AOA spread, will be analyzed; Numerical results on cellular capacity will then be given in Section 5 and finally the paper will be concluded in Section 6.
2. Uplink FDAAA Receiver in Cellular System2.1. System Model
It is assumed that the base station (BS) locating at the center of each cell is equipped with Nr antennas, and there are U mobile stations (MSs, that is, users) in each cell and each user is equipped with a single transmit antenna, as shown in Figure 1. We assume that the 0th user is the desired user, and the other users are interfering users, it is also assumed that the channel remains unchanged during one block transmission. Two cellular structures are shown in Figure 2. On the left is the conventional cellular structure using single FRF (FRF 1), on the right is the cellular structure using hybrid FRF which will be further explained in Section 3.
Uplink transmission in a single cell.
Structure of single FRF and hybrid FRF.
The channel impulse response between the uth user and the BS can be expressed as
(1)hu(τ)=∑l=0L-1hu,lδ(τ-τl),
where hu,l and τl are the channel gain vector and time delay of the lth path, respectively, ∑l=0L-1E{|hu,m,l|2}=1 where hu,m,l is the mth element of hu,l and E{·} denotes statistical expectation. The path delay is assumed to be integer multiples of symbol duration and τl=l. CP is used and its length is assumed to be longer than the maximum path delay so that IBI can be avoided.
The baseband receive signal vector r(n)=[r0(n), r1(n),…,rNr-1(n)]T,(n=1,…,Nc) is given by
(2)r(n)=P0d0-α10-ξ0/10∑l=0L-1h0s0(n-τl)+∑u=1U-1Pudu-α10-ξu/10∑l=0L-1husu(n-τl)+∑i=0I-1∑u=0U-1Pu,idu,i-α10-ξu,i/10×∑l=0L-1hu,isu,i(n-τl)+z(n),
where U and I are the number of users per cell and the number of cochannel cells, respectively; Subscripts u and (u,i) represent the index of the uth user at the desired cell and at the ith cochannel cell, respectively; P represents the transmit power; s is the transmit signal; d represents the normalized distance between the user and the BS of the desired cell; α and ξ represent the path loss exponent and shadowing loss, respectively. z(n)=[z0(n)…zNr-1(n)]T is the vector of complex additive white Gaussian noise (AWGN). In this study, slow transmit power control (TPC) in each cell is assumed so that each user will have the same target receive signal power in average at the corresponding BS. Therefore, the transmit power is given by
(3)Pu=(Ptargetdu-α10-ξu/10)du,0-α10-ξu,0/10,
where Ptarget is the target receive signal power; The frequency domain received signal on the kth frequency is then expressed as
(4)R(k)=H0(k)S0(k)+∑u=1U-1Hu(k)Su(k)+∑i=0I-1∑u=0U-1Hu,i(k)Su,i(k)+Z(k),
where Hu=[Hu,0(k)Hu,1(k)…Hu,Nr-1(k)]T, Su(k), and Z(k)=[Z0(k)Z1(k)…ZNr-1(k)]T are, respectively, the frequency domain channel response, transmit signal, and noise component, given by (5). In the right hand side of (4), the first term comes from the desired user, the second term comes from MUI, the third term comes from CCI, and the last term is the noise component. (5)Su(k)=Pudu-α10-ξu/10∑n=0Nc-1su(n)exp(-j2πnkNc),Hu,m(k)=∑l=0L-1∑n=0Nc-1hu,l,mexp(-j2πnkNc),Zm(k)=∑n=0Nc-1zm(n)exp(-j2πnkNc).
2.2. Propagation Model of Adaptive Antenna Array
Linear antenna array is assumed, and the propagation model is shown in Figure 3. The geometric center of array is denoted by c and the antenna separation is denoted by d. θ0 represents the angle between line of sight (LOS) direction of MS and the BS array plane; The plane waveform of the lth path from the uth user is consisted of a number of unresolvable paths and the angle spread of the unresolvable paths is denoted by δ; In this study, δ is assumed to be zero for simplicity and hu,l represents the plane waveform of lth path from the uth user. The nominal AOA of hu,l observed at array center c is denoted by θu,l and the AOA spread of θu,l is uniformly distributed within a range of Δ. Therefore, the lth path gain of the uth user which is observed at the mth antenna element can be given by
(6)hu,l,m=hu,lexp(-j2π(0.5M-m+0.5)λdcosθu,l),
where m=1,2,3,…,Nr and λ is the carrier wavelength.
Propagation model of linear antenna array.
2.3. FDAAA Receiver
In our previous study, FDAAA receiver has been investigated in [5–7]. The transceiver structure of SC transmission using FDAAA receiver is shown in Figure 4. At the receiver side, the CP is removed and the receive signal at each antenna is transformed to frequency domain by using fast Fourier transform (FFT), then adaptive antenna array (AAA) weight control is then performed on each frequency and the output after AAA weight control is given by [9]:
(7)R^(k)=WFDAAAH(k)R(k),
where WFDAAA(k)=[WFDAAA,0(k),WFDAAA,1(k),…,WFDAAA,Nr-1(k)]T minimizes the mean square error (MMSE) between R^(k) and the frequency domain desired signal S0(k), given by [5–7]:
(8)WFDAAA(k)=X(k)-1p(k),
where X(k)=E{R(k)R(k)H} is the autocorrelation matrix of the received signal vector, p(k)=E{R(k)S0*(k)} is the cross-correlation between the receive signal and the reference signal, superscript H denotes Hermitan transposition, and * denotes the complex conjugate operation. It is assumed that the transmit signals from different users are independent and the noise component is also independent to them. The autocorrelation matrix X(k) is an Nr×Nr square matrix, and the (m,n)th element of X(k) is given by
(9)Xm,n(k)=E{Rm(k)Rn*(k)}=E{H0,m(k)S0(k)S0*(k)H0,n*(k)}+∑u=1U-1E{Hu,m(k)Su(k)Su*(k)Hu,n*(k)}+∑i=0I-1∑u=0U-1E{Hu,i,m(k)Su,i(k)Su,i*(k)Hu,i,n*(k)}+E{Zm(k)Zn*(k)}.
FDAAA uplink transmission.
Since channel state information (CSI) is known for the users within the cell of interest, (9) can be rewritten as
(10)Xm,n(k)=B0H0,m(k)H0,n*(k)+∑u=1U-1PMUI,uHu,m(k)Hu,n*(k)+∑i=0I-1∑u=0U-1diag[PCCI,u,i]Nr×Nr+σ2I,
where B0 is received signal power of the desired user; PMUI,u and PCCI,u represent MUI and CCI power from the uth user, respectively. Similarly, the mth element of the cross correlation matrix can be given by
(11)pm(k)=E{Rm(k)S0*(k)}=B0H0,m(k).
Data decision is then made based on the time domain signal estimate which is obtained by applying inverse FFT (IFFT) to the frequency domain signal component in (7), given by
(12)r^(n)=1Nc∑k=0Nc-1R^(k)exp(j2πknNc).
3. Hybrid FRF Algorithm and Capacity Analysis3.1. Hybrid FRF
Different from traditional cellular system which uses the same FRF for the whole cell, hybrid FRF algorithm adopts FRF = 1 and FRF = 3 adaptively to optimize the cellular structure [10]. According to the location and instantaneous channel status of a user, hybrid FRF algorithm uses FRF 1 for area near the cell center and FRF 3 for area near the edge cell. As a result, two data rates coexist within a cell. Since the same target receive power Ptarget is required for users with different data rates, the relation between energy and power is given as(13a)EsN0=Ptarget×TsN0,(13b)PtargetN0={BW×(EsN0)target,cell center,BW3×(EsN0)target,cell edge,where Ts is symbol period, and BW is the bandwidth. Receive power from the uth user at the desired BS and at the ith cochannel BS are given by (14) and (15), respectively.(14a)BuN0=(Ptarget/N0du-α10-ξu/10)du-α10-ξu/10,(14b)Bu={(Es)target×BW,cell center,(Es)target×BW3,cell edge,
(15a)Bu,iN0=(Ptarget/N0ru,i-α10-ηu,i/10)du,i-α10-ξu,i/10,(15b)Bu,i=(Ptargetr^u,i-α10-ηu,i/10)du,i-α10-ξu,i/10={((Es)target×BWru,i-α10-ηu,i/10)du,i-α10-ξu,i/10,cell center,((Es)target×(BW/3)ru,i-α10-ηu,i/10)du,i-α10-ξu,i/10,cell edge,
where ru,i represents normalized distance between the uth user and its corresponding BS in the ith cochannel cell.
3.2. System Capacity Analysis
Capacity is the highest rate at which information can be sent over the channel with arbitrarily small probability of error [11] and the relation between capacity (bps/Hz) and signal-to-noise-plus-interference ratio (SINR) is given by
(16)C=(1+log2 SINR).
In cellular system, carrier frequency/frequencies will be reused by neighboring cells. Taking FRF into consideration, cellular capacity in bps/Hz/BS is given by
(17)C-=1FRF(1+log2 SINR).
In hybrid FRF cellular system, since FRF 1 and FRF 3 are both used within a cell, cellular capacity depends on user’s location and (17) is rewritten as
(18)C-={log2(1+SINR),FRF1area,13log2(1+SINR),FRF3area.
In order to evaluate the cellular capacity, we are going to derive SINR in the next. For FDAAA, the signal power of array output can be calculated by
(19)PFDAAA=E{1Nc∑k=0Nc-1R^(k)R^*(k)}=1Nc∑k=0Nc-1WFDAAAH(k)[Rs(k)+RNI(k)]×WFDAAA(k),
where Rs(k) is the autocorrelation matrix of the receive signal from the desired user, and RNI(k) is the autocorrelation matrix of the receive signal from interfering users plus noise. Therefore, SINR can be obtained by
(20)SINR=powerofreceivedsignalpowerofinterference+noisepower=∑k=0Nc-1WFDAAA(k)Rs(k)WFDAAA(k)∑k=0Nc-1WFDAAAH(k)RNI(k)WFDAAA(k).
Note that interference power also depends on the users’ location and two cases should be considered:
Case 1 (<xref ref-type="disp-formula" rid="EEq1">1</xref>) [Desired user is inside FRF 1 area].
MUI power is given by
(21)PMUI,u={(Es)target×BW,cell center,(Es)target×BW3,cell edge,
and CCI power is given by
(22)PCCI,u={((Es)target×BWru,i-α10-ηu,i/10)du,i-α10-ξu,i/10,cell center,((Es)target×(BW/3)ru,i-α10-ηu,i/10)du,i-α10-ξu,i/10,cell edge.
Case 2 (<xref ref-type="disp-formula" rid="EEq2">2</xref>) [Desired user is inside FRF 3 area].
MUI power is given by
(23)PMUI,u={(Es)target×BW3,cell center,(Es)target×BW3,cell edge,
and CCI power is given by
(24)PCCI,u={((Es)target×(BW/3)ru,i-α10-ηu,i/10)du,i-α10-ξu,i/10,cell center((Es)target×(BW/3)ru,i-α10-ηu,i/10)du,i-α10-ξu,i/10,cell edge.
4. Impact of Antenna Placement
FDAAA receiver was proposed as a solution to combat MAI in frequency selective fading environment. In our previous study, it has been proved that when the antennas are considered to be uncorrelated with each other, FDAAA receiver has the ability to accommodate up to Nr users in a single cell and even in cellular environment when the FRF is big enough. However, the noncorrelation assumption is impractical and correlation often occurs and depends on the antenna placement in an array. To understand the impact of antenna placement, two parameters, antenna separation d and AOA spread Δ, are considered in this study. Equation (6) can be rewritten by
(25)hu,l=hu,lw(θu,l),
where w(θu,l) is the steering vector of the linear array, given by
(26)w(θu,l)=[w0(θu,l),w1(θu,l),…,wNr-1(θu,l)]T=[exp(-j2π(0.5Nr-0.5)λdcosθu,l),exp(-j2π(0.5Nr-1.5)λdcosθu,l),…,exp(j2π(0.5Nr-0.5)λdcosθu,l)]T.θu,l is uniformly distributed within a range of Δ and the probability density function of θu,l is given by
(27)f(θu,l)={1Δ;-Δ2+θ0≤θu,l≤Δ2+θ0,0;otherwise.
The spatial correlation between the mth and nth antenna elements can be calculated by [12–15]:
(28)Ds(m,n)=∫θu,lwm(θu,l)wn*(θu,l)f(θu,l)dθu,l=1Δ∫-(Δ/2)+θ0(Δ/2)+θ0exp(j2πm-nλdcosθu,l)dθu,l.
According to (28), the correlation between antenna elements as a function of AOA spread Δ as well as antenna separation d is calculated. The antenna correlation for θ0=60° is shown in Figure 5. It is shown that when d increases, the antenna correlation decreases with vibration and finally converges to zero when d becomes infinite. In the extreme case when d=0, all the antenna elements in the array become completely correlated. On the other hand, to increase the AOA spread Δ will speed up the convergence to zero. Therefore, in order to have less correlation between antennas, two possible ways are to increase d by occupying more space or to increase Δ by introducing more reflectors around the antenna array.
Antenna correlation for θ0=60°.
In addition, the antenna correlation will also be affected by angle θ0 and the relation between antenna correlation and θ0 is shown in Figure 6 where d=λ/2 is used. It is shown that the antenna correlation has the smallest value when θ0=90° or θ0=270°. In other words, in order to reduce the antenna correlation, the third way is to adjust the array plane to be vertical to the incoming waveform.
Relation between antenna correlation and θ0.
5. Numerical Result
In the next, we are going to study the impact of antenna placement on the uplink cellular capacity using FDAAA receiver following (18) by Monte Carlo simulations. The parameters to be used are listed in Table 1.
Simulation parameters.
Transmitter
Data modulation
QPSK
FFT size
Nc=256
TPC
Slow TPC
Number of user per cell
U=1~8
No. of CCI cells
I=18
Target receive Es/N0 per antenna
10 dB
Channel
Channel model
Frequency-selective block Rayleigh fading
Power delay profile
L=16 uniform power delay
Angle spread of resolvable paths (AOA)
Δ = 30°, 60°, 180°, 360°
Path loss exponent
α = 3.5
Standard deviation of shadowing losses
ξ = 6 dB
Channel State Information
Available only for user within the desired cell
Nominal angle
Random
Receiver
No. of antennas
Nr=8
Antenna separation
λ/2,λ,5λ,10λ
Channel estimation
Ideal
In order to calculate the capacity for hybrid FRF cellular system, the hybrid FRF scheme, that is, FRF 1 area and FRF 3 area allocation within each cell should be determined at the first place. The cellular capacity will then be calculated based on the hybrid FRF scheme. In order to optimize the capacity performance, hybrid FRF is determined as
(29)FRFhybrid=arg minFRF={1,3}1FRF(1+log2SINR).
For example, when antenna separation is d=λ/2 and AOA spread Δ = 360°, hybrid FRF scheme with varying number of users is shown in Figure 7 where the FRF 1 area and FRF 3 area are separated by the colored circular curves. It is natural to observe that the FRF 1 area decreases when the number of users in each cell increases in order to optimize the cellular capacity when CCI power increased.
Hybrid FRF scheme with varying number of users.
In the next, impact of antenna placement on the cellular capacity will be studied and our focus is on cellular outage capacity (the value that cellular capacity falls below with the outage probability) [16]. At first, the impact of AOA spread on cellular outage capacity is considered. In order to observe the effect of hybrid FRF algorithm, cellular capacity of single frequency reuse (FRF = 1) system is also calculated to make a comparison. The simulation results of 1% and 10% outage capacity are shown in Figures 8 and 9, respectively. It can be observed that the when AOA spread Δ increases from 30° to 360°, the cellular capacity increases for both hybrid FRF and FRF 1 cases. Recall that in Figure 5, we observed decreased antenna correlation when Δ increases. Actually, when d=λ/2 is used, FDAAA uses the correlation between antennas to generate beams in the directions of desired user and nulls in the directions of interfering users. When antenna correlation increases, the radiation pattern of the array will not be good enough and nonzero array gain will occur in the should-be-null directions, as shown in Figure 10. However, it is also observed that the capacity increase by increasing the AOA spread is quite limited, and the residue MAI should be the limiting factor.
Cellular outage capacity with 1% outage probability.
Cellular outage capacity with 10% outage probability.
FDAAA Radiation pattern change from Δ = 0° to Δ = 180° where nonzero array gain appears in should-be-null direction.
The impact of antenna separation d is considered in the following. Assuming AOA spread Δ = 30°, the cellular outage capacity is calculated for d=λ/2,d=λ,d=5λ and 10λ. The results corresponding to 1% and 10% outage probability are shown in Figures 11 and 12, respectively. It is observed that the cellular capacity can be obviously increased when the antenna separation increases. Recall that in Figure 5 we observed decreased antenna correlation when d increases and when d becomes larger than 5λ, antenna correlation drops to below 0.1, and the antenna elements can be treated as independent. In this situation, no beams or nulls will be generated (as shown in Figure 13) and diversity gain of multiple antennas can be utilized to combat MAI, and therefore maximize the achievable SINR.
Cellular outage capacity with 1% outage probability.
Cellular outage capacity with outage probability of 10%.
FDAAA radiation pattern when antenna separation d = 10λ.
In addition, it can be observed from Figures 8-9 and Figures 11-12 that the cellular outage capacity can be increased by using hybrid FRF and the increase in percentage is summarized in Tables 2 and 3 for d=λ/2 and Δ = 30°, respectively. It is now obvious that by using hybrid FRF, cellular outage capacity, especially when the outage probability is low, can be greatly increased by using hybrid FRF. As we know that outage capacity is usually contributed by the users near the cell edge and the quality of service (Qos) of these users always suffers from strong CCI. Therefore, hybrid FRF together with FDAAA receiver is an effective solution to improve the Qos of the cell edge users and therefore can help to improve the fairness among users as well.
Cellular capacity increase by using hybrid FRF (d=λ/2).
Outage probability
Capacity increase
Δ=30°
Δ=60°
Δ=180°
Δ=360°
1%
110%
129%
156%
168%
10%
19%
33%
61%
67%
Cellular capacity increase by using hybrid FRF (Δ=30°).
Outage probability
Capacity increase
d=λ/2
d=λ
d=5λ
d=10λ
1%
110%
142%
172%
174%
10%
19%
41%
68%
71%
6. Conclusions
In this paper, the impact of antenna placement on the FDAAA receiver in hybrid FRF cellular system has been studied. Two parameters, antenna separation and AOA spread, have been considered. Taking the hybrid FRF into consideration, cellular capacity is derived and the impact of antenna placement on cellular capacity is then evaluated. It has been shown that increasing the AOA spread can reduce the antenna correlation, and therefore can increase the cellular capacity by using FDAAA receiver. On the other hand, increasing the antenna separation to above 5λ will reduce the antenna correlation to almost zero and can greatly increase the cellular capacity. In addition, the comparison between hybrid FRF and FRF 1 has shown that hybrid FRF algorithm can effectively improve the cellular outage capacity, and therefore hybrid FRF together with FDAAA receiver is a good solution for uplink transmission in cellular system.
Acknowledgment
This work was supported in part by 2011 KDDI Research Grant Program (J110000208).
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