Network Spectral Efficiency and Outage Probability Analysis of MIMO Ad-hoc Networks with Quantized Beamforming

Outage Probability Analysis for MIMO Ad-hoc Network with Quantized BeamformingG.Ananthi, S.J.Thiruvengadam

TIFAC CORE in Wireless Technologies

Department of Electronics and Communication Engineering

Thiagarajar College of Engineering, Madurai 625015, Tamil Nadu, India

E-mail:gananthi@tce.edu,sjtece@tce.edu

Abstract: In this paper, exact closed form expressions are derived for sum throughput/capacity and outage probability for Multiple Input -Multiple Output (MIMO) ad-hoc network with quantized beamforming and finite rate feedback. In quantized beamforming, each receiver sends the label of the best beamforming vector obtained from the codebook to the transmitter through a finite rate limited feedback. The code book is shared between each transmitter-receiver pair to prevent decoding of other users information in a MIMO ad-hoc network comprising of K simultaneous communicating transceiver pairs with each receiver employing single-user detection. In such a practical environment, feedback links can only convey finite number of bits, from which transmit beamformer designs are investigated using either the outage probability or average Signal to Interference Noise Ratio (SINR) as the figure of merit. Hence, by deriving closed form expressions for outage probability and network sum throughput interesting insights in the high SINR regimes are provided. In lieu of validating the obtained simulation results, theoretical results have also been compared. Index Terms - MIMO, Ad-hoc Networks, Network Sum Throughput, Outage Probability, Quantized Beamforming, Finite Rate feedback, Code Book Vector.1. IntroductionMultiple-Input and Multiple-Output (MIMO) wireless systems have attracted much interest since they significantly increase the capacity of band-limited wireless channels to meet the requirements of the future high data rate wireless communications [1],[2]. Research interests on ad-hoc networks have greatly increased in the past decade, mainly due to the challenging task of designing effective and distributed protocols that yield a good throughput, fairness and energy saving abilities [3],[4]. A mathematical model for determining the capacity of fixed multi-hop ad-hoc networks was introduced in [3]. Prior results on the capacity analysis typically focused on scaling laws of network throughput with an asymptotically large number of nodes. This paper focuses on mobile ad-hoc networks that have no fixed base station along with multiple pairs of users intending to communicate with each other. Employing multiple antennas in each transmit- receive pair has shown great promise in providing higher throughputs on wireless links in comparison to the traditional communication systems. However, practical limitations due to channel impairments and distributed access, may decrease the performance of protocols for ad-hoc networks [3],[4].

In [5], MIMO capacity with interference is studied assuming that single-user detection is employed at the receiver. Without channel state information at the transmitter(CSIT), it is shown that the transmitter should either put equal power into each antenna (weak interference mode) or the transmitter should put all the power on a single antenna (singular mode), depending on the interference to noise power ratio. This concept is extended to MIMO ad-hoc networks with simultaneous pair wise transmissions [6]. In MIMO ad-hoc networks without CSIT, the total capacity of an ad-hoc network is fundamentally limited by the receive antenna size and is independent of transmitting antenna size and transmit power, as the number of transmitter- receiver pairs increases [6]. However, with CSIT, throughput depends on both transmitting as well as receiving antenna size. Further, with perfect CSIT, transmit beamforming achieves best throughput [5]-[7]. But, in practical wireless systems, CSIT suffers from imperfections originating from various sources such as quantization effects, feedback delay and feedback errors. The effects of limited feedback in MIMO communication systems have been analyzed in [8], [9]. However, this concept cannot be directly extended to MIMO ad-hoc networks as it involves interference from the other transmit-receive pairs. Recently, a limited feedback beamforming concept for MIMO ad-hoc network was introduced in [10]. In this method, transmit beamformers of all transmit-receive pairs are selected based on a centralized optimization technique to maximize the sum rate. The performance of MIMO beamforming with quantized feedback in ad-hoc network is analyzed in [11]. A random vector quantization (RVQ) method is used to design the codebook and to derive the closed-form expressions for the transmission capacity. It is proved that a moderate number of feedback bits are sufficient to obtain significant transmission capacity gains compared to non-feedback schemes. Although many complex schemes are employed for better performance, RVQ method is used in ad-hoc network for analytical tractability. In [12], code book is designed.using Grassmannian beamforming criterion to minimize the average SNR reduction due to finite-rate feedback [13]. The closed-form expressions have been derived for capacity loss due to quantized feedback in MIMO ad-hoc networks. The objective of this paper is to study the impact of quantized beamforming with limited feedback based on Grassmannian criterion, on the network sum throughput/capacity and outage probability of MIMO ad-hoc networks, per unit time and unit bandwidth for a given user. The beamforming vector is quantized at the receiver using a fixed codebook designed based on Grassmannian beamforming criterion at both the transmitter and the receiver. The receiver is assumed to convey the index of the best beamforming vector from the codebook and is able to feed a finite number of bits back to the transmitter. It is assumed that a low-bandwidth, error-free, zero-delay feedback channel exists between the transmitter and the receiver. The closed form expressions are derived for network sum throughput and outage probability in MIMO ad-hoc networks.

This paper is organized as follows. Section II presents the system model of MIMO ad-hoc network. Section III analyzes outage probability performance of MIMO ad-hoc network with quantized beamforming and finite rate feedback. Simulation results are discussed in section IV and section V concludes the paper.Notations: Boldface capital letters denote matrices and boldface lower case letters denote vectors. , and denote the Hermitian (complex conjugate transpose) and determinant of matrix of. is a identity matrix ,denotes the channel matrix from the transmitter array to the receiver array, andrepresents the channel matrix from the transmitter array to the receiver array. In this paper, throughput and capacity are used interchangeably, as both refer to mutual information per unit time and unit bandwidth for a given system or user.2. System ModelConsider a MIMO ad-hoc network with user pairs. Each user pair is assumed to have transmit antennas and receive antennas. The received signal vector for user, in Rayleigh flat fading channel, is given by

where is user pair beamforming vector of size determined from . is a channel matrix, is the transmitted binary phase shift keying (BPSK) symbol of user, denotes the combined path loss/shadow fading for the user pair channel, is the transmit signal power, is the distance between transmitter and receiver and is the path loss exponent which can take value in the range 2-4 taking values between 2-4. Also, represents the channel matrix between user receiver and user transmitter. is user-pair beamforming vector of size determined from the

,is the transmitted symbol of interferer, denotes the combined path loss/shadow fading between user receiver and user transmitter, is the distance between the transmit and receive array of kth user pair, is the distance between user receiver and user transmitter and is (r x 1) independent and identically distributed (i.i.d) Gaussian noise vector with zero mean and covariance . In (1), the first term is desired signal and the second term represents the multibeam interference from other users. It is further assumed that each receiver implements single user detection.3. Outage Probability Performance with Quantized Beamforming and finite rate feedback

It is assumed that the receiver is able to feed a finite number of B bits back to the transmitter which is designed using Grassmannian beamforming codebook [9]. The feedback link is considered as error free and delay free. With B bits, each transceiver pair has a total of beamforming vectors. The beamforming vector for kth user, is selected such that the capacity is maximized [9]. It is given by,

The received signal vector for kth user is,

The network sum throughput for transmitter-receiver pairs is derived as

Singular Value Decomposition (SVD) of is written as . Thus, where is a unitary matrix and is a diagonal matrix of eigen values, of . Let,, be the eigen vectors corresponding to , respectively. The entries of are assumed to be i.i.d Gaussian with zero mean and unit variance. The statistical average of is given by

Using [9], can be written as

whereand are the maximum eigen value and corresponding eigen vector of respectively. In (6) , first factor indicates the channel quality on average for K users, while the second factor is an indication of the beamforming codebook quality. Substituting (6) in (4), the capacity is expressed as,

It can be noticed that the quantized beamforming vector does not coincide with the eigen vector corresponding to the largest eigen value of the matrix. It leads to quantization distortion measure which is the difference between the perfect channel estimate network sum throughput and quantized network sum throughput [13].

Assuming that a maximal ratio combiner (MRC) is employed at the receiver, the MRC weight vector is given by

is a quantized beamforming vector selected from the codebook according to the Grassmannian beamforming criterion given in . With these weights, the received signal is given by

Where is the received signal vector with quantized beamforming in and substituting this expression in, it is expressed as

whereis the quantized beamforming vector calculated from , is the transmitted symbol of user, is user-pair beamforming vector determined from andis a transmitted symbol of interferer. The received SINR of the MRC output is calculated as

Using, the desired signal term is approximated as [8],[13] and [14].Here,is the eigen vector corresponding to the largest eigen value of using number of transmit antennas and is the codebook size designed using Grassmannian Beamforming criterion[9]. Further, is approximated as. Interference term is approximated as. Substituting all the above said approximations in equation, the received SINR is written as

where is a sum of independent exponential random variables and is treated as an interference from other user pairs. For simplicity, the quantization error parameter is given as. Conditioning on, can be written as

The cumulative distribution function (CDF) of is given by

where and, where refers to the number of transmitting antennas and is the number of receiving antennas. is a gamma function, is a Hankel matrix of size given by

Here, which is an incomplete gamma function, when is a positive integer. It can be observed that the distribution of depends only on and.

Now, solving for the determinant in, the distribution is represented by the following polynomial

where depends only on and. The values of coefficients are obtained fromand. Applying the polynomial of and using binomial theorem, can be rewritten as

Then, the expression for outage probability is obtained by averaging over the distribution of. It is given by,

It shows that the outage probability for a MIMO ad-hoc network depends on the probability density function (PDF) of the multi-user interference power.

The characteristic function of interference power in a MIMO ad-hoc network is expressed as in [15-17]. Where refers to as effective node density and for ground propagation. The closed-form expression for PDF of interference power is given by [17]

Substituting and the integral expression [18] in the closed-form expression for outage probability for MIMO ad-hoc networks is expressed as

where the parameters are given by ,

,is the Bessel function of Imaginary argument.

4. Results and DiscussionIn this section, the impact of quantized beamforming with finite rate feedback on network sum throughput and outage probability is investigated by simulations. We analyze two different schemes such as transmit beamforming with perfect CSIT and quantized beamforming with finite rate feedback. Simulation results are obtained, with the assumption that the channel matrix from the th user transmitter array to the th user receiver array consists of i.i.d. complex Gaussian entries with zero mean and unit variance. The channel matrices are independent across different transmit-receive pairs. The path loss/shadowing effect is summarized using the coefficient following lognormal distribution. The simulation results correspond to the results obtained in simulated ad-hoc transmit receive pairs with Grassmannian codebook. The outage probability performance is simulated for various values of SINR and is compared with the analytical results. The sum throughput is obtained by averaging over 50 sets of independently generated channel matrices and shadowing coefficients for all transmitreceive pairs.

Fig.1, shows the performance of sum throughput as a function of user pairs for transmit beamforming with perfect CSIT and quantized beamforming with finite rate feedback. The code book size for the quantized beamforming is assumed to be 4. The sum throughput for quantized beamforming is always less compared to transmit beamforming with perfect CSIT. Fig.2, shows the performance of sum throughput as a function of SINR when 4 user pairs are active. Each user pair is assumed to have 2 transmit antennas and 2 receive antennas. The code book size for the quantized beamforming is assumed to be 4.It is observed that sum throughput for both quantized beamforming and transmit beamforming with perfect CSIT gets increased substantially when SINR increases. Fig.3, shows the outage probability performance of perfect CSIT and quantized beamforming with finite rate feedback for codebook sizes of and . The data rate of the system is set at 2 bits/s/Hz. As the size of the codebook increases, the outage probability decreases. The simulation results are obtained by modelling the system as in (6) and the theoretical approximation results are obtained using (20) directly. Fig. 4, shows the outage probability performance of a MIMO ad-hoc networks for Quantized beamforming with finite rate feedback for transmit-receive antenna configurations such as (2,2),(2,4),(4,4). The data rate is assumed to be 2 bits/s/Hz and the code book size is 256. As the antenna size increases, the outage probability performance of the system improves. This is due to the improvement in the inherent diversity/array gain of the system. Fig. 5, shows the outage probability performance as a function of data rate for perfect CSIT and quantized beamforming with finite rate feedback of codebook size and Transmit Receive pairs K=2 at a fixed SINR of 10dB. Further, it is observed that, as the distance between each transmitter and each receiver increases by 2 times, outage probability increases from 0.15 to 0.5.5. Conclusion

The closed form expression for network sum throughput/outage probability of MIMO ad-hoc network with quantized beamforming and finite rate feedback has been derived. The outage probability performance of the MIMO ad-hoc network has been analyzed with respect to the parameters such as the size of codebook, size of transmit & receive antenna, number of user pairs and the distance between the transmit receive pairs. The code book is designed using the concept of Grassmannian criterion and it is assumed that the feedback link from the receiver to the transmitter is error and delay free. The simulation results presented in this paper confirm the accuracy of the obtained analytical results. In future, the interference from the other user pairs may be considered for feedback link to obtain more practical solutions.

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Fig. 1. Sum Throughput of a MIMO ad-hoc network with t = r =2, P= 2, N=4 and =1

Fig. 2. Sum Throughput of a MIMO ad hoc network with , N=4 and =1

Fig. 3. Outage Probability Performance of a MIMO ad hoc network with and =1, Data rate=2bits/s/Hz.

Fig. 4. Outage Probability Performance of a MIMO ad-hoc network for antenna configurations (t,r) : (2,2),(2,4),(4,4), K=2 and =1, Data Rate=2bits/s/Hz and codebook size N= 28 .

Fig. 5. Outage Probability Performance of a MIMO ad-hoc network with and =1, K=2 ,SINR=10dB and codebook size N= 4 for different transmit and receive distances. EMBED Equation.DSMT4

EMBED Equation.DSMT4

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