Degrees of Freedom in Wireless Networks

Transcription

1 Degrees of Freedom in Wireless Networks Zhiyu Cheng Department of Electrical and Computer Engineering University of Illinois at Chicago Chicago, IL 60607, USA Abstract This paper did a survey in the area of degrees of freedom (DoF) in wireless network. We give the definitions of degrees of freedom in different areas first. Then we introduce some methods used to determine the degrees of freedom in different scenario. In section IV, we discuss the applications of the degrees of freedom. We state some interesting open questions in the end of this survey. I. INTRODUCTION There is recent interest in the degrees of freedom in wireless networks. For example, message sharing, beamforming, zero forcing, [24] successive decoding, and dirty paper coding [4] techniques may be combined in many different ways across users, data streams, and antennas to establish inner bounds on the degrees of freedom. Besides, time, frequency, and space all offer degrees of freedom [8] [18], spatial dimendions are especially interesting for how they may be accessed with distributed processing. For the various multiuser MIMO systems, several researchers has determined the degrees of freedom. Background information can be found in [1] [11] [9] [23]. Degrees of freedom for linear interference networks with local side information are explored in [22] and cognitive message sharing is found to improve the degrees of freedom for certain structured channel matrices. This paper is a survey of the area of the degrees of freedom. The rest of the paper is organized as follows. Section II gives the definitions of degrees of freedom in different areas. Section III introduces how to determine the degrees of freedom in different scenario. In section IV, we discuss the applications of the degrees of freedom. Section V state some open questions about the degrees of freedom in wireless networks. II. DEFINITIONS OF DEGREES OF FREEDOM A. Definition in statistics In statistics, the number of degrees of freedom is the number of values in the final calculation of a statistic that are free to vary. The value of calculating a statistic of the number of independent variables is equal to the difference between the total number of data points under consideration, and the number of restrictions. The number of restrictions is equal to the number of parameters which are the same in both observed data set and theoretical data set, e.g. total cumulative values, means. Another definition of degrees of freedom in statistics is given by Lane, David. M. We know that estimates of parameters can be based upon different amounts of information. The number of independent pieces of information that go into the estimate of a parameter is called the degrees of freedom. In general, the degrees of freedom of an estimate is equal to the number of independent scores that go into the estimate minus the number of parameters estimated as intermediate steps in the estimation of the parameter itself. For example, if the variance, σ 2, is to be estimated from a random sample of N independent scores, then the degrees of freedom is equal to the number of independent scores (N) minus the number of parameters estimated as intermediate steps (one, λ estimated by R) and is therefore equal to N 1. Mathematically, degrees of freedom is the dimension of the domain of a random vector, or essentially the number of free components: how many components need to be known before the vector is fully determined. B. The maximum multiplexing gain In a wireless network with each node having a symmetric transmit power constraint ρ, we could define degrees of freedom as the maximum multiplexing gain for which a nonzero diversity gain is attainable. [17] [14] We define the DOF of a matrix H as the DOF of the channel y = Hx + w where w is distributed as N(0, I). Given a random matrix H, which is a function of random variables h 1, h 2,..., h N, we define the structural rank of H as the maximum rank attained by H under any possible assignment of h i and denoted by Rank(H) Consider a channel of the form y = Hx + w, where H is a fixed N N channel matrix, x, y, w are N length column vectors representing the transmitted signal, received signal and the noise vector respectively with the noise being white in the scale of interest. Then the degrees of freedom of this channel is given by D = Rank(H). C. Estimate the capacity In the estimation of capacity of wireless networks, we define degrees of freedom D [26] [13] as the ratio of the sum capacity C(P ) of the network to the log of the total transmit power P, in the limit that P, where the local noise at each node is normalized to have unit variance. D = lim P C(P ) log(p ) It is clear that degrees of freedom provide a capacity approximation C(P ) = D log(p )+o(log(p )) whose accuracy

2 approaches 100% as the total transmit power P approaches infinity. III. HOW TO DETERMINE THE DEGREES OF FREEDOM There has been a recent interest in determining the degrees of freedom [1] [15] of wireless multi-antenna networks, which is the maximum multiplexing gain. The DOF for the K user interference channel was derived in [27] and the degrees of freedom of single-source single-sink (ss-ss) layered networks was obtained in [28]. In another direction, the capacity of ss-ss and multicast deterministic wireless networks were characterized in [7]. Intuition drawn from the deterministic wireless networks were used to identify capacity to within a constant for some example networks in [6]. A similar approach was used in [12] for obtaining degrees of freedom for real gaussian interference networks. In the following subsections, we will discuss how to determine the degrees of freedom in K user interference channel, real gaussian interference networks, and single-source single-sink networks. [17] A. K user interference channel First, we consider the K user interference channel, comprised of K transmitters and K receivers. Each node is equipped with only one antenna. The channel output at the kth receiver over the tth time slot is described as follows: Y [k] (t) = H [k1] (t) + H [k2] (t)x [2] (t) H [kk] (t)x [K] (t) + Z [k] (t) where, k {1, 2,..., K} is the user index, t N is the time slot index, Y [k] (t) is the output signal of the kth receiver, X [k] (t) is the input signal of the th transmitter, H [kj] (t) is the channel fade coefficient from transmitter j to receiver k over the tth time-slot and Z [k] (t) is the additive white Gaussian noise (AWGN) term at the kth receiver. We assume all noise terms are independent identically distributed (i.i.d.) zero-mean complex Gaussian with unit variance. To avoid degenerate channel conditions (e.g., all channel coefficients are equal or channel coefficients are equal to zero or infinity) we assume that the channel coefficient values are drawn i.i.d. from a continuous distribution and the absolute value of all the channel coefficients is bounded between a nonzero minimum value and a finite maximum value, 0 < H min H [kj] H max <. We assume that channel knowledge is causal and globally available. Now, we can determine the degrees of freedom for the K user interference channel: T heorem1 : The number of degrees of freedom per user for this K user interference channel is K/2 max D d 1 + d d k = K/2 B. Real Gaussian interference channel In this subsection, we also consider the K user interference network first. Recall the input-output equation for K user interference network: Y [k] (t) = K j=1 H[kj] X [j] (t) + Z [k] (t) All symbols are real and the channel coefficients are fixed. The time index t is suppressed henceforth for compact notation. Note that we are only interested in constant channels as the channel coefficients are not a function of time. For such a constant interference channel with real and nonzero coefficients, we wish to find out if interference alignment can be accomplished in a manner that K/2 degrees of freedom may be achieved. We explore this issue by constructing an interference channel with non-zero channel coefficients that can achieve within a factor (1 ε) of upperbound of K/2 degrees of freedom. In the process, we demonstrate a new kind of interference alignment scheme, that is inspired by the deterministic channel model, but applicable to the AWGN interference network that we consider. We assume a transmit power contraint P at each transmitter so that: E[(X [j] ) 2 ] P The AWGN is normalized to have zero mean and unit variance. C(P ) is the sum capacity of the K user interference channel, and according to the definition in the previous section, the degrees of freedom are: D = lim P C(P ) 1 2 log(p ) The half in the denominator is because we are dealing with real signals only. For this channel model we have the following result: T heorem2 : Given any ε > 0, there exists a fully connected K-user Gaussian interference channel with constant and real coefficients that achieves K/2(1 ε) degrees of freedom. C. Single-source single-sink networks In this subsection, we determine the degrees of freedom for single-source single-sink networks. While for single-source single-sink networks with single-antennas, it is possible to attain the optimal degrees of freedom of 1 by activating one path between the source to the destination either using amplify-and-forward or a decode-and-forward strategy, the degrees of freedom optimal strategy becomes unclear when the number of antennas is greater than 1. Now we explain the amplify-and-forward strategy and the decode-and-forward strategy first. The amplify-and-forward strategy allows the relay station to amplify the received signal from the source node and to forward it to the destination station. The decode-and-forward strategy allows the relay station to

3 decode the received signal from the source node, re-encode it and forward it to the destination station. Now, consider, for example, a two-hop relay network in the absence of a direct link, where the source and sink have both n antennas and all the relays all have a single antenna. The optimal strategy is no longer to utilize one edge disjoint path at a time since the relay participating in the communication limits the degrees of freedom to 1. While this can be remedied by asking all relays to simultaneously forward their received data, for more-complicated multi-antenna networks, the optimal strategy is no longer obvious. T heorem3 : Given a single-source single-sink gaussian wireless network, with Rayleigh fading coefficients, the degrees of freedom of the network is given by D = min ω Ω Rank(H ω ) An amplify-and-forward strategy utilizing only linear transformations at the relays (that do not depend on the channel realization) is sufficient to achieve this degrees of freedom. IV. THE DEGREES OF FREEDOM IN WIRELESS NETWORKS In this section, we will explore some applications of the degrees of freedom and find out how some factors will influence the degree of freedom in wireless networks. A. Propagation delays and degrees of freedom We know that the degrees of freedom can be used to estimate the capacity of wireless networks, which is stated in the second section of this paper. Now we will discuss how can propagation delays influence the degrees of freedom. [26] First, we consider a K user interference network, i.e., an interference channel with K transmitters and K receivers. We write the signal input-output equations as: Y [i] (n) = n j=1 H[ij] X [j] (n) d [ij] + Z [i] (n) where at discrete time index n, Y [n] (n) is the received signal and Z [i] (n) is the additive noise at the ith receiver. X [j] (n) is the signal transmitted by the jth transmitter and H [ij] is the channel coefficient and d [ij] is the propagation delay between between transmitter j and receiver i. Note that this model assumes propagation delays are integer multiples of the basic symbol duration. Non-integer delays will be addressed subsequently in this work. If we set all delays d [ij] equal to zero, we obtain the classical Gaussian interference channel model. Now we discuss the impact on capacity of propagation. The question that we address is whether propagation delays can significantly impact the capacity of a wireless network. Indeed, for the two user Gaussian interference channel, one can verify that neither the innerbounds nor the outerbounds of Gaussian interference channel capacity are affected by the propagation delays. [25] Since the inner and outerbounds are shown to be within 1 bit of each other, one can conclude that propagation delays do not impact the capacity of the 2 user Gaussian interference channel by more than 1 bit. This observation suggests that perhaps propagation delays do not significantly impact the capacity of the Gaussian interference channel. However, as we show next through an example, this is not the case for more than 2 users. Consider a K user interference channel where all channel coefficients are equal to one and all propagation delays are equal to zero, i.e. H [i,j] = 1, n i,j = 0, i, j {1, 2,..., K}. On this interference channel, all receivers observe statistically equivalent signals. Therefore, if a message can be decoded by any receiver, it can be decoded by all receivers. Consequently, the sum capacity of this interference channel is the sum capacity of the multiple access channel from all transmitters to, say, receiver 1. This multiple access channel has only 1 degree of freedom, and therefore, the sum capacity of the interference channel is log(p ) + o(log(p )). Now, suppose with the same channel coefficients as before, we allow non-zero propagation delays. In particular, we assume: d [ij] mod2 = 0, i = j d [ij] mod2 = 1, i = j Thus, all desired signals arrive with an even propagation delay and all interfering signals arrive with an odd propagation delay at each receiver. On this interference channel, suppose all transmitters tranmit only over even time slots and are silent over odd time slots. Then, each receiver is able to hear his desired signals free of interference over the even time slots and all the interference is aligned over the odd time slots. Thus, even though all the channel coefficients are equal, this interference channel achieves a sum capacity of K/2 log(p )+ o(log(p )). We see through this example, that the presence of propagation delays can increase the degrees of freedom, and therefore the high SNR capacity by a factor of K/2 for the K user interference channel. B. Feedback, cooperation, relays, full duplex operation and the degrees of freedom In this subsection, we discuss how will feedback, cooperation, relays and full duplex operation influence the degrees of freedom in wireless networks. First, we generalize the interference network to the X network, a network where there is an independent message from every source node to every destination node. [3] It was shown that the S D X network - the X network with S source nodes and D destination nodes and messages - has a capacity of S+D 1 log(snr) + o(log(snr)), and it has S+D1 degrees of freedom. In order to discuss how will feedback, cooperation, relays and full duplex operation influence the degrees of freedom in

4 Fig. 1. The S R D network [20] wireless networks, we need to generalize the S D X network to the S R D network(figure 1). This network has S full duplex source nodes, R relays, D full-duplex destination nodes, perfect feedback to all source and relay nodes, noisy co-operation among all nodes. [20] The network is assumed to be fully connected, meaning that all channel gains are non-zero. The main result of this paper is that like S D X network, the S R D network also S+D1 has degrees of freedom. While, achievability follows trivially from the interference alignment based scheme of [3], the main contribution of [20] is the converse argument presented in Theorem 4 as follows. T heorem4 : Let D out = {[(d ij )] : (u, v) {1...S} {S + R + 1,..., S + R + D} S+R+D q=s+r+1 d q,u + S p=1 d v,p d v,u 1} Then D D out where D represents the degrees of freedom region of the S R D node X network. D out is the outerbounds. Furthermore, max [(dij)] D dij S+D 1 Equivalently, the sum-capacity C (ρ) may be expressed as C (ρ) = S+D 1 log(ρ) + o(log(ρ)) Therefore, in most cases, feedback, cooperation, relays and full duplex operation cannot increase the degrees of freedom in wireless networks. In other words, the search for improvements of the order of log(sn R) in most wireless networks ends in interference alignment. The techniques of relays, feedback to source/relay nodes, noisy co-operation and full-duplex operation can only improve the capacity upto a o(log(sn R)) term. There are, however, a few exceptions precluded by our system model where these techniques can improve the degrees of freedom. 1) Relays can improve the degrees of freedom if a network is not fully connected. 2) Co-operation can increase the degrees of freedom if the cost of co-operation is not accounted for (i.e. in genie aided networks). 3) Full duplex operation can increase the degrees of freedom if source nodes can also be destination nodes for other messages. 4) Feedback can improve the degrees of freedom if it is provided to a destination node, in which case, it behaves, in effect, like an extra antenna and can be used to null out the interference. C. Generalized degrees of freedom and interference alignment In this subsection, we will introduce the notion of generalized degrees of freedom first, and then discuss the role of interference alignment in wireless networks. Etkin, Tse, and Wang [25] introduced the notion of generalized degrees of freedom (GDOF) to study the performance of various interference management schemes. As its name suggests, the idea of GDOF is a generalization of the concept of degrees of freedom. Unlike the conventional degrees of freedom perspective where all signals are approximately equally strong in the db scale, the GDOF perspective provides a richer characterization by allowing the full range of relative signal strengths in the db scale. A useful technique in the characterization of the GDOF of a wireless network is the deterministic approach [5]. The deterministic approach essentially maps a Gaussian network to a deterministic channel, i.e, a channel whose outputs are deterministic functions of its inputs. The deterministic channel captures the essential structure of the Gaussian channel, but is significantly simpler to analyze. Reference [2] showed that the deterministic approach leads to a GDOF characterization of the 2-user interference network, which leads to a constant bit approximation of its capacity. The X channel is a generalization of the 2-user interference channel with 4 independent messages, one from each transmitter to each receiver. It is the smallest (in terms of the number of nodes) network where the newly discovered concept of interference alignment [19], [16] becomes relevant. Here we give the GDOF characterization for the X channel [21] and the interference channel [25] respectively. For the X channel, d(α) = lim sup ρ C (ρ,α) 1 2 log(ρ) where C (ρ, α) is the sum capacity of the X channel. Figure 2 is the 2-user Gaussian X channel, and the system model is

5 ACKNOWLEDGMENT I would like to thank Prof. Devroye for giving advice on this project. REFERENCES Fig user Gaussian X channel [21] Y 1 (t) = ρx 1 (t) + ρ α X 2 (t) + Z 1 (t) Y 2 (t) = ρ α X 1 (t) + ρx 2 (t) + Z 2 (t) where ρ, ρ α are the channel gain coefficients. For the interference channel, d(α) = lim sup SNR C (SNR,α) 1 2 log(snr) where C (SNR, α) is the sum capacity of the interference channel. In terms of GDOF, both the X channel and the interference channel perform equally well when interference alignment is not applicable. But when interference alignment is applicable, the X channel has larger GDOF and, therefore, higher capacity than the interference channel. V. CONCLUSION AND FUTURE WORK The area of degrees of freedom contains many open problems that appear to hold significant challenges. Now we state some interesting open questions. 1) Although we could use the degrees of freedom to estimate the capacity of certain channel, such as K user interferernce channel, the exact evaluation of capacity for the simplest relay networks remains open. 2) Another open question is although we know that in most cases, feedback, cooperation, relays and full duplex operation cannot increase the degrees of freedom of wireless networks, we could find some other techniques to improve the degrees of freedom. 3) We know that the distributed processing nature of the MIMO interference channel leads to the loss in the degrees of freedom. For example, while a (1, n, n, 1) interference channel has only one degrees of freedom, the point-to-point MIMO system with n + 1 antennas at both transmitter and receiver has 1 + n degrees of freedom. 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