Optimality Certificate of Dynamic Spectrum Management in Multi-Carrier Interference Channels

Transcription

1 Optimaity Certificate of Dynamic Spectrum Management in Muti-Carrier Interference Channes aschais Tsiafais, Chee Wei Tan, Yung Yi, Mung Chiang, Marc Moonen Eectrica Engineering, Kathoiee Universiteit Leuven, Begium Eectrica Engineering, rinceton University, USA Abstract The muti-carrier interference channe where interference is treated as additive white Gaussian noise, is a very active topic of research, particuary important in the area of Dynamic Spectrum Management (DSM for Digita Subscriber Lines (DSL. Here, mutipe users optimize their transmit power spectra so as to maximize the tota weighted sum of data rates. The corresponding optimization probem is however nonconvex and thus computationay intractabe, i.e. a certificate of goba optimaity requires exponentia time compexity agorithms. This paper shows that under certain channe conditions, this nonconvex probem can be soved in poynomia time with a certificate of goba optimaity. The channe conditions are discussed consisting of different interference modes incuding synchronous and asynchronous DSL transmission. Simuations demonstrate its appicabiity to reaistic DSL scenarios. I. INTRODUCTION Digita Subscriber Line (DSL technoogy remains the number one choice of broadband access technoogy in the word []. The corresponding increased demand for greater bandwidth and faster connection has ed to severa technoogica approaches to mitigate the channe impairments. This has become one of the most important appications of the muticarrier interference channe mode in today s Internet. One of the major imitations of DSL performance today is crossta among different ines operating in the same cabe bunde. Dynamic Spectrum Management (DSM refers to a set of soutions to the crossta probem. Basicay these soutions consist of signa eve coordination and/or spectrum eve coordination. For exampe, receiver and transmitter signa eve coordination correspond to a mutipe access channe and broadcast channe respectivey. In this paper the focus is on spectrum eve coordination, which is aso referred to as spectrum management, spectrum baancing, or muti-carrier power contro. Spectrum eve coordination corresponds to a muti-carrier interference channe where a number of different users try to communicate their separate information over a couped (wired channe. The capacity region of the interference channe is sti an open probem, but for practica systems where the crossta channes are typicay weaer than the direct channe, treating interference as additive white Gaussian noise has been the most practica communication strategy in operationa networs. In this case the transmit power spectrum of each user is designed so as to minimize interference to other ines, whie maintaining a satisfactory data rate. Such a technique reduces the impact of crossta and significanty improves the overa data rate beyond those of the current approach of static spectrum management where fixed over-conservative transmit power spectra are used. The probem of optimay choosing the transmit power spectra in order to maximize the tota weighted sum of data rates of the users turns out to be a nonconvex optimization probem. Severa soutions have been proposed ranging from centraized to distributed agorithms, e.g., [] [3] [4] [5] [6]. However, there is an undesirabe gap in the anaysis of these DSM agorithms: we either have agorithms with optimaity certificate but exponentia running-time, or poynomia-time soutions with ony numerica verification of sma suboptimaity gap but no theoretica characterization of optimaity condition. Indeed, on the one hand, poynomia-time agorithms ie IW [5], ISB [3] and ASB [4] have ony sufficient conditions for convergence but not for optimaity. On the other hand, Optima Spectrum Baancing (OSB [], Branch and Bound Optima Spectrum Baancing (BB-OSB [7] and a prismatic branch and bound agorithm (BB [8], provide a certificate of finding the gobay optima soution, up to a certain accuracy depending on the used discretization of the search space. Unfortunatey, the compexity of these agorithms grows exponentiay with the number of users. Furthermore most of the aformentioned agorithms assume that the number of frequency tones is infinitey arge so that the Lagrangian duaity gap can be viewed as zero [9]. In this paper, we give the conditions on the channe, noise, and power constraints, under which this difficut nonconvex probem of DSM can be soved in poynomia time with an optimaity certificate. To the best of our nowedge, our resuts provide the first sufficient condition for optimaity for a finite number of carriers, and it is appicabe for a genera mode incuding synchronous as we as asynchronous [0] DSL transmission. There are two ey innovations in our proofs: the use of geometric programming (G and M-matrix theory. Furthermore, the optimaity anaysis further eads to a new DSM agorithm, with simuations demonstrating the appicabiity of this optimaity certificate to reaistic DSL scenarios. The foowing notations are used. Bodface uppercase etters denote matrices, bodface owercase etters denote coumn vectors and itaics denote scaars. Furthermore x y denotes componentwise inequaity between vectors x and y and X Y denotes componentwise inequaity between matrices X and Y. We aso et (x denote the th eement of x and [X] ij denote the eement of matrix X on row i and coumn j. We denote the identity matrix by I and the spectra radius of T

2 by ρ(t. II. SYSTEM MODEL Most current DSL systems use Discrete Muti-Tone (DMT moduation. The basic idea of DMT is to spit the avaiabe bandwidth into a arge number of frequency bands, aso caed tones. For each tone, a transmit power can be aocated individuay which corresponds to a number of transmitted bits. The tota data rate for each user is then obtained by adding its transmitted bits over a tones. In our mode, no signa coordination is assumed between the transmitting and/or receiving modem for each user. Each user regards the signas from the other users as noise. The ony degree of freedom is choosing the transmit powers of the different users over the different tones. The transmit power is denoted as s n, where n is the user index ranging from to N and is the tone index ranging from to K. The noise power is denoted as σ n. The vector containing the transmit powers of user n on a tones is s n [s n, s n,..., s n K ]T. h denotes the squared channe gain magnitude from user m into user n on tone, where it refers to the squared channe gain magnitude of ine n when m = n. Furthermore int n is the interference caused to user n on frequency tone. The achievabe bit oading for user n on tone can then be expressed as b n og ( + s n Γ(int n + σ n bits/ s/ Hz, ( where Γ denotes the signa-to-noise ratio (SNR gap to capacity, which is a function of the desired bit error ratio (BER, the coding gain and noise margin []. The DMT symbo rate is denoted as f s. The tota data rate for user n and the tota power used by user n are then, respectivey, given by R n = f s b n and n = s n. ( Such a mode can be used to anayze the foowing specia cases. A first case is synchronized DSL transmission. Here it is assumed that a users are aigned in frequency so that each tone is capabe of transmitting data independenty from the other tones. The interference caused to user n on tone can then be expressed as int n = m n h s m. A second case is the asynchronous DSL transmission case [0]. Here there is not ony crossta within one tone but aso between different tones of different users. The interference caused to user n on tone can then be expressed as int n = m n ( K p= h,p sm p where h,p refers to the squared channe gain magnitude from user m tone p into user n tone. A third case is a genera case that incudes a the previous cases and equas int n = m= ( K p= h,p sm p hn,n,, where is the squared channe gain magnitude from user m tone p into user n tone. A these cases can be summarized with the foowing bit oading expression for user n on tone h,p b n = og ( +, sn N m= K p= h,p sm p hn,n, sn +Γσn. (3 For each of the above cases the parameters h,p have their specific vaues. In the rest of this paper we wi focus on this genera case (3 so that the resuts can be used for a three cases. Note that for notationa simpicity, we absorb Γ into the definition of h,p. III. SECTRUM MANAGEMENT ROBLEMS The ey goa in this text is to maximize the data rates of the bunde of interfering DSL ines. To this end, the objective is to find the optima transmit power spectra maximizing a weighted sum of data rates subject to power constraints. Assuming a per-user tota power constraint n for each user n, this is formuated as the foowing probem: max s,...,s N n= w nr n s.t. K = sn n, n, (4 s.t. 0 s n, n,. Instead of per-user tota power constraints, we wi consider one tota power constraint on a users, where = n= n. Using ( this can be formuated as foows: max s,...,s N n= w K nf s = bn s.t. K = n= sn, (5 s.t. 0 s n, n,, where b n is given by the genera interference mode (3. Note that (4 and (5 are both nonconvex, and thus intractabe. Aso note that in (5 the inequaity tota power constraint can be repaced by an equaity. This is justified by emma. Lemma : At optimaity, we have K = n= sn = in (5. roof: Suppose the optima soution to (5 resuts in a tota power t where t <. As this is the optima soution it means that no better soution can exist. This is not true because by scaing a the powers s n by a factor / t, the objective function in (5 increases. Thus, at optimaity, t = in (5. Note that robem (5 can be seen as a reaxation of probem (4. For symmetric crossta scenarios with not too arge crossta (cf. section IV the optima soution of (5 aocates equa powers to a users and so soving (5 and (4 eads to exacty the same soution, i.e. zero reaxation gap. For asymmetric crossta scenarios, the users wi not necessariy be aocated the same amount of tota transmit powers and there may be a reaxation gap between the soution of (4 and (5. The soution of (5 can however provide a usefu upper bound for probem (4. Finay, we note that the nonconvex spectrum management probem with one tota power constraint on a users (5 has aso been considered in other wor []. IV. OTIMALITY CERTIFICATE FOR OLYNOMIAL TIME SECTRUM MANAGEMENT A. oynomia time spectrum management Since (5 is nonconvex, it can have many ocay optima soutions depending on the vaues of the channe and noise coefficients. In order to sove this nonconvex probem with a certificate of goba optimaity, i.e., the soution is gobay optima, one has to resort to exponentia time compexity

3 3 agorithms such as [] []. In Theorem, we provide sufficient conditions on the channe and noise coefficients, and the tota power constraint for which (5 can be soved in poynomia time with a certificate of goba optimaity. In prior to describing Theorem, we first define the matrix B as foows: [B] d, = ( h,p + Γσn. (6,, where = n + ( N, d = m + (p N. Theorem : For the genera interference system mode (3, (5 can be soved optimay in poynomia time when B is nonsinguar and C I B 0. (7 roof: We start from b n of (5 by incorporating the tota power equaity constraint K = n= sn = (see Lemma This eads us to the foowing equivaent optimization probem: as foows: ( min b n = og h + n,n y ((Cy y w, sn N Nm= Kp= s.t. (Cy s m K m= p= h,p sm p hn,n, sn +Γσn p y, =,..., L (9 (8 y =. This is reorganized as foows: This can be reformuated as the foowing standard G [4]: K b n m= p= ( h,p min + Γσn y,t (t w,, = og sm p. (9 s.t. (Cy y t, =,..., L K m= p= ( h,p + Γσn,, sm p s n (Cy y (0, =,..., L y =. Using the symbo simpified by b n = og,p = ( h,p, ( K m= p= m= K p= + Γσn,,p sm p,p sm p, this can be s n. (0 As it is not so easy to represent four-dimensiona matrices, we go to a two-dimensiona matrix by a bijection map {, } {n, m,, p} as foows: N K n= = = n + ( N d = m + (p N [B] d, =,p L=KN,p sn = = Now, (5 can be reformuated as foows: L=KN ( L q= max s...s L = w og [B],q sq s.t. L = s = s.t. s 0 ( [B] d, s, ( Lq= [B],q s q s (3 where w = w rem(,n+ and rem(a, b refers to the remainder after dividing a by b. By using condition (7 we can mae the foowing change of variabes: = y = Bs, s = B y = y Cy. (4 If (4 is substituted in (3 the cost function reduces to L L w og(y /(Cy = og( (y w /(Cy w, (5 and the ast constraint to y (Cy, =,..., n. (6 We thus obtain the foowing optimization probem: min y ((Cy y w s.t. (Cy y, =,..., n (7 (B y =. Note that (5 and (6 are homogeneous in y (and so in s. Thus the constraint (B y = or, equivaenty, L s = acts as a normaization instead of a constraint. As a consequence it can be repaced by another normaization on y without changing the optima vaue of (7, e.g., y =. (8 For a particuar obtained y for (0, we can recover the optimizer s of (5 by first performing s = (I Cy ( and then normaize (scae this s such that it satisfies T s =. So when condition (7 is satisfied, the soution of the nonconvex spectrum optimization probem (5 can be found by soving a G given by (0. A G can be turned into a convex optimization probem so that a oca optimum is aso a goba optimum. Furthermore the goba optimum can be computed very efficienty. Numerica efficiency hods both in theory and in practice: interior point methods appied to G have provaby poynomia time compexity [5]. This concudes the proof of Theorem. B. Optimaity Anaysis In section IV-A, we provided condition (7 under which nonconvex probem (5 can be soved in poynomia time with a certificate of goba optimaity. In this section we wi further study this optimaity condition. To this end, we wi first introduce some definitions from matrix theory. Definition : Any matrix Q of the foowing form: Q = ti T, t > ρ(t, T 0, ( is caed an M-matrix [3]. Definition : A nonsinguar nonnegative matrix Q is said to be an inverse M-matrix if Q is an M-matrix. Using the above definitions, condition (7 can be reformuated as foows: B is an inverse M-matrix with t = where

4 4 t is defined in Definition. Generay this means that the offdiagona eements of B have to be negative and the diagona eements of B have to be smaer than. Characterizing a matrices whose inverses are M-matrices is an active research fied in the mathematica society, but there are ony imited resuts, even for symmetric matrices [6]. In [6] one specia type of matrix is introduced, caed generaized utrametric matrix, which is an inverse M-matrix. However this matrix does not necessariy satisfy the extra condition that t =. Thus for an N-user case with K frequency tones, it is difficut provide an exact meaning of these conditions on the channe and noise coefficients. However, for a -user case with a singe frequency tone, we can deveop a sufficient condition on the channe and noise coefficients, and the tota power, in order for the condition (7 to hod, i.e., B is an inverse M-matrix with t =. It aso gives us an intuitive understanding of condition (7 for the -user case. Theorem : For a -user DSL scenario with a singe frequency tone, if Γh Γh +Γh Γσ Γσ ( +Γh min h Γσ, Γσ h, (3 then condition (7 is satisfied. roof: For a -user DSL scenario with a singe frequency tone, B is a matrix. Now, (7 can be expressed in terms of the eements of B as I B 0 b b b b b b b b b b b b b b b b b b b b 0. Since the eements of B are a positive, (4 impies (4 max(b, b b b b b, (5 which can be rewritten in terms of the channe coefficients, noise and tota power : Γh Γσ Γh +Γh Γσ ( +Γh min h Γσ, Γσ h. Fig.. (6 Intuitivey, condition (7 is satisfied if both the crossta (i.e. h, h and the tota power constraint are not exceedingy arge. The same intuition can be expected to hod for the case where there are mutipe users and frequency tones, where deveoping mathematica conditions is eft as a future wor. C. Agorithm DSM-G Next, we turn from the optimaity anaysis based on G and M-matrix theory to the design of our DSM agorithm, Agorithm DSM-G. Given the channe, noise and power constraint parameters of the DSL scenario, (7 in Theorem can be checed easiy. If (7 in Theorem is true, we can sove (5 with goba optimaity by first soving (0 and then transforming bac to the transmit powers ( after the renormaization step. Suppose that (7 in Theorem is vioated, but there are ony a sma percentage of negative C ij eements. In this case, we can set the negative eements to be zero, resuting in the matrix C ij = max{c ij, 0}. Intuitivey, soving (0 using C shoud sti ead to a near-optima performance when the percentage of negative eements is reasonaby sma. This intuition is shown to be very usefu in the next section, athough proving the continuity of optimized objective function in the perturbation of C ij remains an open issue. The obtained soution can then projected bac to the feasibe domain of (5 by using proportiona adjustment of the transmit power to satisfy Lemma. However, the objective in (5 thus obtained using C is no onger gobay optima, and does not upper bound the goba optima objective of (4. We propose the foowing agorithm to sove (5: Agorithm DSM-G Agorithm Step Cacuate C = I B. Step Force negative eements of C to zero eading to C. Step 3 Sove (0 using a G sover (e.g., [4] with C. Step 4 Transform soution y to transmit powers s using s = (I Cy where negative eements of s are forced to zero. Step 5 Scae s proportionay such that the tota power of a users is equa to (cf. Lemma V. SIMULATION RESULTS In this section simuation resuts wi be shown for Agorithm proposed in section IV for synchronous as we as asynchronous symmetric N-user ADSL downstream scenarios. Centra Office... Symmetric N-user ADSL downstream scenario The ADSL downstream scenario is shown in Figure. The simuations are performed for a two-user case (N = up to an eight-user case (N = 8. The four-user scenario, for exampe, consists of active users,,3,4 where users 5,6,7,8 are inactive. The twisted pair ines have a diameter of 0.5 mm (4 AWG. The maximum transmit power is 0.4 dbm [7]. The SNR gap Γ is.9 db, corresponding to a coding gain of 3 db, a noise margin of 6 db and a target symbo error probabiity of 0 7. The tone spacing f is 4.35 Hz. The DMT symbo rate f s is 4 Hz. The simuations are performed in Matab on a TraveMate 400WLMi with 768 MB of RAM and an Inte entium M processor.60 GHz. In Tabe I the simuation resuts are shown for the synchronous DSL transmission case, i.e. no intercarrier interference (ICI. The first coumn denotes the number of users. The second coumn denotes the performance of Agorithm with respect to the goba optima agorithm OSB with a very fine granuarity. The third and fourth coumn denote the N

5 5 number and percentage of negative C ij eements for condition (7 respectivey. The simuation time of OSB requires exponentia compexity, i.e. 3 minutes, 3 hours and 5 days for,3 and 4 users scenarios respectivey. For more than four users it is infeasibe to execute within acceptabe time. Therefore the performance for more than 4 users can not be compared to OSB. The simuation time of Agorithm just requires a few seconds even for the 8-user scenario. Note that condition (7 is satisfied up to the 6-user scenario. This means that Agorithm succeeds in finding the optima transmit spectra with a certificate of goba optimaity for up to the 6-user scenario in a few seconds. Athough the conditions are not satisfied for the 7- and 8-user scenarios, it has been verified that Agorithm eads to near-optima transmit spectra. That is because the percentage of negative eements is extremey sma and forcing these to zero does not significanty change the probem. However there is no guarantee of goba optimaity. As the percentage of negative C ij eements increases one can expect suboptima transmit spectra. In Tabe II the simuation resuts are shown for the asynchronous DSL transmission case [0], i.e. with intercarrier interference (ICI. Condition (7 is satisfied up to the 6-user scenario. For the 7- and 8-user scenarios there is a sma percentage of negative C ij eements. This percentage is arger than for the synchronous case because there is a arger amount of crossta caused by ICI. As there exists no gobay optima method for asynchronous DSL transmission, the performance of Agorithm cannot be compared to such a benchmar. However for the - up to the 6-user case we sti have a certificate of goba optimaity for Agorithm. This shows another powerfu appication of an optimaity certificate in DSM agorithm anaysis. In fact, since the channe conditions are symmetric, the optima soution computed using Agorithm for (5 is equa to (4. TABLE I ANALYSIS ERFORMANCE ALGORITHM : SYNCHRONOUS DSL TRANSMISSION Users erf. wrt OSB Nb. neg. C eem. erc. neg. C eem. 00.0% % % / / / % 8 / % TABLE II ANALYSIS ERFORMANCE ALGORITHM : ASYNCHRONOUS DSL TRANSMISSION Users Nb. neg. C eem. erc. neg. C eem % % VI. CONCLUSION Maximizing the weighted date rate of a users for a muticarrier interference channe is an important probem that finds appication in DSL. DSL spectrum management has attracted many researchers who produced a arge variety of agorithms. However, since the probem is nonconvex, a the DSM agorithms that produce an optimaity certificate require exponentia-time compexity. Aso, most previous suboptima DSM heuristics give ony sufficient conditions for convergence to a suboptima soution. This paper everages on G and M-matrix theory to generate a poynomia-time optimaity certificate for a set of conditions on channe, noise and power constraints. Our resuts shed new ight on the structure of power aocation across mutipe carriers in DSL interference channes. Furthermore, we propose Agorithm DSM-G that can compute the goba optima transmit power aocation. This paper provides another step towards understanding the tradeoff between DSM agorithm s compexity and performance guarantee. REFERENCES [] DSL Forum. News Reease - DSL Dominates Goba Broadband Subscriber Growth. Technica report, March 007. [] R. Cendrion, W. Yu, M. Moonen, J. Verinden and T. Bostoen. Optima Muti-user Spectrum Management for Digita Subscriber Lines. IEEE Trans. Comm., May 006. [3] R. Cendrion, M. Moonen. Iterative Spectrum Baancing for Digita Subscriber Lines. In Internationa Communications Conference (ICC, May 005. [4] R. Cendrion, J. Huang, M. Chiang, M. Moonen. Autonomous Spectrum Baancing for Digita Subscriber Lines. IEEE Transactions on Signa rocessing, 55(8:44 457, August 007. [5] W. Yu, G. Ginis and J. Cioffi. Distributed Mutiuser ower Contro for Digita Subscriber Lines. IEEE J. Se. Area. Comm., 0(5:05 5, Jun. 00. [6]. Tsiafais, M. Moonen. Low-Compexity Dynamic Spectrum Management Agorithms for Digita Subscriber Lines. In Accepted for IEEE Internationa Conference on Acoustics, Speech and Signa rocessing (ICASS, Apri 008. [7]. Tsiafais, J. Vangorp, M. Moonen, J. Verinden. A Low Compexity Optima Spectrum Baancing Agorithm for Digita Subscriber Lines. Esevier Signa rocessing, 87(7, Juy 007. [8] Y. Xu, S. anigrahi, T. Le-Ngoc. A Concave Minimization Approach to Dynamic Spectrum Management for Digita Subscriber Lines. In IEEE Int. Conf. on Comm., voume, pages 84 89, June 006. [9] W. Yu and R. Lui. Dua Methods for Nonconvex Spectrum Optimization of Muticarrier Systems. IEEE Transactions on Communications, 54(7, Juy 006. [0] V. M. K. Chan, W. Yu. Mutiuser Spectrum Optimization for Discrete Mutitone Systems with Asynchronous Crossta. accepted in IEEE Transactions on Signa rocessing, 007. [] T. Starr, M. Sorbara, J. M. Cioffi, and. J. Siverman. DSL Advances. rentice Ha, USA, st edition, 003. [] Y. Xu, S. anigrahi, T. Le-Ngoc. Optima Spectrum Management with group power constraint for digita subscriber oops. In Canadian Conference on Eectrica and Computer Engineering, pages , May 005. [3] R.J. emmons A. Berman. Nonnegative Matrices in the Mathematica Sciences. SIAM Cassics in Appied Mathematics, 994. [4] S. Boyd, S.-J. Kim, L. Vandenberghe and A. Hassibi. A tutoria on geometric programming. Optimization and Engineering, 8(, March 007. [5] Y. Nesterov and A. Nemirovsy. Interior oint oynomia Methods in Convex rogramming. SIAM ress, 994. [6] J.J. McDonad and et a. Inverse m-matrix inequaities and generaized utrametric matrices. citeseer.ist.psu.edu/04694.htm. [7] ITU G.99.. Asymmetrica digita subscriber ine (ads transceivers, 999.