Theoretica Anaysis of Power Saving in Cognitive Radio with Arbitrary Inputs Ahmed Sohai, Mohammed A-Imari, Pei Xiao, and Barry G. Evans Abstract In Orthogona Frequency Division Mutipexing (OFDM based cognitive radio systems, power optimization agorithms have been evauated to maximize the achievabe data rates of the Secondary User (SU. However, unreaistic assumptions are made in the existing work, i.e. a Gaussian input distribution and traditiona interference mode that assumes frequency division mutipexing moduated Primary User (PU with perfect synchronization between the PU and the SU. In this paper, we first derive a practica interference mode by assuming OFDM moduated PU with imperfect synchronization. Based on the new interference mode, the power optimization probem is proposed for the Finite Symbo Aphabet (FSA input distribution (i.e., M-QAM, as used in practica systems. The proposed scheme is shown to save transmit power and to achieve higher data rates compared to the Gaussian optimized power aocation and the uniform power oading schemes. Furthermore, a theoretica framework is estabished in this paper to estimate the power saving by evauating optima power aocation for the Gaussian and the FSA input. Our theoretica anaysis is verified by simuations and proved to be accurate. It provides guidance for the system design and gives deeper insights into the choice of parameters affecting power saving and rate improvement. Index Terms Cognitive Radio, OFDM, Finite Symbo Aphabet, MMSE, Mutua Information. I. ITRODUCTIO Cognitive Radio (CR technoogy [] pays a significant roe in making the best use of scarce spectrum to meet the increasing demand for emerging wireess appications, e.g., pubic safety, broadband ceuar, and the medica body area networks for medica appications []. CR technoogy aso pays a potentia roe in vehicuar communications in terms of safety appications and fufiing the growing demand and usage of in-car entertainment [3]. In the interweave spectrum sharing scheme of the CR system, where the Primary User (PU and the Secondary User (SU co-exist in adjacent frequency bands, mutua interference is a imiting factor on the performance of both the PU and the SU. This can be deat by dynamic power aocation schemes in Orthogona Frequency Division Mutipexing (OFDM based CR systems by adjusting the transmit power on each subcarrier of the SU. Different power aocation schemes have been presented in the iterature to maximize the SU data rate under the interference constraint, assuming the Gaussian input distribution [4] [6]. The Gaussian input is theoreticay optima for Mutua Information (MI maximization, however, it is not a vaid assumption for practica systems and the power optimized for The authors are with the Centre for Communication Systems Research, University of Surrey, Guidford, Surrey, GU 7XH, UK (Emai:{a.sohai, m.aimari, p.xiao,b.evans}@surrey.ac.uk. the Gaussian input is suboptima when it is used for Finite Symbo Aphabet (FSA transmission. On the other hand, the FSA input distribution is a more reaistic assumption for practica systems. Based on the fact that the MI attained by the FSA input is aways ower than the capacity attained by the Gaussian input, the difference in achievabe rate can be approximated by using a SR gap mode as proposed in [7]. However, the approximation is not vaid at high SRs due to the arge gap. One possibe soution to address this imitation is to derive the optima power with the FSA input, as given in [8], where authors ony considered a non-cognitive scenario. To the best of our knowedge, no work has been done to derive an optima power with FSA inputs in CR systems. Therefore, in [9], we derived the optima power aocation for the FSA input in OFDM based CR systems. Through Monte Caro simuations, we showed that there is a wastage of transmit power if the Gaussian optimized power is used for the FSA transmission. Whereas, the optima power aocation derived by the proposed scheme eads to a significant power saving, i.e., 90%, depending on the moduation scheme (i.e., BPSK, QPSK and 6-QAM used. In the iterature and in [9], interference from the secondary system to the primary system is cacuated based on the assumption that (i the SU and the PU are OFDM and FDM moduated, respectivey, and (ii both the PU and the SU are perfecty synchronized. In practica systems, these assumptions are unreaistic, since current wireess communication systems are OFDM moduated. Aso it is difficut to maintain perfect synchronization. This paper is an extension of our previous work in [9] and the nove contributions of this paper are summarized beow In this paper, Sec. II presents a nove practica interference mode that assumes OFDM moduation scheme for both the PU and the SU with imperfect synchronization. This has not been reported in the iterature. The previous interference mode is ony appicabe to FDM moduation scheme. Based on the proposed interference mode, the optima power is derived for the FSA input distribution by capitaizing on the reationship between MI and MMSE [0] in Sec. III. Motivated by the resuts obtained in Sec. IV, we evauate theoreticay the average optima power for the Gaussian and the FSA input, and accordingy cacuate the power saving, which again has not been reported in the existing iterature. Given channe statistics, the theoretica anaysis can be used to estimate the power saving without running time consuming Monte Caro simuations. In addition, it
provides us a deeper insight into the parameters affecting power saving (i.e., the optima power for the FSA input distribution is inversey proportiona to d, where d is the minimum distance for unit variance consteations. Our theoretica anaysis is vaidated by simuations in Sec. V and proves to be accurate. Furthermore, we compare achievabe data rate for the FSA transmission under the optima power aocation with FSA inputs and uniform power oading scheme [4]. We found that our proposed power aocation scheme outperforms the uniform power oading scheme. The remainder of the paper is organized as foows. Secs. II, III and IV present the interference mode, the optima power aocation poicy and theoretica anaysis of power saving for OFDM based CR systems, respectivey. We present the simuation and theoretica resuts of the proposed scheme in Sec. V. Finay, concusions are drawn in Sec. VI. II. ITERFERECE MODEL The system mode can be found in [9]. We assume OFDM moduation scheme for both the SU and the PU with imperfect synchronization as shown in Fig.. Side-obes are omitted in the figure for simpicity. Consider a frequency offset δf such that ϵ = δf f, where ϵ is the normaized frequency offset and f is subcarrier spacing. The SU sampes with ϵ after Inverse Fast Fourier Transform (IFFT are X s k = n s= The PU sampes after IFFT are X p k = n p =+ e jπ(n s+ϵk, k = {,...}. ( x p n p e jπnpk, k = {,...}, ( where is tota number of subcarriers, subscripts s, p represent SU and PU, respectivey, and k is number of time sampes. Given ϵ and omitting the channe effect and noise, the input of the Fast Fourier Transform (FFT for the PU and the SU is given by y k = n s = e jπ(ns+ϵk + n p =+ x p n p e jπnpk. (3 Consider the th output of the PU FFT, (Y p which corresponds to the symbo received on the th subcarrier. This is given as Y p = k= y k e jπk, = { +,...}. (4 By substituting Eq. (3 into Eq. (4, and after mathematica manipuations, we obtain Y p = + k= n s= e jπ(n s +ϵk k= n p=+;n p + X p x p n p e jπ(n p k. (5 ns= Y... SU ns=- ns=- ns= PU Y... np=+ np=+ np= Fig.. Graphica representation of OFDM moduated PU and SU with imperfect synchronization Based on Eq. (5, it can be easiy shown that Y p = Xp + k= n s= e jπ(n s +ϵk. (6 In Eq. (6, the second term is the net interference to the th subcarrier of the PU from a the SU subcarriers and is denoted by ψ ψ = k= n s = e jπ(ns +ϵk. (7 Define ψ,ns as the interference from the n s th SU subcarrier to the th PU subcarrier, i.e., ψ,ns = xs n s k= e jπ(ns +ϵk. (8 ote that the signa spectra of each subcarrier is a Sinc function, therefore the interference cacuation expressed in Eq. (8 has taken the side-obes of the Sinc function into account. The average ψ,ns can thus be cacuated as ψ,ns = ϵmax 0 xs n s k= After mathematica manipuations, we obtain ψ,ns = jxs n s π k= [ jπ(ns k e k e jπ(n s +ϵk dϵ. (9 e jπ(n s +ϵk ]. (0 Interference power can be cacuated as J,ns = E x s ns ψ,ns = p ns Φ,ns, where p ns is the transmit power of the n s th SU subcarrier and [ j Φ,ns = π k= [e jπ(n s k k ]] e jπ(n s +ϵk. ( III. OPTIMAL POWER ALLOCATIO POLICY The objective of the power optimization is to cacuate an optima power with FSA input that maximizes the MI of the SU under given constraints, which formuated as foows max p n s I(p ns s, ( n s =
3 subject to p ns Φ,ns n s = = τ th Ω, and p n s 0, n s =,,, (3 where τ th is the interference threshod prescribed by the PU, s is the channe gain between the SU transmitter and receiver of the n s th subcarrier and Ω is the path oss between the SU transmitter and the PU receiver. In the rest of the paper, n s, p ns, s and Φ,ns is represented as n, p n, and Φ n, respectivey, whenever no ambiguity arises. In [9], the optima power is derived and is given as ( p mmse λφn if gn n = g n g Φ n > λ, n (4 0 if gn Φ n λ, where λ is the Lagrange mutipier for the interference constraint and can be soved usinumerica methods, (such as bisection, secant, or ewton by soving the foowing equation (, gn Φn >λ mmse ( λφn Φ n τ th = 0. (5 Ω Simuation resuts are presented in Sec. V. We denote the tota transmit optima power (P = p n with Gaussian inputs as PG and with FSA inputs as P F. In Fig. and Fig. 3, we pot optima power aocation and percentage of power saving, [i.e., ((PG P F /P G 00] in CR systems using Monte Caro simuations. In our simuations, we have adopted LTE parameters for the SU transmission and assume that a tota of 0 MHz bandwidth is divided into 50 Resource Bocks (RBs []. We consider a simpified path oss mode, i.e., Q(r 0 /r γ [] for the simuations, where Q is constant, γ is path oss exponent, r 0 (reference distance and r (distance between the SU transmitter and the PU receiver are defined in meters. The vaues of ϵ, γ and r 0 are 0.04,.7 (for urban microces and 50 m, and τ th is assumed to be equivaent to therma noise per RB, respectivey. The interference introduced to the PU changes according to r which is assumed to vary from 50-85 m. We adopt the IEEE 80. mutipath channe mode with root mean square deay spread of 50 ns. The resuts are averaged over 000 snapshots. It can be ceary seen from Fig. 3 that a significant power saving has been achieved by the proposed optima power PF in comparison to PG. The transmit power saving for distances ranging from 50-85 m has found to be 65 90%, 49.5 83% and 60% for BPSK, QPSK and 6-QAM inputs, respectivey. IV. THEORETICAL AALYSIS OF POWER SAVIG Motivated by promising power saving resuts, in this section we theoreticay anayse the power saving. The advantage is that, for given channe statistics, the theoretica anaysis can be used to estimate the power saving without running time consuming Monte Caro simuations. Theorem : The power saving for a Rayeigh channe distribution by using the proposed optima power (P F compared to conventiona power aocation scheme (PG is given by where and P (F P (G = d σ P saving = P (G P (F, (6 + Cσ Γ [ Γ(, Φ n λ [ ( AΓ, Φ nλ ( 3, Φ nλ Γ(, Φ ] n λ, (7 + BσΓ + Dσ 3 Γ (, Φ nλ (, Φ nλ ], (8 where ( A = f(a af (a + a f (a a3 f (a ; 6 ( B = f (a af (a + a f (a ; ( f (a C = af (a ; D = f (a, f(a = W (α n a, (9 6 and f (a denotes the derivative of f evauated at point a, σ is the channe statistic parameter for Rayeigh distribution, Γ(. is the incompete gamma function [3] and d is the minimum distance for unit variance consteations, i.e., d =, and /5 for BPSK, QPSK and 6-QAM, respectivey. Proof: The average optima power for a given λ with arbitrary input distributions can be obtained as P (S = p (, Sh( d, (0 where h( is a pdf of the channe, and for a Rayeigh fading channe h( = ( /σ e ( /. The MMSE reationships for FSA and Gaussian input distributions are given by [8] mmse (F (p n U e d 4 (p n, ( p n mmse (G (p n = + p n, ( where U= π d and for M-PSK and M-QAM, respectivey. To cacuate p n(, F and p n(, G, we substitute Eqs. ( and ( into Eq. (4. After some mathematica manipuations, we obtain e d 4 (p n p n = U, (3 p n(, F = ( U d d W λ Φ, n (4 p n(, G =, (5
4 where W (. is the Lambert W function [4]. From Eq. (0, the optima power for the FSA input can be derived as p (, Fh( d = d σ W ( α n e d, (6 where α n = U d λ Φ. Using Tayor series, the right hand side n of Eq. (6 becomes [ d σ A e d + B e d + C According to [5] u Φ nλ g n e d + D x m e βxr dx = Γ(v, βur rβ v, v = m + r Φ nλ 3 e 3 d ]. (7 [β > 0, v > 0, r > 0 u > 0]. (8 A cosed form of Eq. (7 can be derived as [ ( d AΓ σ, Φ nλ + BσΓ (, Φ nλ ( 3 + Cσ Γ, Φ nλ + Dσ 3 Γ (, Φ nλ ]. (9 By substituting Eq. (9 into Eq. (0, we obtain Eq. (8. To cacuate A, B, C and D in Eq. (9, we need to derive f(a, f (a, f (a, f (a by defining the function and taking its derivatives as foows f (g = f(g = W (α g, (30 W (α g g [W (α g + ], (3 f (g = W (α g [ W (α g + W (α g ] g [W (α g + ] 3, (3 f 4W (α g (g = g 3 [W (α g + ] 5 [ W (α g 3 + 4W (α g + 3W (α g 6 ]. (33 By substituting the vaues of α n in Eqs. (30, (3, (3 and (33, A, B, C and D can be cacuated. By substituting Eq. (5 into Eq. (0, the optima power for Gaussian inputs can be derived as p (, Gh( d = σ σ By appying Eq. (8, the RHS of Eq. (34 becomes = Γ(, Φ n λ e d e d. (34 Γ(, Φ n λ. (35 By substituting Eq. (35 into Eq. (0, we obtain Eq. (7. A. Theoretica Cacuation of λ for FSA and Gaussian Input Distributions In Eq. (8 and (7, k n, d and σ are constant vaues, however, λ is dependent on the channe gain. Therefore, we cacuate λ numericay via the foowing equation p (, SΦ n h( d = τ th Ω. (36 By substituting Eq. (4 into Eq. (36 and after the same manipuations as in Eqs. (6, (7 and (8, we can obtain the vaue of λ for the FSA input using the foowing equation Φn [ ( d AΓ σ, Φ nλ + BσΓ (, Φ nλ ( 3 +Cσ Γ, Φ nλ + Dσ 3 Γ (, Φ nλ ] = τ th Ω. (37 Simiary, by substituting Eq. (5 into Eq. (36 and after the same manipuations as in Eqs. (34 and (35, we can obtain the vaue of λ for the Gaussian input using the foowing equation [ Γ(, Φ n λ λ Φ nγ(, Φ ] n λ = τ th Ω. (38 By substituting the vaues of k n,, τ th, Ω and σ in Eq. (37 and Eq. (38 λ can be cacuated numericay. The theoretica anaysis gives deeper insights on the parameters affecting power saving. For exampe, it can be seen from Eq. (8 that the optima power for FSA input distribution is inversey proportiona to d. As d(bp SK > d(6 QAM, therefore, average optima power for BPSK is ower, eading to more power saving compared to the optima power for 6- QAM. V. EVALUATIO OF OFDM BASED CR SYSTEM A. Simuation Anaysis In Fig., we compare PG and P F versus distance. We observe from this figure that PG is aways greater than P F over the considered distance range. It has been noted that the gap is smaer at shorter distances compared to onger ones. The expanation is provided in [9]. Moreover, it has been observed that at a given distance, PF increases with the moduation order, (i.e., from BPSK to 6-QAM. The optima power aocation is dependent and specific for every moduation scheme. It woud resut in power inefficiency if one tries to transmit BPSK signa with the power which is optimized for 6-QAM. We have presented the resuts in [9] that the proposed optima power aocation scheme achieves higher data rate compared to the Gaussian optimized power. We have shown that the percentage rate gain of the BPSK, QPSK and 6-QAM is 6.8.4%, 3.8% and 3 5.8%, respectivey, for the interference threshod vaues ranging between -3 mw. However in this paper, we compare the achievabe data rate for the FSA transmission under optima power aocation with FSA inputs and uniform power oading scheme (i.e., τ th /(Ω Φ n as shown in Fig. 4. It can be ceary
5 Tota transmit power (Watt.5.5 Gaussian 6QAM QPSK BPSK Tota achievabe rate (Mb/s 6 4 0 8 6 4 0 BPSK (FSA Optimized Power BPSK (Uniform Power Loading Scheme QPSK (FSA Optimized Power QPSK (Uniform Power Loading Scheme 6QAM (FSA Optimized Power 6QAM (Uniform Power Loading Scheme 0.5 8 6 0 Distance (meter 4 Distance (meter Fig.. 90 Tota transmit power under Gaussian and FSA inputs vs distance. Fig. 4. Comparison of achievabe data rate under FSA optimized power and uniform power oading scheme. Percentage of power saving over Gaussian input 80 70 60 50 40 Fig. 3. 30 BPSK(Simuation BPSK(Theoretica QPSK(Simuation 0 QPSK(Theoretica 6QAM(Simuation 6QAM(Theoretica 0 Distance (meter Percentage of power saving vs distance. seen that the proposed scheme outperforms the uniform power oading scheme over considered distance range. Fig. 5 depicts the effect of normaized carrier frequency offset (i.e., ϵ on percentage of power saving over the Gaussian input by keeping the fixed distance (60 m. It has been observed that the percentage of power saving increases by increasing the vaues of ϵ. This is due to the fact that the proposed optima power and optima power assuming the Gaussian input decreases by increasing the vaues of ϵ, but the Gaussian optimized power decreases faster than the proposed optima power. In Fig. 6, we compare percentage of power saving with the proposed and the conventiona interference modes. It has been shown that the percentage of power saving with the proposed interference mode increases with the increased vaues of ϵ, whereas, the percentage of power saving with the conventiona interference mode presented in [9] has a constant vaue because it does not depend upon vaues of ϵ. B. Anaytica Resuts vs. Simuation Resuts As discussed in Sec. III, our simuation study has shown that the proposed optima power aocation scheme achieves Percentage of power saving over Gaussian input 90 80 70 60 50 40 BPSK QPSK 6QAM 30 0.04 0.06 0.08 0. 0. 0.4 0.6 0.8 0. ormaized carrier frequency offset Fig. 5. Effect of ϵ on percentage of power saving at 60 m distance. significant power saving compared to the optima power under the Gaussian input. Fig. 3 shows the comparison of anaytica (soid ine and simuated (dashed ine power saving. The channe static parameter, i.e., σ in Eqs. (8 and (7 has been cacuated from the empirica Rayeigh distribution and impemented in the simuation. One can see that theoretica resuts coincide we with the simuated ones, and the discrepancy is margina. The minor difference foows from the fact that we used approximated vaues of MMSE in Eq. ( and Tayor approximation in Eq. (6 to cacuate the optima power under the FSA input. It can be concuded that for given channe statistics, the theoretica anaysis can be used to derive an average optima power aocation and estimate power saving without running time consuming Monte Caro simuations. To evauate the accuracy of using Tayor expansion, Fig. 7 depicts the optima power of BPSK and the optima power achieved by different degrees of Tayor poynomias. It is cear from the figure that the 5 th degree of Tayor poynomias approximatey match the exact vaue and thus can be used to cacuate the theoretica optima power under arbitrary input
6 Percentage of power saving over Gaussian input 0 00 90 80 70 60 50 40 BPSK (Proposed Interference Mode BPSK (Conventiona Interference Mode QPSK (Proposed Interference Mode QPSK (Conventiona Interference Mode 6QAM (Proposed Interference Mode 6QAM (Conventiona Interference Mode Optima power with BPSK moduation scheme (Watt 0.5 0. 0.5 0. BPSK optima power 5th degree poynomia 4th degree poynomia 3rd degree poynomia nd degree poynomia st degree poynomia 30 0.04 0.06 0.08 0. 0. 0.4 0.6 0.8 0. ormaized carrier frequency offset 0.05 Distance (meter Fig. 6. Effect of ϵ on proposed and conventiona interference modes at 60 m distance. Fig. 7. Performance using Tayor series approximation. distributions as we as the achieved power saving using the proposed power aocation scheme. The same accuracy of Tayor expansion has been noted for other moduation schemes in other figures. VI. COCLUSIO In this paper, we first estabished the practica interference mode that assumes OFDM moduation scheme for both the PU and the SU with imperfect synchronization. Accordingy, the power aocation probem in OFDM based CR systems is derived under the condition of FSA input appicabe to practica systems. Motivated by the promising power saving resut through Monte Caro simuations, a theoretica evauation of the power saving is presented in order to gain deeper insights into power saving capabiity of the proposed scheme. Furthermore, the theoretica resuts revea that (i our optima power with the FSA input significanty outperforms the conventiona power aocation schemes (i.e., the Gaussian optimized power and uniform power oading scheme in terms of transmit power saving and achievabe data rate; (ii with fixed distance metric, the optima transmit power with the FSA input increases as the moduation order increases, and (iii by increasing the vaue of the normaized frequency offset (ϵ, the percentage of power saving increases with the proposed interference mode, whereas, the percentage of power saving with the conventiona interference mode has a constant vaue. Based on the aforementioned findings we concuded that, by using the proposed power aocation scheme, spectrum and energy efficiency can both be improved. Secondy, in order to achieve a desired energy efficiency, the power shoud be optimized according to the empoyed moduation scheme. ACKOWLEDGMET This work has been supported by the India UK Advance Technoogy Centre of Exceence in ext Generation etworks, Systems and Services (www.iu-atc.com and Engineering and Physica Sciences Research Counci (EPSRC. REFERECES [] E. Tragos, S. Zeaday, A. Fragkiadakis, and V. Siris, Spectrum assignment in cognitive radio networks: A comprehensive survey, IEEE Commun. Surveys Tuts., vo. 5, no. 3, pp. 08 35, Third Quarter 03. [] J. Wang, M. Ghosh, and K. Chaapai, Emerging cognitive radio appications: A survey, IEEE Commun. Mag., vo. 49, no. 3, pp. 74 8, Mar. 0. [3] V. Tumuuru, W. Ping, D. iyato, and S. Wei, Performance anaysis of cognitive radio spectrum access with prioritized traffic, IEEE Trans. Veh. Techno., vo. 6, no. 4, pp. 895 906, May 0. [4] G. Bansa, J. Hossain, and V. Bhargava, Optima and suboptima power aocation schemes for OFDM-based cognitive radio systems, IEEE Trans. Wireess Commun., vo. 7, no., pp. 470 478, ov. 008. [5] Z. Hasan, G. Bansa, E. Hossain, and V. Bhargava, Energy-efficient power aocation in OFDM-based cognitive radio systems: A risk-return mode, IEEE Trans. Wireess Commun., vo. 8, no., pp. 6078 6088, Dec. 009. [6] C. Chiuan-Hsu and W. Chin-Liang, Power aocation for OFDM-based cognitive radio systems under primary user activity, in IEEE Vehicuar Technoogy Conference, May 00, pp. 5. [7] B. Deviers, J. Louveaux, and L. Vandendorpe, Bit and power aocation for goodput optimization in coded parae subchannes with ARQ, IEEE Trans. Signa Process., vo. 56, no. 8, pp. 365 366, Aug. 008. [8] A. Lozano, A. Tuino, and S. Verdu, Optimum power aocation for parae gaussian channes with arbitrary input distributions, IEEE Trans. Inf. Theory, vo. 5, no. 7, pp. 3033 305, Ju. 006. [9] A. Sohai, M. A-Imari, P. Xiao, and B. Evans, Optima power aocation for OFDM based cognitive radio systems with arbitrary input distributions, in IEEE Vehicuar Technoogy Conference, Sep. 03, pp. 909 93. [0] D. Guo, S. Shamai, and S. Verdu, Mutua information and minimum mean-square error in Gaussian channes, IEEE Trans. Inf. Theory, vo. 5, no. 4, pp. 6 8, Apr. 005. [] J. Zyren, Overview of 3GPP ong term evoution physica ayer, Freescae Semiconductor, white paper, Ju. 007. [] A. Godsmith, Wireess Communications. Cambridge University Press, 005. [3] G. Arfken, Incompete Gamma function and reated functions, Mathematica Methods for Physicists, pp. 565 57, 985. [4] R. M. Coress, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, On the Lambert W function, Advances in Computationa Mathematics, vo. 5, no., pp. 39 359, 996. [5] I. S. Gradshteyn and I. M. Ryzhik, Tabe of Integras, Series and Products-7th Edition. Academic Press, 007.