Spectral Efficiency of OFDMA Cognitive Radios under Imperfect Cross-Link Knowledge

Transcription

1 1 Spectral Efficiency of OFDMA Cognitive Radios under Iperfect Cross-Link Knowledge H Saki and M Shikh-Bahaei Institute of Telecounications, King s College London, London, WCR LS, United Kingdo Eail: {hadisaki, sbahaei}@kclacuk arxiv: v1 csit] 19 Oct 017 Abstract We analyse the ectral efficiency perforance and liits of orthogonal frequency-division ultiple access OFDMA cognitive radios CRs with iperfect availability of cross-link knowledge In particular, in contrast to the conventional average and worst cases of channel estiation error in the literature, we propose a stochastic approach to itigate the total iposed interference on priary users Channel-adaptive resource allocation algoriths are incorporated to optiize the cognitive syste functionality under transit and interference power constraints An expression for the cuulative density function cdf of the received signal-to-interference-plus-noise ratio SINR is developed to evaluate the average ectral efficiency Analytical derivations and results are confired through coputer siulations I INTRODUCTION Dynaic ectru anageent DSM has attracted a lot of attention recently as easureents by ectru regulators have shown that the radio frequency RF band is severely underutilized According to the Office of Counications Ofco recent easureents 1], the ectru is severely under-utilized as a result of rigid and inefficient anageent policies, and that 90% of locations have around 100 MHz of ectru available for other services Flexible ectru-sharing for supporting CR and ectru co-existence is a priority issue to overcoe the current capacity crunch and thus enabling the deployent of long ter evolution LTE-advanced and beyond wireless systes In recent years, a significant effort has been ade towards iproving the ectral efficiency of cellular networks in order to eet the growing deand and sophistication of wireless applications Several ectral-efficient technologies, such as cross layed design ], achine-to-achine MM counications, sall-cell SC solution, assive ultipleinput ultiple-output MIMO, and cognitive radio CR - each with reective advantages and challenges - are proising candidates in this direction 3], 4] The ter CR can be defined as an intelligent radio syste that has the ability to sense the priary service behaviour and surrounding environent and adjust its ectru usage and paraeters based on the observed inforation 5], 6] Three ain paradigs have been proposed for cognitive radio in regards to the unlicensed users access to the priary frequency band: i underlay ectru access where secondary users silently coexist with priary users, provided they satisfy an interference liit set by a regulatory authority ii overlay ectru access in which secondary users are only allowed to access the vacant parts of the priary ectru, and iii hybrid ectru access, a cobination of the two forer strategies in which the secondary users sense the priary ectru and adjust their transission paraeters based on the detection, whilst avoid iposing harful interference to the priary users 7] In this paper, we consider underlay ectru-sharing, where robust interference anageent is critical for tackling any harful cross-service interference Orthogonal frequency-division ultiplexing OFDM has eerged as a proinent radio access technology for new generation of wireless counication systes including LTE and LTE-advanced 8],9] Unilike CDMA 10], OFDM-based ulti-user applications, ultiple-access can be accoodated through orthogonal frequency-division ultiple-access OFDMA technique 11] In OFDMA systes, different subcarriers ay be assigned to different users in order to exploit the channel quality rando variations of users across each subcarrier OFDMA technology is considered as a de facto standard for CR networks due to its inherent advantages in ters of flexibility and adaptability in allocating ectru resources in shared-ectru environents 1] Radio resource allocation RRA plays a significant role in optiizing the overall ectral efficiency of conventional OFDMA systes 13] In addition, adaptive RRA is an active area of research in the context of OFDMA-based CR networks with the ai of achieving a balance between axiizing the cognitive network perforance and iniizing the inflicted interference on the licensed users Suboptial and optial power allocation policies are studied in 14], where the aggregate capacity of the CR syste is axiized under a priary receiver Rx interference liit In 15], a queue-aware RRA algorith is proposed to axiize the fairness in OFDMAbased CR networks subject to a total power constraint at the base station A Lagrangian relaxation algorith is adopted in 1] to probabilistically allocate resources based on the availability of the priary frequency band via ectru sensing Most of the RRA algoriths on CR networks in the literature assue perfect channel state inforation CSI between the cognitive transitter CTx and Rx, and few have considered iperfect cross-link CSI However, due to technical reasons such as estiation errors and wireless channel delay, obtaining perfect cross-link CSI is difficult in practical scenarios In 16] and 17], the ergodic capacity is derived over fading channels with iperfect cross-link knowledge, however, the analysis is carried out for a single cognitive user CU Furtherore, due to noisy cross-link inforation,

2 it is unrealistic to assue that the secondary network strictly satisfies a deterinistic interference constraint The authors in 18] propose a RRA algorith for axiizing instantaneous rate in downlink OFDMA CR systes subject to satisfying a collision probability constraint However, 18] only considers the individual ipact of probabilistic interference constraint per subcarrier Motivated by the above, we thoroughly investigate different scenarios by analysing the ipact of deterinistic and probabilistic interference constraints depending on perfect and noisy cross-link knowledge In particular, we develop novel RRA algoriths under average case, worst case, and probabilistic case scenarios of channel estiation uncertainty for ulti-user OFDMA CR networks On the other hand, to the best of authors knowledge, enhancing the average ectral efficiency of ulti-user OFDMAbased CR systes has not been addressed in the literature In this work, by exploiting the advantages of channel adaptation techniques, we propose novel joint power, subcarrier, and rate allocation algoriths for enhancing the average ectral efficiency of downlink ulti-user adaptive M-ary quadrature aplitude odulation MQAM/OFDMA 19], 0] CR systes Given the received power restrictions on the CTx in order to satisfy the priary network interference liit and the cognitive network power constraint, the CTx transit power is a function of the cognitive-cognitive direct-link and cognitive-priary cross-link fading states We develop a cuulative distribution function cdf of the CR s received signal-to-interference-plus-noise ratio SINR to evaluate the average ectral efficiency of the adaptive MQAM/OFDMA CR syste The ain novelties and contributions of this paper are suarized as follows: 1 The coprehensive proble of power, rate, and subcarrier allocation for enhancing the average ectral efficiency of downlink ulti-user OFDMA CR systes subject to satisfying total average transission power and peak aggregate interference constraint has been studied A closed-for expression for the cdf of the OFDMA CR s received SINR is derived under liitations iposed on the CTx through the power and interference constraints Consequently, an upper-bound expression for average ectral efficiency of the adaptive ulti-user MQAM/OFDMA CR syste is forulated 3 The critical issue of violating interference liits associated with iperfect cross-link CSI availability is exained by carrying out the analysis for the average case, worst case, and probabilistic case scenarios of channel estiation error 4 The ipact of deterinistic and probabilistic interference constraints on the syste perforance is considered with perfect and iperfect cross-link CSI In particular, we propose a new low-coplexity deterinistic forulation for the probabilistic cross-link interference The organization of this paper is as follows: Section II presents the network odel and operation assuptions In Section III, the resource allocation proble for enhancing average ectral efficiency of the adaptive ulti-user MQAM/OFDMA under perfect cross-link CSI subject to power and deterinistic interference constraints is developed In Section IV, under noisy cross-link knowledge, the ipact of average case and worst case of channel estiation error based on a posterior distribution of the perfect channel conditioned on its estiate is exained Section V investigates the perforance under a collision probability constraint with iperfect cross-link CSI and proposes a deterinistic forulation of the probabilistic aggregate cross-link interference In all of the RRAs derived in the paper, optial power, rate, and subcarrier assignents are obtained Illustrative nuerical results for various scenarios under consideration are provided in Section VI Finally, concluding rearks are presented in Section VII II SYSTEM MODEL AND RELIMINARIES In this section, the ulti-user OFDMA CR network odel, wireless channel, and operation assuptions are introduced Further, interference anageent schees and ectral efficiency of the adaptive MQAM/OFDMA syste under consideration are studied A Network Architecture and Wireless Channel We consider an underlay shared-ectru environent, as shown in Fig 1, where a cognitive network with a single CTx and n = 1,,N cognitive receiver CRxs coexist with a priary network with a priary transitter Tx and = 1,, M Rxs The cognitive network can access a ectru licensed to the priary network with a total bandwidth of B which is divided into K non-overlapping sub-channels subject to not violating the iposed interference constraint set by a regulatory authority The sub-channel bandwidth is assued to be uch saller than the coherence bandwidth of the wireless channel, thus, each subcarrier experiences frequencyflat fading Let Hn,k ss t, Hps n,k t, and H t, at tie t, denote the channel gains over subchannel k fro the CTx to n th CRx, Tx to n th CRx, and CTx to th Rx The channel power gains Hn,k ss t, H ps n,k t, and H t are assued to be ergodic and stationary with continuous probability density functions pdfs f H ss t, f n,k H ps t, n,k and f H t, reectively In addition, the instantaneous values and distribution inforation of secondary-secondary channel power gains is assued to be available at the CTx 17] In this work, we consider different cases with perfect and noisy cross-link knowledge between CTx and Rxs Each sub-channel is assigned exclusively to at ost one CRx at any given tie, hence, there is no utual interference between different cognitive users 1] It should also be noted that by utilizing an appropriate cyclic prefix, the inter-sybolinterference ICI can be ignored ] The received SINR of cognitive user n over sub-channel k at tie interval t is γ n,k t = n,k H ss n,k t σ n +σ ps where n,k is a fixed transit power allocated to cognitive user n over sub-channel k, σ n is the noise power, and σ ps is the received power fro the priary network Without loss of generality, σ n and σ ps are assued to be the sae across 1

3 3 all users and sub-channels 3, 4] We define Υ n,k t as a vector containing γ n,k t of all tie intervals For the sake of brevity, we henceforth oit the tie reference t Due to the ipact of several factors, such as channel estiation error, feedback delay, and obility, perfect crosslink inforation is not available With noisy cross-link CTx to Rxs knowledge, we odel the inherent uncertainty in channel estiation in the following for where over subcarrier k, H H = Ĥ + H is the actual cross-link gain, Ĥ is the channel estiation considered to be known, and H denotes the estiation error H, Ĥ, and H are assued to be zero-ean coplex Gaussian rando variables with reective variances δ H, δ, and Ĥ 16, 5] For robust receiver design, we consider δ H the estiation Ĥ and error H to be statistically correlated rando variables with a correlation factor ρ = δ /δ +δ, where 0 ρ 1 H H H B Interference Manageent In a shared-ectru environent, and particularly for delay-sensitive services, the licensed users quality of service QoS is highly dependent to the instantaneous received SINRs of cognitive users In order to protect the licensed ectru fro harful interference we pose a deterinistic peak total interference constraint between CTx and priary users ϕ n,k Υ n,k Υ H I th, {1,,M} where Υ is a atrix containing all of Υ n,k, n {1,,N} and k {1,,K}, further, ϕ n,k Υ is the tie-sharing factor subcarrier allocation policy, n,k Υ is the allocated transit power, and Ith denotes the axiu tolerable interference threshold However, as a consequence of uncertainties about the shared-ectru environent and priary service operation, it is unrealistic to assue that the CTx always satisfies the deterinistic peak total interference constraint In practical scenarios, probability of violating the interference constraint is confined to a certain value that satisfies the iniu QoS requireents of priary users robabilistic interference constraint is particularly critical for robust interference anageent given noisy cross-link knowledge To iprove overall syste perforance and to itigate the ipact of channel estiation errors, the following allowable probabilistic interference liit violation is considered N ϕ n,k Υ n,k Υ H > I th ǫ 3, {1,,M} 4 where denotes probability, and ǫ is the collision probability constraint of th Rx Fig 1: Scheatic diagra of the shared-ectru OFDMA syste For siplicity purposes, channels of a single cognitive user are drawn On the other hand, itigating the interference between neighbouring cells is a vital issue due to the increasing frequency reuse aggressiveness in odern wireless counication systes 6] As a reedy to inter-cell interference, and to aintain effective and efficient power consuption, we ipose a total average transit power constraint on the cognitive network as follows } E Υ {ϕ n,k Υ n,k Υ 5 where E x denotes the expectation with reect to x, and denotes the total average transit power liit C Spectral Efficiency The focus of this work is ainly on optial power, rate, and subcarrier allocation for enhancing the average ectral efficiency of the adaptive MQAM/OFDMA CR network In a ulti-user scenario, various subcarriers ay be allocated to different users In other words, users ay experience different channel fading conditions over each sub-channel Therefore, any efficient resource allocation schee in OFDMA ust be based on the sub-channel quality of each user Furtherore, in a shared-ectru environent, satisfying the interference constraints is an iportant factor in allocating resources Eploying square MQAM with Gray-coded bit apping, the approxiate instantaneous bit-error-rate BER expression for user n over subcarrier k is given by ξn,k b Υ = log M n,k Υ Mn,k Υ 3Υ n,k Q M n,k Υ 1 where M n,k Υ denotes the constellation size vector of MQAM which each eleent is a function of the instantaneous received SINR of the cognitive user n over subcarrier k, and Q represents the Gaussian Q-function 6

4 4 The aggregate average ectral efficiency of the adaptive ulti-user MQAM/OFDMA syste per subcarrier per user over the fading channel is defined as ASE = { } E Υ log M n,k Υϕ n,k Υ 7 In order to evaluate the ASE, the distribution of the received SINR, a function of secondary-secondary and secondarypriary channels, ust be developed III DETERMINISTIC INTERFERENCE CONSTRAINT WITH ERFECT CROSS-LINK CSI The objective of this paper is to axiize the aggregate average ectral efficiency of cognitive users while satisfying total transission power and peak axiu tolerable interference constraints In this section, we solve the resource allocation proble with the perfect cross-link knowledge and deterinistic interference constraint A roble Forulation Matheatically, the optiization proble can be stated as follows roble O 1 : ax ϕ n,k Υ, n,k Υ s t: { } E Υ log M n,k Υϕ n,k Υ } E Υ {ϕ n,k Υ n,k Υ ϕ n,k Υ n,k Υ H I th, {1,,M} ϕ n,k Υ = 1, k {1,,K} n=1 8a 8b 8c 8d ϕ n,k Υ {0,1}, n {1,,N}, k {1,,K} 8e ξ b n,kυ ξ, n {1,,N}, k {1,,K} 8f where ξ denotes the coon BER-target In the adaptive ulti-user MQAM/OFDMA CR syste under consideration, different transit power and constellation sizes are allocated to different users and subcarriers Using the upper-bound expression for the Gaussian Q-function, ie, Qx 1/exp x /, the instantaneous BER for user n over subcarrier k, subject to an instantaneous constraint ξn,k b Υ = ξ can be expressed as ξn,kυ b 15Υ n,k n,k Υ 03exp 9 M n,k Υ 1in t where = K H With further anipulation, for a BER-target ξ, the axiu constellation size for user n over subcarrier k is obtained as Mn,k Υ = 1+ n,k Υ 10 in t where ζ = 15 lnξ/03 11 According the constraints 8b and 8c in the optiization proble O 1, the cuulative density function cdf of γ n,k can be written t Hn,k ss F γn,k Γ = Kσn +σps Γ, Ith Hss n,k σ n +σps Γ 1 The probability expression in 1 can be further siplified by considering the cases conditioning on t Hn,k ss 1 H ss H ss n,k Kσn +σ ps I th Hss n,k N σ n +σ ps and Kσn +σ ps > Γ, Ith Hss n,k σ n +σ ps > Γ = n,k > KΓσ n +σ ps I th K 1 Hn,k ss > N Γσ n +σ ps > I th K I th 13 Lea 1: For large values of K, given coplex Gaussian rando variables H with eans µ H and equal for all k {1,,K}, the non-central variance δ H Chi-square rando variable H be approxiated as a Gaussian rando variable with reective ean and variance µ N δ N =δ4 H = K 4K+4µ ], where µ = K = δ H µ H δ H can K+µ ] and roof 1: We can write H = δ H G, where G CN µ H δ H H,1 Assuing equal variance for rando variables, K G is a non-central Chi-Square rando variable with degree of freedo K and non-centrality paraeter µ = K For large values of K, central µ H δ H liit theore CLT can be invoked to show that the noncentral Chi-Square rando variable K G, can be approxiated as a Gaussian rando variable as follows G N K+µ,4K+4µ 14 Hence, N = K H can be approxiated by N = whereµ N = δ H Denoting the pdf of N H N µ N,δ K+µ ] andδ N = δ4 H 15 4K+4µ ] with f, and the cdfs of Hss n,k and with F H ss and F n,k, reectively, we write the cdf of γ n,k as F γn,k Γ = 1 A B, 16

5 5 and A = = = = B= I th K 0 H ss n,k > KΓσ n +σ ps H ss n,k > KΓσ n +σ ps H ss n,k > KΓσ n +σ ps 1 F Hss KΓσ n +σ ps I th K I th K 0 Hn,k ss > N Γσ n +σ ps I th f N N dn f N N dn I th K F N I th K 17 f N N d 18 Recall that the cdf of a Norally-distributed rando variable X with ean µ and standard deviation σ is given by F X x = 1 1+erf x µ ] σ, and the cdf of an Exponentiallydistributed rando variable Y is coputed by F Y y = 1 e y/µ, where µ is the ean Suppose that Hn,k ss follows an exponential distribution with ean µ H ss n,k, hence, the integrals in 17 and 18 can be siplified to 19 and 0, reectively Finally, a closed-for expression for cdf of γ n,k is developed in 1 Trivially, through reective differentiation of 1, the pdf of γ n,k is obtained in B Obtaining Solutions It can be observed that the optiization proble, O 1, is convex with reect to the transit power n,k Υ, however, it is non-convex with reect to ϕ n,k Υ as the tie-sharing factor only takes binary values To obtain a sub-optial solution for proble O 1, we eploy the Lagrangian dual decoposition algorith By applying dual decoposition, the non-convex optiization proble, O 1, is decoposed into independent sub-probles each correonding to a given cognitive user The Lagrangian function of proble O 1 is expressed as 1 Lϕ n,k Υ, n,k Υ,λΥ,µ,ηΥ = { } E Υ log 1+ n,k Υ ϕ n,k Υ n=1 in t N λ k Υ ϕ n,k Υ 1 n=1 } µ E Υ {ϕ n,k Υ n,k Υ ηυ K n=1 ϕ n,k Υ n,k Υ H I th 3 where µ, ηυ, and λ k Υ are the non-negative Lagrangian ultipliers Define lϕ n,k Υ, n,k Υ,λΥ,µ,ηΥ = 1 For siplicity, the analysis is carried out for a single priary receiver log 1+ n,k Υ ϕ n,k Υ in t N λ k Υ ϕ n,k Υ 1 n=1 µ ηυ n=1 ϕ n,k Υ n,k Υ K n=1 ϕ n,k Υ n,k Υ H I th 4 Note that the variation of the Lagrangian function, Lϕ n,k Υ, n,k Υ,λΥ,µ,ηΥ, in 3 with reect to the optiization paraeters, ϕ n,k Υ and n,k Υ, is equal to zero if and only if the derivative of lϕ n,k Υ, n,k Υ,λΥ,µ,ηΥ with reect to ϕ n,k Υ and n,k Υ is equal to zero 7] Based on the Karush-Kuhn-Tucker KKT necessary conditions theore, the optiu solutions n,k Υ,ϕ n,k Υ ust satisfy the following conditions: lϕ n,k Υ, n,k Υ,λΥ,µ,ηΥ n,k Υ { = 0, n,k Υ > 0 < 0, n,k Υ = 0 5 lϕ n,k Υ, n,k Υ,λΥ,µ,ηΥ < 0, ϕ n,k Υ = 0 = 0, ϕ n,k Υ 0,1 ϕ n,k Υ > 0, ϕ n,k Υ = 1 6 N λ k Υ ϕ n,k Υ 1 = 0 7 n=1 } µ E Υ {ϕ n,k Υ n,k Υ = 0 8 K ηυ ϕ n,k Υ n,k Υ H Ith = 0 9 n=1 The Lagrangian dual optiization proble associated with 3 is given by in F λυ,µ,ηυ, st:λυ,µ,ηυ 0 30 λυ,µ,ηυ where FλΥ, µ, ηυ denotes the Lagrangian dual function forulated below FλΥ,µ,ηΥ = f n ϕ n,k Υ, n,k Υ+ λ k Υ where n=1 +µ +ηυi th 31 f n ϕ n,k Υ, n,k Υ = K ax log 1+ n,k Υ ϕ n,k Υ ϕ n,k Υ, n,k Υ in t λ k Υϕ n,k Υ µ ϕ n,k Υ n,k Υ

6 6 exp B= I th K 1 erf N Γσ n +σ ps µ H ss n,k I th µ H ss n,k I th µ N πδ N A= 1 KΓσ exp n +σps 1+erf µ H ss n,k N exp µ δ + I th K dn I th K µ N δ N ] 19 1 Γσ exp n +σps µ N µ Hn,k ss I th +δ Γσ n +σps µ H ss n,ki th +δ Γσ n +σ ps ] 0 µ H ss n,k I th δ F γn,k Γ 1 1 KΓσ exp n +σps 1+erf µ H ss n,k 1 erf f γn,k Γ + µ H ss n,k I th I th K µ N δ N ] 1 Γσ exp n +σps µ N µ H ss n,k I th +δ Γσ n +σps µ H ss n,ki th µ N + I th K +δ Γσ n +σ ps ] µ H ss n,k I th δ 1 Kσn +σ ps exp KΓσ n +σ ps I th K µ erf H ss n,k µ H ss n,k µ N δ N σn +σpsδ N exp I th K µ H ss n,k I th Kµ N µ H n,k ss t+kσ n +σ ps tδ Γ+µ N µ H n,k ss t πi th µ H ss n,k 05σ n +σ ps I th µ N µ H ss n,k σ n +σ ps δ N Γexp Ith erf µ H ss I th K n,k µ N δ N I th µ H ss n,k µ H ss n,k t δ +σ n +σ ps δ N +1 σ n +σ psγσ n +σ psδ N Γ I th µ N µ H ss Γ 1 I th µ H ss n,k n,k I th µ H ss n,k ηυ ϕ n,k Υ n,k Υ H 3 To find the optiu solution of proble 3, we differentiate f n ϕ n,k Υ, n,k Υ with reect to ϕ n,k Υ n,k Υ f n ϕ n,k Υ, n,k Υ ϕ n,k Υ n,k Υ = ln ζυ n,k in t 1+ n,k Υ in t µ ηυ H 33 Applying the KKT conditions yields the optial potential power allocation policy for Lagrangian ultipliersµand ηυ n,k Υ = 1 in t lnµ+ηυ H K, ] I th + 34 where x] + ax{x,0} The solution in 34 can be considered as a ulti-level water-filling algorith where each subcarrier has a distinct water-level for a given user Note that the water levels deterine the potential optiu aount of power that ay be allocated to n th CRx over subcarrier k The result in 34 can be used to find the optial subcarrier allocation strategy By differentiating f n ϕ n,k Υ, n,k Υ with reect to ϕ n,k Υ we have f n ϕ n,k Υ, n,k Υ ϕ n,k Υ + ln 1+ n,k Υ in t ln = ln n,k Υ in t 1+ n,k Υ in t λ k Υ 35

7 7 By substituting the optial power policy 34 in 35 and by applying the KKT conditions, the optial subcarrier allocation proble is forulated as: n = argaxλυ n,k, n {1,,N}, k {1,,K} where n is the optial CRx index, and ΛΥ n,k = ln ζυn,kn,k Υ in t 1+ ζυn,k n,k Υ in t + ln 1+ ζυn,k n,k Υ in t ln The optial subcarrier allocation policy is therefore achieved by assigning the k th subcarrier to the user with the highest value of ΛΥ n,k for all correonding γ n,k To ensure optiality, λ k Υ should be between first and second axias of ΛΥ n,k If there are ultiple equal axias, the tie-slot can be identically shared aong the reective users Substituting 34 and 37 in 3, derives f n ϕ n,k Υ, n,k Υ, therefore, the solution for 31 can be obtained To copute the solution for the non-differentiable dual proble in 30, different optiization algoriths can be applied, including subgradient, ellipsoid, and cutting-plane In this work, we use the subgradient-based ethod to update the values of the coefficients λ k Υ, µ, and ηυ, in order to deterine the optial solution to 30 The subgradient ethod has been widely used for solving Lagrangian relaxation probles The aster proble sets the user resource allocation prices, and in order to update the dual variables, in every iteration of the subgradient ethod, the algorith repeatedly finds the axiizing assignent for the sub-probles individually For any optial pair of ϕ n,k Υ, n,k Υ, the dual variables of proble 31 are updated using the following iterations µ i+1 = µ i τ1 i { E Υ ϕ n,k Υn,k Υ} η i+1 Υ = η i Υ τ i N Ith ϕ n,k Υ n,k Υ H where for the iteration nuber i, τ1 i and τi are the step sizes The initial values of dual ultipliers and step size selection are iportant towards obtaining the optial solution, and can greatly effect the optiization proble convergence The potential optiu continuous-rate adaptive constellation size vector for user n over subcarrier k is written as Mn,k Υ = ax 1, ln in µ+ηυ H 40 Algorith 1 Subgradient-based ethod; ASE, Mn,k Υ, ϕ n,k Υ, and n,k Υ, are the optial values of ASE, M n,k Υ, ϕ n,k Υ, and n,k Υ, reectively 1 Assign initial values to λ k Υ, µ, ηυ, τ1, i and τ, i n {1,,N}, and k {1,,K}, reectively Calculate n,k Υ and ϕ n,k Υ, n {1,,N}, and k {1,,K}, using 34 and 36, reectively 3 Update λ k Υ, µ, ηυ, τ1 i, and τi, for any n {1,,N}, and k {1,,K}, according to 38 and 39 4 Repeat steps and 3 until convergence 5 Deterine n,k Υ and ϕ n,k Υ, using 34 and 36, reectively 6 Based on the obtained result fro step 5, calculate Mn,k Υ using 10, hence, copute ASE according to 7 Note that the aforeentioned expression serves as an upperbound for practical scenarios where only discrete-valued constellation sizes are applicable Nevertheless, the real-valued Mn,k Υ in 40 ay be truncated to the nearest integer The correonding axiu aggregate average ectral efficiency of the adaptive MQAM/OFDMA syste is thus derived below { ASE = E Υ log ax 1, lnin t µ+ηυ H ] ϕ n,kυ } 41 According to 40, no transission takes place, ie, Mn,k Υ = 1, when n,k Υ = 0 Consequently, the optiized cut-off threshold, dictated by the channel quality, power constraint, and interference constraint, is given by: Υ th n,k = lnµ+ηυ H in ζ IV INTERFERENCE CONSTRAINT WITH AVERAGE CASE/WORST CASE IMERFECT CROSS-LINK CSI Due to technical reasons such as estiation errors and wireless channel delay, perfect channel inforation is not available In shared-ectru environents, controlling the interference on the priary receivers is highly dependent on the accuracy of the cross-service channel estiation Here, we assue that iperfect cross-link knowledge between CTx and Rxs is available at the secondary transitter The interference anageent at the cognitive base station is based on the noisy estiation of CTx and Rx channel-to-noise-plus-interference ratio CINR by Ĥ in As previously entioned, by considering the correlated case of the estiation Ĥ and error H rando variables, we derive a posterior distribution of the actual channel conditioned on its estiate, to facilitate robust and reliable interference anageent The axiu achievable aggregate ectral efficiency in bits per second per Hertz bps/hz, for the cognitive radio

8 8 syste operating under peak aggregate interference constraint and total average transit power constraint, for a given BERtarget quality, with noisy cross-link CSI, is the solution to the following optiization proble roble O : ax ϕ n,k Υ, n,k Υ { } E Υ ĥ log M n,k Υϕ n,k Υ s t: constraints in 8b, 8d, 8e, and 8f, 4a ϕ n,k Υ n,k Υ H Ĥ I th, {1,,M} 4b where ĥ is defined as a vector containing Ĥ of all tie intervals The objective of this section is to devise an estiation fraework by eploying a posteriori pdf of the channel estiation error given the channel estiation This general fraework enables us to forulate the average case and worst case scenarios of the channel estiation error A Analysis for the Average Case of Estiation Error roposition 1: Given Ĥ and H are statistically correlated rando variables with a correlation factor ρ = δ /δ +δ, where 0 ρ 1, hence, H H H The posterior distribution of covĥ, H = δ H H given Ĥ is a coplex Gaussian rando variable with reective ean and variance of and µ H = E Ĥ H H Ĥ Ĥ = E H H Ĥ E Ĥ Ĥ + cov H,Ĥ δ H δ = var H H Ĥ Ĥ = δ H = 1 ρ δ H +δ H = ρ Ĥ 43 1 cov H,Ĥ ] δ δ H Ĥ 44 Using, the interference constraint in 4b for the average case of estiation error can be written as ϕ n,k Υ n,k Υ Ĥ + H Ĥ I th 45 With further analysis, the above is reduced to ϕ n,k Υ n,k Υ varx denotes the variance of x and covy,z is defined as the covariance of y and z Ĥ Ĥ + H Ĥ I th 46 where Ĥ Ĥ is a constant Thus, by substituting the expectation in 43, we have ϕ n,k Υ n,k Υ Ĥ +ρ Ĥ I th 47 By adopting a siilar approach to that in the previous section, we eploy the Lagrangian dual optiization ethod to obtain ASE for the average case scenario The potential optiu power allocation policy for user n and subcarrier k is given by n,kυ = 1 in t lnµ+ηυ Ĥ 1+ρ ] + 48 where in the average case, = K Ĥ 1+ρ The optial subcarrier allocation policy is the solution to the following proble n = argaxλυ n,k, n {1,,N}, k {1,,K} 49 where n is the optial CRx index, and ΛΥ n,k = ln n,k Υ in t 1+ n,k Υ in t + ln 1+ n,k Υ in t ln 50 Subsequently, the optial continuous-rate solution for the constellation size of user n over subcarrier k is derived Mn,k Υ = ax 1, lnin t µ+ηυ Ĥ 1+ρ 51 Hence, the following axiu aggregate average ectral efficiency for the ectru-sharing syste under iperfect cross-link CSI knowledge for the average case of estiation error can be achieved based on the optial power, rate, and subcarrier allocation policies { ASE = E Υ ĥ log n,k ax 1, lnin t µ+ηυ Ĥ 1+ρ ] ϕ n,kυ where the optiized cut-off SINR is expressed as: Υ th lnin t µ+ηυ Ĥ 1+ρ ζ } 5 n,k =

9 9 B Analysis for the Worst Case of Estiation Error To derive the interference constraint for the worst case scenario, we ust obtain a forulation for the upper-bound of H Recall that H is a Gaussian rando variable Therefore, we proceed by bounding the channel estiation error with a certain probability By eploying the Chebyshev s inequality, for any Y > 0, we have H Ĥ Ω 1 1 Y }{{} 53 pr where Ω = E H +Y Ĥ H Ĥ var H Ĥ 54 With further anipulation, for a given probability of error, pr, the following holds var H Ω = Ĥ +E 1 pr H H Ĥ Ĥ 55 The interference constraint for the worst case scenario of estiation error is expressed as ϕ n,k Υ n,k Υ Ĥ +Ω I th 56 Utilizing one-level dual decoposition ethod, and by applying KKT conditions, the optiu adaptive power allocation schee for user n over subcarrier k is derived as n,k Υ = 1 lnµ+ηυ Ĥ +Ω in ] + 57 where in the worst case, = K Ĥ + Ω To derive the optial subcarrier allocation policy, the following axiization proble is forulated n = argaxλυ n,k, n {1,,N}, k {1,,K} 58 where the optial cognitive user index, n, can be obtained by substituting 57 in 50 and thus solving the optiization proble in 58 The optial continuous-rate solution for the constellation size of user n over subcarrier k is derived Mn,k Υ = ax 1, lnin µ+ηυ Ĥ +Ω 59 The axiu aggregate average ectral efficiency for the adaptive MQAM/OFDMA syste under iperfect cross-link CSI availability for the worst case of estiation error with a given probability of error, pr, is expressed as { ASE = E Υ ĥ log n,k ax 1, lnin t µ+ηυ Ĥ +Ω ] ϕ n,kυ where the optiized cut-off SINR is expressed as: Υ th lnin t µ+ηυ Ĥ +Ω ζ V ROBABILISTIC INTERFERENCE CONSTRAINT } 60 n,k = In a practical ectru-sharing syste, the collision tolerable level is confined by a axiu collision probability allowed by the licensed network The collision tolerable level is highly dependent on the priary service type For exaple, in case of real-tie video streaing, a high collision probability is not desirable, however, delay-insensitive services can tolerate higher packet loss rates In this section, we consider an underlay ectru-sharing scenario where the priary users can tolerate a axiu collision probability ε, {1,,M} We derive optial power, rate, and subcarrier allocation algoriths for the ulti-user OFDMA CR syste under noisy cross-link CSI availability subject to satisfying the iposed peak aggregate power and collision probability constraints The axiization proble can be forulated as follows roble O 3 : ax ϕ n,k Υ, n,k Υ { } E Υ ĥ log M n,k Υϕ n,k Υ 61a s t: constraints in 8b, 8d, 8e, and 8f, N ϕ n,k Υ n,k Υ H Ĥ > Ith ǫ, {1,,M} 61b We proceed by deriving a posteriori distribution of the actual cross-link given the estiated channel gains roposition : The posterior distribution of the actual channel H given the estiation Ĥ is a coplex Gaussian rando variable with reective ean and variance of and µ H = E Ĥ H Ĥ Ĥ + H Ĥ = EĤ Ĥ Ĥ Ĥ +E H H Ĥ Ĥ = 1+ρĤ 6 δ = H varĥ Ĥ + H Ĥ = varĥ Ĥ +var H Ĥ +cov H Ĥ,Ĥ Ĥ = 1 ρ δ H 63

10 10 Cuulative robability K = 3 K = 64 K = Data Approxiated Model Epirical Data Cuulative robability K = 3 K = Data K = 18 Approxiated Model Epirical Data a βk Chi-Square,, δ H = 1 Ĥ b βk Gaa,05,4, δ H = 05 Ĥ Fig : Approxiated Model and Epirical Data cdfs, obtained fro Monte-Carlo siulations Assuing equal variance δ across all users and H Ĥ subcarriers, the collision probability constraint in 61b can be expressed as δ H Ĥ ϕ n,k Υ n,k Υ Ξ k] > Ith ε 64 where Ξ k] is a coplex Gaussian rando variable with variance of one and ean of µ H µ Ξ k] = Ĥ 65 δ H Ĥ It should be noted that in contrast to the su of equal-weighted Chi-Square rando variables in Lea 1, 64 includes a su of non-equal-weighted Chi-Square rando variables In general, obtaining the exact distribution of the linear cobination of weighted Chi-Square rando variables is rather coplex Although several approxiations have been proposed in the literature, eg, 8, 9, 30], ost are not easy to ipleent In this work, we propose a siple approxiation based on the oents of δ N K H Ĥ n=1 ϕ n,kυ n,k Υ Ξ k] Consider the following equality δ H Ĥ ϕ n,k Υ n,k Υ Ξ k] = βk Ξ k] 66 where βk = N n=1 δ ϕ H n,k Υ n,k Υ Ĥ roposition 3: The distribution of the su of nonequal-weighted non-central Chi-Square rando variables, ie, K β k Ξ k], is siilar to that of a weighted non-central Chi-Square-distributed rando variableξχ D, whereδ, D, δ and ξ are reectively the non-centrality paraeter, degree of freedo, and weight of the new rando variable: δ = µ Ξ k] 67 D = K 68 K ξ = β k +µ Ξ k] K + K µ 69 Ξ k] To investigate the above siilarity, or the accuracy of the proposed approxiation, we copare the cdf of the proposed Chi-Square distribution with that of 66, using Monte-Carlo siulations The results in Fig illustrate that the approxiation is accurate over a wide range of practical values for K over randoly-distributed - eg, Chi-Square or Gaa - weights βk Now 64 can be siplified to: δ H Ĥ ϕ n,k Υ n,k Υ Ξ k] > Ith rξχ Dδ > I th 70 According to 30], since the non-centrality paraeter is sall relative to the degree of freedo, we can approxiate the non-central Chi-Square distribution with a central one using the following ξχ D > I δ th χ D 0 > I th /ξ 71 1+δ /D The right hand side RHS of 71 can be forulated using the upper Gaa function 31] as I th /ξ 1+δ /D χ D 0 > I th /ξ 1+δ /D = ΓK, ΓK 7 where Γ, is the upper incoplete Gaa function, and Γ is the coplete Gaa function roposition 4: For all integer values K 1, and all positive I th /ξ 1+δ /D - this condition is always true because,i th, δ β k, and K are positive; consequently, ξ, δ, and D are also positive - the deterinistic inequality δ H Ĥ +µ Ξ k] ϕ n,k Υ n,k Υ n=1 KIth K! 1/K ln 1 1 ε 1/K 73 satisfies the probabilistic inequality 64 Therefore, the constraint 64 can be replaced by 73

11 11 roof: The proof is given in the Appendix A To obtain ASE for the probabilistic interference constraint and probabilistic case of estiation error scenario, we eploy the Lagrangian dual optiization ethod as in the previous sections, where and I th = α k = δ +µ H Ξ Ĥ k] 74 KIth K! 1/K ln 1 1 ε 1/K 75 Therefore, by solving the Lagrangian optiization proble the following potential optial power allocation solution can be obtained for user n over subcarrier k n,kυ 1 in = lnµ+ηυα k ˆN t ˆN ] + 76 where in the probabilistic case is derived in Appendix A, Section C The optial subcarrier allocation policy is the solution to the following proble n = argaxλυ n,k, n {1,,N}, k {1,,K} 77 where n is the optial CRx index, and ΛΥ n,k = ln n,k Υ in t ˆN 1+ n,k Υ in t ˆN + ln 1+ n,k Υ in t ˆN ln 78 By eploying the sub-gradient ethod in Algorith 1, the Lagrangian ultipliers µ and ηυ can be updated by µ i+1 = µ i τ1 i ϕ n,k Υ n,k Υ 79 η i+1 Υ = η i Υ τ i Ith δ +µ H Ξk] ϕ Υn,kΥ Ĥ n=1 80 Subsequently, optial expressions are derived for the constellation size and hence aggregate ectral efficiency under collision probability constraint and iperfect cross-link CSI: M n,k Υ = ax 1, ASE = ln in, lnin t µ+ηυα ˆN k 81 { E Υ ĥ log n,k ax 1, ˆN µ+ηυα k ]ϕ n,k Υ } 8 robability Density Function pdf User 1, = 30 Watts User, = 30 Watts User 3, = 30 Watts User 1, = 0 Watts User, = 0 Watts User 3, = 0 Watts Received SINR db Fig 3: robability density functions of the received SINR for OFDMA users in a given subcarrier k under different average power constraint values Syste paraeters are: K = 64, k = 16, I th = 5 Watts where the optiized cut-off SINR threshold is coputed by: Υ th n,k = lnin ˆN µ+ηυ Ĥ +Ω ζ The ethodologies for deriving the expressions of the cdf of the received SINR given the estiation, for different average case, worst case, and probabilistic case scenarios of estiation error, are elucidated in Appendix B VI DISCUSSION OF RESULTS In this section, we exaine the perforance of the OFDMA CR network operating under total average transit power and deterinistic/probabilistic peak aggregate interference constraints with perfect/iperfect cross-channel estiation using the reective optial resource allocation solutions In the following results, perfect CSI knowledge of the cognitive user link is assued to be available at the CTx through an error-free feedback channel Thus, H ss, {n,k}, are drawn through a n,k Rayleigh distribution Further, the secondary-secondary power gain ean values, µ H ss n,k, {n,k}, are taken as Uniforlydistributed rando variables within 0 to It should be noted that the sub-channels are assued to be narrow-enough so that they experience frequency-flat fading Interfering crosschannel values, H, {}, are distributed according to a coplex Gaussian distribution with ean 005 and variance 01 For the inaccurate cross-link CSI case, the channel estiation and error for all sub-channels are taken as iid zero-ean Norally-distributed rando variables In addition, the AWGN power ectral density is set to -174 db The total average power constraint is iposed on the syste in all cases Discrete-rate cases with real-valued MQAM signal constellations, ie, log M {,4,6,8,10} bits/sybol, are also considered for practical scenarios All results correond to the scenario with three cognitive receivers and a single priary receiver, hence, the subscript is hereafter oitted The approxiated probability distributions of the received SINRs for cognitive users in a randoly taken subcarrier, ie, here k = 16, under different total average power constraint

12 1 Aggregate Average Spectral Efficiency bps/hz Optial Value Dual Value Nuber of Iterations Aggregate Average Spectral Efficiency bps/hz ξ = 10 ξ = 10 3 ξ = 10 4 ξ = 10 5 ξ = I th Watts Fig 4: Optial and dual values versus the nuber of iterations using the sub-gradient ethod Results for the case with deterinistic interference constraint and perfect cross-link CSI knowledge Syste paraeters are: K = 64, = 30 Watts, I th = 10 Watts, ξ = 10 Fig 6: ASE perforance using the proposed RRA algorith versus I th constraint for different BER-target values Results correond to the case with deterinistic interference constraint and perfect crosslink CSI Syste paraeters are: K = 64, = 30 Watts Aggregate Average Spectral Efficiency bps/hz K = 64 K = 3 = 35 Watts = 30 Watts = 5 Watts = 0 Watts I th Watts Fig 5: ASE perforance versus the tolerable interference power threshold level with different values of and K Results for the case with deterinistic interference constraint and perfect cross-link CSI knowledge Syste paraeters are: I th = 10 Watts, ξ = 10 liits is plotted in Fig 3 For a fixed interference constraint of I th = 5 Watts, it can be observed that the probability of higher received SINR iproves as the value of increases For exaple, for user 3, the probability of receivingγ 3,16 = 10 db is 545% higher as the value of is increased fro 0 to 30 Watts Fig 4 illustrates the evolution of the optial and dual values using the sub-gradient ethod over tie The results correond to the axiu deliverable ASE for the case with deterinistic interference constraint and perfect cross-link CSI knowledge The iterative sub-gradient algorith converges quickly and typically achieves a lower-bound at 965% of the optial value within 1 iterations It can easily be shown that the proposed dual decoposition algorith converges fast for different paraeters of syste settings Fig 5 shows the achievable ASE of the adaptive Aggregate Average Spectral Efficiency bps/hz = 5 Watts = 5 Watts = 0 Watts ρ Fig 7: Achievable ASE with iperfect cross-link CSI and average case of estiation error against ρ for different values of Syste paraeters are: K = 64, I th = 5 Watts, ξ = 10, δ Ĥ = 1 k MQAM/OFDMA CR syste versus CTx-Rx interference power threshold levels under total average power and deterinistic interference constraints with perfect cross-link CSI knowledge As expected, greater ASE values are achieved for higher axiu tolerable interference since I th liits the cognitive users transit power The iproved perforance however approaches a plateau in the high I th region as the threshold becoes the doinant power constraint Note that the iproved perforance by increasing I th coes at the cost of increased probability for violating the priary users QoS Further, iposing a higher axiu peak average power setting enhances the achievable ASE in high I th region -, for the particular values taken in this exaple, achieve the sae ASE over sall I th settings Moreover, increasing the nuber of subcarriers results in higher attainable perforance Achievable ASE perforance under different axiu tolerable interference thresholds for reective values of BERtarget with perfect cross-link CSI availability is shown in Fig

13 13 Aggregate Average Spectral Efficiency bps/hz pr = 01 pr = 03 pr = 05 pr = 07 pr = ρ Aggregate Average Spectral Efficiency bps/hz Average Case robabilistic Case Worst Case I th Watts Fig 8: Achievable ASE with iperfect cross-link CSI and worst case of estiation error against ρ with pr Syste paraeters are: K = 64, = 0 Watts, I th = 5 Watts, ξ = 10 3, δ Ĥ = 1 k Fig 10: erforance under iperfect cross-link CSI for different cases of estiation error againsti th Syste paraeters are:k = 64, = 45 Watts,ξ = 10,pr = 095, ρ = 0, ǫ = 5%,δ Ĥ = 01 k Aggregate Average Spectral Efficiency bps/hz I th = 10 Watts I th = 8 Watts I th = 6 Watts I th = 4 Watts ε Fig 9: Achievable ASE with iperfect cross-link CSI and probabilistic case of estiation error againstǫwithi th Syste paraeters are: K = 64, = 40 Watts, ξ = 10 3, ρ = 05, δ Ĥ = 1 k 6 It can be seen that the syste perforance is iproved under less stringent QoS constraints For exaple, a 69% gain in ASE perforance is achieved by iposing ξ = 10 in coparison to ξ = 10 3 However, the gap in perforance becoes less significant for lower BER-target regies Syste perforance with noisy cross-link CSI and average case of estiation error versus the correlation factor between estiation and error variables ρ is depicted in Fig 7 It can be seen that a higher correlation factor increases the axiu likelihood between true and estiated interfering channels, hence, the probability of violating the interference constraint on average is iproved and in turn a lower ASE for the cognitive syste is realized Further, the achievable ASE with iperfect cross-link CSI knowledge and worst case of estiation error against ρ for different probabilities of channel estiation error bound pr is studied in Fig 8 Apart fro the effect of ρ on the perforance, higher values of pr increase the robustness of the interference anageent schee but coe at the cost of lower achievable ectral efficiencies The results indicate that the iproved ASE perforance by decreasing pr in the lower half region ie, pr 05 is not significant yet it ay cause critical interference to the priary service operation For exaple, given ρ = 05, varying the value of pr fro 05 to 01 results in a 40% increase in the probability of error bound violation but only provides an effective gain of 3% in the cognitive syste perforance The achievable perforance with iperfect cross-channel inforation and probabilistic case of estiation error versus the collision probability ǫ with reective I th values is illustrated in Fig 9 Increasing the axiu probability of violating the interference constraint set by the a regulatory authority significantly iproves the ectral efficiency of the cognitive network The tradeoff is however the degradation of the priary service operation which is deeed highly undesirable in practical scenarios Syste perforance with noisy cross-link CSI for different average case, worst case, and probabilistic case of estiation error is deonstrated in Fig 10 The results show that the probabilistic case with 5% collision probability outperfors the achievable ASE under the worst case scenario with an error bound of pr = 05 For exaple, given I th = 6 Watts, the probabilistic case achieves a 67% gain in ASE over the worst case Further, eploying the average case provides higher ectral efficiencies For instance, a 70% increase in perforance in achieved utilizing the average case over the probabilistic case For high values of I th, the total average power constraint becoes the doinant liit and therefore the perforance under different cases of estiation error eventually converge Note that the average case controls the interference based on the average error estiation, therefore, it cannot itigate the potential instantaneous interference violations On the other hand, ipleenting the worst case can guarantee that the interference constraints are obeyed at any given tie, thus, preserving the priary users QoS The proposed probabilistic case of estiation error provides an optial trade-off between the achievable perforance of cognitive syste and anaging the QoS of priary users In

14 14 particular, the probabilistic case is advantageous in ters of perforance and flexibility over the conventional average case and worst case scenarios VII CONCLUSIONS In this paper, we have studied the ectral efficiency perforance of adaptive MQAM/OFDMA underlay CR networks with certain/uncertain interfering channel inforation We derived novel RRA algoriths to enhance the overall cognitive syste perforance subject to satisfying total average power and peak aggregate interference constraints The proposed fraework considers both cases of perfect and iperfect cross-link CSI knowledge at the cognitive transitter In the latter, different average case, worst case, and probabilistic case scenarios of channel estiation error were odeled and analysed To copute the aggregate average ectral efficiency, we developed unique approxiated distributions of the received SINR for given users over different sub-channels in the reective cases under consideration Through siulation results we studied the achievable perforance of the cognitive syste using our proposed RRA algoriths By adapting the power, rate, and subcarrier allocation policies to the tievarying secondary-secondary fading channels and secondarypriary interfering channels, a significant gain in the ectral efficiency perforance of the cognitive syste can be realized, whilst controlling the interference on the priary service receivers Furtherore, the ipact of paraeters uncertainty on overall syste perforance was investigated In particular, siulation results were provided for different cases of error estiation It was understood that the average case results in higher cognitive syste perforance, however, coes at the cost of potential instantaneous interference violations Subsequently, the worst case can guarantee that the power constraints are obeyed at all ties, yet it does not result in desirable cognitive perforance In contrast, the proposed probabilistic case in this paper, which was derived as a low coplexity deterinistic constraint, provided an optial trade-off between the achievable perforance of the cognitive network and preserving the QoS of the priary users In suary, the probabilistic case can replace the conventional average case and worst case scenarios in practical situations as a result of enhanced perforance and flexibility AENDIX A ROOF OF ROOSITION 4 For a Chi-Square rando variable χ D, with a degree of freedo K, the probability ξχ D δ > Ith ξχ D 0 > I th /ξ, δ /D can be forulated using the upper gaa function 3] as rχ D0 > I th /ξ ΓK, ξ1+ = δ /D δ /D ΓK By defining x y = I th I th ξ1+ δ /D, 85 and using the results fro 3], we have ΓK, x y = y x e ty dt, 86 where y = 1/K Given 1/KΓK = 1/ΓK+1, and using the corollary of 3], we can derive the following equalities ΓK, x y ΓK = y x e ty dt x = e ty dt ΓK ΓK The upper-bound of 87 can be expressed as ΓK, x y ΓK = x e ty dt ΓK +1 y 1 1 e ϑx ] 1/y 88 The expression in 88 is valid for all positive x, if and only if, Thus, for all K 1, 0 ϑ in{1,γk +1] 1/K } 89 ϑ = ΓK +1 1/K 90 Subsequently, fro 64, 88, and 90, ΓK, x y ΓK and by replacing 87 in 91, we have ε e ϑxy ] 1/y ε 9 With further anipulation, it can be shown that Replacing 85 in 93 we have: x y = ln1 K 1 ε 93 ΓK +1 1/K I th K ξ1+ δ /D ln1 1 ε, 94 ΓK +1 1/K Finally, by replacing 67, 68, and 69 in 94, we have: δ H Ĥ +µ Ξ k] ϕ n,k Υ n,k Υ n=1 KIth K! 1/K ln 1 1 ε 1/K 95 AENDIX B RECEIVED SINR CDF DERIVATION To derive the cdf of the received SINR given the estiation, F γn,k γ n,k Ĥ Ĥ, for different average case, worst case, and probabilistic case, scenarios