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capacity limits and performance analysis of cognitive radio with imperfect channel knowledge


IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 59, NO. 4, MAY 2010

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Capacity Limits and Performance Analysis of Cognitive Radio With Imperfect Channel Knowledge

/>Himal A. Suraweera, Member, IEEE, Peter J. Smith, Senior Member, IEEE, and Mansoor Sha, Fellow, IEEE
Abstract—Cognitive radio (CR) design aims to increase spectrum utilization by allowing the secondary users (SUs) to coexist with the primary users (PUs), as long as the interference caused by the SUs to each PU is properly regulated. At the SU, channel-state information (CSI) between its transmitter and the PU receiver is used to calculate the maximum allowable SU transmit power to limit the interference. We assume that this CSI is imperfect, which is an important scenario for CR systems. In addition to a peak received interference power constraint, an upper limit to the SU transmit power constraint is also considered. We derive a closed-form expression for the mean SU capacity under this scenario. Due to imperfect CSI, the SU cannot always satisfy the peak received interference power constraint at the PU and has to back off its transmit power. The resulting capacity loss for the SU is quantied using the cumulative-distribution function of the interference at the PU. Additionally, we investigate the impact of CSI quantization. To investigate the SU error performance, a closed-form average bit-error-rate (BER) expression was also derived. Our results are conrmed through comparison with simulations. Index Terms—Average bit error rate (BER), channel capacity, cognitive radio (CR), partial channel-state information (CSI), quantized feedback.

I. I NTRODUCTION HE RADIO spectrum is one of the most valuable resources for wireless communications. Conservative spectrum policies employed by regulatory authorities have created the perception of a spectrum shortage that has resulted in underutilization of the overall available spectrum for communications. However, measurements performed by agencies such as the Federal Communications Commission has revealed that, at any given time, large portions of spectrum are sparsely occupied. Given this fact, new insights into the use of spectrum have challenged the traditional approaches to spectrum-management motivating research in cognitive radio (CR) technology for opportunistic use of the spectrum [1]–[5].

T

Manuscript received May 21, 2009; revised October 19, 2009. First published February 22, 2010; current version published May 14, 2010. This work was supported by the Australian Research Council under Discovery Grant DP0774689. This paper was presented in part at the IEEE Global Communications Conference, Honolulu, HI, November/December 2009. The review of this paper was coordinated by Prof. H. Zhang. H. A. Suraweera is with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 119260 (e-mail: elesaha@nus.edu.sg). P. J. Smith is with the Department of Electrical and Computer Engineering, University of Canterbury, Christchurch 8140, New Zealand (e-mail: p.smith@elec.canterbury.ac.nz). M. Sha is with Telecom New Zealand, Wellington 6011, New Zealand (e-mail: mansoor.sha@telecom.co.nz). Digital Object Identier 10.1109/TVT.2010.2043454

The CR concept, which was rst introduced by Mitola [1], refers to a smart radio that can sense the external electromagnetic environment and adapt its transmission parameters according to the current state of the environment. According to the quantity, reliability, and type of information available to a CR system, it can adopt three different spectrum-sharing paradigms [6]. CRs can be designed to access parts of the primary user (PU) spectrum for their information transmission, provided that they cause minimal interference to the PUs in that band [2], [3]. This can be achieved in several ways. For example, according to one of the paradigms widely referred to as the interweave approach in the literature, CRs can sense the spectrum and access it when an unused primary slot is detected. In another model, known as the underlay approach, CRs can simultaneously coexist with the PUs, provided that they operate under a certain interference level as imposed by a regulatory agency. Limits on this received interference level at the primary receiver can be imposed with a long-term average or short-term peak constraint, e.g., [7]. Capacity analysis is very useful in understanding the performance limits and, thus, the potential applications of CR systems. Several interesting results on the capacity, outage probability, and throughput of CR systems have recently emerged. See, for example, [7]–[14], [16], and the references therein. In [8], the capacity of nonfading additive white Gaussian noise (AWGN) channels under an average receivedpower constraint at a primary receiver is derived. In [9], it was shown that, with the same limit on the received-power level, the channel capacity for several different fading models (e.g., Rayleigh, Nakagami-m, and lognormal fading) exceeds that of the nonfading AWGN channel. In [10], the ergodic, the outage, and the minimum-rate capacity gains offered by a spectrumsharing approach under average and peak interference constraints in Rayleigh fading environments have been studied. It has been shown in [10] that imposing a constraint on the peak received power on top of the average received-power constraint does not yield a signicant impact on the ergodic capacity as long as the average received power is constrained. In [11], optimal power-allocation strategies to achieve the mean capacity and the outage capacity of the secondary user (SU)1 fading channel under different types of power constraints and channelfading models have been investigated. The authors show that the SU capacity achieved is higher under the average constraint compared with the peak interference power constraint and that

1 In the following, “cognitive radio” and SU will both be used to identify the node that seeks access to the PU’s licensed spectrum.

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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 59, NO. 4, MAY 2010

fading in the channel between the SU transmitter and the PU receiver is benecial for enhancing the SU ergodic and outage capacities. In [7], the author has compared the PU capacity loss under average and peak interference constraints. For the scenario considered in [7], the average interference constraint provides better PU performance. Considering that, in some situations, the PU spectral activity in the vicinity of the CR transmitter may differ from that in the vicinity of the cognitive receiver, in [12], the capacity of opportunistic spectrum acquisition has been investigated. The links of a primary/secondary radio environment could also experience different types of fading such as Rayleigh and line-of-sight (LoS) Rician fading. Under such LoS scenarios, in [13], we have investigated the ergodic capacity of spectrum sharing under average and instantaneous interference constraints. It has been shown that the SU mean capacity is sensitive to the type of fading on the SU–SU and SU–PU links, and depending on the fading type on either link, the capacity can be either larger or smaller compared with the case of symmetric non-LoS Rayleigh fading. In [14], assuming a pathloss shadow-fading model with multiple PUs and SUs, the system-level capacities of CR networks under an average interference power constraint have been investigated. Their results have shown that the uplink ergodic channel capacity of a CR-based central access network can be relatively large when the number of PUs is small. Moreover, the authors have demonstrated the benets of employing multipleinput–multiple-output (MIMO) technology for SU networks targeting urban area deployments where a large number of coexisting PUs are expected. In [15], the allowable transmit powers for single- and multiantenna SU systems are evaluated under different types of fading (Rayleigh and Rician) for the PU–PU link and assuming that a target outage performance is applied in the PU system. Specically, it has been found in [15] that, for PU-PU paths with signicant LoS, the total power allowed for a multiantenna SU system is higher than the power allowed for a single-antenna SU system. Hence, the multiantenna SU system achieves power and diversity gains. References [7]–[14] have all assumed that the SU has full channel-state information (CSI) knowledge of the link between its transmitter and the PU receiver. However, in practice, obtaining full CSI is difcult, and often, only partial CSI information can be acquired. This important situation has been studied in [16] under certain conditions. While [16] looks at the impact of partial CSI on the capacity, it only does so under an average interference constraint. The use of such a constraint is relevant when a long-term interference-induced degradation is to be considered. This may involve modeling both fast-fading and shadow-fading components of the radio channel. When only fast fading (Rayleigh) is considered, an interference constraint based on peak interference is more relevant. Furthermore, our approach to the CSI imperfections is different from that in [16], as our model caters for a range of solutions from near-perfect to seriously awed channel estimates. Even if a genie provides perfect CSI at the receiver, it must be quantized into a limited number of levels before being fed back to the SU transmitter. This process effectively converts the perfect CSI into an imperfect CSI scenario. Therefore, analyzing the impact of CSI imperfections on the SU capacity

is the key motivation of this paper. We assume partial CSI knowledge of the SU–PU link possibly due to a combination of channel estimation error, mobility, feedback delay, and limited feedback. As in [11], we assume that the SU has a maximum transmit power threshold since all real power ampliers have an upper limit on their transmit power. In this paper, we make several contributions. 1) We develop a closed-form expression for the SU mean capacity when it is required to work under a peak interference constraint imposed by the primary. We determine the impact of imperfect CSI of the SU–PU link by examining the effect of this on the interference constraint and the SU mean capacity. 2) Compared with perfect channel knowledge, under imperfect CSI, the SU transmissions may result in a higher than acceptable interference to the PU. Consequently, the PU may demand lowering of the SU transmit power and, in turn, cause the SU to absorb a capacity loss. We relate this loss to the extent of the CSI imperfections. To quantify this SU capacity loss, the cumulative distribution function (cdf) of the received interference at the PU is derived. 3) We enhance the aforementioned result by including the impact of quantization levels on the CSI and determine the number of levels before a regime of diminishing gains sets in. 4) Given the interference constraints, we develop a closedform expression for the uncoded SU average bit error rate (BER) that can be extended to different modulation schemes. This allows us to compare the BER behavior versus peak interference with the corresponding trend for SU mean capacity. This paper is organized as follows: Section II introduces the system model. In Section III, we investigate the mean SU capacity, the statistics of the PU interference, and the quantization effects of the CSI. The average BER of the SU system is analyzed in Section IV. In Section V, numerical results supported by simulations are presented and discussed. Finally, we conclude in Section VI. II. S YSTEM M ODEL In this section, the system and channel models considered in the paper are briey outlined. The system model is shown in Fig. 1. We assume that the PU and SU communication links share the same narrow-band frequency with bandwidth B for transmission. Moreover, point-to-point at Rayleigh fading channels are assumed. Let gsp = |hsp |2 , gss = |hss |2 , and gps = |hps |2 denote the instantaneous channel gains from the secondary transmitter to the primary receiver, from the secondary transmitter to the secondary receiver, and from the primary transmitter to the secondary receiver, respectively. Furthermore, we denote the exponentially distributed probability density functions (pdfs) of the random variables (RVs) gsp , gss , and gps by fgsp (x), fgss (x), and fgps (x), respectively. These pdfs are governed by the parameters λsp = E(gsp ), λss = E(gss ), and λps = E(gps ), respectively, where E() is the expectation operator. The AWGN at the PU receiver and the SU receiver are denoted by np and ns , respectively, and

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receiver’s noise oor. Hence, we are considering situations where the primary’s quality-of-service would be limited by the instantaneous SNR at the primary receiver [9]. Furthermore, a maximum SU transmit power constraint Pm is assumed. In practice, such a limitation arises due to the power amplier nonlinearity [11], resulting in an upper transmit power limit. Now, based on the channel estimate, the cognitive transmitter selects its transmit power Pt as Pt = min Ip , Pm . gsp (2)

Fig. 1.

System model.

Therefore, at the SU, the signal-to-interference-and-noise ratio (SINR) γ can be written as γ= Pt gss Pp gps + σ 2 (3)

have the common distribution CN (0, σ 2 ) (circularly symmetric complex Gaussian variables with zero mean and variance σ 2 for bandwidth B). Perfect knowledge of the SU–SU channel is assumed at the SU receiver. However, the SU is only provided with partial channel knowledge of hsp . There are several mechanisms where this can occur. For example, information about hsp could periodically be measured by a band manager. Next, using a nite bandwidth channel, this information could be provided to the SU. Another example is primary secondary collaboration and exchange, where information about hsp could directly be fed back from the PU receiver to the SU transmitter, as proposed in [17]. A further extension of this work will examine the combined effect of imperfection in the SU–SU channel. With partial CSI of the SU–PU link at the SU transmitter, we have an estimate of the channel hsp of the form hsp = ρhsp + ( 1 ρ2 ) (1)

where Pp is the PU transmit power. The mean capacity of the secondary system can be calculated from


C =B
0

log2 (1 + x)fγ (x)dx


B = loge (2)

1 Fγ (x) dx 1+x

(4)

0

where B is the bandwidth, fγ (x) is the pdf, and Fγ (x) is the cdf of the RV γ. The second equality in (4) follows from integration by parts. Note that, to evaluate the SU mean capacity, an expression for the cdf of the RV γ must be developed. This is derived in the succeeding discussion. The cdf of γ is given by Fγ (x) = Pr


where hsp is the channel estimate available at the secondary transmitter, and is CN (0, λsp ) and is uncorrelated with hsp . The correlation coefcient 0 ≤ ρ ≤ 1 is a constant that determines the average quality of the channel estimate over all channel states of hsp . This model is well established in the literature, which investigates the effects of imperfect CSI [18]. Note that ρ can be used to assess the impact of several factors on the CSI, including channel-estimation error, mobility, and feedback delay. As shown in Section III, the same formulation can be extended to incorporate quantization effects. In [19] and [20], a very similar model is used, where ρ is calculated for a particular training-based channel-estimation scheme. It is shown that ρ is a function of the length of the training sequence, SNR, and Doppler frequency. III. S ECONDARY U SER M EAN C APACITY In this section, we obtain the mean capacity of the SU under a peak interference power constraint. Previous work on the channel capacity of CR has assumed two types of interference constraints at the PU receiver, namely, an average interference constraint and a peak interference power constraint. In this paper, we adopt the latter and assume that the maximum peak interference that the primary receiver can tolerate is Ip . The interference level is measured with respect to the victim

Pt gss <x Pp gps + σ 2 Fτ x(Pp y + σ 2 ) fgps (y)dy (5)

=
0

where Pr() denotes probability, and τ = Pt gss . Therefore, to nd Fγ (x), we rst need an expression for the cdf of τ , Fτ (x) = Pr(τ < x). The cdf of τ is given by Fτ (x) = 1 Pr Pm gss > x, = 1 Pr gss > Ip gss >x gsp (6)

x x , gss > gsp . Pm Ip

Note that (6) can further be simplied by considering the g cases (x/Pm ) (x/Ip )sp and conditioning on gsp . This approach gives the following equation: Pr gss > x x , gss > gsp |sp g Pm Ip Pr gss > = Pr gss >

x Pm

, ,

gsp < gsp >

x Ip gsp

Ip Pm Ip Pm .

(7)

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Now, averaging over gsp , Fτ (x) can be expressed as
Ip /Pm

Substituting (12) into (5), using the pdf of gps and a change of variable gives 1e Fγ (x) = 1
∞ λspp m P
I

Fτ (x) = 1
0

Pr gss >


x Pm

fgsp (y)dy x > y fgsp (y)dy. Ip

e λss Pm x

σ2

Pp λps e

x λsp Pm


Ip /Pm

Pr gss

(8)

×
0

+ Pp 1 ps y λ ∞

dy e

x λss Pm

We can easily simplify the rst integral in (8) as
Ip /Pm



e

σ λspp m λss Pm x P

I

2

+ Pp 1 ps y λ

Pp λps
0

1+

λsp σ 2 x λss Ip

+

λsp xy λss Ip

dy. (14)

Pr gss
0

x > Pm

fgsp (y)dy
Ip /Pm

Simplifying (14) with the help of the identities [23, eqs. (3.310) and (3.383.10)] yields 1e Fγ (x) = 1 . (9)
λspp m P
I

= Pr gss

x > Pm
0

fgsp (y)dy Ip Pm

e λss Pm x

σ2

1+

Pp λps λss Pm x

= Pr gss >

x Pm

Pr gsp <

Since the pdf and the cdf of the exponentially distributed RV gsp are given by 1 y/λsp e fgsp (y) = λsp we reexpress (8) as Fτ (x) = 1 Pr gss > x Pm 1 λsp Pr gsp <


λss Ip σ2 λss Ip + e Pp λps λsp λps Pp x λsp λps Pp x x 1 λss Ip × Γ 0, + σ2 + λss Pm Pp λps λsp x

(15) where Γ(a, x) = x ta1 et dt is the upper incomplete gamma function [23, eq. (8.350.2)]. Alternatively, (15) and later results can be rewritten in terms of the exponential integral E1 (.) using the relation Γ(0, x) = E1 (x). Now, substituting (15) into (4) yields the following equation: C = B 1e
λspp m P
I



Fgsp (y) = 1 ey/λsp

(10)

Ip Pm e dy. (11)



e λss Pm x (1 + x) 1 +
∞ Pp λps x λss Pm
λss Ip

σ2

loge (2)
σ2

dx

e
Ip /Pm

x λss Ip y λ1 y sp

0

λss Ip e Pp λps × + λsp λps Pp loge (2) × Γ 0,

e λsp λps Pp x x(1 + x) σ2 + λss Ip λsp x dx. (16)

0

Finally, after simplifying the integral in (11), we obtain the closed-form cdf of γ as Fτ (x) = 1 e λss Pm
x

x 1 + λss Pm Pp λps

1e

λspp m P

I

1 1+
λsp λss Ip x

e

x λspp m λss Pm P

I

.

(12)

By decomposing into partial fractions, the rst integral in (16) can be evaluated with the help of [23, eq. (3.383.10)]. Unfortunately, to the best of the authors’ knowledge, the second integral in (16) cannot be evaluated in closed form. Therefore, (16) is expressed as 1 e λsp Pm C = Pp λ B loge (2) 1 λss Pps m × Γ 0, σ2 λss Pm
σ2



Ip

The pdf of τ can also be obtained trivially by differentiating Fτ (x) with respect to x. Therefore, the pdf of τ is 1e λss Pm
I λspp m P

fτ (x) =

e λss Pm +
x

1 e λss Pm 1 +
I

I λspp m P

x λss Pm

e λss Pm Γ 0,

λss Ip

σ2

σ2 Pp λps

σ2

e Pp λps

λsp λss Ip x

λss Ip e Pp λps + λsp λps Pp loge (2) (13) × Γ 0,

e λsp λps Pp x x(1 + x) σ2 + λss Ip λsp x dx. (17)

λsp e + λss Ip

x λspp m λss Pm P

0

1+

λsp λss Ip x

2

.

x 1 + λss Pm Pp λps

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We next consider the case where the primary interference at the SU receiver is negligible. This scenario arises when the primary transmitter to the secondary receiver is deeply shadowed. The mean SU channel capacity in this case is given by


Note that, when λsp = λss = 1 is assumed, this is the mean capacity evaluated in [9] for Rayleigh fading channels. As a double check, this further veries the correctness of our mean capacity expression.

C=B
0

log2 1 +

x fτ (x)dx. σ2

(18)

A. Statistics of the Interference at the PU Receiver In this section, we will study the statistics of the interference at the primary receiver due to availability of partial CSI at the cognitive transmitter. The interference inicted at the primary receiver can be found from Pt gsp = min Ip gsp , Pm gsp . gsp (24)

Substituting (13) in (18), the SU mean capacity can be expressed as 1 e λsp Pm C = B λss Pm + e λss Pm
I λspp m P



Ip



log2 1 +
0 ∞

x x e λss Pm dx σ2

log2 1 +
0 ∞

x σ2

e λss Pm
x

1+

λsp λss Ip x

dx
x

I λsp λspp m P e + λss Ip

log2
0

x 1+ 2 σ

e λss Pm 1+
λsp λss Ip x

2 dx.

(19) The integrals in (19) can be evaluated in closed form using integration by parts. Therefore, the SU mean capacity can be expressed in closed form as
σ2 C 1 e λsp Pm σ2 = Γ 0, e λss Pm B loge (2) λss Pm 1 Ip Γ 0, λsp σ 2 λsp Pm loge (2) 1 λss Ip



Ip

Since gsp = gsp , we note that in the presence of partial chan nel information, the interference at times may not be limited to Ip . Hence, the PU’s protection cannot be guaranteed. As such, it is important to analyze the interference statistics under imperfect CSI. A suitable measure for this is the interference cdf. Based on this, we assume that the PU will request the SU to use a reduced level of Ip , e.g., Ip , so that the interference remains below Ip with a desired probability (e.g., 95% or 99%). This strategy, in turn, results in a capacity loss for the SU. Let Z = Pt gsp so that Z represents the interference produced by the SU transmitter. The cdf of Z is given by Pr(Z < z) = 1 Pr(Z > z) = 1 Pr gsp > z1 , gsp > z2 gsp (25)

+

e

λspp m P λsp λss Ip σ2

I

loge (2) 1

Γ 0,

σ2 λss Pm

e λss Pm .

σ2

(20)

where z1 = z/Pm , and z2 = z/Ip . Moreover, we can write gsp > z1 , > z2 = gsp
∞ x/z2

If the tolerable interference at the primary receiver Ip is high, (20) simplies to 1 σ2 C ≈ Γ 0, B loge (2) λss Pm e λss Pm
σ2

Pr gsp

fgsp ,sp (x, y)dy dx g
z1 0

(26)

(21)

as eT → 0 and Γ(0, T ) → 0 as T = (Ip /λsp Pm ) → ∞. In addition, note that in the special case where no constraint upon the maximum allowable transmit power is imposed, i.e., Pm → ∞, the pdf in (13) simplies to fγ (x) = λsp λss Ip 1 +
λ0 λss Ip x 2

where fgsp ,sp (x, y) is the joint density function of the RVs gsp g √ and gsp . Using the joint pdf of r1 = gsp and r2 = gsp [21] and a simple transformation of variables gives fgsp ,sp (x, y) = g
x+y 1 e (1ρ2 )λsp I0 2 )λ2 (1 ρ sp

√ 2ρ xy (1 ρ2 )λsp (27)

(22)

and the SU mean capacity is given by λsp C = B λss Ip


where I0 () is the zeroth-order modied Bessel function of the rst kind [23, eq. (8.431.1)]. Substituting the joint density function fgsp ,sp (x, y) in (26) gives g 1 Pr(Z < z) = 1 (1 ρ2 )λ2 sp
x/z2 ∞

log2 1 + 1+
λss Ip λsp σ 2 λsp σ 2 λss Ip

x σ2

0

λsp λss Ip x

2 dx

e
z1



x (1ρ2 )λsp

=

loge

loge (2) 1

.

(23)

×
0

e



y (1ρ2 )λsp

I0

√ 2ρ xy (1 ρ2 )λsp

dy dx.

(28)

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√ Using the variable transform t = y, and with the help of [22, eq. (10)], the inner integral in (28) can be solved to give Pr(Z < z) = 1 1 λsp


B. Effect of Quantized CSI Feedback In the previous section, we assumed that the sources of imperfect CSI (channel estimation error, mobility, and feedback delay) result in the estimate hsp given in (1). This is reasonable and is widely used in the literature. However, when quantization effects are considered, it is not clear whether such a model is accurate. In practice, CSI will be fed back to the SU transmitter using a nite number of bits representing gsp ranges. If the channel information is quantized, at the secondary transmitter, we have the estimate g gsp = D(sp ) (36)

e
z1

λx sp

dx +

1 λsp



e
z1

λx sp

Q

× where

2ρ2 x , λsp (1 ρ2 )

2x λsp (1 ρ2 )z2

dx

(29)



Q(a, b) =
b

xe

x2 +a2 2

I0 (ax)dx

(30)

is the rst-order Marcum Q-function. Again, applying the √ variable transfrom t = x and using [22, eq. (55)], the second integral in (29) can be simplied. Therefore, we nally obtain the cdf of Z in closed form as Pr(Z < z) =1e +e
z λsp Pm

where the quantization law is generically described by a staircase function D(x). The quantizer output is limited to the range [0, L]. Hence, D(x) takes both clipping and quantization into account. In the case where the number of quantization levels is N (i.e., a log2 (N ) bit representation), the quantization function can be expressed as a generic staircase function in the following way:
N

D(x) = 2ρ2 z 2Ip λsp Pm (1 ρ2 )
i=1

qi g(x, Ti1 , Ti )

(37)

z λsp Pm

Q

t + Q r where 1 2 1+

, λsp Pm (1 ρ2 ) (s r)z (s + r)z , 2Pm 2Pm e 2Pm I0
sz

where Ti represents the ith quantization threshold value, and qi is the amplitude representing the ith quantization interval. The function g(x, α, β) is the rectangular function given by g(x, α, β) = 1, 0, α≤x<β otherwise. (38)

t r

2ρ Ip z (1 ρ2 )λsp Pm

(31)

In the case of midriser uniform quantization [24] with step size Δ = L/N , one has Ti = 0, i=0 i Δ, 0 < i < N +∞, i = N Δ qi = i Δ . 2 (39) (40)

2 s= λsp t= 2 λsp

Ip ρ2 + 1+ 2 1ρ (1 ρ2 )z 1+ Ip ρ2 1 ρ2 (1 ρ2 )z

(32) (33)

r=

s2

16ρ2 Ip . 2 (1 ρ2 )2 z λsp

(34)

Note that the Marcum Q-function can be evaluated using the Marcumq function in MATLAB. In the special case of innite SU transmit power Pm → ∞, the cdf of Z in (31) simplies to Pr(Z < z) = 1 2 1+ t r (35)

using e0 = 1, Q(a, 0) = 1, and I0 (0) = 1 in (31). The cdf of Z can be used to evaluate the SU transmit power back-off in the following way. Noting that the cdf in (31) is a function of the constant Ip , we write Pr(Z < z) = FZ (z|Ip ). To ensure that Z < Ip with probability p under the modied power constraint Ip , we require FZ (Ip |Ip ) = p. Evaluating Ip requires a numerical solution of the equation FZ (Ip |Ip ) p = 0 since (31) is not invertible in clo