Channel Capacity Limits of Cognitive Radio with Imperfect Channel Knowledge Himal A. Suraweera∗ , Peter J. Smith† , Mansoor Shafi‡ and Michael Faulkner§
∗ Department of Electrical and Computer Engineering, National University of Singapore, Singapore 119260 † Department of Electrical and Computer Engineering, University of Canterbury, Private Bag 4800, Christchurch, New Zealand ‡ Telecom New Zealand, PO Box 293, Wellington, New Zealand § Center for Telecommunications and Microelectronics, Victoria University, Melbourne, Victoria 8001, Australia E-mail:
[email protected],
[email protected],
[email protected],
[email protected]
Abstract—Cognitive radio 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 PU-SU CSI is imperfect as it is the practical case. 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 can not always satisfy the received peak interference power constraint at the PU and has to back off its transmit power. The resulting capacity loss for the SU is quantified using the cumulative distribution function of the interference at the PU. Additionally, we investigate the impact of CSI quantization. Our results are confirmed through comparison with simulations.
I. I NTRODUCTION The cognitive radio concept, first introduced in [1], refers to a smart radio which can sense the external electromagnetic environment and adapt its transmission parameters according to the current state of the environment. Cognitive radios can be designed to access parts of the primary user (PU) spectrum for their information transmission, provided that they cause minimal interference to the primary users in that band [2], [3]. Limits on this received interference level at the primary receiver can be imposed with an average/peak constraint [4]. The capacity of wireless systems has been extensively studied under fixed spectrum access. Several interesting results on the capacity, outage probability and throughput of cognitive radio systems have recently emerged. See for example, [4]– [10]. In [5], Gastpar derived the capacity of different nonfading additive white Gaussian noise (AWGN) channels with the average received-power at a primary receiver being constrained. In [4], Ghasemi and Sousa showed that with the same limit on the received-power level, channel capacity for a range of fading models (e.g., Rayleigh, Nakagami-m and log-normal fading) exceeds that of the non-fading AWGN channel. In [6], the ergodic and the outage capacity gains offered by a spectrum-sharing approach under average and This research is supported under the Australian Research Council’s Discovery funding scheme (DP0774689).
peak interference constraints in Rayleigh fading environments have been studied. In [7], optimal power allocation strategies to achieve the ergodic capacity and the outage capacity of the SU fading channel under different types of power constraints and fading channel models have been studied. Considering that in some situations, primary user spectral activity in the vicinity of the cognitive radio transmitter may differ from that in the vicinity of the cognitive receiver, in [8], the capacity of opportunistic spectrum acquisition has been investigated. The links of a primary/secondary radio environment could experience different types of fading such as Raleigh and lineof-sight Rician fading. Under such scenarios, in [9], we have investigated the ergodic capacity of spectrum sharing under average and instantaneous interference constraints. References [4], [6]–[9] have all assumed that the secondary user (SU)1 has full channel state information (CSI) knowledge of the link between its transmitter and the PU receiver. However, in practice, obtaining full CSI is difficult. This important situation has been studied in [10]. Whilst [10] looks at the impact of partial CSI, it only does so under an average interference constraint. The use of this constraint is relevant when a long term interference induced degradation is to be considered. This involves modeling the radio channels as a product of Rayleigh and lognormal fading. 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 to [10], as our model caters for a range of solutions from near-perfect to seriously flawed channel estimates. Even if a genie provides perfect CSI, it must be quantized into a limited number of levels before feeding 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 PU-SU link possibly due to a combination of channel estimation error, mobility, feedback delay and limited feedback. Similar to [7], we assume that the SU has a maximum transmit power threshold. 1 In the following, “cognitive radio” and SU will be used to identify the node which seeks access to the PU’s licensed spectrum.
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In this paper we make the following contributions: • 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 capacity. • Compared to 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 a 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. • We enhance the above 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. This paper is organized as follows. Sec. II provides the system model. In Sec. III, we investigate the mean SU capacity, the statistics of the PU interference and quantization effects of the CSI. In Sec. IV, numerical results supported by simulations are presented and discussed. Finally, we conclude in Sec. V. II. S YSTEM AND C HANNEL M ODEL In this section, the system and channel model considered in the paper are briefly outlined (cf. [7, Fig. 1]). Point-to-point flat Rayleigh fading channels are assumed. Let g0 = |h0 |2 and g1 = |h1 |2 denote the instantaneous channel gains from the secondary transmitter to the primary and secondary receivers respectively. Furthermore, we denote the exponentially distributed probability density functions (pdfs) of the random variables (RVs) g0 and g1 by fg0 (x) and fg1 (x) respectively. These pdfs are governed by the parameters λ0 = E(g0 ) and λ1 = E(g1 ) where E(·) is the expectation operator. We assume that the interference from the PU transmitter to the SU receiver can be ignored or can be considered in the AWGN at the SU receiver [7]. The AWGN at the PU receiver and SU receiver is n0 and n1 respectively with the distribution CN (0, N0 ) (circularly symmetric complex Gaussian variables with zero mean and variance N0 ). Perfect knowledge of the SU-SU channel is assumed at the SU receiver. However, the SU is only provided with partial channel knowledge of h0 . There are several mechanisms where this can occur. For example, information about h0 could be periodically measured by a band manager. Next, using a finite bandwidth channel, this information could be provided to the SU. Another example is primary secondary collaboration and exchange, where information about h0 could be directly fed back from the PU receiver to the SU transmitter as proposed in [12]. Further extension of this work will examine the combined effect of imperfection in the SU-SU channel. With partial CSI information of the SU-PU link at the SU transmitter, we have an estimate of the channel h0 of the form ˆ 0 = ρh0 + 1 − ρ2 u, h (1)
ˆ 0 is the channel estimate available at the secondary where h transmitter and u is a complex Gaussian RV with zero mean and unit variance uncorrelated with h0 . The correlation coefficient, 0 ≤ ρ ≤ 1, is given by ρ=
cov(g0 , gˆ0 ) var(g0 )var(ˆ g0 )
,
(2)
ˆ 0 |2 . The correlation coefficient, ρ, determines where gˆ0 = |h the average quality of the channel estimate over all channel states of h0 . This model is well established in the literature which investigates the effects of imperfect CSI [11]. 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 we show in Sec. III the same formulation can be extended to incorporate quantization effects. III. SU M EAN C APACITY In this section, we obtain the mean capacity of the SU under a peak interference power constraint. Previous work on channel capacity of cognitive radio has assumed two types of interference constraints at the PU receiver: 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 Q [4]. Furthermore, a maximum SU transmit power constraint, Pm is assumed. In practice, such a limitation arises due to the power amplifier nonlinearity [7]. Now based on the channel estimate, the cognitive transmitter selects its transmit power, Pt , as Q (3) Pt = min , Pm . gˆ0 Therefore, the mean SU channel capacity is given by [7] ∞ x fγ (x)dx, log2 1 + (4) C=B N0 B 0 where γ = Pt g1 and B is the bandwidth. In order to find the mean capacity in (4), the pdf of γ needs to evaluated. We begin by finding the cdf of γ, Fγ (x), given by Fγ (x) = 1 − Pr (γ > x) Qg1 = 1 − Pr Pm g1 > x, >x gˆ0 x x = 1 − Pr g1 > , g1 > gˆ0 . Pm Q Simplifying further we can express Fγ (x) as Q/Pm x pgˆ0 (y)dy Pr g1 > Fγ (x) = 1 − P m ∞ 0 x − Pr g1 > y pgˆ0 (y)dy. Q Q/Pm
(5)
(6)
The last line follows as the integral depends on whether Pxm < Q x 1 −y/λ0 and Fgˆ0 (y) = ˆ0 (y) = λ0 e Q y, i.e, y > Pm . Since fg
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1 − e−y/λ0 we can write (6) as x Q Fγ (x) = 1 − Pr g1 > Pr g0 < Pm Pm ∞ 1 − λ xQ y − λ1 y − e 1 e 0 dy. λ0 Q/Pm Finally, after simplifying the integral we get Q x Fγ (x) = 1 − e− λ1 Pm 1 − e− λ0 Pm −
1 1+
Q
λ0 λ1 Q x
(7)
(8)
x
e− λ0 Pm − λ1 Pm .
Now the pdf of γ can be obtained by differentiating Fγ (x) with respect to x and is given by fγ (x) =
Q 0 Pm
−λ
1−e λ 1 Pm
Q
x
e− λ1 Pm +
Q 0 Pm
−λ
1 e λ 1 Pm 1 +
x 1 Pm
−λ
λ0 λ1 Q x
(9)
x
λ0 e− λ0 Pm − λ1 Pm + 2 . λ1 Q 1 + λλ10Q x Substituting (9) in (4), the normalized mean capacity, C˜ = can be expressed as
C B,
and the normalized capacity C˜ is given by ∞ λ0 1 x C = log2 1 + 2 dx B λ1 Q 0 N0 B 1 + λλ10Q x 1Q loge λ0λN 0B . = 0B loge (2) 1 − λ0λN 1Q
(13)
Note that when λ0 = λ1 = 1 is assumed, this is the capacity evaluated in [4] for Rayleigh fading channels. A. Statistics of the Interference at the PU Receiver The interference at the primary receiver is given by Qg0 , Pm g0 . Pt g0 = min gˆ0
(14)
Since gˆ0 = g0 , we note that in the presence of partial channel information, the interference at times may not be limited to Q. Hence, the PU’s protection can not 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 ˜ so that the interference remains reduced level of Q, say Q, below Q with a desired probability (95% or 99% say). This results in a capacity loss for the SU. Let Z = Pt g0 . The cdf of Z is given by
Pr(Z < z) = 1 − Pr(Z > z) (15) Q ∞ 1 − e− λ0 Pm x C − λ xPm g0 = e 1 dx (10) log2 1 + = 1 − Pr g0 > z1 , > z2 , B λ 1 Pm N0 B 0 gˆ0 Q − λ xP ∞ z e 1 m x e− λ0 Pm where z1 = Pzm and z2 = Q . Moreover, we can write dx log2 1 + + λ λ 1 Pm 0 N0 B 1 + λ 0Q x ∞ x/z2 1 g0 x ∞ Pr g0 > z1 , > z2 = fg0 ,ˆg0 (x, y)dydx, Q e− λ1 Pm x λ0 − λ P g ˆ0 z1 0 e 0 m + log2 1 + 2 dx. λ1 Q N0 B (16) 0 1 + λλ10Q x where fg0 ,ˆg0 (x, y) is the joint density function of the RVs√g0 √ and gˆ0 . Using the the joint pdf of r1 = g0 and r2 = gˆ0 The integrals in (10) can be evaluated in closed-form using and a simple transformation of variables gives integration by parts. Therefore, C˜ can be expressed as √ x+y 2ρ xy 1 − (1−ρ 2 )λ Q 0 I , e fg0 ,ˆg0 (x, y) = 0 N0 B 1 − e− λ0 Pm N0 B C (1 − ρ2 )λ20 (1 − ρ2 )λ0 = Γ 0, e λ1 Pm (11) (17) B loge (2) λ 1 Pm 1 Q where I0 (·) is the zeroth-order modified Bessel function of the Γ 0, − λ0 N0 B λ first kind. Substituting (17) in (16) gives 0 Pm loge (2) 1 − λ1 Q ∞ x 1 Q − N0 B e (1−ρ2 )λ0 (18) Pr(Z < z) = 1 − e− λ0 Pm N0 B 2 2 (1 − ρ )λ0 z1 Γ 0, e λ1 Pm , + λ 1 Pm 0B x/z2 √ loge (2) 1 − λ0λN y 2ρ xy 1Q − dydx. e (1−ρ2 )λ0 I0 (1 − ρ2 )λ0 ∞ a−1 −t 0 √ e dt is the upper incomplete where Γ(a, x) = x t Using the variable transform t = y, and using [13, Eq. (10)] gamma function. the inner integral in (18) can be solved. Therefore we get Note that in the special case as Pm → ∞, the pdf in (9) ∞ x 1 simplifies to e− λ0 dx (19) Pr(Z < z) = 1 − λ0 z1
∞ λ0 1 2ρx 2x (12) fγ (x) = 2 , − λx + , dx, e 0Q λ1 Q 1 + λλ10Q x λ0 z1 λ0 (1 − ρ) λ0 (1 − ρ)z2
978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings.
where
Q(a, b) =
∞
2 2 − x +a 2
xe b
I0 (ax)dx,
(20)
is the first-order Marcum √ Q-function. Again applying the variable transfrom t = x and using [13, Eq. (55)] the second integral in (18) can be simplified. Therefore, finally we obtain the cdf of Z in closed-form as z
(21) Pr(Z < z) = 1 − e− λ0 Pm
z 2ρ2 z 2Q , + e− λ0 Pm Q 2 λ0 Pm (1 − ρ ) λ0 Pm (1 − ρ2 ) ⎛
⎞
t ⎝ (s − r)z (s + r)z ⎠ + Q , r 2Pm 2Pm √ sz t 2ρ Qz 1 − 2P m 1+ e , − I0 2 r (1 − ρ)λ0 Pm where
2 ρ2 Q s= , 1+ + λ0 1 − ρ2 (1 − ρ2 )z 2 t= λ0
and
ρ2 Q 1+ − 1 − ρ2 (1 − ρ2 )z
r=
s2 −
16Qρ2 . λ20 (1 − ρ2 )2 z
(22)
,
(23)
(24)
Note that the Marcum Q-function can be evaluated using the Marcumq function in MATLAB. In the special case of infinite SU transmit power, Pm → ∞, the cdf of Z simplifies to t 1 1+ . (25) P (Z < z) = 2 r B. Effect of Quantized Feedback In the previous section we assumed that the sources of imperfect CSI (channel estimation error, mobility and feedback ˆ 0 given in (1). This is reasonable delay) result in the estimate h and is widely used in the literature. 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 finite number of bits representing gˆ0 ranges. If the PU-SU channel information is quantized, at the secondary transmitter we have the estimate g0 ), g˜0 = Q(ˆ
capacity. To compute ρ˜, the mean squared error between the exact g0 and g˜0 is used. As we illustrate in Section IV, the results obtained from this simple approximation method are accurate in most cases of interest. Let us consider E[|g0 − g˜0 |2 ] given by E[|g0 − g˜0 |2 ] = E[(|h0 |2 − Q(|ρh0 + 1 − ρ2 u|2 ))2 ] (27) = E[|h0 |4 ] − 2E[|h0 |2 Q(|ρh0 + 1 − ρ2 u|2 )] + E[Q2 (|ρh0 + 1 − ρ2 u|2 )]. ˆ 0 it Using the moments of |h0 |2 and rewriting in terms of h can be shown that ˆ 0 |2 Q(|h ˆ 0 |2 )] E[|g0 − g˜0 |2 ] = 2λ20 − 2 ρ2 E[|h (28) 2 ˆ ˆ 0 |2 )]. + λ0 (1 − ρ )E[Q(|h0 | )] + E[Q2 (|h 2
So we need to find E[Y Q(Y )], E[Q(Y )] and E[Q2 (Y )] where Y is an exponentially distributed RV with rate parameter λ0 . First, consider the evaluation of E[Y Q(Y )] given by ∞ yQ(y)pY (y)dy (29) E[Y Q(Y )] = 0
Ti+1 N y 1 qi ye− λ0 dy. = λ0 i=0 Ti Simplifying (29) we get E[Y Q(Y )] = λ0
N
qi
i=0
Ti λ0
Ti
e− λ0 − 1 +
Ti+1 λ0
(30) Ti+1 e− λ0 .
Similarly E[Q(Y )] and E[Q2 (Y )] are given by Ti +1 N y 1 qi e− λ0 dy E[Q(Y )] = λ0 i=0 Ti N Ti+1 Ti = qi e− λ0 − e− λ0 ,
(31)
i=0
and N 1 2 Ti +1 − λy qi e 0 dy λ0 i=0 Ti N Ti+1 T − λi − λ 2 0 = qi e 0 − e
E[Q2 (Y )] =
(26)
where Q(·) is the quantization operator. An exact mathematical analysis of the effects of quantization on the statistics of the interference at the PU receiver and subsequently the mean SU capacity is complex. To overcome this difficulty, we propose to use an approximate equivalent correlation ρ˜ to mimic quantization effects. This allows us to conveniently use the developed cdf expression to investigate the impact of quantization on the SU
1+
(32)
i=0
respectively. With no quantization and assuming a model of the form given in (1) with an equivalent correlation, ρ˜, we calculate E[|g0 − gˆ0 |2 ] as g02 ] − 2E[g0 gˆ0 ] (33) E[|g0 − gˆ0 |2 ] = E[g02 ] + E[ˆ = 2 2λ20 − E[|h0 |2 |˜ ρh0 + 1 − ρ˜2 u|2 ] = 2λ20 (1 − ρ˜2 ).
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3
1
Theoretical Monte Carlo Simulation
0.9
SU SNR = 10 dB 2.5
0.8
c1 = 10
2
0.6
c1 = 0.1 0.5
c1 = 5 0.4
Mean SU Capacity
Pr ( γ < x )
0.7
SU SNR = 5 dB
1.5
SU SNR = 0 dB
1
0.3
c1 = 1 0.2 0.5
0.1 0
Monte Carlo Simulation Theoretical −2
−1
10
0
10
1
10
10
0 −20
−15
Fig. 1.
The cdf of γ for different c1 .
Therefore, to mimic both the channel estimation error and the quantization effects, we equate (28) and (33) which gives ρ2 E[Y Q(Y )] + λ0 (1 − ρ2 )E[Q(Y )] − 12 E[Q2 [Y ]] . ρ˜ = λ0 (34) IV. N UMERICAL R ESULTS In this section, we confirm the derived analytical results in Sec. III, through comparison with Monte Carlo simulations. In the following results, we assume N0 B = 1, Pm = 1, PU transmit power, Pp = 1 and assume that the signal-to-noise ratio (SNR) at PU and SU are given by 5 dB. The parameters c1 and c2 are defined as c1 = λλ01 and cQ2 = PU SNR. Hence, c1 is the ratio of the SU-PU link strength, which is usually less than 1 in the common scenario of long PU links and short SU links. Parameter c2 is a proportionality factor so that the acceptable interference, Q, is c2 times the PU SNR, and c2 < 1. We start by comparing the theoretical and simulated cdf of γ in Fig. 1. All four plots correspond to different values of c1 while c2 = 0.5. We see that (8) exactly matches the simulated cdfs. The cdf can also be trivially used to obtain the outage SU capacity. Figure 2 shows the mean capacity of the SU in bits/s/Hz against Q in dB under the instantaneous peak received power constraint and SU transmit power constraint. In all plots, c1 = 0.1. As expected, the SU capacity is low when the maximum received power at the PU is small. The reason is that the SU transmit power is limited in this regime of Q and is approximately Q/ˆ g0 . However, as we see, the capacity increases when the tolerable Q is increased. In the high Q regime, the capacity approaches a plateau behavior. In this region, the maximum transmit power of the SU, Pm largely dominates (3). Furthermore, as the PU SNR is increased, a higher capacity can be obtained. This observation is also intuitive. The theoretical results from (11) are perfectly verified by computer simulations.
−10
−5
0
5
10
15
20
Q in dB
x
Fig. 2.
The SU mean capacity versus Q for different SU SNRs.
Under a peak received power constraint, with SU transmitter employing partial CSI, at times the actual interference caused to the PU receiver exceeds the level of Q. This is not acceptable and a solution from the PU point of view is to ˜ < Q. In Fig. 3 we show the resulting demand a new Q percentage capacity loss of the SU against ρ due to such a demand. In order to illustrate this, we set a new peak ˜ < 0.05 and c1 = 0.1. The capacity constraint, Pr(Z > Q) loss for the SU is defined as C˜Original − C˜New , (35) C˜Loss = C˜Original where C˜Original and C˜New are the mean SU capacities obtained ˜ into (11) respectively. First using (21), by plugging Q and Q we check whether the set target Pr(Z > Q) < 0.05 is guaran˜ is substituted into (21) until the teed. If not satisfied, a new Q condition is satisfied. When the error in the PU-SU channel estimate is high, i.e., for a small ρ and c2 = 0.1, the capacity loss is roughly 65%. However, when ρ increases the capacity loss becomes insignificant. For example when ρ = 0.99 it is less than 2%. Interestingly, no capacity loss is observed for c2 = 0.3. That is, the condition, Pr(Z > Q) < 0.05, is not violated for all ρ. Figure 3 is also useful to determine the accuracy of channel knowledge required at the SU, in order to reap the capacity gains offered by the shared spectrum concept. Figures 4 and 5 illustrate results related to quantization effects of the CSI. Figure 4 shows the actual and theoretically approximated cdf of the PU interference due to quantization. All plots correspond to 16 quantization levels. We can see that the cdf plots obtained by plugging the equivalent correlation, ρ˜, into (21) match reasonably well with the exact simulated plots. This confirms that the equivalent ρ˜ is a convenient parameter for studying the impact of CSI quantization on the SU mean capacity. Results not shown here for other different number of levels also showed a good match. Figure 5 shows how the equivalent correlation, ρ˜, varies against ρ for three different quantization levels. An appreciable difference between the cases of 4 and 8 levels can be observed. However,
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70
1
c2=0.1 60
c2=0.2
0.9
c =0.3
0.8
2
Equivalent Correlation
Capacity Loss (%)
50
40
30
20
0.7 0.6 0.5 0.4 0.3 0.2
N=16 N=8 N=4
10 0.1
0
0
0
0.2
Fig. 3.
0.4
ρ
0.6
0.8
1
Percentage SU capacity loss against ρ.
0
0.2
Fig. 5.
0.4
ρ
0.6
0.8
1
Equivalent correlation (˜ ρ) against ρ.
1
that the quantization effects can also be incorporated into the simple flexible CSI model considered. To this end, we proposed an approximate correlation coefficient to mimic the quantization of the CSI. The accuracy of this simple yet useful approach was confirmed from simulations.
Monte Carlo Simulation Theoretical
0.9 0.8 0.7
Pr ( Z
