A Lower Bound to the Receiver Operating Characteristic of a Cognitive Radio Network
11 A Lower Bound to the Receiver OperatingCharacteristic of a Cognitive Radio Network (submitted to the
IEEE Transactions on Information Theory , July 2010)
Giorgio Taricco
Abstract
Cooperative cognitive radio networks are investigated by using an information-theoretic approach.This approach consists of interpreting the decision process carried out at the fusion center as a binary(asymmetric) channel, whose input is the presence of a primary signal and output is the fusion centerdecision itself. The error probabilities of this channel are the false-alarm and missed-detection proba-bilities. After calculating the mutual information between the binary random variable representing theprimary signal presence and the set of sensor (or secondary user) output samples, we apply the data-processing inequality to derive a lower bound to the receiver operating characteristic. This basic idea isdeveloped through the paper in order to consider the cases of full channel and signal knowledge andof knowledge in probability distribution. The advantage of this approach is that the ROC lower boundderived is independent of the particular type of spectrum detection algorithm and fusion rule considered.Then, it can be used as a benchmark for existing practical systems.
Index Terms
Cognitive radio networks, Data-Processing inequality, Spectrum sensing, Sensor networks, ReceiverOperating Characteristic.
I. I
NTRODUCTION
Cognitive Radio (CR) technologies have gained considerable interest in the last few years becauseof two factors: i ) the increasing demand for wireless spectrum from a large number of applications; ∗ Giorgio Taricco is currently with Politecnico di Torino (DELEN), corso Duca degli Abruzzi 24, 10129, Torino, Italy(e-mail: [email protected]).
November 13, 2018 DRAFT a r X i v : . [ c s . I T ] J u l and ii ) the fact that many portions of licensed spectrum are neglected or underutilized by the regularlicensees [1]–[6].The concept of CR depends considerably on the application context [7]. Nevertheless, an officialdefinition has been given by the Global Standards Collaboration (GSC) group within the ITU [8]: “A radioor system that senses its operational electromagnetic environment and can dynamically and autonomouslyadjust its radio operating parameters to modify system operation, such as maximize throughput, mitigateinterference, facilitate interoperability, access secondary markets.” According to this definition, a CRdevice should be able to autonomously exploit unused portions spectrum to increase its own signallingrate without limiting the use of the radio spectrum from licensed users. Thus, the most important featureof a CR device is the ability to detect the availability of spectrum holes [7], which can be accomplishedby suitable spectrum sensing techniques. The key role of spectrum sensing has been recognized in thetechnical literature as the enabling technique for CR systems. Different strategies have been envisagedto effectively implement this feature and a comprehensive taxonomy can be found in [7].A simple statement of the CR detection problem can be given as follows. In a CR network thereare two classes of users: i ) primary users , i.e., those users who have license rights of some other formof priority with respect to the radio channel access; ii ) secondary users , i.e., those users who have nolicence rights or have more limited priority to the channel access than the primary users. Secondary usersare those who need CR capabilities, such as spectrum sensing, in order to avoid causing interferenceto primary users. Thus, secondary users have to estimate the radio channel condition before attemptinga transmission, i.e., they need to assess whether the channel is idle or busy (hypotheses H and H ,respectively). This estimation is usually affected by error and characterized by two error probabilities: • The false-alarm probability P fa , corresponding to the detection of hypothesis H when H is true. • The missed-detection probability P md , corresponding to the detection of hypothesis H when H istrue.Ideally, secondary users should operate toward reaching the goal of having P fa = P md = 0 . However,radio channel impairments prevent to attain this operating level, and a suitable tradeoff has to be sought.Typically, secondary users are allowed a maximum level of interference to the primary users, whichtranslates into a maximum probability of missed detection. Then, the CR users can maximize theirthroughput by maximizing the false-alarm probability P fa under the constraint of a given P md . Typicalvalues of these probabilities have been set to P md = P fa = 0 . in the contest of the developing standardIEEE 802.22 [6]. A complete picture of the performance of a CR system is provided by the receiver November 13, 2018 DRAFT operating characteristic (ROC) plot. The ROC is a plot of the missed-detection probability P md versus thefalse-alarm probability P fa . Its derivation depends on the radio channel parameters (fading, noise power)and on the type of decision process implemented to detect the presence of a primary signal.It has been widely recognized in the literature (see, e.g., [7] and references therein) that user cooperation enhances the performance of a CR system, both in terms of ROC, and of avoiding the hidden primary userproblem . This problem is considered one of the major challenges to the implementation of a CR system,and is similar to the hidden node problem experienced in Carrier Sense Multiple Accessing (CSMA) [7].The hidden primary user problem derives from the shadowing of secondary users, occurring while sensingthe primary signal transmission. More precisely, a secondary user can be in the range of a primary user receiver but out of the range of another primary user transmitter . Then, the secondary user senses thechannel idle, because it cannot capture the primary user signal, and then starts its transmission. However,since it is in the range of the other primary user receiver, it eventually interferes with the reception ofthe primary signal. Having multiple secondary users sensing the channel reduces the chances of fallinginto this situation.Fig. 1 illustrates the block diagram of a CR system based on user cooperation. We can see that theprimary signal is present if ξ = 1 . This signal is received by a set of K secondary users (or sensors) whichsample it during a certain observation window. Secondary users can exploit individually this informationin order to make a decision on the spectrum availability. Otherwise, they can share it by sending a suitablesignal through a control channel to a central processing unit (i.e., implementing user cooperation). Thisunit provides for the fusion of the user information and is then called fusion center (FC) [7].The goal of this paper is to analyze, by using information-theoretic results, the behavior of a cooperativeCR system. Our approach is based on the observation that appending the decision process implementedat the FC to the primary signal transmission channel yields an equivalent binary channel with input ξ ,the random variable indicating the signal presence, and output ˆ ξ , the FC decision. According to thisinterpretation, the false-alarm and missed-detection probabilities correspond to the two error probabilitiesof this binary channel (conditioned to ξ = 0 and ξ = 1 ). In general, this channel turns out to be anasymmetric binary channel because the error probabilities are different.Then, by using the data-processing inequality, we can calculate an upper bound to the channel capacity,which translates into a lower bound to the ROC. This basic idea is developed through the paper in order toderive the lower bound of a cooperative CR system ROC, which is independent of the spectrum detectionand fusion strategy used. This lower bound can be applied to assess the validity of specific combinationsof spectrum sensing and fusion strategy. November 13, 2018 DRAFT ξ ∈ { , } × Primarysignal S S ... S K y ( n ) y ( n ) y K ( n ) Fig. 1. Block diagram of a cognitive radio system with input ξ , denoting the primary signal presence or absence, and outputgiven by the set of sensor outputs y k ( n ) for k = 1 , . . . , K and n = 1 , . . . , N . Dotted lines represent the fading channelsconnecting the primary transmitter to the sensors (secondary users). The remainder of the paper is organized as follows. The system model is illustrated in Section II, wherethe key concept of applying the data-processing inequality to the cooperative CR system is introducedand analyzed in detail. Section III deals with the derivation of the mutual information of the cooperativeCR system without the FC channel and detection. This section considers the case of known channelgains and signal, as a baseline, and the case of known channel gain and signal distribution, as a furtherdevelopment. Relevant asymptotic cases are also studied, in order to mitigate the numerical difficultiesin the derivation of the results. Section IV illustrates the analytic results through numerical examplesincluding. Lower bounds to the ROC are reported in this section along with a comparison of these resultswith an energy detection estimator. Finally, our conclusions are collected in Section V.II. S
YSTEM MODEL AND OPTIMUM
ROCWe consider a CR system (illustrated in Fig. 1) equipped with K sensors sensing the wireless spectrumover N sampling times in order to provide information about the channel availability to secondary users.Intentionally, the diagram does not show the terminal part consisting in the collection of the sensormeasurements, their compacting, their transmission to the FC through a control channel, and the FCprocessing block providing the output decision about the signal presence.We assume a block fading channel where the n th sampled signal received by sensor k is given by y k ( n ) = z k ( n ) ξ = 0 h k s ( n ) + z k ( n ) ξ = 1 (1) November 13, 2018 DRAFT for k = 1 , . . . , K and n = 1 , . . . , N . Here, z k ( n ) ∼ N c (0 , σ k ) are the iid received noise samples, h k arethe block fading gain coefficients, s ( n ) are the primary user’s symbols, and ξ denotes the the randomvariable indicating that the primary signal is present ( ξ = 1 ) or absent ( ξ = 0 ). The variances σ k areknown parameters. We can interpret ξ as the imponderable primary user decision to convey informationthrough the channel at the time the CR system is trying to check the existence of a spectrum hole .In the following we assume that the random variable ξ is not necessarily equiprobable but rather wehave P ( ξ = 0) = α . Then, α represents the a priori probability of primary signal absence.As already mentioned, in this framework we do not consider the remaining part of the communicationsystem beyond the block diagram of Fig. 1. This part consists of a distributed algorithm at the K sensorsand at the fusion center (FC) aimed at condensing the available channel sensing information (at thesensors), sending it to the FC, and jointly processing in order to make a reliable decision on the presenceof a primary transmitted signal.On the contrary, we regard the block diagram in Fig. 1 as a binary input-continuous output vectorchannel, which we study in order to derive the mutual information I (cid:44) I (cid:16) ξ ; { y k ( n ) } K,Nk =1 ,n =1 (cid:17) . (2)Using the data processing inequality (DPI) [9], we can see that the mutual information I upper boundsthe mutual information of the channel corresponding to the completion of the transmission chain to theFC by any conceivable distributed algorithm.Completing the transmission chain up to the FC’s output yields a binary-input binary-output channel.Denoting the FC’s output by ˆ ξ , we have from the DPI: I ( ξ ; ˆ ξ ) ≤ I . (3)Now, by the definition of the false-alarm and missed-detection probabilities (denoted by P fa and by P md ,respectively), we have: P fa = P ( ˆ ξ = 1 | ξ = 0) P md = P ( ˆ ξ = 0 | ξ = 1) . In general, we have a binary asymmetric channel whose transition probability matrix can be written as P = − P fa P fa P md − P md . Notation z ∼ N c ( µ , Σ ) denotes a circularly symmetric complex Gaussian distributed vector with mean µ , covariancematrix Σ = E [ zz H ] − µµ H , and pdf det( π Σ ) − exp[ − ( z − µ ) H Σ − ( z − µ )] . November 13, 2018 DRAFT −5 −4 −3 −2 −1 −5 −4 −3 −2 −1 fa P m d Fig. 2. ROC lower bound curves corresponding to α = 0 . and mutual information I indicated by the labels. The mutual information, assuming P ( ξ = 0) = α , is given by [9]: I ( ξ, ˆ ξ ) = H b ( α (1 − P fa ) + ¯ α P md ) − αH b ( P fa ) − ¯ αH b ( P md ) , (4)where ¯ α (cid:44) − α and H b ( p ) (cid:44) − p log p − (1 − p ) log (1 − p ) is the binary entropy function [9].Finally, inserting (4) into inequality (3), we obtain a relationship between the false-alarm and missed-detection probabilities, which represents a lower bound to the ROC for the given CR system.The parametric dependence of the ROC lower bound on the mutual information is illustrated in Fig.2. As expected, as I ↑ , the ROC lower bounds decrease monotonically to P fa = P md = 0 . November 13, 2018 DRAFT
III. C
ALCULATION OF I Let us define for convenience the following matrices and vectors: Y (cid:44) ( y k ( n )) K,Nk =1 ,n =1 Z (cid:44) ( z k ( n )) K,Nk =1 ,n =1 h (cid:44) ( h , . . . , h K ) T s (cid:44) ( s (1) , . . . , s ( N )) H . . Then, we can simplify (1) by writing it as follows: Y = ξ hs H + Z , (5)and hence the mutual information (2) becomes I = h ( Y ) − h ( Y | ξ ) , where h ( · ) denotes the differential entropy [9]. First, it is plain to see that h ( Y | ξ ) = h ( Z ) = N K (cid:88) k =1 log ( π e σ k ) . The evaluation of h ( Y ) is more difficult. We distinguish among different assumptions concerning thedistribution of the secondary channel gain vector h and the signal vector s . In the following we considerthe cases of i ) known gains and signal at the receiver, and of ii ) known gain and signal distribution atthe receiver. A. Known gains and signal at the receiver
In order to equalize the noise variances, we transform the channel equation (5) by pre-multiplying bythe inverse of the square root of the noise covariance matrix Σ z (cid:44) diag( σ , . . . , σ K ) . We obtain Y = ξ A + Z , (6)where A (cid:44) Σ − / z hs H and the entries of Z are then iid as N c (0 , . This linear transformation isinvertible and does not change the mutual information I . In order to calculate the mutual information,we resort to Theorem A.1 (Appendix A). Since ξ = 0 , , in order to use this result we can subtract A / November 13, 2018 DRAFT and obtain symmetric input. Theorem A.1 tells us that the channel is equivalent to a binary-input realadditive Gaussian channel with SNR (cid:107) A (cid:107) / . We obtain I = H b ( α ) − α E (cid:20) log (cid:18) αα e Z −(cid:107) A (cid:107) (cid:19)(cid:21) − ¯ α E (cid:20) log (cid:18) α ¯ α e Z −(cid:107) A (cid:107) (cid:19)(cid:21) , (7)where Z ∼ N (0 , (cid:107) A (cid:107) ) Remark III.1
It is plain to see that (7) is invariant to the mapping α (cid:55)→ − α , i.e., to exchanging the a priori probabilities of primary signal presence and absence. The ROC performance improves as theseprobabilities get closer to or to , as illustrated in the following. The symmetry of the resulting ROClower bound suggests to define an equilibrium point corresponding to P fa = P md , which is referred to as equilibrium probability and denoted by P eq in the sequel. Under these operating conditions, the binarychannel ξ → ˆ ξ is symmetric. Remark III.2
It is worth noting that the mutual information I , and hence the lower bound to the ROC,depend only on (cid:107) A (cid:107) (in this case). This parameter can be written as SNR (cid:44) (cid:107) A (cid:107) = K (cid:88) k =1 | h k | (cid:107) s (cid:107) σ k , (8)which corresponds to the sum of the secondary users’ receive SNR’s. For this reason, we refer to it inthe following by the term additive SNR .
1) Limiting behavior for α → : Expanding (7) for α → we obtain: I = E [1 − Z + SNR − e Z − SNR ] α log e+ E [1 − SNR − Z + e Z − SNR ]2 log(2) α + O ( α )= (cid:20) SNR α −
12 (e SNR − α (cid:21) log e + O ( α ) . We can see that the first- and second-order approximations represent upper and lower bounds, respectively,to the mutual information I . These lower bounds are illustrated in Fig. 3, plotting the ratio I /α versus α and its second-order approximation (lower bound) for SNR = 1 (0 dB). Notation Z ∼ N ( µ, σ ) represents a real Gaussian random variable with mean µ and variance σ . November 13, 2018 DRAFT −4 −3 −2 −1 α I / α exact2nd order (lower bound)1st order (upper bound) Fig. 3. Plot of the ratio I /α and of its second-order approximation [ SNR − . SNR − α ] log e versus α for SNR = 1 (0 dB).
2) Limiting behavior for
SNR → ∞ : Applying Theorem B.1 (Appendix B), we obtain the followingbounds: H b ( α ) − √ πα ¯ α ln 2 1 SNR / e − SNR / ≤ I≤ H b ( α ) − √ πα ¯ α ln 2 1 SNR / e − SNR / + √ πα ¯ α [ π + 8 + (ln( α/ ¯ α )) ]2 ln 2 1 SNR / e − SNR / . These bounds yield, for
SNR → ∞ , the following asymptotic approximation:
I ∼ H b ( α ) − √ πα ¯ α ln 2 1 SNR / e − SNR / . (9) November 13, 2018 DRAFT0
3) Limiting behavior for α → and SNR → ∞ : Finally, we can also expand (4) for α → at theequilibrium point of the ROC, and obtain I ( ξ ; ˆ ξ ) = (1 − P eq ) log − P eq P eq α − (1 − P eq ) P eq (1 − P eq ) α + O ( α ) . When both α and P eq → , we obtain the approximation P eq ≈ e − SNR . (10) B. Known gain and signal distribution at the receiver
The case of known gain and signal distribution at the receiver can be handled by exploiting the resultsderived in Section III-A. First, we notice that A = Σ − / z hs H is a random matrix whose joint pdf ofthe entries depends on the distributions of the channel gain vector h and of the signal vector s . Then,starting from (7), we can apply the chain rule for the mutual information and the independence between ξ and the vectors h , s in order to obtain the following result: I = I ( ξ ; A ) + I ( ξ ; Y | A )= I ( ξ ; Y | A )= H b ( α ) − E (cid:20) α log (cid:18) αα e √ Z − Γ (cid:19) + ¯ α log (cid:18) α ¯ α e √ Z − Γ (cid:19)(cid:21) . (11)where Z ∼ N c (0 , is independent of Γ (cid:44) (cid:107) A (cid:107) , and the average is with respect to both Z and Γ . Inaccordance with eq. (8), we have Γ = K (cid:88) k =1 | h k | (cid:107) s (cid:107) σ k , but in this framework Γ is a random variable whose mean value is defined as the additive SNR, i.e., SNR (cid:44) E [Γ] . The lower bound to the ROC depends on the distribution of Γ . Some examples illustrate this dependencein Section IV. IV. N UMERICAL EXAMPLES
In this section we illustrate the ROC bound obtained by numerical examples in order to compare thelower bounds obtained with some real estimation scheme.
November 13, 2018 DRAFT1 −6 −5 −4 −3 −2 −1 −6 −5 −4 −3 −2 −1 P fa P m d Fig. 4. ROC lower bound in the case of known channel gains and signal for different values of the additive SNR (reported onthe plot) and a priori probability of primary signal absence α = 0 . . A. Known gains and signal at the receiver
The first example reported in Fig. 4, which consider the case of known channel gains and signal(Section III-A) with a priori probability of primary signal absence α = 0 . . It can be noticed that thecurves are symmetric with respect to exchanging the probabilities P fa and P md . We can also notice athreshold behavior with respect to the SNR , which is better illustrated in Fig. 5. The SNR thresholdlies between and dB: below the threshold, the equilibrium probability decreases slowly; above thethreshold, the decrease rate becomes faster. The curves in Fig. 5 are lower bounds to the equilibriumprobability versus the additive SNR for different values of the probability of signal absence (or presence)and in the asymptotic case of α → , which is given by eq. (10). November 13, 2018 DRAFT2 −5 0 5 10 1510 −4 −3 −2 −1 Additive SNR (dB) P eq −2 −3 asympt Fig. 5. Lower bound to the equilibrium probability P eq versus the additive SNR with a priori probability of primary signalabsence (or presence) α = 0 . , . , − , − (solid curve). The dashed curve corresponds to α → (asymptotic case). B. Comparison with an energy detection scheme
A simple spectrum sensing scheme based on energy detection corresponds to the following estimationrule: ˆ ξ = , (cid:107) Y (cid:107) < θ + ln α , (cid:107) Y (cid:107) > θ + ln ¯ α (12)From the equivalent channel equation (6), the resulting false-alarm and missed-detection probabilities aregiven by: ˆ ξ = P fa = P ( (cid:107) Z (cid:107) > θ + ln ¯ α ) P md = P ( (cid:107) A + Z (cid:107) < θ + ln α ) (13)Since (cid:107) Z (cid:107) and (cid:107) A + Z (cid:107) are central and noncentral χ -distributed random variables, we can findexplicit expressions of the two probabilities. In fact, the cdf’s can be found in standard textbooks, such November 13, 2018 DRAFT3 as [10]. We have: P ( (cid:107) Z (cid:107) < u ) = γ ( KN, u ) , where γ ( n, x ) (cid:44) Γ( n ) − (cid:82) ∞ x u n − e − u du is the normalized upper incomplete Gamma function, and P ( (cid:107) A + Z (cid:107) < u ) = 1 − Q KN ( (cid:112) (cid:107) A (cid:107) , √ u ) , where Q m ( a, b ) (cid:44) (cid:82) ∞ b x ( x/a ) m − e − ( x + a ) / I m − ( ax ) dx is the generalized Marcum’s Q function de-fined in [10].Figures 6 and 7 show the ROC corresponding to an energy detector spectrum sensing scheme fortwo values of the product KN and SNR = 0 and dB, respectively. The diagrams also report theinformation-theoretic lower bound derived in Section III-A. We can see that increasing the product KN for a fixed SNR degrades the resulting ROC. This can be understood by observing that the variances of (cid:107) A + Z (cid:107) and (cid:107) Z (cid:107) are proportional to KN (they are (1 + 2 SNR ) KN and KN , respectively), whilethe mean value difference is equal to SNR . Therefore, as the KN increases, the overlapping of the twopdf’s increases, and hence the probabilities of false-alarm and missed-detection. Remark IV.1
It is worth noting that the previous results hold for fixed additive SNR. Then, increasingeither K or N implies that the individual secondary user SNR’s | h k | (cid:107) s (cid:107) /σ k must decrease to keepthe overall additive SNR constant. On the contrary, if one fixes the individual SNR’s, the additive SNRincreases and both the lower bound and the energy-detector ROC improve. Thus, the fact that the ROCcurves decrease as KN increases shall be interpreted by saying that the energy detector performancewould improve if we could concentrate all the available sensors in a single one by keeping the totaladditive SNR constant. C. Known gain and signal distribution at the receiver
Here we consider the case of iid Rayleigh fading gains, where γ k (cid:44) E [ | h k | ] /σ k , and (cid:107) s (cid:107) hasprobability distribution P ( (cid:107) s (cid:107) = S m ) = p m for m = 1 , . . . , M . If we assume that all the γ k aredifferent, the pdf of Γ can be derived as follows: p Γ ( G ) = M (cid:88) m =1 p m K (cid:88) k =1 exp( − G/ ( γ k S m )) γ k S m (cid:89) (cid:96) (cid:54) = k − γ (cid:96) /γ k . November 13, 2018 DRAFT4 −2 −1 −2 −1 P fa P m d lower boundKN=1KN=10 Fig. 6. ROC obtained with an energy detector with
SNR = 0 dB, KN = 1 and , and α = 0 . . Solid lines are obtainedanalytically and markers correspond to Monte-Carlo simulation results. The lowest dashed curve corresponds to the information-theoretic lower bound. We can use this result to calculate the double integral I = H b ( α ) − (cid:90) ∞−∞ exp( − z / √ π (cid:90) ∞ p Γ ( G ) (cid:20) α log (cid:18) αα e √ Gz − G (cid:19) + ¯ α log (cid:18) α ¯ α e √ Gz − G (cid:19)(cid:21) dG dz. (14)As an illustrative example, we consider the following scenario: • K = 4 secondary users. • γ k = (4 + k ) dB for k = 1 , . . . , K . • (cid:107) s (cid:107) = 1 . November 13, 2018 DRAFT5 −4 −3 −2 −1 −4 −3 −2 −1 P fa P m d lower boundKN=1KN=10 Fig. 7. Same as Fig. 6 but
SNR = 10 dB.
The ROC curves are reported in Fig. 8. The lowest curve corresponds to the lower bound calculated byusing (14). The other curves correspond to the implementation of a spectrum sensing algorithm basedon energy detection for different combinations of the number of secondary users K and sampling times N . In all cases, the same additive SNR is assumed, SNR = (cid:80) Kk =1 γ k , namely,
10 log (10 . + 10 . + 10 . + 10 . ) = 12 .
66 dB . As already noticed in Remark IV.1, the best operating condition for the energy detector corresponds tothe case of K = N = 1 (at fixed additive SNR).V. C ONCLUSIONS
In this work we proposed an information-theoretic method to derive a lower bound to the receiveroperating characteristic of a cognitive radio network based on cooperative sensors. The bound stems fromthe application of the data-processing inequality to the binary asymmetric channel arising by considering
November 13, 2018 DRAFT6 −3 −2 −1 −2 −1 P fa P m d lower boundED K=N=1ED K=4,N=1ED K=4,N=5 Fig. 8. ROC curves corresponding to α = 0 . , K = 4 secondary users, and Rayleigh fading. The solid curve is the lowerbound. The other curves correspond to energy detection (ED) with different combinations of K and N and constant additiveSNR ( . dB). the primary signal presence as a binary input and the fusion center decision on the signal presence as abinary output. The bound takes into account the possible knowledge of the a priori probability of primarysignal presence and applies to every kind of single-input multiple-output channel connecting the primarysignal to the multiple cooperative sensors (i.e., the secondary users of the cognitive radio system).Key advantages of this approach are: i ) independence from the implementation of the connectionbetween the sensors and the fusion center; and ii ) independence from the fusion rule. Both featuresderive from the information-theoretic method we have followed, based on the equivalence between theactual channel model (connecting the primary transmitter to the sensors and then to the fusion center)and the binary asymmetric channel with error probabilities corresponding to the false-alarm and missed-detection events.In order to illustrate this basic idea, we considered two scenarios of interest for cognitive radio networks: November 13, 2018 DRAFT7
1) The case of full channel gain and primary signal information at the fusion center.2) The case of full distribution information about the channel gain and the primary signal at the fusioncenter.The first case has been investigated in full detail by deriving the mutual information I between theprimary signal presence variable ξ and the set of sensor observations Y . This expression has beenanalyzed asymptotically, both for large additive SNR and for vanishing α (probability of signal absence).The asymptotic expression for large additive SNR, eq. (9), is based on an integral which is an extensionof the one calculated in [11, Prob. 4.12]. In essence, this expression can lead to an asymptotic expansionof the mutual information of the binary symmetric channel for large SNR. Full details on the derivationare reported in Appendix B.In the second case, a general expression of the mutual information I required to obtained the receiveroperating characteristic lower bound has been derived as the average value of an expression dependingon two random variables (the random additive SNR Γ and the auxiliary Gaussian random variable Z ).This average leads to a double integral, which has been expanded in detail in the case of iid Rayleighdistributed channel gains. In the numerical results section, the distribution of Γ is reported in a fairlygeneral case, and numerical results are included for illustration purposes.Finally, it is worth mentioning that the series expansion of the mutual information of a binary inputadditive Gaussian channel (Theorem B.1) is also a novel contribution of this paper. It extends the seriesexpansion of the capacity of the same channel, which can be found in [11, Prob. 4.12]. In the presentcontext, this series expansion is needed to account for the possible knowledge of the a priori probabilityof primary signal presence, which is available in several cognitive radio system and worth being used toimprove the quality of the decision rule at the fusion center.A PPENDIX AE QUIVALENCE OF BINARY INPUT G AUSSIAN CHANNELS
In this appendix we show that a binary-input, vector-output additive Gaussian channel is equivalent, asfar as concerns the mutual information, to a binary-input additive Gaussian channel with scalar output.Let the binary input be X = ± with α (cid:44) P ( X = − and ¯ α (cid:44) − α = P ( X = +1) .Let the channel equation be y = X a + z , (15)where a ∈ R n × is a given constant vector and z is a vector of iid Gaussian random variables distributedas N (0 , . November 13, 2018 DRAFT8
The mutual information is given by I ( X ; y ) = h ( y ) − h ( y | X ) = h ( y ) − h ( z ) . We know that h ( z ) = ( n/
2) log (2 π e) . To calculate h ( y ) , we note that p y ( y ) = α exp( −(cid:107) y + a (cid:107) /
2) + ¯ α exp( −(cid:107) y − a (cid:107) / π ) n/ . Hence, I ( X ; y )= − E (cid:20) log ( α e −(cid:107) y + a (cid:107) / + ¯ α e −(cid:107) y − a (cid:107) / ) (cid:21) − n log e2= H b ( α ) − α E (cid:20) log (cid:18) αα e a T z − (cid:107) a (cid:107) (cid:19)(cid:21) − ¯ α E (cid:20) log (cid:18) α ¯ α e a T z − (cid:107) a (cid:107) (cid:19)(cid:21) . (16)The scalar product a T z is a real Gaussian random variable with zero mean and variance (cid:107) a (cid:107) .Similarly, we can find the mutual information of the channel Y = aX + Z, (17)where a ∈ R and Z ∈ N (0 , . In this case, I ( X ; Y )= − E (cid:20) log ( α e − ( Y + a ) / + ¯ α e − ( Y − a ) / ) (cid:21) − log e2= H b ( α ) − α E (cid:20) log (cid:18) αα e aZ − a (cid:19)(cid:21) − ¯ α E (cid:20) log (cid:18) α ¯ α e aZ − a (cid:19)(cid:21) . (18)Then, the mutual information eqs. (16) and (18) coincide provided that a = (cid:107) a (cid:107) . These results aresummarized by the following theorem. Theorem A.1
A binary-input vector-output additive Gaussian channel y = X a + z , where the entriesof z are iid and distributed as N (0 , , is equivalent, in terms of mutual information, to a binary-inputreal additive Gaussian channel with SNR γ = (cid:107) a (cid:107) . The mutual information is: I ( X ; Y ) = H b ( α ) − α E (cid:20) log (cid:18) αα e √ γZ − γ (cid:19)(cid:21) − ¯ α E (cid:20) log (cid:18) α ¯ α e √ γZ − γ (cid:19)(cid:21) , (19) November 13, 2018 DRAFT9 where Z ∼ N (0 , . Remark A.1
The equivalence between the scalar and n -vector channels in terms of mutual informationcan be predicted by observing that the vector channel is a combination of n parallel Gaussian channels.Then, provided the receiver knows the vector a , the useful part of the signal can be combined coherentlyat the receiver while the noise is combined incoherently. This implies an n -fold increase of the SNR, fora given mutual information, when passing from the scalar to the n -vector channel.A PPENDIX BA SYMPTOTIC APPROXIMATION OF (19)In this appendix we derive an asymptotic approximation of the mutual information (19). This resultextends [11, Prob. 4.12], corresponding to the equiprobable input case.
Theorem B.1
The asymptotic expansion of (19) for γ → ∞ is given by I ( X ; Y ) = H b ( α ) − ∞ (cid:88) n =0 ( − n k n ( α ) γ n +1 / e − γ/ , (20) where k n ( α ) = 2 √ πα ¯ αn ! ln 2 n (cid:88) k =0 (cid:18) n k (cid:19) π n − k ) | E n − k ) |× k (cid:88) m =0 m (2 k ) (2 m ) (ln( α/ ¯ α )) k − m ) . (21) Consecutive partial sums in (20) are lower and upper bounds to I ( X ; Y ) .Proof: We start by considering the integral (cid:90) ∞−∞ ln(1 + ρ e z − γ )e − z / (8 γ ) dz √ πγ = (cid:90) ∞−∞ ln(1 + ρ e z )e − ( z +2 γ ) / (8 γ ) dz √ πγ = (cid:90) ∞−∞ ln(1 + ρ e z )e − z / (8 γ ) − z/ − γ/ dz √ πγ = 2 √ π ∞ (cid:88) n =0 ( − n c n ( ρ )(8 γ ) n +1 / e − γ/ (22)where c n ( ρ ) (cid:44) π (cid:90) ∞−∞ ln(1 + ρ e z ) z n n ! e − z/ dz. (23) November 13, 2018 DRAFT0
Since consecutive partial sums of the series expansion of e − x are lower and upper bound of the limit,also consecutive partial sums of (22) are lower and upper bounds of the lhs.To calculate the coefficients c n ( ρ ) , we notice that c (cid:48) n ( ρ )= 14 π n ! √ ρ (cid:90) ∞−∞ z n cosh(( z + ln ρ ) / dz = 12 π n ! √ ρ (cid:90) ∞−∞ (2 u − ln ρ ) n cosh u du = 12 π n ! √ ρ n (cid:88) k =0 (cid:18) n k (cid:19) k (ln ρ ) n − k ) (cid:90) ∞−∞ u k cosh u du = 12 n ! n (cid:88) k =0 (cid:18) n k (cid:19) π k (ln ρ ) n − k ) √ ρ | E k | (24)by using the integral [12, 3.523-4] (cid:90) ∞−∞ u k cosh u du = 2 (cid:18) π (cid:19) k +1 | E k | , for every integer k ≥ , where E n is the n th Euler number ( E = 1 , E = − , E = 5 , E = − , . . . ).Since we have: (cid:90) x (ln u ) n √ u du = 2 √ x n (cid:88) k =0 ( − k k n ( k ) (ln x ) n − k , for every integer n ≥ and with n ( m ) (cid:44) ( n ! / ( n − m )!) , we can integrate (24) term by term and obtainthe following final expression: c n ( ρ ) = 12 n ! n (cid:88) k =0 (cid:18) n k (cid:19) π k | E k | (cid:90) ρ (ln u ) n − k ) √ u du = √ ρn ! n (cid:88) k =0 (cid:18) n k (cid:19) π n − k ) | E n − k ) |× k (cid:88) m =0 ( − m m (2 k ) ( m ) (ln ρ ) k − m . (25)In the special case of ρ = 1 , we have: c n (1) = 1 n ! n (cid:88) k =0 k (2 n ) (2 k ) π n − k ) | E n − k ) | . November 13, 2018 DRAFT1
Then, in accordance with [11, Prob. 4.12], where c n corresponds to c n (1) / n , we have: c (1) = 1 c (1) = 8 + π c (1) = 384 + 48 π + 5 π c (1) = 46080 + 5760 π + 600 π + 61 π ...Finally, we apply (25) to derive the asymptotic expansion (20) and the coefficients (21).R EFERENCES
IEEE J. Select. Areas Commun. , vol. 23, no. 2,pp. 201–220, Feb. 2005.[4] R.S. Blum, S.A. Kassam, and H.V. Poor, “Distributed detection with multiple sensors: Part II - Advanced topics,”
Proc.IEEE , vol. 85, no. 1, pp. 64–79, Jan. 1997.[5] S. Srinivasa and S.A. Jafar, “The throughput potential of Cognitive Radio: A theoretical perspective,”
IEEE Communica-tions Magazine , vol. 45, no .5, pp. 73–79, May 2007.[6] C.R. Stevenson, G. Chouinard, Z. Lei, W. Hu, S.J. Shellhammer, and W. Caldwell, “IEEE 802.22: The first cognitiveradio wireless regional area network standard,”
IEEE Communications Magazine , vol. 47, no .1, pp. 130–138, Jan. 2009.[7] T. Y¨ucek and H. Arslan, “A survey of spectrum sensing algorithms for cognitive radio applications,”
IEEE Commun.Surveys & Tutorials , vol. 11, no. 1, pp. 116–130, First quarter 2009.[8] B. Fette,
Cognitive Radio Technology, 2nd Ed.
Burlington, MA: Academic Press, 2009.[9] T.M.Cover and J.A.Thomas,
Elements of Information Theory.
New York: Wiley, 1991.[10] J.G. Proakis,
Digital Communications , Third Ed. New York: Mc-Graw-Hill, 1995.[11] T. Richardson and R. Urbanke,
Modern Coding Theory , New York: Cambridge University Press, 2008.[12] I.S. Gradshteyn and I.M. Ryzhik,
Table of Integrals, Series, and Products , th edition, Elsevier, 2007.edition, Elsevier, 2007.