Simultaneous Sensing and Transmission for Cognitive Radios with Imperfect Signal Cancellation
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(2) 2. (a) Quiet spectrum sensing.. (b) Simultaneous spectrum sensing and data transmission.. Fig. 1: Frame structure. transmitter activity. In [21] a two-phase concurrent sensing and • We evaluate how the imperfect signal cancellation due transmission scheme was proposed employing a suitable control to decoding errors, affects the sensing performance. In power mechanism. Nevertheless, an important drawback of addition, we derive the sensing performance parameters these techniques is the requirement for extra dedicated hardware i.e., probability of detection (PD ) and probability of false (antennas) for sensing that increases the cost of the system. alarm (PF A ) for BPSK, QPSK, and general M-QAM SU signals. Moreover, as mentioned earlier, these approaches introduce • It is shown that the remaining signal, and consequently the phenomenon of self-interference, which degrades the sensing performance, and hence, self-interference cancellation its energy follows a truncated distribution. Applying the schemes have to be adopted, which however, increase the power concept of truncated distribution, we derive the mean consumption of the system with their turn. and variance of a truncated central or non-central chisquared variable. Further, in combination with central On the other hand, there are works which propose simullimit theorem (CLT), the distribution, mean and variance taneous sensing and transmission using an inactive SU. In of the sum of N truncated central or non-central chi[16] a cognitive radio system was proposed, which performs squared variables are derived. This is used to model the spectrum sensing through an inactive SU, while an active SU distribution of the energy detection test statistics. Finally, is transmitting. A similar analysis is proposed in [22], where the approximated expressions are evaluated by numerical a cognitive base station transmits data to some SUs using zero results which verify the accuracy. forcing, while some other SUs carry out spectrum sensing. The rest of the paper is organized as follows. In Section However, again, these approaches face challenges, such as the extra power consumption and waste of resources by using an II, the system model and the proposed method are described. Section III presents the derivation of the PF A and PD for BPSK inactive SU or some other SUs for spectrum sensing. In [23], a different concept was proposed, where the simul- and QPSK SU modulated signals, while Section IV provides taneous spectrum sensing and data transmission is obtained the expression of PF A for any M-QAM SU modulated scheme. through the collaboration of the SU transmitter and the SU Numerical results are illustrated in Section V. Finally, Section receiver, which perform in different nodes. The SU transmitter VI concludes the paper. Notation: Bold-face lower case letters are used to declare is responsible for the data transmission, while the SU receiver decodes the signal from the secondary transmitter, removes it vectors. R {·} and I {·} denote the real and the imaginary from the total received signal and carries out spectrum sensing part of {·}, respectively, while E {·} and V {·} represent in the remaining signal. The two main advantages of this the expectation and the variance of {·}, respectively. The T technique compared to the approach at the transmitter’s side are superscript (·) represents the transpose of (·). · denotes that i) it does not use extra antennas for the spectrum sensing, the standard vector norm, while |·| is the absolute value. The hence, it can be easily implemented in the current systems with chi-squared distribution with q degrees of freedom is denoted 2 no additional hardware change and also, ii) it does not face the by χq . problem of self-interference that described earlier. Furthermore, II. S YSTEM MODEL this approach offers much better detection performance than that of using inactive SUs, if we assume that the adopted A. Signal Model detection scheme is the Energy Detector. The reason is that We consider a cognitive radio system as shown in Figure 2, the decoding and cancellation of the SU transmitted signal is where the primary user transmitter (PU-Tx) and the secondary almost impossible by using the inactive SU, because the latter user transmitter/receiver (SU-Tx/Rx) are equipped with one needs information about the channel, modulation and coding antenna. The goal of this system is to detect if the PU is and etc., which are hardly available in practice. However, the active or idle following the concept of simultaneous spectrum work of [23] is under the ideal assumption of perfect signal sensing and data transmission through the cooperation of the decoding. SU-Tx and the SU-Rx. Therefore, the detection problem can In this context, the contributions of this paper are three-fold: be formulated as the following binary hypothesis test, which is a baseband symbol sampled model: • Unlike existing works that assume perfect signal cancellation, in this paper, we investigate simultaneous sensing and transmission taking the imperfect signal cancellation into account. Energy detection is then applied on the remaining signal to detect the presence or absence of the PU.. H0 : H1 :. y=hs+w,. (1). y=xp +hs+w,. (2). where h denotes the scalar flat fading channel from the SUTx to the SU-Rx, which is assumed to be known at the.
(3) 3. \. Ͳ. WhͲddž. KVÖ. WhZdž. ^hͲZdž. ^hͲddž. Fig. 2: System model. secondary users as in [23], and it is also assumed to be real after the phase compensation with channel power γ, T s = [s (1) · · · s (N )] denotes an N × 1 vector, which is the signal transmitted by the SU-Tx with power Ps and it T is a modulated signal, xp = [xp (1) · · · xp (N )] denotes an N × 1 vector, referred to as the received (faded) signal from T the PU-Tx, w = [w (1) · · · w (N )] denotes an N × 1 vector, which is the additive noise at the receiving antenna of the SU-Rx, modelled as an independent and identically distributed (i.i.d.) complex Gaussian vector zero mean and covariance with 2 matrix given by E wwH = σw IN , where IN denotes an T identity matrix of size N , and y = [y (1) · · · y (N )] denotes an N × 1 vector, referred to as the total received signal at the SU-Rx. As mentioned, in this paper we assume that the channel is known. However, in reality, the channel should be estimated. Therefore, the channel estimation error is an important factor which has to be taken into account in our analysis, constituting a valuable idea for future studies. Nevertheless, in Section V, we present preliminary results how this uncertainty affects the sensing performance of our proposed detector. B. Method Description In a cognitive radio network, the goal of the SU-Tx is to access the spectrum when it is not occupied by the PU and thus avoiding interference to the PU network (this is obtained by considering a high target probability of detection). Therefore, in the beginning, namely during the very first frame of cognitive data transmission, the SU-Tx divides the frame in two time slots: i) one sensing period (τ units of time) and ii) one data transmission period (T units of time). If the PU is detected to be idle during the sensing period, the SU-Tx changes mode and starts the data transmission to the SU-Rx. Then, in the following frames the SU-Rx decodes the signal of the SU-Tx and removes it from the total received signal. Subsequently, a detector is applied to the remaining signal for spectrum sensing. At the end of the frame, if the SU detects a change in the state of the PU (e.g. the PU starts the transmission after the sensing period), then the SU-Rx informs the SU-Tx via a control channel and the latter stops the transmission in order to avoid causing interference to the PU. Hence, in the next frame, the SU-Tx, again, divides the frame in two time slots and the above process is repeated. However, at the end of the frame, if the SU detects that the frequency band is idle (absence of the. ĞĐŽĚĞƌ. \. H o + ŶĞƌŐLJ ĞƚĞĐƚŽƌ. ! H o +. VÖ. Fig. 3: Receiver structure of the proposed method for spectrum sensing and data transmission at the same time.. Fig. 4: Frame structure of the proposed simultaneous sensing and transmission method. PU), there is no need to devote a period for sensing during the next frame, and then, the idea of simultaneous spectrum sensing and data transmission is applied improving the SU’s throughput. The structure of the secondary receiver is depicted in Figure 3, while the frame structure of this method is presented in Figure 4. C. Proposed Algorithm The aforementioned methodology can be applied for any modulation scheme, but in this step, for simplicity, we consider that the transmitted signal from the SU is BPSK modulated and the noise is a real (not complex) Gaussian vector. Later, we will show how the proposed algorithm can be applied for QPSK and any M-QAM modulated signal with complex Gaussian noise. Hence, following these assumptions the detection problem of (1)-(2) can be reformulated via the following procedure. 1) After the initial stage of sensing, the SU Rx tries to decode the signal transmitted by the SU Tx, using the BPSK demodulator, which is based on the Euclidean distance√[24]. Note that given that the transmitted symbol √ is s = Ps , the correctly decoded signal √ is sˆ = Ps , while the wrongly decoded signal is sˆ = − Ps or more generally +s → correct decoding, sˆ = (3) −s → wrong decoding..
(4) 4. 2) Then, the decoded signal is removed from the total received signal and the new hypothesis test is expressed as follows: H00B : y (n) = w (n) , H0B = (4) : y (n) = 2hs (n) + w (n) , H 01B H10B : y (n) = xp (n) + w (n) , H1B = (5) H11B : y (n) = xp (n) + 2hs (n) + w (n) ,. PkB is the probability of false alarm for the case that k bits are decoded wrongly, which can be approximated as follows:. ε − μ H 0B , (8) Pk B = Q
(5) VH 0B where μH0B and VH0B are the mean and variance of the test statistic T (y |H0B ), respectively.. Proof: The Binomial distribution [25] is used for the proof where n = 0, 1, ..., N − 1, the index B in the hypothesis of the first part of Theorem 1, due to the fact that considering test denotes the BPSK scenario, H00B and H10B correindependent experiments, the number of wrong decoded bits spond to the hypothesis of correct decoding of the received is different in each of them, ranging from 0 to N with a signal, while the PU is idle and active, respectively, while corresponding probability. For the second part, the probability H01B and H11B represent the wrong decoding case, while of false alarm P of the ED of (6) is determined by PkB (ε) = kB the PU is idle and active, respectively. Now, it is clear ) > ε|H ), where the derivation of the distribution Pr (T (y 0B that without considering the decoding errors (hypotheses of the test statistic T (y |H 0B ) is required. H01B and H11B ), the new hypothesis test is the same as the one of quiet spectrum sensing, with the difference that Focusing on H00B , we can notice that the remaining signal, here, the whole duration of the frame is used for spectrum after the cancellation of the correct decoded signal, consists sensing instead of a small quiet period. only of noise. This vector includes the set of noise values, 3) The last step is the application of the ED in the remaining which let the BPSK demodulator to decide correctly about signal, examining how the decoding errors affect the the transmitted symbol. Then, the Cumulative Distribution sensing performance. We selected the ED as detection Function (CDF) of the remaining signal under H00B can be technique due to the fact that it does not require knowledge written as follows: of the PU characteristics (modulation type, pulse shaping, {y ≤ g|H00B } etc.), which are often unknown. The ED is shown in (6) Fy (g |H00B ) = P
(6) .
(7) = P y ≤ g|cB , s = + Ps P s = + Ps N −1 2 2 < ε → H0B
(8)
(9) . T (y) = y = |y (n)| , (6) > ε → H1B + P y ≤ g|cB , s = − Ps P s = − Ps n=0
(10) 1 where ε denotes a properly defined threshold, responsible = P w ≤ g|g ≥ − Ps 2 for the decision about the presence or absence of the PU.
(11) 1 + P w ≤ g|g ≤ + Ps , 2 III. P ROBABILITY OF FALSE ALARM AND P ROBABILITY OF where the symbols are assumed√to be of equal probability DETECTION √ (s = + P ) = P (s = − Ps ) = 1/2), cB denotes (P s The calculation of the detection threshold ε is independent the correct decision event under the BPSK scenario and from the transmitted primary signal and hence, the evaluation F (g|H ) represents the CDF of y under H00B . As it y 00 B of our proposed detection scheme is obtained via the derivation can be seen, the CDF is related to two cases: i) correct √ of the PF A . Therefore, in this section, we mainly derive the decoding given that the transmitted symbol is s = + Ps PF A , first for BPSK and then for QPSK SU modulated signals. √ decoding given that the transmitted symbol Furthermore, for the same SU modulated signals, we derive or ii) correct s = − Ps . Then, the distribution of noise for both is the PD under the assumption that the primary user is Gaussianscenarios is depicted in Figures 5a and 5b. A very interesting distributed. information which can be extracted by these two figures is that the received signal is always decoded correctly in the A. Probability of false alarm for BPSK signals γPs s region where − γP ≤ w ≤ + 2 σw σ 2 , while it is sometimes In this subsection, the probability of false alarm for the decoded correctly and some otherwwrongly in the region. BPSK case (PF AB ) is determined by Theorem 1, which is s s where w ≥ γP or w ≤ − γP 2 2 , based on the fact if the σ w subsequently proved. √ σw √ Theorem 1: Consider a secondary user with one receive transmitted symbol is s = Ps or s = − Ps . This analysis antenna, which collects a large number of samples N . The shows that the remaining noise under H00B follows a truncated distribution√[26]-[27] in √ the following intervals: SU-Rx decodes the received samples, removes them from the normal √ √ i) w ≤ − P , ii) − P ≤w ≤ + P and iii) w ≥ + Pt , t t t total received signal and applies an ED in the remaining signal. γPs P = where for the rest of this paper, we assume that 2 . t Then, the PF AB is defined by σw Now, it becomes clear that we should determine what is the N distribution of the sum of N truncated central and/or nonN N −k PF AB = , (7) PkB PekB (1 − PeB ) central chi-squared random variables. However, the closed form k k=0 expression of this distribution is not mathematically tractable. where k denotes the number of wrong decoded bits, PeB = Therefore, we should examine if the distribution of the test. γPs Q is the probability of bit error for BPSK [24] and statistic T (y |H0B ) can be approximated by using the CLT 2 σw.
(12) 5. 0.5. 0.5. √. 0.45. P(w/c,s= Ps ). 0.4. 0.4. 0.35. 0.35. 0.3. region of wrong decoding. 0.25. 0.3. region of correct decoding. 0.25. 0.2. 0.2. 0.15. 0.15. 0.1. 0.1. 0.05. 0.05. 0√ −4 Ps. √ −3 Ps. √ −2 Ps. √ −1 Ps. 0. √ 1 Ps. (a) Correct decoding of s =. √ P(w/c,s=- Ps ). 0.45. √ 2 Ps. √. √ 3 Ps. √ 4 Ps. 0√ −4 Ps. region of correct decoding. √ −3 Ps. √ −2 Ps. region of wrong decoding. √ −1 Ps. √ 1 Ps. 0. √ 2 Ps. √ 3 Ps. √ 4 Ps. √ (b) Correct decoding of s = − Ps = −1.. Ps = 1.. Fig. 5: Distribution of noise under the hypothesis H00B . [28]. The test statistic T (y |H0B ) consists of variables under one of the following cases: i) always correct decoding, ii) always wrong decoding and iii) sometimes correct and some other wrong decoding. The first two categories include a sequence of i.i.d random variables and hence, assuming a large number of samples the CLT can be applied. However, an independent but not identically distributed sequence is involved in the third case. Nevertheless, even in this case, the CLT can be used since the Lyapunov’s and Lindeberg’s conditions for non-identical variables [29] are satisfied. Thus, the mean and variance of T (y |H0B ) are respectively given by μH0B = (N −k)μH00B + kμH01B and VH0B = (N − k)VH00B + kVH01B , where μH00B , μH01B , VH00B and VH01B are the mean and variance of the test statistic T (y |H00B ) and T (y |H01B ), respectively, with y meaning only one sample. Hence, the derivation of the mean and variance for T (y |H00B ) and T (y |H01B ) is required. However, the calculation of these parameters is obtained with the help of the following three lemmas. Lemma 1 is valid for all truncated central chi-squared variables, Lemma 2 is valid, only when the non-central chi-squared variable is truncated to the interval [a, b] where 0 ≤ a ≤ b < ∞, while when b = ∞, Lemma 3 is used to evaluate the truncated non-central chi-squared variable. [a,b] [a,b] Lemma 1: The mean μc and variance Vc of a central chi-squared variable with one degree of freedom, truncated to the interval [a, b] where 0 ≤ a ≤ b ≤ ∞, is given by afχ2 (a) − bfχ21 (b) [a,b] 1 , (9) μc = 1 + 2 Fχ21 (b) − Fχ21 (a) Vc[a,b] = 2 − 4 +2. afχ21 (a) − bfχ21 (b). 2. Fχ21 (b) − Fχ21 (a). a2 fχ21 (a) + afχ21 (a) − b2 fχ21 (b) − bfχ21 (b) Fχ21 (b) − Fχ21 (a). variable with one degree of freedom. Proof: Let F [a,b] and f [a,b] denote respectively the CDF χ2 χ2 1T 1T and PDF of a central chi-squared variable with one degree of freedom, truncated to the interval [a, b] where 0 ≤ a ≤ b ≤ ∞. For the untracated case, i.e. a = 0 and b = ∞, we drop the superscript and note only Fχ21 and fχ21 . Then, for x ∈ [a, b], F [a,b] (x) = 2 χ. 1T. Fχ2 (x)−Fχ2 (a) 1. 1. Fχ2 (b)−Fχ2 (a) 1. 1. and f [a,b] (x) = 2 χ. , (10). where Fχ21 and fχ21 denote respectively the CDF and the Probability Density Function (PDF) of a central chi-squared. 1. 1. 1. According to [30], the moment generating function (MGF) for a truncated central chi-squared random variable X ∼ χ21T is given as follows: Fχ21 (b (1 − 2t)) − Fχ21 (a (1 − 2t)) −1/2 (1 − 2t) . MX (t) = Fχ21 (b) − Fχ21 (a) (11) Then, we can use the MGF to calculate the mean and variance of a truncated central chi-squared variable with one degree of freedom as follows: [a,b] μc = E {X |a ≤ X ≤ b } = M (t) |t=0 and 2 X [a,b] Vc = V {X |a ≤ X ≤ b } = E X |a ≤ X ≤ b − 2 (E {X |a ≤ X ≤ b }) , where E X 2 |a ≤ X ≤ b = M X (t) |t=0 afχ21 (a) − bfχ21 (b) =3+4 Fχ21 (b) − Fχ21 (a) 2 2 a fχ1 (a) + afχ21 (a) − b2 fχ21 (b) − bfχ21 (b) +2 . (12) Fχ21 (b) − Fχ21 (a) [a,b]. . 1T. fχ2 (x). Fχ2 (b)−Fχ2 (a) .. [a,b]. Lemma 2: The mean μc and variance Vc of a noncentral chi-squared variable with one degree of freedom and non-centrality parameter λ, truncated to the interval [a, b] where 0 ≤ a ≤ b < ∞, is given by Fχ23,λ (b) − Fχ23,λ (a) + λ Fχ25,λ (b) − Fχ25,λ (a) , μ[a,b] nc = Fχ21,λ (b) − Fχ21,λ (a) (13).
(13) 6. [a,b] Vnc =. + +. 3 Fχ25,λ (b) − Fχ25,λ (a). by. [−∞,a] μnc inf = E x2 |x ≤ a 2 = E x2 |x ≤ a = E |μ + w| |μ + w ≤ a = μ2 + 2μE {w |w ≤ a − μ } + E w2 |w ≤ a − μ , (19). Fχ21,λ (b) − Fχ21,λ (a) 6λ Fχ27,λ (b) − Fχ27,λ (a) Fχ21,λ (b) − Fχ21,λ (a) λ2 Fχ29,λ (b) − Fχ29,λ (a). . 2 − μ[a,b] , nc. (14). while the variance is given by [−∞,a] Vnc inf = V x2 |x ≤ a where Fχ21,λ denotes the CDF of a non-central chi-squared vari 2 2 4 able with one degree of freedom and non-centrality parameter = E x |x ≤ a − E x |x ≤ a 2 λ. 4 2 = E |μ + w| |μ + w ≤ a − E |μ + w| |μ + w ≤ a Proof: Let F [a,b] denotes the CDF of a non-central chiχ2 1T ,λ = μ4 + 4μ3 E {w |w ≤ a − μ } + 6μ2 E w2 |w ≤ a − μ 3 squared variable with one degree of freedom, truncated to w |w ≤ a − μ + E w4 |w ≤ a − μ the interval [a, b] where 0 ≤ a ≤ b < ∞ and non-centrality + 4μE. 2 parameter λ. For the non-truncated case, i.e. a = 0 and b = − μ[−∞,a] . (20) nc inf ∞, we drop the superscript and note only Fχ21,λ . Then, for Fχ2 (y)−Fχ2 (a) Then, according to [32], the expression of E {um |u ≤ a }, for 1,λ 1,λ y ∈ [a, b], F [a,b] (y) = . Fχ2 (b)−Fχ2 (a) some fixed a is given by χ2 1T ,λ 1,λ 1,λ m According to [31], the moments of the truncated non-central m m E {u |u ≤ a } = (21) μm−r σ r Ir , chi-squared distribution can be calculated as follows r r=0 h (i, p, λ) , (15) where E Ti = h (0, p, λ) fx (d) + (r − 1) Ir−2 , Ir = −dr−1 (22) Fx (d) where i ∈ {0, 1, 2, ...}, h (0, p, λ) = Fχ2 (b) − Fχ2 (a) and Fχ21,λ (b) − Fχ21,λ (a). p,λ. p,λ. with the following initial conditions: i) I0 = 1 and ii) I1 = − Ffxx(d) (d) . Therefore, using (21) and (22) and the fact that h (i, p, λ) = 2 the noise has been assumed to be Gaussian with zero mean, k=0 the first moments distribution are derived of the truncated × h (0, p + 2i + 2k, λ) . (16) as follows: E w0 |w ≤ a = I0 = 1, E w1 |w ≤ a =. Then, the mean and variance of a truncated non-central chi- σI1 = −σ fx (d) , E w2 |w ≤ a = σ 2 I2 = σ 2 1 − d fx (d) , Fx (d) F (d) x . squared variable Y with one degree of freedom is derived 3 fx (d) 3 3 2 fx (d) [a,b] E w |w ≤ a = σ I = σ − 2 and −d 3 Fx (d) Fx (d) as follows μnc = E {Y |a ≤ Y ≤ b } = E {Y |a ≤ Y ≤ b } . 4 [a,b] 2 f (d) f (d) x x 4 4 3 and Vnc = V {Y |a ≤ Y ≤ b } = E Y |a ≤ Y ≤ b − E w |w ≤ a = σ I4 = σ −d − 3d + 3 . Fx (d) Fx (d) 2 (E {Y |a ≤ Y ≤ b }) . Finally, using (19)-(22), we can prove the mean and variance [−∞,a] [−∞,a] Lemma 3: The mean μnc inf and Vnc inf of a squared of (17) and (18), respectively. Then, with the results of Lemma 1, we can develop Theorem Gaussian variable, where the Gaussian variable is truncated to the interval [a, ∞] or [−∞, α], with mean μ and variance σ 2 , 2, which defines the mean μH00B and variance VH00B , required for the calculation of μH0B and VH0B . is given by Theorem 2: Consider a secondary user with one receive fx (d) fx (d) [−∞,a] + σ2 1 − d μnc inf = μ2 − 2μσ , (17) antenna, which collects a number of samples N . The SU-Rx Fx (d) Fx (d) decodes the samples, removes them from the total received signal and applies the Energy Detector in the remaining fx (d) [−∞,a] 4 3 fx (d) 2 2 signal. Then, the mean and variance of T (y |H00B ) can be + 6μ σ 1 − d Vnc inf = μ − 4μ σ Fx (d) Fx (d) respectively defined as follows: fx (d) √ √ √ 3 2 fx (d) [ − Pt , P t ] [ Pt ,∞] −2 + 4μσ −d t] t ,∞] Pw + μ[P Pw , (23) μH00B = μ[0,P Fx (d) Fx (d) c c . 2 f f (d) (d) x x [−∞,a] √ √ √ − 3d + 3 − μnc inf , + σ 4 −d3 [ − P t , Pt ] [ Pt ,∞] Fx (d) Fx (d) V H00B = Vc[0,Pt ] Pw + Vc[Pt ,∞] Pw √ (18) √ . 2 [ Pt ,∞] [ Pt ,∞] [Pt ,∞] + μc 1 − Pw Pw where d = a − μ and Fx , fx denote the CDF and DPF of a √ √ √ √ normally distributed variable x.. 2 [ − Pt , P t ] [ − Pt , Pt ] [0,Pt ] + μc 1 − Pw Pw Proof: Let x = μ + w be a normally distributed random √ √ √ variable with mean μ and variance σ 2 , truncated to the interval [0,Pt ] [Pt ,∞] [− Pt , Pt ] [ Pt ,∞] 2 − 2μ μ P P , (24) w w c c [−∞, a]. Then, the mean of the truncated variable x is given i. i . k p Γ 2 +i λ i × k 2 Γ p2 + k.
(14) 7. [0,Pt ]. [0,Pt ]. [P ,∞]. [P ,∞]. , μc√t √ and Vc t √ are defined [ − Pt , Pt ] [ Pt ,∞] and Pw are exby Lemma 1, while Pw√ √ √ √ P (− Pt ≤w≤ Pt ) [ − Pt , P t ] pressed as follows: Pw = = P cB √ √ √ √ √ Fy ( Pt )−Fy (− Pt ) P (w≥ Pt ) F y ( − Pt ) [ Pt ,∞] and Pw = = , P cB P cB P cB where Fy (·) is the CDF of a normally distributed variable y and PcB is the probability of correct decision for BPSK [33]. where μc. , Vc. Proof: The proof of (23) and (24) is described in Appendix A and B, respectively. In a similar manner, following the results of Lemma 2 and 3, we develop Theorem 3, which defines the mean μH01B and variance VH01B , required for the calculation of μH0B and VH 0B . Theorem 3: Consider a secondary user with one receive antenna, which collects a number of samples N . The SU-Rx decodes the samples, removes them from the total received signal and applies the Energy Detector in the remaining signal. Then, the mean and variance of T (y |H01B ) can be respectively defined as follows: μH01B =. √ √ [ Pt ,3 Pt ] t] μ[0,P Pw nc. +. √ Pt ,∞] [3 Pt ,∞] μncinf Pw ,. [3. √. √. √. √. (25). √. [3 Pt ,∞] [3 Pt ,∞] [0,Pt ] [ Pt ,3 Pt ] V H01B = Vnc Pw + Vncinf Pw √ 2 √ √ [3 Pt ,∞] [3 Pt ,∞] [3 Pt ,∞] + μncinf 1 − Pw Pw √ √ √ √ . 2 [ Pt ,3 Pt ] [ Pt ,3 Pt ] t] + μ[0,P ] 1 − P Pw w nc √. √. [3 Pt ,∞] [ t] − 2μ[0,P μncinf Pw nc √. √. √. √ Pt ,3 Pt ,]. √. [3 Pw. √. Pt ,∞]. √. ,. (26) √. P ( Pt ≤w≤3 Pt ) Fx (3 Pt )−Fx ( Pt ) [ Pt ,3 Pt ] where Pw = = , Pc B Pc B √ √ √ P (w≥3 Pt ) Fx (−3 Pt ) [3 Pt ,∞] [0,Pt ] [0,Pt ] Pw = = and μnc , Vnc , Pc B Pc B [Pt ,∞] [Pt ,∞] μncinf , Vncinf are defined by Lemma 2 and Lemma 3. Proof: The proof of (25) and (26) is similar to that presented in Appendix A and B. . B. Probability of false alarm for QPSK signals In this subsection, we derive the PF A under the QPSK scenario. If we apply the algorithm proposed in Section II in under the QPSK scenario, the hypothesis test of (1) can be reformulated as in (28) at the top of the next page, where n = 0, 1, ..., N − 1, the index Q corresponds to the QPSK scenario, the hypothesis H00Q represents the case that the received signal is decoded correctly, while the hypotheses H01Q , H02Q and H03Q correspond to the wrong decoding case and more specifically: i) H01Q : the real part is decoded wrongly and the imaginary part is correctly decoded, ii) H02Q : the real part is correctly decoded and the imaginary part is wrongly decoded and iii) H03Q : both the real and imaginary part are wrongly decoded. Based on Theorem 1, for the derivation of the probability of false alarm under the BPSK scenario, the extension to QPSK case is straightforward and it is given as follows 2N 2N −k 2N , (29) PkQ PekQ 1 − PeQ PF AQ = k k=0. where PkQ = PkB and PeQ is the probability of bit error for QPSK, same as for BPSK, namely PeQ = PeB , while the factor 2 is due to the fact that a QPSK signal consists of two orthogonal BPSK ones. Furthermore, the corresponding approximated PF AQapr is given by (30) at the top of the next page, where μH00Q = 2μH00B and VH00Q = 2VH00B because both the real and imaginary part follow a truncated central chi-squared distribution, μH01Q = μH00B + μH01B and VH01Q = VH00B + VH01B , because the real part follows a truncated non-central chi-squared distribution, while the imaginary part follows a truncated central chi-squared distribution, μH02Q = μH00B + μH01B and VH02Q = VH00B + VH01B , because the real part follows a truncated central chi-squared distribution, while the imaginary part follows a truncated noncentral chi-squared distribution and finally, μH03Q = 2μH01B and VH03Q = 2VH01B , because both the real and imaginary part follow a truncated non-central chi-squared distribution.. Therefore, all the parameters of (7) have been derived. How- C. Probability of detection In this subsection, we derive the probability of detection ever, as mentioned, the required condition for the evaluation of the detector is the proper calculation of the detection threshold assuming that the primary signal follows a Gaussian distribution ε, which is significantly complex through (7), particularly as N with zero mean and variance σP2 U . This is a valid assumption, increases. For this reason, an approximated expression for the e.g. with an OFDM signal, where independent data streams are probability of false alarm is derived in this paper as follows: used for the modulation of each carrier [34]. Furthermore, it is ⎞ ⎛ customary to assume that xp is Gaussian-distributed, because ε − N (1 − PeB ) μH00B − N PeB μH01B the modulation and generally the symbols of the primary user ⎠, PF ABapr = Q ⎝ are unknown [13]. Similar signal modeling is frequently used N (1 − PeB ) V H00B + N PeB V H01B in cognitive radio literature, e.g. [35]-[37]. Therefore, under the (27) scenario that the SU signal is BPSK modulated, the theoretical where the index Bapr denotes approximation under the BPSK expression for the PD is also given by (27) by substituting scenario. This equation simplifies (7), based on the fact that 2 2 σw with σw + σP2 U in the related parts. Specifically, the PD for a large number of samples, the expected number of correct is given by and wrong decoded bits can be approximated with the help ⎞ ⎛ of the probability of correct (1 − PeB ) and wrong bits PeB , ε − N 1 − PeB μH10B − N PeB μH11B ⎠, PDBapr = Q ⎝ respectively. Now, according to (27) the computation of ε N 1 − PeB V H10B + N PeB V H11B requires the inverse Q function which can be computed directly (31) in most of the mathematical software packages..
(15) 8. H 0Q :. ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ⎛. H00Q H01Q H02Q H03Q . : y (n) = R {w (n)} + I {w (n)} , : y (n) = 2hR {s (n)} + R {w (n)} + I {w (n)} , : y (n) = 2hI {s (n)} + R {w (n)} + I {w (n)} , : y (n) = 2hR {s (n)} + 2hI {s (n)} + R {w (n)} + I {w (n)} , 2. . . . (28) ⎞. N Pe2Q μH03Q. ⎜ ε − N 1 − PeQ μH00Q − N 1 − PeQ PeQ μH01Q + μH02Q − ⎟ ⎟. PF AQapr = Q ⎜ . ⎝ ⎠ 2 2 N 1 − PeQ V H00Q + N 1 − PeQ PeQ V H01Q + V H02Q + N PeQ V H03Q. (30). where the mean μH10B , μH11B and variance VH10B , VH11B are defined by (23), (24), (25) and (26), respectively, with (33) μH0,k,t = μ R{sk }−R{st } + μ I{sk }−I{st } , √ √ Ps Ps the difference being that now, the probability of bit error and (34) VH0,k,t = V R{sk }−R{st } + V I{sk }−I{st } , the interval. of interest are not anymore related to PeB = √ √ Ps Ps γPs γPs Q and Pt = σ2 +σ2 , but they are based on the 2 σw w P U. where μ R{sk }−R{st } , μ I{sk }−I{st } , V R{sk }−R{st } , γPs γPs √ √ √ Ps Ps Ps following parameters: PeB = Q , P = . 2 2 d 2 2 σw +σP U σw +σP U V I{sk }−I{st } , are derived by Theorems 4 and 5. √ Similarly, the PD under the scenario that the SU signal is Ps 2 For the case of correct decoding, namely when k = t, it is QPSK modulated is given by (30) by substituting again σw 2 obvious that (33) and (34) are expressed as follows: μH0,k,t = with σw + σP2 U in the related parts. As mentioned, in this paper, we derive the PD under the μ0 + μ0 and VH0,k,t = V0 + V0 . Therefore, the derivation of assumption that the primary signal is Gaussian-distributed. the mean and variance under the scenario of correct decoding Deriving the PD for the case that the primary signal has for M-QAM signals is required and it is given by Theorem 4. different distribution is a valuable idea for future studies. Theorem 4: Consider a secondary user with one receive antenna, which collects a number of samples N . The SU-Rx decodes the samples, removes them from the total received IV. P ROBABILITY OF FALSE ALARM FOR M-QAM signal and applies the Energy Detector in the remaining signal. Then, the mean and variance of T (y |H0,k=t,t=k ) can be In the previous section, we discussed the derivation of the respectively defined as follows: PF A and PD under the BPSK and QPSK scenarios. However, 2 μH0,k=t,t=k = 2μ0 = E |y | |H 0,k=t,t=k in this section, we generalize our work to higher modulation schemes, and particularly, we examine the PF A for M-QAM = 2E R2 {y } |0 ≤ R2 {w} ≤ Pt P1 modulated signals. The derivation of the PD is straightforward # 2 as in Section III. Furthermore, we should note that here we + 2 √ E R2 {y } #R2 {w} ≥ Pt P2 , (35) M focus on the derivation of the approximated PF A , because as mentioned earlier, the calculation of the detection threshold 2 ε through the accurate PF A is complicated, especially as N VH0,k=t,t=k = 2V0 = V |y | |H 0,k=t,t=k increases and the situation becomes more complex under the = 2V R2 {y } |0 ≤ R2 {w} ≤ Pt P1 M-QAM scenario due to the fact that under the wrong decoding # 2 case the estimated symbol can be anyone of M − 1 possible + 2 √ V R2 {y } #R2 {w} ≥ Pt P2 symbols. M 2 The approximated probability of false alarm for the M-QAM + 2 E R2 {y } |0 ≤ R2 {w} ≤ Pt }) P1 P2 case (PF AM −Qapr ) is defined by 2 2 2 ## 2 E R {y } R {w} ≥ Pt + 2√ P1 P2 PF AM −Qapr = M ⎞ ⎛ 4 M ! M − 2 √ E R2 {y } |0 ≤ R2 {w} ≤ Pt ! ⎜ε − N P (s = sk ) P (ˆ s = st |s = sk ) μH0,k,t ⎟ # M ⎟ ⎜ k=1 t=1 ⎟ (36) × E R2 {y } #R2 {w} ≥ Pt P1 P2 , " = Q⎜ ⎟ ⎜ M M ! ! ⎠ ⎝ where H0,k=t,t=k represents the case of correct decoding for N P (s = sk ) P (ˆ s = st |s = sk ) VH0,k,t k=1 t=1 M-QAM signals under H0 hypothesis, M denotes the size (32) of the constellation, Pc corresponds to the probability of M−Q where sk denotes the transmitted symbol, st denotes the esti- correct symbol for M-QAM modulation [24] and P1 , P2 are mated symbol, while μH0,k,t and VH0,k,t represent, respectively, respectively obtained by (37) and (38) at the top of the next the mean and variance of the test statistic of (6) under each case page. of correct or wrong decoding under H0 hypothesis, expressed Proof: The proof of (35) is presented in Appendix C, while as follows: the proof of (36) is similar to that of (35). .
(16) 9.
(17)
(18) $ P −√Pt ≤ I {w} ≤ √Pt + % & √ √ Pc M−Q , P1 = P − Pt ≤ R {w} ≤ Pt +2P I {w} ≥ Pt / M √ √ %& $
(19) P − Pt ≤ I {w} ≤ Pt + √ √ PcM−Q , P2 = P R {w} ≥ Pt +2P I {w} ≥ Pt / M. Theorem 5: Consider a secondary user with one receive antenna, which collects a number of samples N . The SU-Rx decodes the samples, removes them from the total received signal and applies the Energy Detector in the remaining signal. Then, the mean and variance of T (y |H0,k=t,t=k ) are defined by (33) and (34), respectively, where μ R{sk }−R{st } = μ2l and √ √ √ Ps M V R{sk }−R{st } = V2l with l = 1 − 1 , ..., M − 1. The same √. Ps. range is also valid for μ I{sk }−I{st } and V I{sk }−I{st } . More √. √. Ps. Ps. specifically the mean μ2l , for the whole range, is derived as follows: 2r √ μ2r = 1 + M# ' #
(20) #2 # # # 2# × E |R {y }| #0 ≤ #2r Pt + R {w}# ≤ Pt P3 ## ' #2 #
(21) 2 # 2 ## + √ E |R {y }| ##2r Pt + R {w}# ≥ Pt P4 , M (39) μ2d = #
(22) ' #
(23) #2 ## #2 # # # # = E #2d Pt + R {w}# ##0 ≤ #2d Pt + R {w}# ≤ Pt × P5 .. (40). # ' #
(24) #2 2# # # μ2f = E |R {y }| ##0 ≤ #2f Pt + R {w}# ≤ Pt P6 ## ' #2 #
(25) 2 # 2 ## + √ E |R {y }| ##2f Pt + R {w}# ≥ Pt P7 , M (41) 2q μ2q = 2 − √ M# ' #
(26) #2 # # # 2# × E |R {y }| #0 ≤ #2q Pt + R {w}# ≤ Pt P8 ## ' #2 #
(27) 2 # 2 ## + √ E |R {y }| ##2q Pt + R {w}# ≥ Pt P9 , M (42) √. √. √. where√ r = 1 − 2M , ..., −1, d = 1, ..., 2M − 1, f = 2M and √ q = 2M + 1, ..., M − 1, while the variance V2r is obtained as follows: 2r V2r = 1 + √ M. (38). # ' #
(28) #2 # # # # × V |R {y }| #0 ≤ #2r Pt + R {w}# ≤ Pt P3 ## ' #2 #
(29) 2 # 2 ## √ V |R {y }| ##2r Pt + R {w}# ≥ Pt P4 + M 2r + 1+ √ M # '2 #
(30) #2 2# # # × E |R {y }| ##0 ≤ #2r Pt + R {w}# ≤ Pt P 3 P4 ## '2 #2 #
(31) 2 # 2 ## +√ P3 P 4 E |R {y }| ##2r Pt + R {w}# ≥ Pt M 2 2r √ −2 1+ √ M # M# ' #2 2# #
(32) # × E |R {y }| ##0 ≤ #2r Pt + R {w}# ≤ Pt ## ' #2 #
(33) # 2 ## (43) ×E |R {y }| ##2r Pt + R {w}# ≥ Pt P3 P4 , . Furthermore, the derivation of the mean and variance under the scenario of wrong decoding for M-QAM signals is necessary and it is given by Theorem 5.. (37). . 2. √. where r = 1 − 2M , ..., −1, while the probabilities P3 , P4 , P5 , P6 , P7 , P8 and P9 are given by √ √ √ √ P − Pt − 2r Pt ≤ R {w} ≤ Pt − 2r Pt P3 = PwM −Q √ √ P R {w} ≥ Pt − 2r Pt P4 = PwM −Q √ √ √ √ P − Pt − 2d Pt ≤ R {w} ≤ Pt − 2d Pt P5 = PwM −Q √ √ √ √ P − Pt − 2f Pt ≤ R {w} ≤ Pt − 2f Pt P6 = PwM −Q √ √ P R {w} ≥ Pt − 2f Pt P7 = PwM −Q √ √ √ √ P − Pt − 2q Pt ≤ R {w} ≤ Pt − 2q Pt P8 = PwM −Q √ √ P R {w} ≥ Pt − 2q Pt P9 = , (44) PwM −Q while H0,k=t,t=k represents the case of wrong decoding for M-QAM signals under H0 hypothesis and PwM −Q denotes the probability of wrong symbol detection for M-QAM modulation [24]. Finally, we should mention that the derivation of the variances V2d , V2f and V2q is not provided in this paper due to lack of space, but they can be defined following the same methodology as for (43). Proof: The proof of Theorem 5 is presented in Appendix D. .
(34) 10. . T (y|H0B ) ∼ N N (1 − PeB ) μH00B + N PeB μH01B , N (1 − PeB ) V H00B + N PeB V H01B .. 4. 10 -3. 6 Gaussian approx. Histogram of H 0. 3.5. 10 -3. 9 Gaussian approx. Histogram of H 0. 5. B. (45). 10 -3 Gaussian approx. Histogram of H 0. 8. Q. 16QA M. 7. 3 4. 2.5. 6 5. 2. 3 4. 1.5. 2. 3. 1. 2 1. 0.5 0 3200. 1. 3400. 3600. 3800. 4000. T(y|H ) 0. (a). 4200. 4400. 0 3500. 3600. 3700. 3800. T(y|H. B. (b). 3900 0. 4000. 4100. 0 1800. 4200. 1850. ). 1900. 1950. 2000. T(y|H. Q. 2050. 2100. 2150. 2200. ) 16QAM. (c). Fig. 6: Theoretical and empirical distributions of (a) T (y|H0B ), (b) T y|H0Q and (c) T y|H016−QAM with N = 2000, 2 Es = 10 dB and σw = 2 dB. V. N UMERICAL RESULTS In this section, we present simulation results in order to i) verify the derived expressions of (27), (30) and (32) for the PF A , ii) compare our proposed system with the conventional one in terms of the throughput, iii) evaluate the detection reliability of the “ED with imperfect signal cancellation (EDISC)” for the case of perfect channel estimation and iv) examine the PD of our proposed method taking into account the channel estimation error.. . . 0. . As we follow the same model as in [23], we can use the same approach for the evaluation of the throughput, replacing the PF A and PD under the case of perfect signal cancellation with our deriving expressions for the PF A and PD under the case of imperfect signal cancellation. For conciseness, we refer the readers to [23] for the details of the throughput derivation. The frame duration is T = 100 ms, the sampling frequency f = 6 MHz and for both cases, we assume a target probability of detection PD = 0.9, while the secondary transmit and primary received SNR are considered as SN RSU = 7 dB and SN RP U = −20 dB. Figure 7 depicts the average achievable throughput versus the sensing time τ , where it is obvious that the proposed cognitive radio systems exhibits higher average achievable throughput than the conventional one despite the fact of considering imperfect signal cancellation.. A. Evaluation of the probability of false alarm We should mention that under the assumption of a large number of samples N and the use of the CLT, the relationship between the PF A and the corresponding distribution of the test statistic. is the following: T (y) ∼ N (μ, V ) => PF A = √ Q ε−μ , where N (μ, V ) denotes a Gaussian distribution with V mean μ and variance V . Therefore, the verification of the PF A under the BPSK (27), QPSK (30) and M-QAM (32) scenarios C. Performance analysis with respect to PD can be obtained through the verification of the distribution of In this paper, we assume that the PU is either absent or the teststatistic under the BPSK (T (y|H 0B )), QPSK T y|H0Q and 16-QAM T y|H016−QAM scenarios. The present for a long period as in fixed networks, e.g. TV channels distribution of T (y|H0B ) is given by (45) at the top of the and backhaul networks. Therefore, in this section, we present page. derive the distribution of simulation results to analyze the detection reliability of the With similar manner, we can T y|H0Q and T y|H016−QAM , which we will not show “ED with imperfect signal cancellation (EDISC)” for the frame here due to lack of space. that the simultaneous spectrum sensing and data transmission Figures 6a, 6b and 6c depict the histogram obtained from takes place. We compare this scheme with the “ED with perfect 10000 Monte-Carlo realizations of T (y|H0B ), T y|H0Q signal cancellation (EDPSC)” as proposed in [23] and with the and T y|H016−QAM , respectively, where the number of case that the SU-Rx informs the SU-Tx about the status of the measurement samples is set to N = 2000. In this figure, PU by using a “Conventional ED (CED)” as given by (5) in we show that the CLT provides for the [38], where there is no need for decoding, and hence signal good approximation distribution of T (y|H0B ), T y|H0Q and T y|H016−QAM , cancellation. For simplicity, we assume that the PU follows verifying with this way the Gaussian approximations of (27), a Gaussian distribution with zero mean and variance σP2 U , (30) and (32) for the PF A . while the transmitted signal from the SU-TX is BPSK, QPSK, 16-QAM or 64-QAM modulated. Furthermore, the channel is B. Performance analysis with respect to throughput assumed to be of unit power, the probability of false alarm is In this subsection, we compare our proposed scheme with te set to PF A = 0.1, while the received PU Signal to Interference 2 σP U conventional one in terms of the average achievable throughput. plus Noise Ratio (SINR) at the SU-Rx (SIN RP U = Ps +σ 2 ) w.
(35) 11. 2.5. 1.6. conventional CR system proposed CR system. Achievable throughput (bits/sec/Hz). Achievable throughput (bits/sec/Hz). 1.4 1.2 1 0.8 0.6 0.4. conventional CR system proposed CR system. 2. 1.5. 1. 0.5. 0.2 0. 0. 0.02. 0.04. 0.06. 0.08. 0. 0.1. 0. 0.02. Sensing time (ms). 0.04. 0.06. 0.08. 0.1. Sensing time (ms). (a) Probability of PU being active P (H0 ) = 0.5.. (b) Probability of PU being active P (H0 ) = 0.8.. Fig. 7: Average achievable throughput of the proposed and conventional cognitive radio system versus the sensing time with secondary transmit SNR, SN RSU = 7dB, received primary SNR, SN RP U = −20 dB and target detection probability PD = 0.9. 1. 1. 0.9. 0.9. 0.7. EDISC Pfa B CED. 0.6 0.5 0.4 0.3. 0.6 0.5 0.4 0.3 0.2. 0.1. 0.1 -20. -15. -10 SINR(dB). -5. 0. 5. Fig. 8: PD versus the SINR of the PU, under the BPSK scenario, for N = 100, Es = 10 dB and 2 σw = 2 dB.. EDISC Pfa Q CED. 0.7. 0.2. 0 -25. EDPSC EDISC Pfa Qapr. 0.8. EDPSC EDISC Pfa Bapr. Probability of detection. Probability of detection. 0.8. 0 -25. -20. -15. -10 SINR(dB). -5. 0. 5. Fig. 9: PD versus the SINR of the PU, under the QPSK scenario, for N = 100, Es = 10 dB and 2 σw = 2 dB.. varies from −25 to 5 dB. Here, we should mention that by the reliability of (27) and (30), respectively. saying interference we mean the presence of the SU-Tx. Finally, from Figure 12 is observed that the difference Figures 8, 9, 10 and 11 depict the PD as a function of the in the detection performance between the simulation and SINR of the PU comparing the aforementioned three techniques: approximated results decreases as the number of samples i) CED, ii) EDPSC and iii) EDISC. The number of samples increases. The reason is that our theoretical approximated is set N = 100 and it is observed that our proposed detection expression is based on the CLT which requires a large number scheme provides significantly better detection performance than of samples for better reliability. the CED for all the cases. Furthermore, it is observed that the EDISC approaches the detection performance of the EDPSC performing slightly worse, under the BPSK and QPSK scenario, D. Performance analysis with respect to channel estimation while it presents inferior performance for 16-QAM and 64- error In the following experiment, we evaluate the performance QAM compared to ones for BPSK/QPSK, but still better. This can be explained by the fact that our technique considers a more of the EDISC with respect to the channel uncertainty under ˆ can be modeled as realistic scenario taking into account the decoding errors and the BPSK case. The estimated channel h ˆ the probability of correct decision deteriorates as we employ h = h + , where the channel estimation error follows a higher order constellations (especially for low SINR values). Gaussian distribution with zero mean and variance σ2 [38]Moreover, Figures 8 and 9 show that the performance of the [39]. Therefore, in the simulations, the channel estimation error PD based on the calculated decision threshold related to the varies in each realization. approximated PF A (i.e. (27) and (30) for Figure 8 and Figure Figure 13 depicts the PD as a function of the SINR of the PU 9, respectively) is very close to the PD based on the calculated for the BPSK case considering imperfect channel estimation. decision threshold related to the more accurate PF A (i.e. (7) This figure shows that the channel uncertainty degrades the and (29) for Figure 8 and Figure 9, respectively), validating detection performance of the EDISC. Furthermore, we see that.
(36) 12. 1. 1 EDPSC EDISC Pfa. 0.9. EDPSC EDISC Pfa. 0.9. 16QAMapr. CED. Probability of detection. Probability of detection. 0.8. 0.7 0.6 0.5 0.722 0.72. 0.4. 0.718 0.3. 0.716. 0.7 0.6 0.5. 0.72. 0.4 0.3. 1.8 dB. 0.71 0.2. 0.712 -12 -20. -15. -10. -11 -5. -10 0. 0.1 -25. 5. -11.5 -11 -10.5 -10 -9.5 -20. -15. SINR(dB). 1 0.9. 0.8. 0.8. N=100. Probability of detection. Probability of detection. 1. N=20. 0.5 0.4 0.3 0.2. EDISC (approximated) EDISC (simulation). 0.1 0 -25. -20. -15. -10 SINR(dB). -5. 0. 5. Fig. 11: PD versus the SINR of the PU, under the 64-QAM scenario, for N = 100, Es = 10 dB and 2 σw = 2 dB.. 0.9. 0.6. -10. SINR(dB). Fig. 10: PD versus the SINR of the PU, under the 16-QAM scenario, for N = 100, Es = 10 dB and 2 σw = 2 dB.. 0.7. 1.2 dB. 0.715. 0.714 0.2 0.1 -25. 64QAMapr. CED. 0.8. -5. 0. 5. Fig. 12: Comparison between the approximated and simulated performance for the proposed ED, under 2 the QPSK scenario for Es = 7dB and σw = 0dB. when the channel estimation is more accurate σ2 = 0.002, the effect of the channel estimation error reduces, but our detector, still, performs well for moderate channel uncertainty σ2 = 0.02. VI. C ONCLUSIONS In this paper, we investigated the idea of simultaneous spectrum sensing and data transmission considering imperfect signal cancellation in the data domain. We analyzed how the decoding errors affect the detection reliability of the system and derived the analytical expressions for the probability of false alarm assuming digitally modulated signals (i.e. BPSK, QPSK, M-QAM). Furthermore, we presented a detailed analysis around the distribution of the sum of N truncated central or non-central chi-squared random variables. Finally, the numerical results showed that the detection performance of our proposed scheme is considerably better than the conventional ED, verifying in this way the accuracy of the proposed study. Future works include i) the derivation of the analytical expressions for the PF A under the channel estimation errors, ii) the derivation of the PD for the case that the primary signal has different distribution than Gaussian and iii) the adaptation of our scheme in a more dynamic traffic pattern.. EDISC with perfect h EDISC with. 2. EDISC with. 2. =0.002 =0.02. 0.7 0.6 0.5. 0.86. 0.4. 0.84. 0.3. 0.82. 0.2. 0.8. 0.1. 0.78. 0 -25. -14 -20. -15. -10. -13 -5. -12. -11 0. -10 5. SINR(dB). Fig. 13: PD versus the SINR of the PU, for the BPSK case considering channel uncertainty, for 2 N = 100, Es = 10 dB and σw = 2 dB. A PPENDIX A DERIVATION OF THE MEAN. μH00B. For the derivation of the mean μH00B , we use the law of total expectation [40] as follows: 2 E |y | |H 00B = #
(37)
(38) 2# = E |y | #s = + Ps , sˆ = + Ps.
(39)
(40) × P s = + Ps P sˆ = + Ps |cB #
(41)
(42) 2 # + E |y | #s = − Ps , sˆ = − Ps .
(43)
(44) × P s = − Ps P sˆ = − Ps |cB ,. (46). where the expression of (46) can be extended as follows: 2 E |y | |H 00B =.
(45)
(46) 1 2 = E |y | |w ≥ − Pt P w ≥ − Pt |cB 2.
(47)
(48) 1 2 + E |y | |w ≤ + Pt P w ≤ + Pt |cB . (47) 2.
(49) 13. However, the remaining noise under H00B follows a truncated √ normal √ distribution √ in the following√intervals: i) w ≤ − Pt , ii) − Pt ≤ w ≤ Pt and iii) w ≥ Pt . Thus, exploiting the symmetry of the BPSK constellation, (47) can be further expressed by 2 E |y | |H 00B = #
(50)
(51) 2# = E |y | #− Pt ≤ w ≤ Pt
(52).
(53) × P − Pt ≤ w ≤ Pt |cB
(54) 2 + E |y | |w ≥ Pt.
(55) (48) × P w ≥ + Pt |cB , where according to Bayes’ Theorem √ √[41] √ √ √ P (− Pt ≤w≤ Pt ) = P − Pt ≤ w ≤ Pt ≤ Pt |cB P cB √ √ P (w≥ Pt ) and P w ≥ + Pt |cB = . Then, in terms of Pc B. 2. Lemma 1, we are interested in finding the interval of |w| , and hence, (48) is given by 2 μH00B = E |y | |H00B # P − √P ≤ w ≤ √ P t t 2# 2 = E |w| #0 ≤ |w| ≤ Pt Pc B # P w ≥ √ P t 2# 2 + E |w| #|w| ≥ Pt . (49) Pc B Finally, according to Lemma 1, (49) takes its final shape given by (23). A PPENDIX B DERIVATION OF THE VARIANCE. VH00B. The derivation of the variance VH00B is based on the law of the total variance [40] and can be written as follows: #
(56)
(57) 2 2# |y | |H 00B = V |y | #s = + Ps , sˆ = + Ps.
(58)
(59) × P sˆ = + Ps P sˆ = + Ps |cB #
(60)
(61) 2# + V |y | #s = − Ps , sˆ = − Ps.
(62)
(63) × P sˆ = − Ps P sˆ = − Ps |cB #
(64)
(65) 2 2 # + E |y | #s = Ps , sˆ = Ps .
(66)
(67) × 1 − P s = Ps P sˆ = Ps |cB .
(68)
(69) × P sˆ = Ps P sˆ = Ps |cB #
(70)
(71) 2 2# + E |y | #s = − Ps , sˆ = − Ps .
(72)
(73) × 1 − P s = − Ps P sˆ = − Ps |cB .
(74)
(75) × P sˆ = − Ps P sˆ = − Ps |cB #
(76)
(77) 2 # − 2E |y | #s = Ps , sˆ = Ps .
(78)
(79) × P s = Ps P s = Ps |cB . V. . Ɛϭ. ƐϮ. Ɛϴ. Ɛϵ ƐϮϮ ƐϮϵ. ƐϯϬ
(80). Ɛϱϳ. Ɛϲϰ. Fig. 14: Constellation for 64-QAM modulation. #
(81)
(82) 2 # × E |y | #s = − Ps , sˆ = − Ps .
(83)
(84) × P s = − Ps P s = − Ps |cB .. (50). Then, following the same methodology as the one of the derivation of the mean μH00B , the expression of (50) is simplified to (24). A PPENDIX C DERIVATION OF THE MEAN μH0,k=t,t=k FOR SIGNALS. M-QAM. The derivation of μH0,k=t,t=k under any M-QAM scenario is obtained considering the example of 64-QAM modulation. The constellation of 64-QAM is depicted in Figure 14, where the red symbols represent the inner-constellation symbols, the black symbols represent the outer-constellation symbols, while the blue symbols correspond to the outermost-constellation symbols. • For the inner-constellation symbols, the received signal is decoded correctly according to the following constraints C1 , C2 : √ √ C1 : −√Pt ≤ R {w} ≤ √ Pt , (51) C2 : − Pt ≤ I {w} ≤ Pt . •. For the outer-constellation symbols the received signal is decoded correctly according to the following constraints C1 , C2 : √ √ C1 : − Pt ≤ R {w} ≤ +√ Pt horizontal up, C2 : I {w} ≥ − Pt √ √ C1 : − Pt ≤ R {w} ≤ +√ Pt horizontal down, C2 : I {w} ≤ + Pt √ C1 : √ R {w} ≥ −√ Pt vertical right, C2 : − Pt ≤ I {w} ≤ + Pt √ C1 : √ R {w} ≤ +√ Pt vertical left. C2 : − Pt ≤ I {w} < + Pt (52). •. For the outermost-constellation symbols the received signal is decoded correctly according to the following constraints C1 , C2 : √ C1 : R {w} ≥ −√ Pt right up, C2 : I {w} ≥ − Pt.
(85) 14. √ C1 : R {w} ≥ −√ Pt C2 : I {w} ≤ + Pt √ C1 : R {w} ≤ +√ Pt C2 : I {w} ≥ − Pt √ C1 : R {w} ≤ +√ Pt C2 : I {w} ≤ + Pt. right down, left up, left down.. (53). We can generalize by saying that the intervals of interest for any M-QAM modulation scheme, under the case of correct √ decoding, √ are the following √ for the real part: i) R√{w} ≤ − Pt , ii) − Pt ≤ R {w} ≤ Pt and iii) R {w} ≥ Pt . The same intervals are valid for the imaginary part. Then, the mean value for any M-QAM modulation scheme can be expressed as follows: M 2 2 E |y | |H 0,k=t,t=k = E |y | |s = sk , sˆ = sk. √ √ that the estimated symbol is sˆ = s30 = 3 Ps + j Ps is given as follows:
(86)
(87)
(88) 2 Pt ≤ Pt + R {w} ≤ 4 Pt =>
(89)
(90) Pt ≤ +R {w} ≤ 3 Pt . (57) Following the steps of the proposed algorithm in Section II, it can be shown that R (y ) = R {s29 } + R {w} − R {s30 }
(91)
(92)
(93) = Pt + R {w} − 3 Pt = −2 Pt + R {w} . (58) Because we apply the ED in the signal of (58), we are interested in the √ constraints of this quantity. Therefore, adding the factor −2 Pt in (57), the constraint C3 is proved.. Our goal is to derive the total mean of the symbols which follow the constraint C1 , the total mean of the symbols which × P (ˆ s = sk |cM −Q ) P (s = sk ) , follow the constraint C2 and etc. For example, the constraint (54) C is validated when the transmitted symbol is s and the 3 29 where cM −Q represents the correct decision for M-QAM. decoded symbol is one of them in the green circle of Figure Based on the constraints of (51)-(53) and following the 14, or when the transmitted symbol is s30 and the decoded same methodology as in Appendix A, (54) can be further symbol is one of them in the same column with s31 and etc. determined by (55) at the top of the next page, where the Now, for the case that the transmitted symbol is s29 and the factor 2 expresses the symmetry between the imag- decoded symbol is sˆ30 , we can write that √ √ √ real and √ P (− Pt ≤R{w}≤ Pt ) P (− Pt ≤I{w}≤ Pt ) inary part, Pint = , E R2 y 29,30 = PcM−Q √ √ √ # P (− Pt ≤R{w}≤ Pt )P (I{w}≥ Pt ) #2 # |R {s} = R {s29 } , R {ˆ Pext1 = , and Pext2 = = 1 E R #y29,30 s} = R {s30 } PcM −Q √ √ M P (R{w}≥ Pt )P (I{w}≥ Pt ) × P (R {ˆ s} = R {s30 } |wM −Q ) , (59) . The first item of (55), defined as PcM −Q A, is valid for all the transmitted symbols of (54), the second where w M −Q represents the wrong symbol decision for Mand third item, defined as B and C, respectively, are valid for QAM, while y 29,30 denotes the remaining signal when the all the symbols except the inter-constellation symbols and for transmitted symbol is s and the decoded symbol is sˆ . 29 30 this reason are weighted by the factor √2M , while the fourth However, (59) can be further expressed as follows: item, defined as D, is only valid for the outermost-constellation 4 1 symbols and that’s why is weighted by the factor M . Based E {E} E R2 y 29,30 = on this analysis, the proof of (35) is obtained. M √ √ √ √ P Pt ≤ R {w} ≤ 3 Pt P − Pt ≤ I {w} ≤ Pt × . PwM −Q (60) A PPENDIX D k=1. DERIVATION OF THE MEAN. μH0,k=t,t=k FOR M-QAM. SIGNALS. where# √ #2 # √ √ √ E = #−2 Pt + R {w}# #− Pt ≤ −2 Pt + R {w} ≤ Pt .. √ Furthermore, if the transmitted symbol is s29 and the decoded √Let’s assume that the symbol of interest is s29 = Ps + j Ps . Then, the received signal is wrongly decoded according symbol is s20 , the final result is to the following constraints for the real part of noise: 1 E {E} E R2 y 29,30 = √ √ M C1 : √ −6√ Pt + R {w} ≥ −√ Pt √ √ √ √ P Pt ≤ R {w} ≤ 3 Pt P Pt ≤ I {w} ≤ 3 Pt C2 : −√Pt ≤ −4√Pt + R {w} ≤ +√Pt × . PwM −Q C3 : −√Pt ≤ −2√Pt + R {w} ≤ +√Pt (61) C4 : −√Pt ≤ +2√Pt + R {w} ≤ +√Pt (56) C5 : −√Pt ≤ +4√Pt + R {w} ≤ +√Pt Taking into account all these symbols, the final expression is C6 : − Pt ≤ +6√ Pt + R {w} ≤ +√ Pt given by (s29 and sˆ14 , sˆ22 , sˆ30 , sˆ38 , sˆ46 , sˆ54 , sˆ62 ): C7 : +8 Pt + R {w} ≤ + Pt . 1 E {E} = E R2 y −2,29,C3 First, we show why the constraint C3 takes this shape and then, M√ √ the rest constraints are proved similarly. The constraint C3 is P Pt ≤ R {w} ≤ 3 Pt × , (62) valid when we decide wrongly that the transmitted symbol is Pw M −Q the symbol s30 or any another symbol in the same column denotes the mean when the with s30 (Figure 14). Thus, the condition which guarantees where E R2 y −2,29,C3.
(94) 15. #
(95)
(96) 2 # E |y | |H 0,k=t,t=k = 2E R2 {y } #− Pt ≤ R {w} ≤ Pt Pint )* + ( A. #
(97) #
(98)
(99) 2 # # + 2E R {y } #− Pt ≤ R {w} ≤ Pt Pext1 + 2 √ E R2 {y } #R {w} ≥ Pt Pext2 ( )* + ( M )* + . 2. . B. #
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