Spectrum sensing based on cumulative power spectral density

b _m are the symbols to be transmitted; g(n) is the shaping window; N _s satisfies the Nyquist criterion; \(N_{s}=\frac {F_{s}}{B}\geq 2\) samples per symbol (sps), where F _s is the sampling frequency and s _p (n) and s _q (n) are respectively the real and imaginary parts of s(n). s(n) has a bandwidth B and an even power spectral density (PSD). s(n) is assumed to be complex-valued zero mean unknown deterministic² signal [11, 12, 17–19].

where h is the complex channel gain. w(n) is \(\mathcal {N}(0,\sigma _{w}^{2})\), where \(\mathcal {N}(m,V)\) stands for a normal distribution of a mean m and a variance V. Further, w(n)=w _p (n)+jw _q (n) is an i.i.d complex circular symmetric, i.e.,

A wrong decision about the channel status can affect either the PU transmission or the efficient use of the channel. In fact, a missed detection can cause a harmful interference by the transmission of the SU in the same band of the PU. A false alarm, however, decreases the profit of the opportunity of the channel. Therefore, the probability of detection (p _d ) should be increased as much as possible, by keeping the probability of false alarm (p _fa ) low. Neyman-Person’s detection method consists in a trade-off between a high p _d and a low p _fa .

where \(\hat {P}_{y}(u)\) is the estimated PSD of y(n) using Eq. (11). Unlike ED, which makes the sum of energy on all frequency points of PSD, CPSD makes the sum of the energy on a given frequency points interval. The sense of variation of CPSD will be tested in order to make a decision on the PU signal existence as detailed below.

The last value shows that ψ _w (k) increases linearly on the frequency interval [v,l] with k till its highest value \((l-v+1)\sigma _{w}^{2}\).

Under H ₁, this constraint is not satisfied, since the PSD of s(n) is not a constant, and Γ(k) has higher values than the one obtained under H ₀ due to the additional power of s(n).

Distributions of test statistics are essential in order to find analytically the probability of false alarm p _fa and the detection probability p _d . T _p and T _a have the same statistical distribution since W(k) is i.i.d. and S(k) is deterministic as s(n) is assumed to be deterministic. In the following, we develop the distribution of T _p under H ₀ and H ₁ over a Gaussian channel, where the channel effect h is assumed to be constant. Similarly, the distribution of T _a can be found. T _or made its decision by applying the logical OR on the decisions of T _p and T _n which have the same distribution.

4.3 Probabilities of T _av

T _av can be developed following similar steps as Eq. (30).

$$\begin{array}{*{20}l} T_{av}&=\frac{2}{N^{2}\sigma_{w}^{2}}\sum\limits_{k=1}^{\frac{N}{2}}\left(\frac{N}{2}-k+1\right)\frac{|Y(k)|^{2}+|Y(-k+1)|^{2}}{2}\\ &\quad-\frac{N+2}{4} \end{array} $$

(43)

Under H ₀, Y(k)=W(k), then T _av becomes the sum of independent terms. Based on CLT, T _av asymptotically follows \(\mathcal {N}(\mu _{0}^{av},V_{0}^{av})\) under H ₀.

Under H ₁, Y(k)=hS(k)+W(k) and S(k) is deterministic, so T _av is still following a normal distribution: \(\mathcal {N}(\mu _{1}^{av},V_{1}^{av})\) under H ₁.

The probability of false alarm, \(p_{fa}^{av}\), and detection, \(p_{d}^{av}\), of T _av are expressed as follows:

$$\begin{array}{*{20}l} p_{fa}^{av}=Q\left(\frac{\lambda-\mu_{0}^{av}}{\sqrt{V_{0}^{av}}}\right) \end{array} $$

(44)

$$\begin{array}{*{20}l} p_{d}^{av}=Q\left(\frac{\lambda-\mu_{1}^{av}}{\sqrt{V_{1}^{av}}}\right) \end{array} $$

(45)

Since W(k) is i.i.d. and P _s (k) is even, \(\mu _{0}^{av},V_{0}^{av}\), \(\mu _{1}^{av}\), \(V_{1}^{av}\) can be found by following similar steps to μ ₀, V ₀, μ ₁ and V ₁, we can find that \(\mu _{0}^{av}=\mu _{0}\), \(V_{0}^{av}=\frac {V_{0}}{2}\), \(\mu _{1}^{av}=\mu _{1}\), and \(V_{1}^{av}=\frac {V_{1}}{2}\).

The theoretical and the simulated ROC curves of proposed detectors are with good agreement as shown in Figs. 3 a, b. Simulations were done under following conditions: 16-QAM modulation, γ=−12 dB, N=1000 samples, and N _s =4 sps.

As shown in Fig. 3 a, T _av is the most efficient detector for both considered shaping filters. For the simulations of Section 6 under Gaussian channel, only T _av and T _or are compared to other well-known detectors. The rest of simulations in this paper are done with a rectangular pulse-shaping filter.

where \(\bar {\gamma }\) is the average SNR.

Concerning our detectors, T _p , T _a , and T _av , their detection probabilities over Rayleigh fading channel can be derived similarly, since they have a similar probability of detection: \(p=Q(\frac {\lambda -\delta \gamma }{V_{H_{0}} + \beta \gamma })\), where δ, \(V_{H_{0}}\), and β are constants.

Hereinafter, we only derive \(p_{dr}^{p}\), the detection probability of T _p over a Rayleigh channel. Once \(p_{dr}^{p}\) is derived, \(p_{dr}^{a}\) and \(p_{dr}^{av}\), the probability of detection of T _a and T _av , can be easily found.

where g ₁(γ) is the first-order Taylor series approximation of g(γ).

where \(\theta _{av}=-\frac {b}{\sqrt {V_{0av}+\frac {c\lambda }{2b}}}\).

γ ₀ is not modified in the expression of \(\hat {p}_{dr}^{av}\), since T _p and T _av have the same mean under H ₁.

As there is no analytic solution of Eq. (54), we solve it in numerical way.

In Fig. 4, analytical results are very closed to simulation results. T _av and T _or both achieve the best performance among the proposed detectors, thus they will be compared to the well-known ED and CSD under Rayleigh channel.

In this section, we compare the performance of our detectors to that of the energy detector (ED) [8] and the cyclostationary detector (CSD) [10], under Gaussian and Rayleigh channels. Throughout the upcoming simulations, A 16-QAM baseband-modulated PU signal with a rectangular pulse-shaping filter is considered.

6.1 Performance analysis over Gaussian channel

Figure 5 presents the ROC curves under a Gaussian channel of various numbers of samples per symbol N _s . Simulations are done using N=1500 samples and γ=−12 dB. According to that figure, the performance increases with N _s . For the limiting case, i.e., when N _s =2 sps, CSD shows a poor performance relatively to T _av , T _or , and ED, while T _av outperforms all other detectors. However, T _av and T _or outperform ED and CSD for different values of N _s .

Figure 6 shows the variation of the probability of detection with respect to SNR for a constant p _fa =0.1. The number of samples is fixed N=1000 samples, and various values are assigned for N _s . T _av and T _or reach higher probabilities of detection than ED and CSD for similar SNR and different values of N _s . In addition, increasing N _s leads to enhancing the performance of T _or , T _av , and CSD. For example, T _av reaches p _d =0.9 at SNR ≃−7 dB for N _s =2 sps, while the same probability of detection is reached for SNR ≃−9 dB and SNR ≃−10 dB at N _s =4 sps and N _s =8 sps respectively.

6.2 Performance analysis over Rayleigh channel

Figure 7 shows the ROC curves under Raleigh fading channel for N _s =4 and N _s =8 sps, with N=1000 samples and an average SNR of −5 dB.

Over Rayleigh fading channel, T _av and T _or are still outperforming ED and CSD. The same fading suffered by the negative and the positive frequency parts of PSD affects the performance of our detectors. This fact makes the performances of ED and CSD closed to the proposed detectors, as shown in Fig. 7, where the gap of performance among our detectors, ED and CSD becomes smaller comparing it to a Gaussian channel.

6.3 Complexity analysis

According to Eq. (29), T _p needs \(\frac {N}{2}\) operations to obtain |Y(k)|² and \(\frac {N}{2}\) multiplication operations to evaluate the product \(\left (\frac {N}{2}-k+1\right)|Y(k)|^{2}\). Moreover, T _p performs N addition operations: \(\frac {N}{2}\) operations to calculate \(\left (\frac {N}{2}-k+1\right)\), \(\frac {N}{2}-1\) operations to compute the overall sum and, at the end, one addition operation is required to subtract \(\frac {N+2}{4}\). Furthermore, to compute the Y(k), we need Nlog ₂(N) operations using the fast Fourier transform (FFT) algorithm. The overall number of operations needed by T _p is \(\mathcal {C}(T_{p})\):

$$ \mathcal{C}(T_{p})=N\left(2+\text{log}_{2}(N)\right) $$

(55)

To obtain the decision of T _or , we have to previously find T _p and T _n . Since those two test statistics have the same complexity, therefore, the complexity of T _or becomes the double of that of T _p except for the calculus of the FFT which should be evaluated once. Then the required number of operations of T _or , \(\mathcal {C}(T_{or})\), becomes

$$ \mathcal{C}(T_{or})=N\left(4+{log}_{2}(N)\right) $$

(56)

For T _av , we can find the corresponding number of operations, \(\mathcal {C}(T_{or})\), similarly to T _or :

$$ \mathcal{C}(T_{av})=N\left(3+{log}_{2}(N)\right) $$

(57)

To compare the complexity of our detectors to that of ED and CSD, we should emphasize that ED needs N operations of multiplication and N−1 for addition:

$$ \mathcal{C}(ED)=2N-1 $$

(58)

For CSD, the corresponding number of operations, \(\mathcal {C}(CSD)\), can be derived as follows [25]:

$$\begin{array}{*{20}l} \mathcal{C}(CSD)&=(N_{s}-1)N(L+1)+4(N_{s}-1)L^{2} \\ &\ \ \ \ +8(N_{s}-1)^{3}+6(N_{s}-1)^{2}+2(N_{s}-1) \end{array} $$

(59)

where L is an odd number and stands for the length for the unit window used in the detection process of CSD [10].

According to Eqs. (56) and (57), the complexity T _or and T _av is independent of the number of samples per symbol N _s , contrary to CSD where the complexity depends on N _s as shown in Eq. (59). As shown previously, increasing the oversampling rate leads to obtain a robust performance for CSD, T _or and T _av , while the performance of ED is not affected by the oversampling rate.

Figure 8 shows the number of samples and the complexity of the various detectors in terms SNR for a target (p _fa ;p _d )=(0.1;0.9) under a Gaussian channel. As shown in Fig. 8 a, T _or and T _av need a number of samples less than that of ED and CSD, in order to reach the target probabilities. In addition, the required number of samples decreases with an increasing N _s . Figure 8 b shows the number of performed operations corresponding to the number of required samples given in Fig. 8 a. The complexity of T _or and T _av decreases if N _s increases due to the fact that the number of required samples decreases with an increasing N _s . Contrariwise, the complexity of CSD grows with N _s even if the total number of samples decreases, this is because the complexity of CSD depends on N _s . On the other side, our proposed detectors are slightly more complicated than ED.

To summarize, ED is less complicated than the proposed detectors. However, T _av and T _or require shorter observation time to fulfill the equivalent performance of ED and CSD.

Due to many factors (such as thermal noise, ambient interference, receiver non-linearity, etc.), the variance of the white noise cannot be estimated accurately. This fact introduces the noise uncertainty (NU) problem which prevents the detector from reaching a target (p _fa ,p _d ) even with a large observation time. According to Eq. (15), the noise variance should be pre-estimated in order to perform the normalization. That means our proposed detectors are sensitive to the estimation of the noise variance. In this section, the impact of the NU on the robustness of our proposed detectors is evaluated.

where \(s=10\log _{10}(\sigma _{w}^{2})\), \(\hat {s}=10\log _{10} (\hat {\sigma }_{w}^{2})\), and ρ=10 log10(r).

In a conventional detection mechanism, a suitable threshold λ is fixed according to a target p _fa . In our proposed detectors, the choice of λ depends on setting the estimated value of the noise variance and the observation time (i.e. the number of the received samples). A wrong estimation of \(\sigma _{w}^{2}\) may lead to an inappropriate λ. This fact increases the false alarm rate and deteriorates the spectrum sensing performance.

In Fig. 9, the variation of p _fa with ρ is illustrated for the proposed detectors and the conventional ED. All considered detectors are assumed to have p _fa =0.1 when no NU exists (i.e. ρ=0), then the threshold of each detector is chosen according to p _fa =0 with a perfect estimation of the noise variance. Figure 9 shows that the p _fa of ED ia higher than the ones of the proposed detectors. Indeed, our detectors exploit both the energy and the CPSD shape of the received signal unlike ED which exploits only the energy. On the other hand, T _av and T _a have the same p _fa as these two detectors are based on linear combination of |W(k)|² which is i.i.d.. T _or has a superior p _fa than T _p since these two detectors are related to each other by a non-linear Eq. (41).

To show the effect of the noise uncertainty on the ROC curves, we evaluate numerically the performance loss Δ p _d =p _d (0)−p _d (ρ) where p _d (0) stands for the case where there is no NU, and p _d (ρ) stands for probability of detection for the case where NU is equal to ρ dB. Figure (10) presents Δ p _d of our detectors and ED for p _fa =0.1, N=1000 samples, SNR of -10 dB, and N _s =3 sps. As shown in this figure, our detectors are less sensitive to the noise uncertainty than the energy detector for p _fa =0.1. We observe that T _a is most affected among the proposed detectors. T _av and T _or have the same detection loss since they both are related to both T _p and T _n , which exhibit the same NU.

For \([v;l]=\left [-\frac {N}{2}+1;\frac {N}{2}\right ]\), \(\psi _{y}\left (\frac {N}{2}\right)=\frac {1}{N}{\sum \nolimits }_{k=-\frac {N}{2}+1}^{\frac {N}{2}}|Y(k)|^{2}\) becomes the estimated energy of the received signal.

Such a normalization makes η _y (k) not vulnerable to the NU as its calculus does not depend on the estimation of the noise variance (as presented in Eq. (62)), and its distribution and its statistical parameters (mean and variance) are also independent of the noise variance (see Appendix 7).

This normalization is equivalent to a scaling and does not change the shape form of the CPSD.

Figure 11 shows η _y (k) under H ₀ and H ₁ and a SNR =0 dB for various values of N _s . We have under H ₀ a shape like a straight line, but a curved shape under H ₁. As shown in this figure, the difference between η _y (k) and the straight line increases as N _s increases, which means the detection becomes more reliable with the increasing of N _s .

Figure 12 shows the performance of proposed detectors under a NU of 0, 0.75, and 1.5 dB with N _s =4 sps, SNR of -10 dB, and N=1000 samples under Gaussian and Rayleigh flat-fading channels. For a NU =0 dB, T _av and T _or outperform \(T_{av}^{s}\) and \(T_{or}^{s}\) respectively for both Gaussian and Rayleigh channels. Beside that, when the NU grows, the performance of \(T_{av}^{s}\) and \(T_{or}^{s}\) is not affected, whereas the detectors T _av , T _or , and ED suffer a performance degradation and become less robust than the self-normalization detectors, as shown in Figs. 12 b, c and 13 b, c. However, T _av and T _or have a superior performance relative to ED for different values of ρ.

¹ This work was presented in part as a book chapter in Springer book [27]

² According to [18] and [19], almost-periodic signals can be considered as deterministic in a fraction of time (i.e., for our application, this is the time of sensing). As the linear modulated signal s(n) is almost cyclostationary signal according to [19], then it is almost-periodic [18, 19] and consequently it can be considered as deterministic during a period of spectrum sensing. On the other hand, in [28] the author motivates the use of the deterministic assumption of the cyclostationary signals in a fraction of time while the purely stationary noise signals (do not have any cyclic features) are considered random.

³ for simplicity, we refer to S _N (k) by only S(k) in the forthcoming sections

⁴ \(Q(x)=\frac {1}{\sqrt {2\pi }}\int _{x}^{+\infty }{e^{\frac {-t^{2}}{2}}dt}\) [22]

Since W(k) is Gaussian and i.i.d. then:

Since W _p (k) and W _q (k) are independent and \( E\left [W_{p}(k_{1})W^{2}_{p}(k_{2})\right ]=E\left [W^{2}_{q}(k_{1})W_{q}(k_{2})\right ]=0 \forall k_{1} \) and k ₂, since W _q (k) and W _p (k) are Gaussian, then Eq. (78) becomes zeros.

where f ⁽ⁿ⁾(t ₀) is the nth order derivative of f(t) at t ₀.

Figure 14 a, b presents a comparison between \(p_{d}^{p}=Q(g(\gamma))\) and its approximations Q(g ₁(γ)) and Q(g ₂(γ)) under different conditions. Figure 14 a shows the variation of \(p_{d}^{p}\) and its approximations in terms of SNR for different p _fa . The number of samples is fixed to 1500 and N _s =4 sps, while Fig. 14 b shows under SNR of 10 dB and N _s =4 sps, the variation of \(p_{d}^{p}\) and its approximations in terms of N for different p _fa . As shown, the analytic and the approximated curves are closed to each others under the various conditions. As expected, g ₂(γ) leads to a more robust approximation, since Q(g ₂(γ)) is almost colinear with Q(g(γ)). Even though, g ₁(γ) results are very closed to exact ones. As no important loss is obtained when g ₁(γ) is used, and since it is more simple to deal with the first-order Taylor series, g ₁(γ) will be considered to approximate the detection probability under Rayleigh fading channel.