Adaptive Spectrum Sensing with Noise Variance Estimation for Dynamic Cognitive Radio Systems Deepak R. Joshi and Dimitrie C. Popescu
Octavia A. Dobre
Department of Electrical and Computer Engineering Old Dominion University Norfolk, Virginia Email: {djosh002,dpopescu}@odu.edu
Faculty of Engineering and Applied Science Memorial University of Newfoundland St. John’s, Canada Email:
[email protected]
CR represents an emerging technology designed to enable dynamic access to the frequency spectrum and reuse of licensed frequencies under the specific conditions that no harmful interference be caused to the incumbent licensed users of the spectrum [1], [2], [3]. An important characteristic of the CR systems is the efficient and reliable sensing of the electromagnetic environment, which includes estimation of the spectrum used by other radio systems that operate at the same time in the same frequency bands [3]. The efficiency is critical since the CR systems will be operating on a wide range of frequencies, whereas reliability is crucial as interference to Primary Users (PU) must be avoided or maintained below a certain threshold level. Various methods have been proposed for spectrum sensing in CR systems [4], [5] among which we note the ones based on cyclostationarity [6], [7], pilot signals and matched filter [7], and polyphase filter banks [8], [9]. These methods have been developed and investigated in static scenarios, where the spectrum usage and background noise statistics do not vary in time. However, in practical systems, spectrum usage changes in time as the number of active transmissions and/or their corresponding parameters change, and dynamic scenarios need to be investigated. Furthermore, most of these methods require knowledge of the noise variance, which is assumed a 978-1-4244-7417-2/10/$26.00 ©2010 IEEE
II. S YSTEM M ODEL AND P ROBLEM S TATEMENT We consider the filter bank spectrum sensing system described schematically in Fig. 1, where the received signal {X(n)} at some time instant n is given by: H \ H 0,0 1,0
y (n) 0
M E0(Z )
X(n)
NE Z−1
X(n−1)
−1 Z
M E1(Z )
y (n) 1
M− Point IDFT W*
NE
EM−1(ZM )
prototype filter (0th band) 1st band 2nd band
0
H \H 0,1 1,1 X 1
H0,M−1\H1,M−1
NE y (n) X(n−M+1) Polyphase Branch Filters M−1 (a) M Band Polyphase DFT Filter Bank H(f)
X 0
Sensed Information
I. I NTRODUCTION
priori available. Nonetheless, noise varies due to temperature changes, ambient interference, etc., and its variance needs to be estimated for accurate spectrum sensing. This motivates the work presented in this paper in which spectrum sensing using DFB is studied in a dynamic scenario where the noise variance is estimated from the received noisy signal. The rest of the paper is organized as follows: Section II introduces the system model and formally states the problem. Sections III and IV discuss the sensing threshold adaptation along with the estimation of the unknown noise variance. Section V formally states the proposed algorithm which is illustrated with numerical results obtained from simulation in Section VI. Final remarks and conclusions are given in Section VII.
Threshold Detection on Group of Subbands
Abstract—Cognitive radios (CR) are regarded as a viable solution to enabling flexible use of the frequency spectrum in future generations of wireless networks. An important aspect of spectrum management in CR systems is adaptation of the spectrum sensing methods employed by CRs in order to accurately detect the changing patterns of spectrum use and to update the spectrum and interference constraints under which CR terminals operate. In this paper we study adaptation of the spectrum sensing threshold in CR using discrete Fourier transform (DFT) filter bank (DFB) method in a dynamic scenario where the sensing threshold is adapted to minimize the spectrum sensing error in the presence of noise. We present an algorithm for spectrum sensing threshold adaptation using DFB with estimated noise variance which we illustrate with numerical examples obtained from simulations. These show the effectiveness of the proposed method in dynamic scenarios with varying noise variance. Index Terms—Cognitive radio, dynamic threshold adaptation, filter bank, noise estimation, spectrum sensing.
(2*PI/M) (4*PI/M)
X M−1
i th band
(2*PI*i/M)
f
(b) Frequency Response of Filter Bank
Fig. 1.
Block diagram of DFB system for spectrum sensing.
X(n) = S(n) + V (n),
(1)
with S(n) being the active radio signal at the location of the CR system, and V (n) the additive white Gaussian noise (AWGN) corrupting the active signal with zero mean and
variance σV2 . The received signal is passed through a polyphase DFB whose output is used for spectrum sensing. The computational complexity is reduced by using the polyphase property of filter banks, which yields a decrease in the sampling rate by a factor of M (the number of subbands) [8]. The basic building block of the DFB is the prototype (or zero-th band) low pass filter. Other band filters are realized by shifting the prototype filter in frequency. The prototype filter is designed to minimize the spectrum leakage and its length is chosen as the smallest multiple of M that can satisfy some desired stopband attenuation requirements. The frequency response of the filter bank is written as [10], ∞
H(z) =
h(n)z −n .
(2)
From equations (8) and (9) Hk (z) = H0 (zW k ) M −1 (z −1 W −k )l El (z M ) = and thus, Yk (z) = Hk (z)X(z) M −1 W −kl (z −l El (z M )S(z)) = +
l=0 M −1
This can be separated into even and odd numbered coefficients as, = Eo (z 2 ) + z −1 Ee (z 2 ) ∞ h(2n)z −2n = n=−∞
∞
+ z −1
h(2n + 1)z −2n .
(3)
n=−∞
Applying similar approach to represent H(z) into M th polyphase filter bank, H(z) =
∞
h(nM )z −nM + z −1
n=−∞ ∞
∞
h(nM + 1)z −nM +
h(nM + M − 1)z −nM ,
(4)
n=−∞
where M is any integer. Thus, the filter bank system can be represented using polyphase components as H(z) =
M −1
z −l El (z M ),
(5)
l=0
where El (z) =
∞
el (n)z −n
(6)
and el (n) is defined as el (n) = h(nM + l), 0 ≤ l ≤ M − 1.
(7)
The polyphase can be efficiently implemented using DFT filter bank as [10], (8) Hk (z) = H0 (zW k ), where W = exp(−j2π/M ) and H0 (z) corresponds to the prototype filter and is expressed as 1 + z −1 + z −2 + · · · + z −(M −1) M −1 = z −l El (Z M ). =
l=0
where S(z) and V (z) are the Z-transforms of the signal S(n) and AWGN V (n) respectively. When M is power of two, i.e. M = 2m with m a positive integer, the DFT can be calculated efficiently using the Fast Fourier Transform (FFT). Because the output bandwidth of the DFT filter bank is approximately M times narrower than that of X(n), usually decimated uniform DFT banks are used. The output of the k-th subband of the DFB can be expressed as the time average Yk (n) =
n 1 | yk (i) |2 . M
(12)
This provides information on the active signal spectrum in the k-th subband and may be used for detecting PU spectrum. The computational complexity of the system is equivalent to that of the realization of the prototype filter and one IDFT since the M bands of filter bank share the same structure. Multi-channel detection for DFB spectrum sensing can be done by extending the energy detection results obtained for a single channel. A given spectrum band is considered to be vacant if there is only noise and the subband is considered to be occupied by the PU if there is PU signal and noise present. Thus, the following binary hypothesis testing is performed at any given time instant n to find the occupancy of the k-th subband: H0,k : Xk (n) = vk (n)
n=−∞
H(z)
(11)
i=n−M
n=−∞
· · · + z −(M −1)
W −kl (z −l El (z M )V (z)),
l=0
n=−∞
H(z)
(10)
l=0
H1,k : Xk (n) = yk (n) + vk (n)
k = 0, 1, . . . , M − 1,
(13)
where the hypotheses H0,k and H1,k respectively indicate the absence and presence of the primary user signal in the k-th subband, and vk (n) is a Gaussian random process with zero (k)2 mean and variance σv . Under the assumption of absence of coherent detection, signal samples may also be modeled as a (k)2 Gaussian random process with variance σy . The decision rule corresponding to the k-th subband is given by H1,k
(9)
Yk (n) γk (n), k = 0, 1, . . . , M − 1, H0,k
(14)
where γk (n) is the threshold. By using (14), the probability of detecting a PU signal in subband k is [7] (k)
Pd (γk (n)) = Pr [Yk (n) > γk (n)|H1,k ] ⎡ =
(k)2 M (σv
⎤
(k)2 σy ) ⎦
γk (n) − 1 + erfc ⎣ 2 (k)2 (k)2 2 M (σv + σy )2
,
(k)
Pmd (γk (n)) = 1 − Pd (γk (n)).
(16)
The probability of false alarm for subband k is [7] (k)
Pf a (γk (n))
= Pr[Yk (n) > γk (n)|H0,k ] =
(k)2 γk (n) − M σv 1 erfc , (k)2 √ 2 2σv M
III. S ENSING T HRESHOLD A DAPTATION Since both probability of missed detection and probability of false alarm are important for accurate spectrum sensing, the optimum value of the threshold level for the test statistic in a given subband k is chosen to minimize the sensing error defined as E (k) (γk (n)) =
+ (1 −
(k) δ)Pmd (γk (n)),
(18)
where 0 < δ < 1 is a given constant weighting the probability of false alarm relative to that of missed detection. The optimum threshold for subband k is found by solving the constrained optimization problem min E (k) (γk (n)) subject to
(k)
γk (n)
(k)
Pf a (γk (n)) ≤ α, and Pmd (γk (n)) ≤ β,
(19)
where α and β are the threshold values corresponding to false alarm and missed detection. The constrained optimization
(20)
−β ,
where λ1 and λ2 are the Lagrangian multipliers. In order to find the necessary conditions for the constrained optimization problem (19) we take the partial derivatives of the Lagrange function (20) with respect to γ, λ1 , and λ2 , respectively, and after some algebraic manipulation we obtain (k)2 γk (n)−M σ2 2 exp −( ) (k)2 √ 2σ2 M λ1 − δ = 0, − (21) (k)2 λ2 + δ − 1 γ (n)−M σ exp −( k (k)2 √ 1 )2 2σ1
(17)
where erfc(·) is the error complementary function [11]. The performance of the spectrum sensing at any time instant n depends mainly upon the values of these probabilities, which are functions of the thresholds γk (n). We note that a large probability of missed detection implies a higher chance that the CR will not detect the presence of a PU transmission in subband k, whereas a low probability of false alarm implies better chances for the CR system to occupy subband k. Our goal in this paper is to study adaptation of the sensing threshold γk (n) in dynamic scenarios, where the variances of (k)2 (k)2 the active signal σy and/or noise σv change in time. For accurate sensing we do not assume that the noise variance is known and we estimate its value from the noisy received signal X(n) by using an autoregressive (AR) model for the received signal.
(k) δPf a (γk (n))
(k) −λ2 Pmd (γk (n))
(15)
and, thus, the probability of missed detection of a PU signal in subband k is (k)
problem (19) can be solved using the Lagrange multipliers method. For this we form the Lagrangian function
(k) L(γ, λ1 , λ2 ) = E (k) (γk (n)) − λ1 Pf a (γk (n)) − α
M
(k)2 γk (n) − M σ1 1 erfc − α = 0, (k)2 √ 2 2σ1 M
(22)
(k)2 γk (n) − M σ2 1 1 − β − erfc = 0, (k)2 √ 2 σ2 M
(23)
(k)2
(k)2
(k)2
(k)2
(k)2
where σ1 = σv and σ2 = σv + σy . The optimal value of γ, λ1 , and λ2 is found solving (21), (22) and (23) simultaneously. Since these equations are transcendental and have no closed-form solutions they may be solved numerically through iterative methods, such as the Newton’s method. In dynamic scenarios the optimal value of the threshold γk can be adapted by using a gradient-based update γk (n + 1) = γk (n) − μk ∇E (k) (n),
(24)
where E (k) (n) is given by equation (18), μk is a suitably chosen step size, and the gradient ∇E (k) (n) is calculated as [11] ⎡ 2 ⎤ (k)2 γ −δ (n) − M σ k 1 ⎦ √ exp ⎣− ∇E (k) (n) = (k)2 √ 2σ12 πM 2σ1 M ⎡ 2 ⎤ (k)2 (1 − δ) γk (n) − M σ2 ⎦ + (1 − δ) + (k)2 √ exp ⎣− (k)2 √ 2σ2 πM 2σ2 M (25)
IV. N OISE VARIANCE E STIMATION The noise variance is estimated from the received contaminated noisy signal as discussed in [12]. We assume that the uncontaminated PU signal yk (n) follows a p-th order AR model with transfer function H (k) (z) =
1+
p
1 (k)
−j j=1 aj z
, k = 1, . . . , M − 1. (26)
(k)
The coefficients aj p
satisfy the set of Yule-Walker equations
(k)
(k) ai R(k) y (|j − i|) = −Ry (j), j > 0, k = 1, . . . , M − 1,
i=1
(27) (k) where Ry (j) are the autocorrelation coefficients of the uncontaminated signal yk (n). These are related with the (k) autocorrelation coefficients RX (j) of the noisy signal Xk as [13] (k)
(k)
(k)2
Ry (0) = RX (0) − σv , k = 0, . . . , M − 1, (k) (k) Ry (j) = RX (j), j > 0,
(28)
(k)2
with the estimate of noise variance, σ ˆv , given by
p (k) ˆ (k) (k) ˆ (k) p a { R (j) + a (|j − i|)} R X X j=1 j i=1 i 2 , σ ˆv(k) = p (k)2 j=1 aj (29) ˆ (k) (j) are the estimates of the autocorrelation coeffiwhere R X cients of the noisy signal Xk (n).
VI. S IMULATIONS AND N UMERICAL R ESULTS In order to illustrate the proposed algorithm we performed simulations for a DFB spectrum sensing system with M = 32 polyphase branch filters in a dynamic scenario with noise variance that changes in time. The simulation parameters are: the threshold/probability constraints α = 0.1 and β = 0.2, the gradient constant μ = 0.5, the AR model order parameters p = 2 and q = 78, and the algorithm tolerance = 10−3 . For the sensing error weight parameter δ, several values in the interval [0.1, 0.9] were tested during the simulations. In a first simulation experiment we have studied the dependence of the spectrum sensing error on the optimal threshold value for SNR values to −3, 0, and 3 dB, with actual and estimated noise variance. Results of this experiment are shown in Figures 2 and 3 which are typical for all the simulations we ran. From these figures we note that the spectrum sensing error is dominated by the probability of false alarm at low threshold values, and by the probability of missed detection 0.5 With actual noise variance 0.45
With estimated noise variance
0.4
V. A LGORITHM FOR DYNAMIC T HRESHOLD A DAPTATION WITH N OISE VARIANCE E STIMATION
0.35 SNR −3 dB
•
Input parameters: – Number of subbands M , step size μ and tolerance , random initialization of threshold value in all subbands (γk (0)).
•
For each subband – calculate the average energy using equation (12). – calculate unbiased estimate of the autocorrelation ˆ X (j)} from observed noisy signal, coefficients {R – Compute the AR parameters using a least squares procedure [13], – calculate the noise variance using equation (29), – While (γk (n + 1) − γk (n))> , ∗ update ∇E (k) (n) using (25), ∗ update γk (n) using (24), ∗ check the constraints conditions.
We note that, in order to estimate the noise variance, the received signal is modeled using an AR model whose parameters are computed from an overdetermined set of q > p high order Yule-Walker equations using a least squares procedure [13]. Using the estimated noise variance, the gradient-based update is then applied to adapt the sensing threshold incrementally in the direction of the optimal value which minimized the sensing error (18).
Error
0.25 SNR 0 dB 0.2 0.15 0.1 0.05 SNR 3 dB 0
0
0.05
0.1
0.15 Threshold γ
0.2
0.25
0.3
Fig. 2. Spectrum sensing error for sensing threshold values less than 0.3 and sensing error weight δ = 0.5. 0.7
With actual noise variance 0.6
With estimated noise variance
0.5
Error E(γ)
The proposed algorithm for sensing threshold adaptation combines the gradient based updates of the sensing threshold presented in Section III with the estimation of noise variance discussed in Section IV. The algorithm is formally stated as follows:
E(γ)
0.3
SNR 3 dB
0.4
SNR 0 dB 0.3
0.2 SNR − 3 dB 0.1
0
0.3
0.4
0.5
0.6 0.7 Threshold γ
0.8
0.9
1
Fig. 3. Spectrum sensing error for sensing threshold values larger than 0.3 and sensing error weight δ = 0.45.
VII. C ONCLUSION
0.8 With actual noise variance With estimated noise variance
0.7 0.6 Threshold
Scenario 3, SNR = − 3 dB 0.5
Scenario 2, SNR =0 dB
0.4 0.3 0.2 Scenario 1, SNR =3 dB 0.1
0
200
400
Fig. 4.
600
800 1000 1200 Number of iterations
1400
1600
1800
Dynamic Threshold Adaptation.
at high threshold values. This behavior is explained by the fact that the probability of false alarm is high at low threshold values while the probability of missed detection is high at high threshold values. The spectrum sensing error achieves its minimum value for threshold values in the interval [0.1, 0.3] with lower optimal threshold values (around 0.1) for higher SNR and higher optimal threshold values (around 0.3) for lower SNR. From the first simulation experiment we have also observed that using the actual noise variance results in lower spectrum sensing error than using the estimated noise variance when the SNR is above 0 dB. When the SNR is below 0 dB we have observed cases where using the estimated noise variance results in lower sensing error than using the actual noise variance. This can be explained by the fact that, at low SNR values, when the estimate of the noise variance is lower than the actual noise variance, it will imply an artificial increase of the SNR when the estimated noise variance is used to evaluate the sensing error instead of the actual noise variance. In a second simulation experiment we have studied the threshold adaptation performed by the proposed algorithm when the noise variance changes in time. Results of this experiment are illustrated in Fig. 4 in which we start with the SNR = 3 dB (scenario 1), for which the algorithm adjusts the threshold to the optimal value. Once this is reached, the noise variance changes such that the SNR = 0 dB (scenario 2). After the threshold is adapted to the new optimal value corresponding to scenario 2, the noise variance changes again such that in scenario 3 we have the SNR = −3 dB. Once again we notice that the algorithm adapts the sensing threshold to its new optimal value for the new scenario. For all the figures we note that there is a difference between actual and estimated noise variance cases. This difference is smaller for higher SNR values, which imply better variance estimates, and may be minimized by increasing the estimation interval. However, increasing the duration of the estimation interval will also increase the overall sensing time.
A novel algorithm for adaptation of the spectrum sensing threshold in dynamic CR systems is presented in the paper. Spectrum sensing uses a DFB approach and the proposed algorithm adapts the threshold to minimize the spectrum sensing error using gradient-based updates. For accurate adaptation in dynamic scenarios, an estimate of the noise variance is used in the threshold calculation. The proposed algorithm is illustrated with numerical examples obtained from simulations, which confirm its effectiveness in optimizing the sensing threshold as well as in adapting it in dynamic scenarios. The proposed algorithm uses the energy detection method as a basic building block, and may not be effective at very low SNR values. We note that the performance of energy detection can be enhanced by taking more samples up to a certain SNR value, after which further increases in the number of samples will not improve its performance [14], and finding this SNR value in the context of the proposed application will be the object of future research. R EFERENCES [1] W. Krenik, A. M. Wyglinsky, and L. Doyle, “Cognitive Radios for Dynamic Spectrum Access,” IEEE Communications Magazine, vol. 45, no. 5, pp. 64–65, May 2007. [2] S. Haykin, “Cognitive Radio: Brain-Empowered Wireless Communications,” IEEE Journal on Selected Areas in Communications, vol. 23, no. 2, pp. 201–220, February 2005. [3] I. F. Akyildiz, W.-Y. Lee, M. C. Vuran, and S. Mohanty, “NeXt Generation/Dynamic Spectrum Access/Cognitive Radio Wireless Networks: A Survey,” Computer Networks, vol. 50, no. 13, pp. 13–18, September 2006. [4] T. Yucek and H. Arslan, “A Survey of Spectrum Sensing Algoriths for Cognitive Radio Applications,” in IEEE Communications Surveys and Tutorials, vol. 11, no. 1, 2009. [5] S. Haykin, D. J. Thomson, and J. H. Reed, “Spectrum Sensing for Cognitive Radio,” Proceedings of the IEEE, vol. 97, no. 5, pp. 849– 877, May 2009. [6] O. A. Dobre, S. Rajan, and R. Inkol, “Joint Signal Detection and Classification based on First-Order Cyclostationarity for Cognitive Radios,” EURASIP Journal on Advances in Signal Processing, 2009. [7] D. Cabric, “Cognitive Radios: System Design Perspective,” Ph.D. dissertation, University of California, Berkeley, 2007. [8] F. J. Harris, C. Dick, and M. Rice, “Digital Receivers and Transmitters Using Polyphase Filter Banks For Wireless Communications,” IEEE Transaction On Microwave Theory And Techniques, vol. 51, no. 4, pp. 1395–1412, April 2003. [9] B. Farhang-Boroujeny and R. Kempter, “Multicarrier Communication Techniques for Spectrum Sensing and Communication in Cognitive Radios,” IEEE Communication Magazine, vol. 46, no. 12, pp. 80–85, April 2008. [10] P. P. Vaidyanathan, Multirate Systems and Filter Banks. Prentice Hall, 1993. [11] M. Abramowitz and I. A. Stegun, Eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publication, Inc., 180 Varick Street, New York, N.W. 10014, 1972. [12] K. K. Paliwal, “Estimation of Noise Variance from the Noisy AR Signal and Its Application in Speech Enhancement,” in Proceedings Acoustics, Speech, and Signal Processing, IEEE International Conference on ICASSP ’87., vol. 12, 1987, pp. 297–300. [13] J. A. Cadzow, “Spectral Estimation: An Overdetermined Rational Model Equation Approach,” in Proceedings of the IEEE, vol. 70, no. 9, September 1982, pp. 907–939. [14] R. Tandra and A. Sahai, “Fundamental limits on detection in low SNR under noise uncertainty ,” in Proceedings of the International Conference on Wireless Networks, Communications, and Mobile Computing, vol. 1, June 2005, pp. 464–469.
