1. Spectrum Sensing In Cognitive Radio Networks

2. Spectrum Scarcity
Rapid development in wireless communication applications has increased the demand on available wireless spectrum
Traditionally the available spectrum is statically allocated by frequency regulation bodies
However, spectrum occupancy measurements shows the underutilization of some licensed bands
However, spectrum occupancy measurements showed that some licensed bands are significantly underutilized. For example,
This indicates that limitations in spectrum availability is due to inefficient static allocation techniques, rather than physical spectrum scarcity.

3. Cognitive Radio Networks
Accordingly, dynamic spectrum allocation paradigms emerged, called cognitive radio (CR) networks
Cognitive Radio (CR): capable of sensing operating environment and dynamically utilize available radio resources

4. Cognitive Radio Networks
Two types of users:
Primary users :users of high priority or legacy rights to use the spectrum.
Secondary users :users with lower priority, who are not assigned a specific frequency band to operate at.
Secondary users can only use the spectrum such that they do not cause interference to primary users

5. Cognitive Radio Networks
Three main functions to facilitate CR operation:
Spectrum sensing
Spectrum management and hand off
Dynamic spectrum allocation and sharing
Spectrum sensing is a fundamental in CR operation. Its performance decides the level of interference with PUs and spectrum utilization efficiency

6. The performance of a spectrum sensing techniques can be evaluated using two metrics:
Detection probability: probability that a CR correctly decides that the spectrum is busy, when primary transmission is taking place
False alarm probability: probability that the CR makes a wrong decision that the spectrum is occupied while it is actually not
Note: For the spectrum sensing algorithm to be efficient, it should satisfy both high spectral utilization as well as minimal interference with primary users. In other words, the sensing techniques needs to achieve high detection probability and a low false alarm probability, a requirement that seems to be contradicting based on the above graph.

7. Spectrum Sensing Techniques
Blind no prior information is required about the primary signal to be detected
Energy based spectrum sensing
Spectrum sensing based on statistical covariance
Eigenvalue based sensing
Knowledge Based information on primary signal needs to be available a priori at the cognitive user end
Cyclostationary feature detection
Coherent detection/matched filtering

8. 1.Energy Based Sensing
Most common sensing technique with least computational complexity
Secondary user decides whether or not there is primary signal transmitted based on the energy of the received signal x(t). Hence the detection statistics can be defined as follows:
Where x(n) is the received signal and Ns is the number of samples over which energy is computed

9. 1.Energy Based Sensing Pd = Pr(T > λ | H1)= Pf = Pr(T > λ| H0)=
Let the received signal has the following hypothesis
H0 represents the null hypothesis meaning that there is no primary signal and only AWGN noise exists , H1 describes the existence of a primary user’s signal in addition to AWGN noise
The detection statistics T is compared with a threshold λ to know whether the primary user’s signal exists or not. The primary user’s signal exists only if the detection statistics T is larger than the threshold .
The probability of detection Pd , at hypothesis H1 ,and the probability of false alarm Pf ,at hypothesis H0 ,can therefore be defined as follows:
Pd = Pr(T > λ | H1)=
Pf = Pr(T > λ| H0)=

10. 2. Spectrum Sensing Based On Statistical Covariance
The received signal under H1 and H0 can be given by:
Considering L samples, define the following vectors (Parameter L is called the smoothing factor:
x(n)= [ x(n) x(n-1)…x(n-L-1)]T
s(n)= [ s(n) s(n-1)…s(n-L-1)]T
η(n)= [η(n) η(n-1)… η(n-L-1)]T
Hence, the statistical covariance matrices of the received signal can be defined as:
It can be shown that

11. 2. Spectrum Sensing Based On Statistical Covariance
If the signal s(n) is not present Rs=0. Hence, the off-diagonal elements of Rx are all zeros. If there is a signal and the signal samples are correlated, Rs is not a diagonal matrix. Hence, some of the off-diagonal elements of Rx should be nonzero. Denote rnm as the element of matrix Rx at the nth row and mth column, and let
Then, if there is no signal, T1 / T2 = 1. If the signal is present, T1 / T2 > 1. Hence, ratio T1 / T2 can be used to detect the presence of the signal.

12. 2. Spectrum Sensing Based On Statistical Covariance
In practice, the statistical covariance matrix can only be calculated using a limited number of available signal samples at the receiver. Hence, we can compute the sample autocorrelations of the received signal as
where Ns is the number of available samples

13. 2. Spectrum sensing based on statistical covariance
Statistical covariance matrix Rx can be approximated by the sample covariance matrix defined as
Based on the above sample covariance matrix we can use the Covariance Absolute Value(CAV) detection algorithm

14. Covariance Absolute Value(CAV) Detection Algorithm
Sample the received signal
Choose a smoothing factor L and a threshold γ1, where γ1 should be chosen to meet the requirement for the probability of false alarm
Compute the autocorrelations of the received signal λ(l), l = 0, 1, , L − 1, and form the sample covariance matrix
Compute
Determine the presence of the signal based on T1 (Ns), T2(Ns), and threshold γ1. That is, if T1(Ns)/T2(Ns) > γ1, the signal exists; otherwise, the signal does not exist.

15. Theoretical Analysis for the CAV Algorithm
The above proposed CAV algorithm is valid based on the assumption that the transmitted primary signal samples are correlated, i.e., Rs is not a diagonal matrix (Some of the off-diagonal elements of Rs should be nonzeros)
Obviously, if signal samples s(n) are i.i.d., then
in this case, the assumption is invalid, and the algorithm cannot detect the presence of the signal.

16. Theoretical Analysis for the CAV Algorithm
However, usually, the signal samples should be correlated due to three reasons:
The signal is oversampled
The propagation channel has time dispersion;
The original signal is correlated. In this case, even if the channel is a flat-fading channel and there is no oversampling, the received signal samples are correlated.

17. Threshold Selection
As mentioned earlier, for a good detection algorithm, a high Pd and low Pf should be achieved. The choice of threshold γ is a compromise between Pd and Pf
Usually, the threshold is chosen such that a certain value of false alarm probability Pf is achieved
The threshold selection can be based on either theoretical derivation or computer simulation:
1- If threshold selection based on computer simulation:
We first set a value for Pf
We find a threshold γ to meet the required Pf : to do so we can generate white Gaussian noises as the input (no signal) and adjust the threshold to meet the Pf requirement. Note that the threshold here is related to the number of samples used for computing the sample autocorrelations and the smoothing factor L

18. Threshold Selection 2- If theoretical derivation is used:
We need to find the statistical distribution of T1(Ns)/T2(Ns) which is generally a difficult task. In [1], using the central limit theorem, the distribution of this random variable is approximated and the theoretical estimations for the two probabilities Pd , Pf and the threshold associated with these probabilities are derived as follows (equations (74)(76)(77) in [1]):

19. References
[1] Yonghong Zeng and Ying-Chang Liang, “Spectrum-sensing algorithms for cognitive radio based on statistical covariances”, IEEE transactions on vehicular technology. Vol. 58. no. 4, pp , May 2009
[2] Z. Quan,, S. Cui, and A. H. Sayed, "Optimal linear cooperation for spectrum sensing in cognitive radio networks," IEEE Journal of Selected Topics In Signal Processing, vol.2, pp , 2008.