3. Proposed Method
3.1 Log-likelihood ratio test
Typically, the received samples commonly exhibit rich correlation information that can improve the performance of spectrum sensing. Here, we present the feature detection when received samples are correlated. Denoting 𝐻1 and 𝐻0 as the hypotheses with and without signal presence. Notably, under 𝐻0 hypothesis, received samples are all uncorrelated Gaussian random variables, and, under 𝐻1 hypothesis, received samples are correlated for 𝑛 ∈ Ω𝐾 . The optimum detector is quite complicated if a straightforward high-dimensional search is applied. On the contrary, a fine-grained approach is adopted here. Firstly, let’s assume that we have a perfect knowledge of all unknowns except 𝜃 and 𝐿. Considering complex Gaussian and jointly-Gaussian, after necessary manipulations, we can formulate the decision metric by the log-likelihood ratio as
(4)
3.1.1 Original detector
To derive a practical detector from above derivations and observations, the detection scheme is designed below. For each trial value of (𝜃, 𝐿), i.e., (𝜃̃, 𝐿̃), we propose to estimate 𝜉 as
(5) With
(6)
and
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(7) , 𝜉 is replaced by its estimate 𝜉̂. Since 𝛾(𝑛) and 𝜙(𝑛) are sample estimates of 𝛾̅(𝑛, 𝑁) = 𝐸[𝑦(𝑛)𝑦∗(𝑛 + 𝑁)] and 𝐸[|𝑦(𝑛)|2] , respectively, the estimation of correlation coefficient and total power can be given by
(8) Because correlation coefficient equals the cross correlation divided by the auto correlation, where 𝑁Ω̃𝑘 = ∥ Ω̃𝑘 ∥ = 𝑁𝑟 + 𝐿̃ denotes the cardinality of the set Ω̃𝑘.
Finally, the above estimates are substituted into (4) to make a decision on hypotheses.
The proposed method reduces the brute-force high dimensional search to a two-dimensional search algorithm. Though many unknown parameters are mutually related, the proposed scheme decomposes the original joint detection and estimation problem without necessitating any iteration and is summarized below.
Algorithms 1: Original detector input : received samples 𝑦(𝑛)
1 for each (𝜃̃, 𝐿̃)
output : detection result 𝑇/𝐹
3.1.2 Modified detector via restoring to time- and frequency-nonselective
correlation
The estimation (and hence detection) performance is limited at a low SNR or doubly-selective fading channel environment. Therefore, we will modify the original detector by eliminating the effects of doubly-selective fading channels. The impacts of multipath fading channels are removed by employing the complement property of correlation profile. Moreover, to cancel the effects of high-speed vehicles, i.e., 𝜅, we will try to
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explicitly compensate it to recover the observed correlation from the impacts generated by time-selectivity and mobility. Another advantage of the modified detector (MD) comes up with that different 𝜌𝑛’s for ∀𝑛 ∈ Ω𝑘 reduce to a single 𝜌.
Remember that, under 𝐻1 hypothesis, if we alias the complementary correlations for 𝑛 ∈ Ω1,𝑘 and 𝑛 ∈ Ω3,𝑘, a flat correlation profile can be achieved for 𝑛 ∈ Ω1,𝑘 ∪ 𝑛 ∈ Ω2,𝑘. Besides, 𝛾(𝑛) asymptotically approaches 𝛾̅(𝑛, 𝑁) = 𝐸[𝑦(𝑛)𝑦∗(𝑛 + 𝑁)] if 𝜉 is compensated. We denote the aliased version of sample correlation for 𝑛 ∈ Ω1,𝑘 as
(9)
Notice that 𝜉 still produces a phase rotation in 𝛾𝑎(𝑛). Therefore, 𝜉 still should be estimated. To eliminate the effects of the mobility, i.e., 𝜅, the zeroth-order Bessel function of the first kind produced by κ must be estimated. Owing to the complement property, 𝐸[𝛾𝑎(𝑛)] (𝑓𝑜𝑟 𝑛 ∈ Ω1,𝑘) = 𝐸[𝛾(𝑛)] (for 𝑛 ∈ Ω2,𝑘 = 𝐽0(κN) ). Hence, Notice that only one 𝜌 is estimated for the aliased version of correlation in that it is flat for ∀𝑛 ∈ Ω1,𝑘∪ Ω2,𝑘. The range of samples used to estimate 𝜎2 shortens accordingly. The derived feature detector allows us to employ the stationarity of received signals; therefore, under this situation of constant plateau in the region exhibiting correlation, i.e., similar to the cases of AWGN or single-path channels, the decision metric (4) reduces to (12)
(12)
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where 𝜂𝑎 denotes the decision threshold of the modified detector. Most importantly, even though radio signals propagate through complicated doubly-selective fading channels, the correlation characteristics of received signals can still be simply recovered from impacts of channels based on the proposed techniques, without applying any channel equalization techniques on received signals.
However, under 𝐻0 hypothesis, the ensemble average of 𝛾(𝑛) is zero, so is 𝐽̂0(𝜅𝑁). Therefore, the condition of division by zero happens for 𝛾𝑎′(𝑛) (or 𝛾(𝑛)). Such a condition must be avoided because 𝐽̂0(𝜅𝑁) should not be compensated under 𝐻0 hypothesis. This condition can be prevented when 𝐽̂0(𝜅𝑁) is smaller than 𝜒, which is a parameter to be determined by simulations.
3.1.3 Complexity reduction and analysis
Most computationally intensive tasks of digital signal processing are preferably implemented using hardware. Hence, the hardware complexities of the proposed algorithms are analyzed in this section. Besides, we will propose counterparts of the proposed OD and MD with reduced complexity, called reduced OD (reOD) and MD (reMD), respectively. As shown in Table I, the proposed OD and MD have a complexity of 𝑂(𝑁𝑟3), while their reduced counterparts, reOD and reMD, have a complexity of 𝑂(𝑁𝑟2).
Table I Complexity comparison of the proposed and conventional detectors
Hence, the complexity of the reduced versions is reduced by an order of Nr.
Additionally, the proposed MD/reMD has less complexity than the proposed OD/reOD in terms of the required number of complex multiplications. The conventional detectors AD and SD respectively have complexities 𝑂(𝑁𝑟) and 𝑂(𝑁𝑟3) in terms of the complex multiplication, while the SD needs no division. The complexity of AD is proportional to Nr because the unknown parameter θ is not searched by it. As will be presented in Performance Evaluation Section, with additional computational cost paid for the proposed methods, the detection performance of the proposed methods can be enhanced over doubly-selective fading channels as well.
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Algorithms 2: Modified detector input : received samples 𝑦(𝑛)
1 for each (𝜃̃, 𝐿̃)
output : detection result 𝑇/𝐹
3.2 Proposed detectors over line-of-sight multipath fading channels
Similar to NLOS channels, the correlation of received samples 𝑦(𝑛) also exists for the proposed OD so that the proposed OD can also be applied for LOS time-variant multipath fading channels. Moreover, the complement property also holds for the LOS channels. Based on the complement property, the proposed MD can also be suited for LOS time-variant multipath fading channels. Notably, the estimation (10) of the proposed MD over LOS channels does not give an estimate of 𝐽0(·). Instead, (10) estimates the constant plateau value,
(13) , and the proposed MD compensates it, so that the correlation profile over LOS time-variant multipath fading channels can still be restored to the one without time and frequency-selectivity.
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