6 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 31, NO. 3, MARCH 2013 (**for** continual pilots) or LNɛ (**for** scattered pilots). This will gives us special advantage to design the algorithm. If we use known methods to estimate ɛ first and then get Nɛ or LNɛ, the estimation error is effectively multiplied by N or LN times. Thus in the following we will propose a new method that directly estimates Nɛ (**for** continual pilots) or LNɛ (**for** scattered pilots) using the ACs of the received signal. A. Frequency offset estimation We consider continual pilot case of model (14). Let ϑ = Nɛ mod 1. We further define δ = ϑ if 0 ≤ ϑ ≤ 0.5 and δ = ϑ−1 if 0.5

ZENG et al.: SPECTRUM SENSING FOR **OFDM** SIGNALS USING PILOT INDUCED AUTO-CORRELATIONS 7 where f k is a weight and ∑ K k=1 f k =1. Theorem 4: Assume that the sensing time is long and the initial CFO estimation is sufficiently accurate. The optimal linear estimator based on (61) should have the coefficients f k = k 2 (N s − kN)/c, (62) where c = ∑ K k=1 k2 (N s − kN) is a constant. Proof: To obtain the optimal estimator, we need the best combining coefficients f k **for** BLUE. As discussed above, we need the variance of θ1(k) 2πk .Letˆζ(k) =χ 1 (k)+jχ 2 (k). Then ( ) μ sin(2πˆδk)+χ 2 (k) φ(kN) = arctan μ cos(2πˆδk)+χ 1 (k) ⎛ = arctan⎝ tan(2πˆδk)+ ⎞ χ2(k) μ cos(2πˆδk) ⎠ . (63) 1+ χ1(k) μ cos(2πˆδk) Assume that μ>>|ˆζ(k)| (valid at long sensing time) and the remaining error ˆδ is very small. **Using** the first order Taylor approximation, we have Thus φ(kN) ≈ 2πˆδk + χ 2 (k) μ cos(2πˆδk) ≈ 2πˆδk + χ 2(k) μ . (64) φ(kN) 2πk ≈ ˆδ + χ 2(k) 2πkμ . (65) That is, at H 1 , we approximately have θ 1 (k) 2πk ≈ χ 2(k) 2πkμ . (66) Note that χ 2 (k) is the imagine part of ˆζ(k). Thus 1 Var(χ 2 (k)) = Var(ˆζ(k))/2 =Var(ζ(k))/2 = 2(N s−kN) σ4 1 . Hence, finally we have ( θ1 (k) ) 1 Var = 2πk 8π 2 μ 2 k 2 (N s − kN) σ4 1. (67) Thus the BLUE should have combining coefficients f k = k 2 (N s − kN)/c. □ The final estimation of the CFO δ is: δ 1 = δ 0 + ˆδ 0 . If there are scattered pilots, we can use them to estimate the composite CFO (LNɛ mod 1) based on model (39). The derivation and the algorithm are similar. We can simply replace N by LN in all the equations above. However, we can only estimate LNɛ using the scattered pilots. It may not be possible to obtain Nɛ mod 1 from LNɛ mod 1. On the other hand, if we have estimated Nɛ mod 1 based on the continual pilots, it is easy to get the estimation **for** LNɛ mod 1. Itmaybe possible to use both the continual and scattered pilot **for** the estimation of Nɛ mod 1. B. Search the frequency offset After the estimation, we first subtract the normalized CFO in the test statistics. The remaining (residue) of the CFO is much smaller than be**for**e. Let the remaining (residue) of the normalized CFO is in interval (−Δ, Δ], whereΔ ≪ 0.5. We then search the CFO within this interval. For each CFO in the interval, we obtain a value **for** the test statistic. The maximum of all the values are then used **for** the final decision. We use continual pilot case as an example to describe the details of the method in the following. First the CFO is compensated, that is, we multiply r(kN) by e −j2πkδ1 to obtain ¯r(kN) =r(kN)e −j2πkδ1 .Let¯ζ(k) = ζ(k)e −j2πkδ1 ,and¯η(k) =η(k)e −j2πkδ1 .Wehave H 0 :¯r(kN) =¯η(k) H 1 :¯r(kN) =e j2πk¯δμ + ¯ζ(k), (68) k =1,...,K, where ¯δ = δ − δ 1 is the remaining CFO (the estimation error). The ALRT test based on the compensated system is there**for**e ¯T ALRT −CP ( ) 1 K∑ =Re (N s − kN)e −j2πk¯δ ¯r(kN) . (69) r(0) k=1 Although ¯δ is still unknown, but it is assumed to be much small than δ = Nɛ mod 1. So we can search ¯δ in the interval (−Δ, Δ] with limited searching steps. Let M be the number of searching steps and ¯δ m = −Δ+2mΔ/M be the searching points, m =1, 2, ··· ,M. For each searching point, we compute the test ¯T (m) ALRT −CP ( 1 K∑ =Re (N s − kN)e −j2πk¯δm ¯r(kN) r(0) k=1 Let the maximum of all the tests be ) . (70) ¯T (max) ALRT −CP = max ¯T (m) ALRT −CP . (71) 1≤m≤M The searching complexity is only O(KM)=O(MN s /N ). This complexity is usually much lower than the complexity **for** computing the AC r(kN), which is O(Ns 2/N ), asN s is usually much larger than M. Hence, the searching only increases the complexity marginally. V. SUB-OPTIMAL APPROACHES At very low SNR, the frequency offset estimation may not be very accurate. We should also consider alternative approaches without frequency offset estimation. One such approach is simply taking absolute value in the ALRT. We get three methods **for** continual pilot, scattered pilots and their combination as follows. T ALRT −CP−A = 1 r(0) K∑ (N s − kN)|r(kN)|. (72) k=1 T ALRT −SP−A = 1 K/L ∑ (N s − kLN)|r(kLN)|. (73) r(0) T ALRT −CSP−A = k=1 K/L λ ∑ (N s − kLN)|r(kLN)| r(0) + 1 r(0) k=1 K∑ (N s − kN)|r(kN)|. (74) k=1