In non-coherent approach, we contrive to two CUSUM-based strategies without any esti-mation of realization about the unknown channel fading coefficients in the receiver end.
One is called classical CUSUM algorithm, and another is termed to be weighted CUSUM algorithm. In classical CUSUM algorithm, we treat the unknown channel factor as a ran-dom variable with known prior statistics and calculate the likelihood ratio between joint probability density functions of observations under the conditions before and after change occurs, while in weighted CUSUM algorithm we take the unknown channel coefficient as deterministic but unknown constant during the detection process and then weight the
likelihood ratio by applying prior information as weighting function.
After restricting ourselves to purely Rayleigh-fading channels, we may assume that the flat fading channel is modeled as h ∼ CN (0, σh2). The more general case under frequency-selective channel will be further discussed in later subsection.
3.2.1 Classical CUSUM Algorithm
Givenσ2h at the receiver end, the joint distribution of received signals in specific interval of observation time before or after change can be fully specified with the channel effect averaged. Whereas, the observation sequence after the change time {yt0, yt0+1, . . .} is dependent due to the presence of fading channel. Due to the non-homogeneous feature of observations after reoccupying time t0, it seems implicit in the validity of applying CUSUM algorithm. Thus, we need to study the natural trend of the log-likelihood before and after change.
First, before the change timet0, it can be verified from the Kullback information that Eθ0[lnpΘ(k,i)(yi, yi+1,· · · , yk)
pθ0(yi, yi+1,· · · , yk) ]≤ Eθ0[lnpΘ(k,j)(yj, yj+1,· · · , yk)
pθ0(yj, yj+1,· · · , yk) ]≤ 0,
∀i ≤ j ≤ k ≤ t0,
(3.3)
where pΘ(k,j) denotes the joint distribution of received signals from time j to k given t0 = j and pθ0 denotes the joint distribution before change occurs. From the above inequality, we can observe a negative drift of the expected log-likelihood before change, which indicates the absence of the primary signaling. Similarly, we have
0≤EΘ(k,j)[ln pΘ(k,j)(yj, yj+1,· · · , yk) pθ0(yj, yj+1,· · · , yk) ]≤ EΘ(k,i)[lnpΘ(k,i)(yi, yi+1,· · · , yk)
pθ0(yi, yi+1,· · · , yk) ],
∀k ≥ j ≥ i ≥ t0,
(3.4)
which indicates the positive tendency as the change has occurred. Therefore, we might apply the idea of CUSUM algorithm to detect the beginning of the reoccupying signals by the discrimination property. Although the log-likelihood ratio here is not additive due to
the dependency among the observations after change, we still adopt the term “CUSUM”
to represent the increasing amount on expected log-likelihood ratio. Then, the decision rule is given by
where Kkj represents the covariance matrix of received signals from time j to k under t0 = j, and ykj collects observationsyj, yj+1, . . . , yk.
To be more specific, we can view the received signals from timej to k under t0 = j alternatively as real part and imaginary part are collectively jointly Gaussian. Further, it is a circular symmetric complex Gaussian random vector with its joint density function denoted as CN (0, Kkj), where
In conclusion, we treat the unknown channel factor as a random variable with known prior statistics and calculate the likelihood ratio between joint probability density func-tions of observafunc-tions under the condifunc-tions before and after change occurs. At each time instant, we search for the time at which the backward accumulated likelihood ratio is
maximum, in other words, the instant the reoccupying most likely takes place. Then, we raise alarm to declare the change at the first time the resultant accumulated likelihood ratio is larger than a well-chosen threshold. Appropriate threshold ~ can be determined by numerical simulation in advance of detection.
3.2.2 Extension to Frequency-selective Fading Case
Under the concern of frequency-selective fading channel between cognitive user and un-derlying primary user or base station, the received signal in time-domain could be rela-tively modeled as
yk = θ(k, t0) + nk, where θ(k) =
Υ(k, t0) ⊛ hef f(k), as k ≥ t0
0, as k < t0
(3.10)
with
Υ(k, t0) =
s((k− t0)modNs), as k≥ t0 0, as k < t0.
(3.11) and
hef f(k) =
h(k), as 0≤ k ≤ L − 1 0, Otherwise.
(3.12) Similar to previous flat fading case, t0 denotes the unknown presence time instant of primary signal,Nsdenotes the length of the repeated segment of the preamble signal,nk
models the AWGN with varianceσ2nat timek, and s(i) is retrieved from the ith element of the periodic fragment of preamble symbol s= [s(0), s(1), . . . , s(Ns− 1)]T. Note that the fading effects are caused by the channel h consists ofL uncorrelated taps h(l), l = 0, 1, . . . , L− 1, where each element of h are modeled as purely Rayleigh-fading with varianceσl2 with uniform power constraintΣLl=0σ2l = 1.
Now, since the natural tendency is still reserved in the case of frequency-selective fading, we could design the corresponding decision strategy from the idea of CUSUM algorithm in a similar way for detecting the beginning of the reoccupying of underlying primary system in multipath environments. Specifically, the decision rule is in the form
of of received signals convoluted withL channel taps from time j to k under t0 = j.
In detail, ykj conditioned ont0 = j can be decomposed as where theL by 1 vector xicollects the symbols convoluted with h at time instanti. After such rearrangement, we could assure that yjk|t0=j is also a random vector whose real part and imaginary part are collectively jointly Gaussian with its joint density function denoted asCN (0, Ckj), where that the elements of covariance Ckj only depends on the valuek− j and can be calculated and stored in advance.
3.2.3 Modified Window-limited Version
Although the resultant accumulated likelihood ratio of the proposed classical CUSUM decision strategy could not be calculated in recursive way, we could resort to examine
the required length of backward observations that keeps comparable efficacy with the one without any curtailment of observational window. On the other hand, we also curious about the influence on the case with regard to limited data storage in the receiver equip-ment.
Thus, we can simply replace the decision strategy in previous proposed classical CUSUM algorithm with a modified window-limited version, which is thus given as
gkW L = max
max (1,k−W +1)≤j≤klnpΘ(k,j)(yj, yj+1,· · · , yk)
pθ0(yj, yj+1,· · · , yk) (3.18) tW La = min{k : gkW L ≥ ~}, (3.19) whereW is predetermined window size according to available temporary buffer.
Surprisingly, the effectiveness of classical CUSUM algorithm after truncating the length of needed backward observations is pretty nearly comparable with the original one with full memory of past observations, which lowers the complexity and required storage during implement and is demonstrated by simulation results.
3.2.4 Weighted CUSUM Algorithm
Consider another view about the unknown fading factor, we take the unknown channel coefficient h as deterministic but unknown constant during the detection process in our proposed weighted CUSUM algorithm. The main idea is to weight the likelihood ra-tio with respect to all possible values of the fading coefficient by using a well-chosen weighting function and take the resultant weighted likelihood ratio as an indicator about whether the reoccupying has occurs or not. Once the resultant weighted likelihood ex-ceeds a particular predetermined threshold, which reveal a distinct possibility of primary user’s activity, we stop taking observations and raise an alarm to declare that the change has very likely occurred.
Specifically, the form of the decision strategy of weighted CUSUM algorithm is as following:
gkweighted
= max
1≤j≤kln Z ∞
−∞
pΥ(j,j)|h(yj)· · · pΥ(k,j)|h(yk)
pθ0(yj,· · · , yk) ph(h)dh (3.20)
and
tweighteda = min{k : gweightedk ≥ ~} (3.21) That is, for every time instantk, we calculate the weighted likelihood ratio from time j = 1, 2, . . . , k to determine the most possible change point and compare the resultant log-likelihood ratio to some determined threshold. Once exceeding, we raise an alarm to declare the reoccupying of underlying primary user.
Provided statistical information about the fading coefficient, it is fairly reasonable to choose the weighting function ph as CN (0, σh2). So, we could calculate the weighted likelihood ratio as following After integrating manipulations, we can get the close form of the weighted log-likelihood ratio
Then we could represent the resultant weighted log-likelihood ratio at time instantk as
gkweighted
= max
1≤j≤k
σn2
2(2σn2l + 1)[(fSa k
j)2+ ( eSb k
j)2]− ln(σh2l + 1) (3.24) and the stopping time to raise an alarm is
tweighteda = min{k : gweightedk ≥ ~} (3.25) From the closed-form of weighted log-likelihood ratio (3.23), we can see that the computing complexity of weighted CUSUM algorithm is less than the one of classical CUSUM algorithm, which involves more multiplications during detection process. On the other hand, at low-SNR region, the manipulation of directly weighting over the log-likehood ratio might alleviate the impact of low resolution due to fixed noisy power before and after change.