In this section some noncommutative ring theory notions are used [16]. We start by giving an overview of the mathematical background leading to the algebraic detection technique. First let's suppose that the frequency range available in the wireless network is B Hz; so B could be expressed as B = [f_{0}, f_{
N
}]. Saying that this wireless network is cognitive, means that it supports heterogeneous wireless devices that may adopt different wireless technologies for transmissions over different bands in the frequency range. A CR at a particular place and time needs to sense the wireless environment in order to identify spectrum holes for opportunistic use. Suppose that the radio signal received by the CR occupies N spectrum bands, whose frequency locations and PSD levels are to be detected and identified. These spectrum bands lie within [f_{1}, f_{
K
}] consecutively, with their frequency boundaries located at f_{1} < f_{2} < ··· < f_{
K
}. The nth band is thus defined by: B_{
n
}: {f ∈ B_{
n
}: f_{n1}< f < f_{
n
}, n = 2,3,..., K}. The PSD structure of a wideband signal is illustrated in Figure 1. The following basic assumptions are adopted:

(1)
The frequency boundaries f_{1} and f_{
K
}= f_{1} + B are known to the CR. Even though the actual received signal may occupy a larger band, this CR regards [f_{1}, f_{
K
}] as the wide band of interest and seeks white spaces only within this spectrum range.

(2)
The number of bands N and the locations f_{2},..., f_{K1}are unknown to the CR. They remain unchanged within a time burst, but may vary from burst to burst in the presence of slow fading.

(3)
The PSD within each band B_{
n
}is smooth and almost flat, but exhibits discontinuities from its neighboring bands B_{n1}and B_{n+1}. As such, irregularities in PSD appear at and only at the edges of the K bands.

(4)
The corrupting noise is additive white and zero mean.
The input signal is the amplitude spectrum of the received noisy signal. We assume that its mathematical representation is a piecewise regular signal:
Y\left(f\right)=\sum _{i=1}^{K}{\chi}_{i}\left[{f}_{i1},{f}_{i}\right]\left(f\right){p}_{i}\left(f{f}_{i1}\right)+n\left(f\right)
(4.1)
where: χ_{
i
}[f_{i1}, f_{
i
}]: the characteristic function of the interval [f_{i1}, f_{
i
}], (p_{
i
})_{i∈[1,K]}: an N th order polynomials series, (f_{
i
})_{i∈[1,K]}: the discontinuity points resulting from multiplying each p_{
i
}by a χ_{
i
}and n(f): the additive corrupting noise.
Now, let X(f) the clean version of the received signal given by:
X\left(f\right)={\sum}_{i=1}^{K}{\chi}_{i}\left[{f}_{i1},{f}_{i}\right]\left(f\right){p}_{i}\left(f{f}_{i1}\right)
(4.2)
And let b, the frequency band, given such as in each interval I_{
b
}= [f_{i1},f_{i}] = [ν,ν + b], ν ≥ 0 maximally one change point occurs in the interval I_{
b
}.
Now denoting X_{
ν
}(f) = X(f + ν), f ∈ [0,b] for the restriction of the signal in the interval I_{
b
}and redefine the change point which characterizes the distribution discontinuity relatively to I_{
b
}say f_{
ν
}given by:
{y}_{n}=\left\{\begin{array}{cc}{f}_{v}=0\hfill & \text{if}\phantom{\rule{2.77695pt}{0ex}}{X}_{v}\phantom{\rule{2.77695pt}{0ex}}\text{is}\phantom{\rule{2.77695pt}{0ex}}\text{continuous}\hfill \\ 0<{f}_{v}\le b\hfill & \text{otherwise}\hfill \end{array}\right.
Now, in order to emphasis the spectrum discontinuity behavior, we decide to use the N th derivative of X_{
ν
}(f), which in the sense of distributions theory is given by:
\frac{{d}^{N}}{d{f}^{N}}{X}_{v}\left(f\right)={\left[{X}_{v}\left(f\right)\right]}^{\left(N\right)}+\sum _{k=1}^{N}{\mu}_{Nk}\delta {\left(f{f}_{v}\right)}^{\left(k1\right)}
(4.3)
where: μ_{
k
}is the jump of the k th order derivative at the unique assumed change point:f_{
ν
}
{\mu}_{k}={X}_{\nu}^{\left(k\right)}\left({f}_{\nu}^{+}\right){X}_{\nu}^{\left(k\right)}\left({f}_{\nu}^{}\right)
with μ_{
k
}= 0⌋_{k = 1...N}if there is no change point and μ_{
k
}≠ 0⌋ _{k = 1...N}if the change point is in I_{
b
}.
[X_{
ν
}(f)]^{(N)}is the regular derivative part of the N th derivative of the signal.
The spectrum sensing problem is now casted as a change point f_{
ν
}detection problem. Several estimators can be derived from the previous equations equation. For example any derivative order N can be taken and depending on this order the equation is solved in the operational domain and back to frequency domain the estimator is deduced. In a matter of reducing the complexity of the frequency direct resolution, those equations are transposed to the operational domain, using the Laplace transform:
\begin{array}{c}L\left({X}_{\nu}{(f)}^{(N)}\right)={s}^{N}{\widehat{X}}_{\nu}(s){\displaystyle \sum _{m=0}^{N1}{s}^{Nm1}}\frac{{d}^{m}}{d{f}^{m}}{X}_{\nu}(f){\rfloor}_{f=0}\\ ={e}^{s{f}_{\nu}}\left({\mu}_{N1}+s{\mu}_{N2}+\mathrm{\cdots}+{s}^{N1}{\mu}_{0}\right)\end{array}
(4.4)
Given the fact that the initial conditions, expressed in the previous equation, and the jumps of the derivatives of X_{
ν
}(f) are unknown parameters to the problem, in a first time we are going to annihilate the jump values μ_{0},μ_{1},..., μ_{N1}(Appendix 1) then the initial conditions (Appendix 2). After some calculations steps detailed, we finally obtain:
{\displaystyle \sum _{k=0}^{N1}\left({}_{k}^{N}\right).{f}_{\nu}^{Nk}.{\left({s}^{N}{\widehat{X}}_{\nu}(s)\right)}^{\left(N+k\right)}=0}
(4.5)
In the actual context, the noisy observation of the amplitude spectrum Y(f) is taken instead of X_{
ν
}(f). As taking derivative in the operational domain is equivalent to highpass filtering in frequency domain, which may help amplifying the noise effect. It is suggested to divide the whole previous equation by s^{l}which in the frequency domain will be equivalent to an integration if l > 2N, we thus obtain:
{\displaystyle \sum _{k=0}^{N1}\left({}_{k}^{N}\right).{f}_{\nu}^{Nk}.\frac{{\left({s}^{N}{\widehat{X}}_{\nu}(s)\right)}^{\left(N+k\right)}}{{s}^{l}}=0}
(4.6)
Since here is no unknown variables anymore, the previous equation is now transformed back to the frequency domain, we obtain the polynomial to be solved on each sensed subband:
{\displaystyle \sum _{k=0}^{N1}\left({}_{k}^{N}\right).{f}_{\nu}^{Nk}.{L}^{1}\left[\frac{{\left({s}^{N}{\widehat{X}}_{\nu}(s)\right)}^{\left(N+k\right)}}{{s}^{l}}\right]=0}
(4.7)
And denoting:
{\phi}_{k+1}={L}^{1}\left[\frac{{\left({s}^{N}{\hat{X}}_{\nu}\left(s\right)\right)}^{\left(N+k\right)}}{{s}^{l}}\right]={\int}_{0}^{+\infty}{h}_{k+1}\left(f\right).X\left(\nu f\right).df
(4.8)
where: {h}_{k+1}\left(f\right)=\left\{\begin{array}{cc}\frac{{\left({f}^{l}{\left(bf\right)}^{N+k}\right)}^{\left(k\right)}}{\left(l1\right)!}\hfill & 0<f<b\hfill \\ 0\hfill & \text{otherwise}\hfill \end{array}\right.
To summarize, we have shown that on each interval [0, b], for the noisefree observation the change points are located at frequencies solving:
{\displaystyle \sum _{k=0}^{N}\left({}_{k}^{N}\right).{f}_{\nu}^{Nk}.{\phi}_{k+1}=0}
(4.9)
To summarize, we have shown that on each interval [0, b], for the noisefree observation the change points are located at frequencies solving:
{\displaystyle \sum _{k=0}^{N}\left({}_{k}^{N}\right).{f}_{\nu}^{Nk}.{\phi}_{k+1}=0}
(4.10)
In [17], it was shown that edge detection and estimation is analyzed based on forming multiscale pointwise products of smoothed gradient estimators. This approach is intended to enhance multiscale peaks due to edges, while suppressing noise. Adopting this technique to our spectrum sensing problem and restricting to dyadic scales, we construct the multiscale product of N + 1 filters (corresponding to continuous wavelet transform in [17]), given by:
Df=\u2225\prod _{k=0}^{N}{\phi}_{k+1}\left({f}_{\nu}\right)\u2225
(4.11)
4.1. Implementation issues
The proposed algorithm is implemented as a filter bank which is composed of N filters mounted in a parallel way. The impulse response of each filter is:
{h}_{k+1}\left(f\right)=\left\{\begin{array}{cc}\frac{{\left({f}^{l}{\left(bf\right)}^{N+k}\right)}^{\left(k\right)}}{\left(l1\right)!}\hfill & 0<f<b\hfill \\ 0\hfill & \text{otherwise}\hfill \end{array}\right.
(4.12)
where k ∈ [0 ... N  1] and l is chosen such as l > 2 × N. The proposed expression of h_{k+1}⌋_{k∈[0...N1]}was determined by modeling the spectrum by a piecewise regular signal in frequency domain and casting the problem of spectrum sensing as a change point detection in the primary user transmission. Finally, in each stage of the filter bank, we compute the following equation:
{\phi}_{k+1}\left(f\right)=\underset{0}{\overset{+\infty}{\int}}{h}_{k+1}\left(\nu \right).X\left(f\nu \right).d\nu
(4.13)
Then, we process by detecting spectrum discontinuities and to find the intervals of interest.
4.2. Algorithm discrete implementation
The proposed algorithm in its discrete implementation is a filter bank composed of N filters mounted in a parallel way. The impulse response of each filter is:
{h}_{k+1,n}=\left\{\begin{array}{cc}\frac{{\left({n}^{l}{\left(bn\right)}^{N+k}\right)}^{\left(k\right)}}{\left(l1\right)!}\hfill & 0<n<b\hfill \\ 0\hfill & \text{otherwise}\hfill \end{array}\right.
(4.14)
where k ∈ [0 ... N  1] and l is chosen such as l > 2 × N. The proposed expression of h_{k+1,n}⌋_{k∈[0...N1]}was determined by modeling the spectrum by a piecewise regular signal in frequency domain and casting the problem of spectrum sensing as a change point detection in the primary user transmission. Finally, in each detected interval \left[{n}_{{\nu}_{i}},{n}_{{\nu}_{i+1}}\right], we compute the following equation:
{\phi}_{k+1}=\sum _{m={n}_{{\nu}_{i}}}^{{n}_{{\nu}_{i+1}}}{W}_{m}{h}_{k+1,m}{X}_{m}
(4.15)
where W_{
m
}are the weights for numeric integration defined by:
\begin{array}{ll}\hfill {W}_{0}& ={W}_{M}=0.5\phantom{\rule{2em}{0ex}}\\ \hfill {W}_{m}& =1\phantom{\rule{1em}{0ex}}\text{otherwise}\phantom{\rule{2em}{0ex}}\end{array}
In order to infer whether the primary user is present in the detected intervals, a decision function is computed as following:
Df=\u2225\prod _{k=0}^{N}{\phi}_{k+1}\left({n}_{\nu}\right)\u2225
(4.16)