## 4 Mathematical Modelling and the IA-SPM Algorithm
### The IA-SPM Algorithm  
The state equation is analyzed by the discrete Markov process, and the quantitative analysis of the system detection probability of the IA-SPM is obtained.  
According to the state of the PU, when the SU sends the data, the signal stored in the frequency band can be represented by the formula  
\[y(k)=\left\{\begin{array}{cc}n(k)&H_{0}\\ n(k)+s(k)&H_{1}\end{array}\right. \tag{9}\]  
where \(k\) represents the sampling time, \(y(k)\) is the signal stored in the frequency band, \(n(k)\) indicates the signal transmitted by the PU, \(s(k)\) indicates the signal transmitted by the PU, and \(s(k)\) is the existence of a probabilistic problem. In order to facilitate the study, we assume that it is a single-carrier modulated, phase-modulated signal and that the channel it is passed through is an additive Gaussian white noise channel. \(H_{0}\) indicates that the PU is in an idle state and \(H_{1}\) indicates that the PU is in an active state. In order to reduce the complexity of the analysis, it is assumed that there is no channel noise. When the system is in the \(H_{1}\) state, there is a conflict.  
Assuming that \(n(k)\) and \(s(k)\) are random processes \((i,i,d)\) with a Gaussian distribution, their mean is 0 and their variance is \(\delta_{n}^{2}\) and \(\delta_{s}^{2}\), respectively. For an SU that receives the signal, \(n(k)\) is a useful signal and \(s(k)\) is an interfering noise. So \(\gamma=\delta_{s}^{2}/\delta_{n}^{2}\) is defined as the signal-to-noise ratio, which represents the power ratio of the PU transmitting signal to the SU transmitting signal.  
It is the key to how to adaptively adjust the parameters of the silent period under the condition that the sensing period is determined. The minimum length of the silence period is specified as \(N_{min}\) and the maximum length is \(N_{max}\). The IA-SPM is divided into the following two conditions:  
Condition 1: If an SU judges that the PU is not present based on the perceived result during the current silence period, it will send the data and then wait for the response from the receiver.  
If the SU sender receives the feedback signal successfully, it is judged that the PU does not exist. So the self-detection time of the next period can be reduced. The sampling points are reduced by \(\mu^{1/2}(\mu\geq 1)\) times, expressed by the formula  
\[N_{i+1}=max\Big{\{}round\Big{(}N_{i}\div\mu^{\frac{1}{2}}\Big{)},N_{min}\Big{\}} \tag{10}\]  
If the SU sender does not receive the feedback signal, it judges that the PU is present. Then the next period of self-detection time increases. So that the energy detection can be greater than the threshold, the number of sampling points is expanded by \(\mu^{1/2}(\mu\geq 1)\) times, expressed by the formula  
\[N_{i+1}=min\Big{\{}round\Big{(}N_{i}\times\mu^{\frac{1}{2}}\Big{)},N_{max} \Big{\}} \tag{11}\]  
Condition 2: If an SU judges that the PU is present based on the perceived results during the current silence period, it will automatically increase the length of the silence until it is determined that the PU has disappeared.  
\[N_{i+1}=\left\{\begin{array}{cc}N_{i}&\text{PU exists}\\ max\Big{\{}round\Big{(}N_{i}\times\mu^{\frac{1}{2}}\Big{)},N_{max}\Big{\}}& \text{PU not exists}\end{array}\right. \tag{12}\]Then, the silent period of the next sensing cycle remains unchanged until the SU judges that the PU does not exist. This will then go to Condition 1.