In Stage I, the data transmission time is adaptively adjusted subject to the interference time constraint. Here, we assume that the SU Tx know the statistical information of the PU's activity, including the mean, the variance and the PDF of PU's ON and OFF periods.

### 4.1 Adaptive algorithm for adjusting the transmission time {T}_{1}^{i} in Stage I

The aim of the adaptive algorithm is to limit the interference time generated in the scenario where the licensed channel switches from OFF state to ON state while SU Tx is transmitting data on the channel.

Given that the channel is in OFF state at *t*_{
i
}, the conditional probability that the channel keeps being in OFF state during {T}_{1}^{i} is

\mathsf{\text{Pr}}\left\{{\mathcal{H}}_{0}\left({t}_{i+1}\right)|{\mathcal{H}}_{0}\left({t}_{i}\right)\right\}=\frac{{\int}_{{t}_{i}+{T}_{1}^{i}-{t}_{\mathsf{\text{sp}}1}}^{\infty}{f}_{1}\left(s\right)ds}{{\int}_{{t}_{i}-{t}_{\mathsf{\text{sp}}1}}^{\infty}{f}_{1}\left(s\right)ds}=\frac{1-{F}_{0}\left({t}_{i}+{T}_{1}^{i}-{t}_{\mathsf{\text{sp}}1}\right)}{1-{F}_{0}\left({t}_{i}-{t}_{\mathsf{\text{sp}}1}\right)}

(2)

where {\mathcal{H}}_{\mathsf{\text{0}}}\mathsf{\text{(}}t\mathsf{\text{)}} denotes that the channel is in OFF state at time *t*, and {\mathcal{H}}_{1}\mathsf{\text{(}}t\mathsf{\text{)}} denotes that the channel is in ON state at time *t*. Then, the probability that the channel turns from OFF state to ON state at the time of *t*_{i+1}is

\mathsf{\text{Pr}}\left\{{\mathcal{H}}_{1}\left({t}_{i+1}\right)|{\mathcal{H}}_{0}\left({t}_{i}\right)\right\}=1-\mathsf{\text{Pr}}\left\{{\mathcal{H}}_{0}\left({t}_{i+1}\right)|{\mathcal{H}}_{0}\left({t}_{i}\right)\right\}=\frac{{F}_{0}\left({t}_{i}+{T}_{1}^{i}-{t}_{\mathsf{\text{sp}}1}\right)-{F}_{0}\left({t}_{i}-{t}_{\mathsf{\text{sp}}1}\right)}{1-{F}_{0}\left({t}_{i}-{t}_{\mathsf{\text{sp}}1}\right)}

(3)

Similarly, the probability that the channel switches from OFF state to ON state in the middle of the *i* th transmission is

\mathsf{\text{Pr}}\left\{{\mathcal{H}}_{1}\left({t}_{i}+s\right)|{\mathcal{H}}_{0}\left({t}_{i}\right)\right\}=1-\mathsf{\text{Pr}}\left\{{\mathcal{H}}_{0}\left({t}_{i}+s\right)|{\mathcal{H}}_{0}\left({t}_{i}\right)\right\}=\frac{{F}_{0}\left({t}_{i}+s-{t}_{\mathsf{\text{sp}}1}\right)-{F}_{0}\left({t}_{i}-{t}_{\mathsf{\text{sp}}1}\right)}{1-{F}_{0}\left({t}_{i}-{t}_{\mathsf{\text{sp}}1}\right)}

(4)

where 0 *< s <* {T}_{1}^{i}. The PDF of the residual time *s* of channel's OFF state is the derivative of (4) with respect to *s*:

\begin{array}{cc}\hfill {f}_{R1}\left(s\right)=\frac{{f}_{0}\left({t}_{i}+s-{t}_{\mathsf{\text{sp}}1}\right)}{1-{F}_{0}\left({t}_{i}-{t}_{\mathsf{\text{sp}}1}\right)},\hfill & \hfill 0<s<{T}_{1}^{i}\hfill \end{array}

(5)

Then, the average interference time is calculated as follows:

I\left({T}_{1}^{i},{c}_{1}\right)=\underset{0}{\overset{{T}_{1}^{i}}{\int}}\left({T}_{1}^{i}-s\right){f}_{R1}\left(s\right)ds=\frac{\left({T}_{1}^{i}+{c}_{1}\right)\left[{F}_{0}\left({T}_{1}^{i}+{c}_{1}\right)-{F}_{0}\left({c}_{1}\right)\right]-{\int}_{{c}_{1}}^{{T}_{1}^{i}+{c}_{1}}s{f}_{0}\left(s\right)ds}{1-{F}_{0}\left({c}_{1}\right)}

(6)

where *c*_{1} = *t*_{
i
} - *t*_{sp1} is the time interval starting from the most recent channel switch point *t*_{sp1} to the *i* th SU transmission.

It can be seen from (6) that the interference time I\left({T}_{1}^{i},c\right) is related to the transmission time interval {T}_{1}^{i} and the time interval *c*_{1}. While the second variable *c*_{1} can't be controlled, the interference time I\left({T}_{1}^{i},{c}_{1}\right) is limited by adjusting the transmission time interval {T}_{1}^{i}. It is noticed that the interference time I\left({T}_{1}^{i},{c}_{1}\right)is a monotonically increasing function of {T}_{1}^{i}, because the derivative of (6) with respect to {T}_{1}^{i} is positive, that is to say

\frac{\partial I\left({T}_{1}^{i},{c}_{1}\right)}{\partial {T}_{1}^{i}}=\frac{{F}_{0}\left({T}_{1}^{i}+{c}_{1}\right)-{F}_{0}\left({c}_{1}\right)}{1-{F}_{0}\left({c}_{1}\right)}>0

(7)

Let *α* represents the prescribed interference parameter, considering I\left({T}_{1}^{i},{c}_{1}\right) is a monotonically increasing function of {T}_{1}^{i}, the optimal transmission time interval {T}_{1,\mathsf{\text{opt}}}^{i} satisfying the interference limit is

\begin{array}{c}\mathsf{\text{max}}{T}_{1}^{i}\hfill \\ \begin{array}{cc}\hfill \mathsf{\text{s}}.\mathsf{\text{t}}.\hfill & \hfill I\left({T}_{1}^{i},{c}_{1}\right)=\frac{\left({T}_{1}^{i}+{c}_{1}\right)\left[{F}_{0}\left({T}_{1}^{i}+{c}_{1}\right)-{F}_{0}\left({c}_{1}\right)\right]-{\int}_{{c}_{1}}^{{T}_{1}^{i}+{c}_{1}}s{f}_{0}\left(s\right)ds}{1-{F}_{0}\left({c}_{1}\right)}\le \alpha \hfill \end{array}\hfill \end{array}

(8)

What is worth mentioning is that larger *α* results in larger interference time I\left({T}_{1}^{i},{c}_{1}\right), larger transmission time {T}_{1}^{i} and hence larger throughput of secondary system. Therefore, *α* can be seen as a tradeoff parameter between the interference time and the throughput of secondary system.

Equation (8) shows that {T}_{1,\mathsf{\text{opt}}}^{i} is a function of *c*_{1}, and {T}_{1,\mathsf{\text{opt}}}^{i} declines dramatically to a very small value with the increase of *c*_{1}. However, in reality every system has a minimum transmission time interval *T*_{1, min}. Besides, the maximum transmission time interval *T*_{1, max} is given to further limit the interference. Here, we suggest setting *T*_{1, max} to be max_{
x
}*f*_{0}(*x*). Therefore, the adaptive transmission time {T}_{1,\mathsf{\text{adp}}}^{i} is set to be

{T}_{1,\mathsf{\text{adp}}}^{i}=\mathsf{\text{max}}\left\{{T}_{1,\mathsf{\text{min}}},\mathsf{\text{min}}\left\{{T}_{1,\mathsf{\text{max}}},{T}_{1,\mathsf{\text{opt}}}^{i}\right\}\right\}

(9)

### 4.2 Algorithm for estimation of *t*_{sp1}

It is clear from (6), (8), and (9) that the calculation of the adaptive transmission time {T}_{1,\mathsf{\text{adp}}}^{i} depends on the information of the switch point time *t*_{sp1}. Thus, it is very important to estimate *t*_{sp1} for the proposed ICASST algorithm. Denote {t}_{\mathsf{\text{sp}}1}^{\prime} and {\widehat{t}}_{\mathsf{\text{sp}}1} as the old channel switch point and the estimated new channel switch point, respectively, and the real value of the new switch point is *t*_{sp1} = *t*_{
i
} + *s*.

In this article, minimum mean square error (MMSE) principle is adopted to estimate the new channel switch point *t*_{sp1}. Given that the channel state is ON at *t*_{i+1}and is OFF at *t*_{
i
}, the conditional probability density function of the residual time *s* of channel's OFF state is:

{f}_{R1|{t}_{i}}\left(s\right)=\frac{{f}_{R1}\left(s\right)}{\mathsf{\text{Pr}}\left\{{\mathcal{H}}_{1}\left({t}_{i}+{T}_{1}^{i}\right)|{\mathcal{H}}_{0}\left({t}_{i}\right)\right\}}=\frac{{f}_{0}\left({t}_{i}+s-{{t}^{\prime}}_{\mathsf{\text{sp}}1}\right)}{{F}_{0}\left({t}_{i}+{T}_{1}^{i}-{{t}^{\prime}}_{\mathsf{\text{sp}}1}\right)-{F}_{0}\left({t}_{i}-{{t}^{\prime}}_{\mathsf{\text{sp}}1}\right)},0<s<{T}_{1}^{i}

(10)

Then the mean squared estimation error *ε* between the estimated value {\widehat{t}}_{\mathsf{\text{sp}}1}and real value *t*_{sp1} is

\begin{array}{cc}\hfill \epsilon & =E\left[{\left({\widehat{t}}_{\mathsf{\text{sp}}1}-{t}_{\mathsf{\text{sp}}1}\right)}^{2}\right]=\epsilon {\left[{\widehat{t}}_{\mathsf{\text{sp}}1}-\left({t}_{i}+s\right)\right]}^{2}={\int}_{0}^{{T}_{1}^{i}}{({\widehat{t}}_{sp1}-({t}_{i}+s))}^{2}{f}_{R|{t}_{i}}\left(s\right)ds\hfill \\ ={\left({\widehat{t}}_{\mathsf{\text{sp}}1}-{{t}^{\prime}}_{\mathsf{\text{sp}}{1}^{\prime}}\right)}^{2}-\frac{2\times \left({\widehat{t}}_{\mathsf{\text{sp}}1}-{t}_{\mathsf{\text{sp}}1}^{\prime}\right)\times {\int}_{{c}_{1}^{\prime}}^{{c}_{1}^{\prime}+{T}_{1}^{i}}s{f}_{0}\left(s\right)ds}{{F}_{0}\left({c}_{1}^{\prime}+{T}_{1}^{i}\right)-{F}_{0}\left({c}_{1}^{\prime}\right)}+\frac{{\int}_{{c}_{1}^{\prime}}^{{c}_{1}^{\prime}+{T}_{1}^{i}}{s}^{2}{f}_{0}\left(s\right)ds}{{F}_{0}\left({c}_{1}^{\prime}+{T}_{1}^{i}\right)-{F}_{0}\left({c}_{1}^{\prime}\right)}\hfill \end{array}

(11)

where {c}_{1}^{\prime}={t}_{i}-{t}_{\mathsf{\text{sp}}1}^{\prime} is the time interval between the old channel switch point {t}_{\mathsf{\text{sp}}1}^{\prime} and the *i* th transmission.

The optimal estimate of the new channel switch point {t}_{\mathsf{\text{sp}}1}^{*} is the one which minimizes the mean squared estimation error *ε*, and the expression of {t}_{\mathsf{\text{sp}}1}^{*} is

{t}_{\mathsf{\text{sp}}1}^{*}=\mathsf{\text{arg}}\underset{{t}_{i}<{\widehat{t}}_{\mathsf{\text{sp}}1}<{t}_{i}+{T}_{1}^{i}}{\mathsf{\text{min}}}\left\{\epsilon =E\left[{\left({\widehat{t}}_{\mathsf{\text{sp}}1}-{t}_{\mathsf{\text{sp}}1}\right)}^{2}\right]\right\}

(12)