### 6.1 Case I: ideally no noise or negligible noise

In order to measure spectrum utilization, and compare it to the traditional periodical sensing, it is necessary firstly to figure out how much time the “sensing” stage occupies and how much time the “transmitting” stage occupies during transmission. In our new full duplex TSU, it is also necessary to estimate the durations of the “sensing” stage “transmitting and sensing”. Assuming that *D*
_{
H
} is the total duration of the spectrum holes in an observation interval, such as in Fig. 4, it can be written as \(D_{H} = \sum _{i=0}^{N_{H}-1} D_{i}\), where *N*
_{
H
} denotes the number of holes in the observation interval, and *D*
_{
i
}, indicates the time duration of the *i*
^{th} spectrum hole *h*
_{
i
}. Denoting \({T_{H}^{d}}\) as the total duration of the data transmission (i.e., the“transmitting ” stage in periodical sensing and the “transmitting and sensing” stage both at TSU) of all detected spectrum holes in an observation interval, it can be written as \({T_{H}^{d}} = \sum _{i=0}^{{N_{H}^{d}} - 1} T_{i}\), where \({N_{H}^{d}}\) denotes the total number of spectrum holes detected by SU during the observation interval, while *T*
_{
i
}, indicates the duration of the real data transmitting stage for the *i*
^{th} spectrum hole *h*
_{
i
}. Denote *η* as the utilization of the spectrum holes. Spectrum utilization, *η*, can thus be calculated as

$$ \eta = \frac{{T_{H}^{d}}}{D_{H}} = \frac{\sum_{i=0}^{{N_{H}^{d}} - 1} T_{i}}{\sum_{i=0}^{N_{H}-1} D_{i}}. $$

(39)

#### 6.1.1 Ideally no noise or negligible noise in periodical spectrum sensing

In the traditional periodical spectrum sensing, the duration of a spectrum hole, *D*
_{
i
}, which can be regarded as a random variable, as in [16], follows an exponential distribution with an assumed mean *μ*. Its cumulative distribution function (CDF) can therefore be given as

$$ F_{D_{i}}(D) = 1 - exp\left(-\frac{D}{\mu}\right), $$

(40)

and its probability density function (PDF) can be described as

$$ f_{D_{i}}(D) =\frac{1}{\mu}exp\left(-\frac{D}{\mu}\right). $$

(41)

In addition, *T*
_{
i
} can also be regarded as a random variable, since:

$$ T_{i} = D_{i} - N_{i}W=D_{i} - D_{i}f_{sens}W=D_{i}(1-f_{sens}W), $$

(42)

where *f*
_{
sens
} is the frequency of periodical spectrum sensing. It is a fixed value for a CR spectrum sensing system. *N*
_{
i
} is the sensing instants in the *i*th spectrum hole.

In order to compute the CDF of *T*
_{
i
}, for an arbitrary *T*, we have

$$ {}\begin{aligned} P\left(T_{i} \leq T \right) = P\left(D_{i} \leq \frac{T}{1-f_{sens}W}\right) = F_{D}\left(\frac{T}{1-f_{sens}W}\right). \end{aligned} $$

(43)

Therefore, its CDF and PDF can be obtained, respectively, as

$$ \left\{\begin{array}{lcr} F_{T_{i}}(T) = 1 - exp\left(-\frac{T}{\mu\left(1-f_{sens}W\right)}\right) \\ f_{T_{i}}(T) = \frac{1}{\mu\left(1-f_{sens}W\right)}exp\left(-\frac{T}{\mu\left(1-f_{sens}W\right)}\right) \end{array} \right.. $$

(44)

Furthermore, we have the expectation of *T*
_{
i
}:

$$\begin{array}{@{}rcl@{}} \bar{T} &=& E\{T_{i}\}\\ &=& \int_{0}^{+\infty} T \frac{1}{\mu\left(1-f_{sens}W\right)} exp\left(-\frac{T}{\mu\left(1-f_{sens}W\right)}\right) dT \\ &=& \mu \left(1-f_{sens}W\right) \end{array} $$

(45)

One must note that *f*
_{
sens
}
*W*≤1 because the sensing period \(\frac {1}{f_{sens}}\) is always greater or equal to the length *W* of the sensing window. It is therefore reasonable to assume that when the sensing frequency *f*
_{
sens
} increases, the duration of data transmission decreases.

If there is no noise or negligible noise, each valid spectrum hole is assumed to be detected. So, Eq. (39) can be written as

$$ \eta_{ideal} = \frac{{T_{H}^{d}}}{D_{H}} = \frac{\sum_{i=0}^{N_{H} - 1} T_{i}}{\sum_{i=0}^{N_{H}-1} D_{i}} = \frac{\bar{T}_{H}^{d}}{\bar{D}_{H}}, $$

(46)

where \(\bar {T}_{H}^{d} = \bar {T}\) and \(\bar {D}_{H} = \mu \). So spectrum utilization *η*
_{
period
} in an ideal periodical spectrum sensing system is obtained as:

$$ \eta^{period}_{ideal} = \frac{\bar{T}_{H}^{d}}{\bar{D}_{H}} = 1-f_{sens}W, $$

(47)

where W is a fixed value when the licensed channel is sensed by a sensing window with a fixed sized. In Eq. (47), one can conclude that the utilization of the spectrum decreases when the size of the sensing window becomes larger. This result makes sense because the wasted time when the spectrum is not used is equal to the size of the sensing window during the sensing stage.

#### 6.1.2 Ideally no noise or negligible noise when sensing and transmitting at the same time

Similar to traditional periodical spectrum sensing, the duration *D*
_{
i
} of a spectrum hole, at a full duplex TSU, can be regarded as a random variable foll an exponential distribution with an assumed mean *μ* whose cumulative distribution function (CDF) and probability density function (PDF) are shown in Eqs. (40) and (41).

Similarly, *T*
_{
i
} can also be regarded as a random variable, indicating the duration of the “sensing and transmitting ” stage. In each spectrum hole, data transmission always happens except during the first spectrum sensing window. Thus, the transmission duration can be described as:

$$ T_{i} = D_{i} - W. $$

(48)

In order to compute the CDF of *T*
_{
i
}, for an arbitrary *T*, we have

$$ P\left(T_{i} \leq T \right) = P\left(D_{i} \leq T+W\right) = F_{D}\left(T+W\right). $$

(49)

Therefore, its CDF and PDF can be obtained, respectively, as

$$ \left\{\begin{array}{lcr} F_{T_{i}}(T) = 1 - exp\left(-\frac{T+W}{\mu}\right) \\ f_{T_{i}}(T) = \frac{1}{\mu}exp\left(-\frac{T+W}{\mu}\right) \end{array} \right.. $$

(50)

Furthermore, we have the expectation of *T*
_{
i
} as

$$\begin{array}{@{}rcl@{}} \bar{T} &=& E\{T_{i}\} = \int_{0}^{+\infty} T \frac{1}{\mu}exp\left(-\frac{T+W}{\mu}\right) dT \\ &=& \mu exp\left(-\frac{W}{\mu}\right) \end{array} $$

(51)

If there is no noise or negligible noise, each valid spectrum hole is assumed to be detected. So, Eq. (39) can be written as

$$ \eta^{duplex}_{ideal} = \frac{{T_{H}^{d}}}{D_{H}} = \frac{\sum_{i=0}^{N_{H} - 1} T_{i}}{\sum_{i=0}^{N_{H}-1} D_{i}} = \frac{\bar{T}_{H}^{d}}{\bar{D}_{H}}= exp\left(-\frac{W}{\mu}\right), $$

(52)

where \(\bar {T}_{H}^{d} = \bar {T}\) and \(\bar {D}_{H} = \mu \). In Eq. (52), *W* represents the size of the first sensing window in one spectrum hole. The utilization of the spectrum also decreases when the size of the sensing window *W* becomes larger. The size of the first sensing window is adaptive and changeable. Its range, *W*
_{
adaptive
} should be *W*
_{
min
}<*W*
_{
adaptive
}<*W*
_{
max
}. The aim of having an adaptive window is to decrease *W* and improve spectrum utilization. It regulates the trade-off between the probability of detection and spectrum utilization because the probability of detection increases with *W*, while spectrum utilization decreases with *W*.

### 6.2 Case II: noisy environment

In general, there is non-negligible noise which increases the probability of false alarms. False alarms cause spectrum holes not to be used. Thus, spectrum utilization is affected by the probability of false alarm.

First, when spectrum sensing is carried out only at TSU, spectrum utilization in Eq. (39) can be expressed as

$$ {}\begin{aligned} \eta_{noise} = \frac{{T_{H}^{d}}}{D_{H}} = \frac{\sum_{i=0}^{N_{H} - 1} \left(T_{i}-T_{i}^{loss}\right)}{\sum_{i=0}^{N_{H}-1} D_{i}} = \frac{\bar{T}_{H}^{d}}{\bar{D}_{H}}= \frac{\bar{T}-\bar{T}^{loss}}{\bar{D}_{H}}, \end{aligned} $$

(53)

where \({T}_{i}^{loss}\) denotes the wasted durations in the *i*th spectrum hole *h*
_{
i
} which are caused by false alarms while \(\bar {D}_{H} = \mu \), \(\bar {T}_{H}^{d} = \bar {T}-\bar {T}^{loss}\), \(\bar {T}^{loss}\) is defined as the expected value of the wasted spectrum duration \(E\{T_{i}^{loss} \}\) in *i*th spectrum hole.

#### 6.2.1 Noisy environment in periodical spectrum sensing

In a periodical spectrum sensing scheme, \(E\left \{T_{i}^{loss} \right \}\) comes entirely from false-alarms during the “sensing” stage. It can be expressed as the expected value \(E\left \{T_{i}^{loss} \left | \text {sensing stage} \right.\right \}\) of all wasted durations in one period of spectrum sensing which denotes as \(\bar {W}_{loss}\):

$$ E\left\{T_{i}^{loss} \right\}= \bar{N}E\left\{T_{i}^{loss}\left|\text{sensing stage}\right.\right\} = \bar{N} \bar{W}_{loss}, $$

(54)

where \(\bar {N}\) is the expected value of the number of spectrum sensing times in each spectrum hole. According to Eq. (42), we can obtain the expression below:

$$ \bar{D}_{H} = \mu= \bar{N}\left(W+\bar{W}_{loss}\right)+\bar{T}= \bar{N}\left(\frac{1}{f_{sens}}+\bar{W}_{loss}\right), $$

(55)

Thus, the expected value of the number of spectrum sensing times in each spectrum hole \(\bar {N} \) is:

$$ \bar{N} = \frac{\mu}{\frac{1}{f_{sens}}+\bar{W}_{loss}}, $$

(56)

When spectrum sensing is only based on energy detection at TSU, \(E\{T_{i}^{loss} | \text {sensing stage} \}\) depends on the probability of false alarm \(P_{f_{1}}\). at TSU. It can be derived as follows:

$$ {}\begin{aligned} E\left\{T_{i}^{loss} \left| \text{sensing stage} \right.\right\} &= \bar{W}_{loss}\\ &= WP_{f_{1}}{\sum_{r=0}^{+\infty} \left((r+1)P_{f_{1}}^{r}\right)} \\ &= WP_{f_{1}}\left(1+2P_{f_{1}}+3P_{f_{1}}^{2}+\ldots\ldots\ldots nP_{f_{1}}^{n-1}\right) \\ &= {\lim}_{n\rightarrow +\infty} {WP_{f_{1}} \left[{\frac{1-P_{f_{1}}^{n}}{(1-P_{f_{1}})^{2}}-\frac{nP_{f_{1}}^{n}}{1-P_{f_{1}}}}\right]} \\ &\approx {\frac{WP_{f_{1}} }{(1-P_{f_{1}})^{2}}} \end{aligned} $$

(57)

Here, *W* is the size of the spectrum sensing window. Its value is *W*=*W*
_{
max
}. This is because, in our adaptive window algorithm the size of the sensing window does not change when PU is inactive.

According to Eqs. (55) and (56), \(E\{T_{i}^{loss} \}\) is expressed as:

$$ \bar{T}= \mu-\bar{N}(W+\bar{W}_{loss})=\mu-\frac{\mu(W+\bar{W}_{loss})}{\frac{1}{f_{sens}}+\bar{W}_{loss}}, $$

(58)

Then, according to Eqs. (57) and (58), we can derive the utilization of the spectrum \(\eta ^{period}_{noise}\) in a periodical spectrum sensing system in a noisy environment as:

$$ \eta^{period}_{noise} = \frac{\bar{T}}{\bar{D}_{H}}=\frac{\frac{1}{f_{sens}W}-1}{\frac{1}{f_{sens}W}+\frac{P_{f_{1}} }{(1-P_{f_{1}})^{2}}}. $$

(59)

From Eq. (59), one can conclude that spectrum utilization \(\eta ^{period}_{noise}\) decreases when the probability of false-alarm \(P_{f_{1}}\) increases. On the other hand, the utilization \(\eta ^{period}_{noise}\) becomes lower when the size of the sensing window *W* becomes larger which implies that Eq. (59) makes sense.

#### 6.2.2 Noisy environment when sensing and transmitting at the same time

When sensing and transmitting at the same time in a full duplex TSU, the expected value of the wasted durations \(\bar {T}^{loss}_{duplex}\) in each spectrum hole consists of two components. The first is the expected value of the wasted spectrum durations during the sensing stage. The other one is the wasted spectrum durations during the transmitting and sensing stage. In other words, we have

$$ {}\begin{aligned} \bar{T}^{loss}_{duplex}&= E\left\{T_{i}^{loss}\right\} \\ &= E\left\{T_{i}^{loss} \left| \text{sensing stage} \right.\right\}\\ &\quad + N_{duplex}E\left\{T_{i}^{loss} \left| \text{sensing and transmitting stage}\right.\right\},\\ \end{aligned} $$

(60)

where *N*
_{
duplex
} represents the number of sensing times in transmitting and sensing stage. In Eq. (60), \(E\{T_{i}^{loss} | \text {sensing stage} \}\) is the expected value of the wasted spectrum durations during the spectrum sensing in the sensing stage. Its expression is shown in Eq. (57). The sensing stage occurs once at the beginning of the spectrum hole.

In Eq. (60), \(E\left \{T_{i}^{loss} \left | \text {sensing and transmitting stage}\right.\right \}\) denotes the expected value of the wasted spectrum durations during the transmitting and sensing stage.

Similar to Eq. (57), we can obtain the expected value of the wasted spectrum durations during the transmitting and sensing stage as:

$$ {}\begin{aligned} E\left\{T_{i}^{loss} \left| \text{sensing stage and transmitting stage} \right.\right\} &\,=\, WP_{f_{2}}{\sum_{r=0}^{+\infty} \left((r+1)P_{f_{1}}^{r}\right)} \\ &\,=\, WP_{f_{2}}\left(1+2P_{f_{1}}+3P_{f_{1}}^{2}\right.\\&~\,\qquad\qquad\left.+\ldots\ldots\ldots nP_{f_{1}}^{n-1}\right) \\ &\,=\, {\lim}_{n\rightarrow +\infty} \!{WP_{f_{2}} \!\!\left[\!{\frac{1\,-\,P_{f_{1}}^{n}}{(1\,-\,P_{f_{1}})^{2}}\,-\,\frac{nP_{f_{1}}^{n}}{1\,-\,P_{f_{1}}}}\!\right]} \\ &\!\approx\! {\frac{WP_{f_{2}} }{(1-P_{f_{1}})^{2}}} \end{aligned} $$

(61)

By comparing Eq. (57) with Eq. (61), one can see that the only difference between \(E\left \{T_{i}^{loss} \left | \text {sensing stage} \right.\right \}\) and \(E\{T_{i}^{loss} | \text {transmitting and sensing stage}\}\) is that the probability of false alarm in hypothesis *H*
_{0} is different from the corresponding false alarm in *H*
_{2}.

In addition, in order to derive the utilization of a spectrum hole, we need to know *n*
_{
duplex
} since the average duration of the “transmitting and sensing” stage \(\bar {T}\) is

$$ \bar{T}= {\frac{WP_{f_{1}} }{(1-P_{f_{1}})^{2}}}+ \bar N_{duplex}\left[{\frac{WP_{f_{2}} }{(1-P_{f_{1}})^{2}}}+W\right], $$

(62)

Thus, \(\bar N_{duplex}\) can be expressed as:

$$ \bar N_{duplex}= {\frac{\bar{T}-{\frac{WP_{f_{1}} }{\left(1-P_{f_{1}}\right)^{2}}} }{{\frac{WP_{f_{2}} }{\left(1-P_{f_{1}}\right)^{2}}}+W}}, $$

(63)

In Eq. (62), the duration of the “transmitting and sensing” stage includes the wasted durations in both the sensing stage and the transmitting and sensing’ stage as well as the used spectrum in hypothesis *H*
_{2}. From Eq. (63), we can obtain the following:

$$\begin{array}{@{}rcl@{}} \bar{T}-\bar{T}^{loss} &=& \bar{T}- {\frac{WP_{f_{1}} }{\left(1-P_{f_{1}}\right)^{2}}}-\bar N_{duplex}{\frac{WP_{f_{2}} }{\left(1-P_{f_{1}}\right)^{2}}} \\ &=& \bar N_{duplex}W \\ &=& {\frac{\bar{T}-{\frac{WP_{f_{1}} }{\left(1-P_{f_{1}}\right)^{2}}} }{{\frac{P_{f_{2}} }{\left(1-P_{f_{1}}\right)^{2}}}+1}} \end{array} $$

(64)

We substitute Eqs. (51) and (64) into Eq. (53), to obtain an expression for the spectrum utilization \(\eta _{noise}^{duplex}\):

$$\begin{array}{@{}rcl@{}} \eta_{noise}^{duplex} &=& \frac{{T_{H}^{d}}}{D_{H}} = \frac{\bar{T}-\bar{T}^{loss}}{\bar{D}_{H}} \\ &=& {\frac{\mu exp\left(-\frac{W}{\mu}\right)-{\frac{WP_{f_{1}} }{\left(1-P_{f_{1}}\right)^{2}}} }{\mu\left({\frac{P_{f_{2}} }{\left(1-P_{f_{1}}\right)^{2}}}+1\right)}} \end{array} $$

(65)

In Eq. (65), when the probabilities of false alarm \(P_{f_{1}}\) and \(P_{f_{2}}\) increase, the utilization \(\eta _{noise}^{duplex}\) becomes smaller. On the other hand, the utilization *η* also becomes lower when the size of the sensing window *W* becomes larger. Thus, we can conclude that Eq. (65) makes senses.

### 6.3 Spectrum utilization in cooperative spectrum sensing between TSU and RSU

Next, we introduce the utilization of a spectrum hole when we use our new spectrum sensing scheme, i.e., when we combine BER estimation with energy detection to realize spectrum sensing. When we use the new spectrum sensing method, all relevant expressions are the same as Eqs. (60)–(65) except that we use \(P_{f_{1}}^{coop}\) and \(P_{f_{2}}^{coop}\) to replace the original \(P_{f_{1}}\) and \(P_{f_{2}}\). Moreover, the duration of the “transmitting and sensing” stage not only includes the wasted duration as well as the used spectrum in hypothesis *H*
_{2}, but also includes the length of the training sequence which is used in BER estimation. So Eq. (62) can be rewritten as:

$$ {}\begin{aligned} \bar{T}&= {\frac{(W+\bar{W_{ts}})P_{f_{1}}^{coop} }{\left(1-P_{f_{1}}^{coop}\right)^{2}}} +\bar{W_{ts}}+ \bar{N}_{coop}\left[{\frac{(W+\bar{W_{ts}})P_{f_{2}}^{coop} }{\left(1-P_{f_{1}}^{coop}\right)^{2}}}+W+\bar{W_{ts}}\right] \\ &= {\frac{\left[W+(1-P_{f_{1}})W_{ts}\right]P_{f_{1}}^{coop} }{\left(1-P_{f_{1}}^{coop}\right)^{2}}} +(1-P_{f_{1}})W_{ts}\\&\quad+ \bar{N}_{coop}\left[{\frac{\left(W+\left(1-P_{f_{1}}\right)W_{ts}\right)P_{f_{2}}^{coop} }{\left(1-P_{f_{1}}^{coop}\right)^{2}}} \right. \\ &\quad+\left.W+\left(1-P_{f_{2}}\right)W_{ts}\right] \end{aligned} $$

(66)

Here, \(\bar {W_{ts}}\) is the expected value of the length of the training sequence and *W*
_{
ts
} is the length of training sequence in each estimation of BER. Thus, the number of sensing times \(\bar N_{coop}\) can be expressed as:

$$ {}\begin{aligned} \bar N_{coop} &= {\frac{\bar{T}- {\frac{(W+(1-P_{f_{1}})W_{ts})P_{f_{1}}^{coop} }{\left(1-P_{f_{1}}^{coop}\right)^{2}}}-\left(1-P_{f_{1}}\right)W_{ts}}{{\frac{\left(W+\left(1-P_{f_{1}}\right)W_{ts}\right)P_{f_{2}}^{coop} }{\left(1-P_{f_{1}}^{coop}\right)^{2}}}+W+\left(1-P_{f_{2}}\right)W_{ts}}} \\ &= {\frac{\bar{T}\left(1-P_{f_{1}}^{coop}\right)^{2}- {\left(W+\left(1-P_{f_{1}}\right)W_{ts}\right)P_{f_{1}}^{coop} }-\left(1-P_{f_{1}}^{coop}\right)^{2}\left(1-P_{f_{1}}\right)W_{ts}}{{\left(W+\left(1-P_{f_{1}}\right)W_{ts}\right)P_{f_{2}}^{coop} }+\left(W+\left(1-P_{f_{2}}\right)W_{ts}\right)\left(1-P_{f_{1}}^{coop}\right)^{2}}} \\ \end{aligned} $$

(67)

Then \(\bar {T}^{loss}\) can be expressed according to Eqs. (66) and (67) as

$$ {}\begin{aligned} \bar{T}^{loss}&= {\frac{WP_{f_{1}}^{coop} }{\left(1-P_{f_{1}}^{coop}\right)^{2}}}+ \bar{N}_{coop}{\frac{WP_{f_{2}}^{coop} }{\left(1-P_{f_{1}}^{coop}\right)^{2}}} \\ &= {\frac{WP_{f_{1}}^{coop} }{\left(1-P_{f_{1}}^{coop}\right)^{2}}}+W {\frac{\bar{T}- {\frac{\left(W+\left(1-P_{f_{1}}\right)W_{ts}\right)P_{f_{1}}^{coop} }{\left(1-P_{f_{1}}^{coop}\right)^{2}}}-\left(1-P_{f_{1}}\right)W_{ts}}{{W+\left(1-P_{f_{1}}\right)W_{ts} }+{\frac{\left(W+\left(1-P_{f_{2}}\right)W_{ts}\right)\left(1-P_{f_{1}}^{coop}\right)^{2}}{P_{f_{2}}^{coop}}}}} \\ &= {\frac{WP_{f_{1}}^{coop} }{\left(1-P_{f_{1}}^{coop}\right)^{2}}}+ {\frac{\bar{T}- {\frac{\left(W+\left(1-P_{f_{1}}\right)W_{ts}\right)P_{f_{1}}^{coop} }{\left(1-P_{f_{1}}^{coop}\right)^{2}}}-(1-P_{f_{1}})W_{ts}}{{1+\left(1-P_{f_{1}}\right)\frac{W_{ts}}{W} }+{\frac{\left(1+\left(1-P_{f_{2}}\right)\frac{W_{ts}}{W}\right)\left(1-P_{f_{1}}^{coop}\right)^{2}}{P_{f_{2}}^{coop}}}}} \end{aligned} $$

(68)

Finally, it is easy to derive the utilization of the spectrum \(\eta _{noise}^{coop} \) as

$$\begin{array}{@{}rcl@{}} \eta_{noise}^{coop} &=& \frac{{T_{H}^{d}}}{D_{H}} = \frac{\bar{T}-\bar{T}^{loss}}{\bar{D}_{H}} \\ &=& exp\left(-\frac{W}{\mu}\right)-{\frac{WP_{f_{1}}^{coop} }{\mu\left(1-P_{f_{1}}^{coop}\right)^{2}}} \\ &&-{\frac{exp\left(-\frac{W}{\mu}\right)-{\frac{\left(W+\left(1-P_{f_{1}}\right)W_{ts}\right)P_{f_{1}}^{coop} }{\mu(1-P_{f_{1}}^{coop})^{2}}}-\frac{\left(1-P_{f_{1}}\right)W_{ts}}{\mu}}{{1+\left(1-P_{f_{1}}\right)\frac{W_{ts}}{W} }+{\frac{\left(1+\left(1-P_{f_{2}}\right)\frac{W_{ts}}{W}\right)\left(1-P_{f_{1}}^{coop}\right)^{2}}{P_{f_{2}}^{coop}}}}} \\ \end{array} $$

(69)

From Eqs. (68) and (69), we can conclude that the utilization of the spectrum depends on the probability of cooperative false-alarm \(P_{f_{2}}^{coop}\). The loss of spectrum \(\bar {T}^{loss}\) becomes larger when the cooperative probability of false-alarm at the “transmitting and sensing” stage \(P_{f_{2}}^{coop}\) increases. This is reasonable because the “transmitting and sensing” stage occupies most of the spectrum hole for a CR full duplex system. If \(P_{f_{2}}^{coop}\) increases, it implies that the CR transmitter will spend more time on spectrum sensing instead of sensing and transmitting. In other words, some of the spectrum hole is missed without transmitting data at TSU. Thus, it is reasonable to assume that the utilization of the spectrum \(\eta _{noise}^{coop}\) decreases with the increase in \(P_{f_{2}}^{coop}\) in Eq. (69).

In addition, according to Eqs. (68) and (69), we can also conclude that spectrum utilization \(\eta _{noise}^{coop}\) is larger when the training sequence *W*
_{
ts
} that is used in BER estimation has a larger duration. However, it is possible that the longer length of the training sequence causes interference to PU especially at the end of a spectrum hole when PU might become active.