A. Optimal Sensing Duration and Maximal Achievable Data Rate of Cognitive Radio

For the energy detection based spectrum sensing approach, the statistical test result based on a set of detected samples can be expressed as [1] $$\begin{equation} \mathcal {t}\left ({\mathbf {Y} }\right)=\frac {1}{K}\sum \limits _{k=1}^{K} \left |{ y\left ({k }\right) }\right |^{2} \end{equation}$$ View Source\begin{equation} \mathcal {t}\left ({\mathbf {Y} }\right)=\frac {1}{K}\sum \limits _{k=1}^{K} \left |{ y\left ({k }\right) }\right |^{2} \end{equation} where $$\begin{equation} \mathbf {Y}=\left \{{{y\left ({k }\right)}\vert {y\left ({k }\right)=s\left ({k }\right)+z\left ({k }\right)\!,~k\in \mathbf {K}}}\right \}\!, \end{equation}$$ View Source\begin{equation} \mathbf {Y}=\left \{{{y\left ({k }\right)}\vert {y\left ({k }\right)=s\left ({k }\right)+z\left ({k }\right)\!,~k\in \mathbf {K}}}\right \}\!, \end{equation} Also, $k$ and $\mathbf {K}$ represent the sample-index and the set including those indices with cardinality of $K$ , respectively. $K=f_{\mathrm {s}}\tau$ , where $f_{\mathrm {s}}$ and $\tau$ are the sampling rate and sensing duration, respectively. $y\left ({k }\right)$ is the $k$ th sample for testing. $s\left ({k }\right)$ and $z(k)$ are the $n$ th items of the signal and noise sample-sets that are mutually independent with variances of $\sigma _{s}^{2}$ and $\sigma _{z}^{2}$ , respectively. $z\left ({k }\right)\sim \text {CN}(0,\sigma _{z}^{2})$ (i.e., it follows a circular symmetric complex Gaussian (CSCG) distribution with mean of 0 and variance of $\sigma _{z}^{2}$ ) [13], [14]. For a sensing operation via a Nakagami-$m$ fading channel, the conditional probability density function (PDF) of the instantaneous signal to noise ratio (SNR) is given as follows [14], [15] $$\begin{align} f_{\dot {\gamma }}(x\vert \gamma)=\frac {1}{\Gamma \left ({m }\right)}\left ({\frac {m}{\gamma } }\right)^{m}x^{m-1}e^{\left ({- \frac {m}{\gamma }x }\right)},\quad x\ge 0,~ m\ge 1/2\notag \\ {}\end{align}$$ View Source\begin{align} f_{\dot {\gamma }}(x\vert \gamma)=\frac {1}{\Gamma \left ({m }\right)}\left ({\frac {m}{\gamma } }\right)^{m}x^{m-1}e^{\left ({- \frac {m}{\gamma }x }\right)},\quad x\ge 0,~ m\ge 1/2\notag \\ {}\end{align} where $m$ is the Nakagami fading parameter. A smaller $m$ results in more severe multipath fading. $m=1/2$ , $m=1$ , $m>1$ and $m=\infty$ represent the single-side Gaussian, Rayleigh, Rician and line of sight (LOS) channels, respectively [14], [15]. $\Gamma \left ({\cdot }\right)$ denotes the Gamma function [13], [23], $\dot {\gamma }$ is the instantaneous SNR, and $\gamma$ is the mean of $\dot {\gamma }$ defined as $\gamma =\sigma _{s}^{2}/\sigma _{z}^{2}$ . Formula (3) indicates that $\dot {\gamma }\sim \mathrm {Ga}\left ({m,\gamma /m }\right)$ , where the sign ‘~’ denotes ‘follow’ and $\mathrm {Ga}\left ({k,\theta }\right)$ as a generic sign represents the Gamma distribution with $k$ and $\theta$ being its shape and scale parameter, respectively [13], [23].

Obviously, $\gamma$ is also a random variable (RV) in the sense of longer term fading due to the path loss or shadowing [14]. For the simplification of analysis, the path loss in this instance is assumed to be constant during sensing. Thus, $\gamma$ usually meets a logarithm Gaussian distribution [13], [14]. For the ease of mathematical handling, it is typical to employ the Gamma distribution as its approximate alternative, the PDF of which is given by [16]. $$\begin{equation} f_{\gamma }\left ({y\vert \bar {\gamma } }\right)=\frac {y^{M-1}}{\Gamma \left ({M }\right)\bar {\gamma }^{M}}\exp \left ({- \frac {y}{\bar {\gamma }} }\right)\!,\quad y\ge 0,~M\ge 1/2\quad \end{equation}$$ View Source\begin{equation} f_{\gamma }\left ({y\vert \bar {\gamma } }\right)=\frac {y^{M-1}}{\Gamma \left ({M }\right)\bar {\gamma }^{M}}\exp \left ({- \frac {y}{\bar {\gamma }} }\right)\!,\quad y\ge 0,~M\ge 1/2\quad \end{equation} where $M$ is a parameter inverse-proportionally reflecting the severity of shadow fading and $\bar {\gamma }$ is a parameter identical to the mean of $\gamma$ (i.e., the mean SNR of shadow fading). An overall view of Eqs. (3) and (4) suggests that $\dot {\gamma }$ experiences a Nakagami-Gamma shadowed fading [16], the PDF of which, if taking the form of $f_{\dot {\gamma }}\left ({x\thinspace \vert \thinspace \bar {\gamma }}\right)$ , will contain the modified Bessel function of the second kind known as the generalized-$K$ model [22].

Let us focus on the context that the licensed (i.e., primary) signal does not vary so fast that it is considered to be a constant during sensing. It propagates via an aforementioned fading channel to arrive at the detector of a cognitive (i.e., secondary) system. Therefore, the variation of $s\left ({k }\right)$ depends only on the channel. The temporal spacing of $s\left ({k }\right)\!, k=1\cdots K$ is sufficient so that they can be viewed as uncorrelated. Either $z(k)$ or $\mathrm {s}\left ({k }\right)\!,k=1\cdots K$ have an independent identical distribution (i.i.d.).

Under hypothesis $\mathcal {H}_{0}$ (i.e., $s\left ({k }\right)=0, k=1\cdots K),\mathcal {t}\left ({\mathbf {Y} }\right)$ of (1) can be rewritten as $$\begin{equation} \mathcal {t}\left ({\mathbf {Y}\vert \mathcal {H}_{0}}\right) =\mathcal {t}\left ({\left \{{z\left ({k }\right)\!, k\in \mathbf {K} }\right \} }\right)=\frac {1}{K}\sum \limits _{k=1}^{K} \left |{ z\left ({k }\right) }\right |^{2} \end{equation}$$ View Source\begin{equation} \mathcal {t}\left ({\mathbf {Y}\vert \mathcal {H}_{0}}\right) =\mathcal {t}\left ({\left \{{z\left ({k }\right)\!, k\in \mathbf {K} }\right \} }\right)=\frac {1}{K}\sum \limits _{k=1}^{K} \left |{ z\left ({k }\right) }\right |^{2} \end{equation} which follows the Chi-square distribution with a degree of freedom of $2K$ if $\sigma _{z}^{2}=1$ [13], a special case of Gamma distribution. Here, we still regard it as Gamma distributed, represented as [13] $$\begin{equation} \mathcal {t}\left ({\mathbf {Y}\vert \mathcal {H}_{0}}\right) \sim \mathrm {Ga}\left ({K, \sigma _{z}^{2}/K }\right)\!. \end{equation}$$ View Source\begin{equation} \mathcal {t}\left ({\mathbf {Y}\vert \mathcal {H}_{0}}\right) \sim \mathrm {Ga}\left ({K, \sigma _{z}^{2}/K }\right)\!. \end{equation}

In the other case of hypothesis, $\mathcal {H}_{1}$ (i.e., $s\left ({k }\right)\ne 0, k=1\cdots K)\,\,\mathcal {t}\left ({\mathbf {Y} }\right)$ of (1) can be rewritten as $$\begin{equation} \mathcal {t}\left ({\mathbf {Y}\vert \mathcal {H}_{1}}\right) =\mathcal {t}\left ({\left \{{y\left ({k }\right)\!,k\in \mathbf {K} }\right \} }\right)=\frac {1}{K}\sum \limits _{k=1}^{K} \left |{ s\left ({k }\right)+z\left ({k }\right) }\right |^{2}\quad \end{equation}$$ View Source\begin{equation} \mathcal {t}\left ({\mathbf {Y}\vert \mathcal {H}_{1}}\right) =\mathcal {t}\left ({\left \{{y\left ({k }\right)\!,k\in \mathbf {K} }\right \} }\right)=\frac {1}{K}\sum \limits _{k=1}^{K} \left |{ s\left ({k }\right)+z\left ({k }\right) }\right |^{2}\quad \end{equation} which is also observed to be Gamma distributed. Therefore, a proposition needs to be proposed as follows.

The mean and variance of $\mathcal {t}\left ({\mathbf {Y}\thinspace \vert \thinspace \mathcal {H}_{1}}\right)$ of (7) (denoted $\mu _{\mathcal {H}_{1}}$ and $\sigma _{\mathcal {H}_{1}}^{2}$ , respectively) are also derivable (see Appendix A) based the corresponding formulae of Gamma distribution [13], given by $$\begin{align*} \mu _{\mathcal {H}_{1}}=&\kappa \theta =\left ({1+\gamma }\right)\sigma _{z}^{2},\tag{10.a}\\ \sigma _{\mathcal {H}_{1}}^{2}=&\kappa \theta ^{2}=\frac {\left ({1+\gamma }\right)^{2}\sigma _{z}^{4}}{K\xi \left ({m,\gamma }\right)}. \tag{10.b}\end{align*}$$ View Source\begin{align*} \mu _{\mathcal {H}_{1}}=&\kappa \theta =\left ({1+\gamma }\right)\sigma _{z}^{2},\tag{10.a}\\ \sigma _{\mathcal {H}_{1}}^{2}=&\kappa \theta ^{2}=\frac {\left ({1+\gamma }\right)^{2}\sigma _{z}^{4}}{K\xi \left ({m,\gamma }\right)}. \tag{10.b}\end{align*} Similarly, for $\mathcal {t}\left ({\mathbf {Y}\thinspace \vert \thinspace \mathcal {H}_{0}}\right)$ , its mean and variance are given by $$\begin{align*} \mu _{\mathcal {H}_{0}}=&\sigma _{z}^{2}, \tag{11.a}\\ \sigma _{\mathcal {H}_{0}}^{2}=&\sigma _{z}^{4}/K. \tag{11.b}\end{align*}$$ View Source\begin{align*} \mu _{\mathcal {H}_{0}}=&\sigma _{z}^{2}, \tag{11.a}\\ \sigma _{\mathcal {H}_{0}}^{2}=&\sigma _{z}^{4}/K. \tag{11.b}\end{align*}

Substituting $m=1$ and $m\to \infty$ into (10.b) yields $\sigma _{\mathcal {H}_{1}}^{2}=\sigma _{z}^{4}{(\gamma +1)}^{2}/K$ and $\sigma _{\mathcal {H}_{1}}^{2}=\sigma _{z}^{4}(2\gamma +1)/K$ , coinciding with the 3rd and 1st cases of (8) of [1], respectively. Thus, formula (10) provides a more general form than [1].

Let $p_{\mathrm {d}}$ denote the detection probability that the CR system detects the presence of a licensed signal while it is indeed active, and $p_{\mathrm {f}}$ is the false alarm probability such that the CR system falsely declares the presence of the licensed signal in its absence. In [1], it focuses on a context in which the signal is complex PSK-modulated and is propagated along a LOS path while the noise is CSCG distributed. This paper is focused on a new context in which the licensed system’s transmitted signal does not vary significantly within the cognitive system’s sensing duration so that it can be considered invariant. Nevertheless, it goes through a Nakagami-Gamma shadowed fading channel prior to its arrival. In such a new context, let us re-derive $p_{\mathrm {d}}$ and $p_{\mathrm {f}}$ as follows.

Based on proposition 1, the PDF of $\mathcal {t}\left ({\mathbf {Y}\thinspace \vert \thinspace \mathcal {H}_{1}}\right)$ is given by $$\begin{equation*} f_{\mathcal {t}\left ({\mathbf {Y}\vert \mathcal {H}_{1}}\right)}\left ({x }\right)=\frac {x^{\kappa -1}}{\Gamma \left ({\kappa }\right)\theta ^{\kappa }}\exp \left ({- \frac {x}{\theta } }\right)\!, \quad x\ge 0. \tag{12}\end{equation*}$$ View Source\begin{equation*} f_{\mathcal {t}\left ({\mathbf {Y}\vert \mathcal {H}_{1}}\right)}\left ({x }\right)=\frac {x^{\kappa -1}}{\Gamma \left ({\kappa }\right)\theta ^{\kappa }}\exp \left ({- \frac {x}{\theta } }\right)\!, \quad x\ge 0. \tag{12}\end{equation*}

Accordingly, $p_{\mathrm {d}}$ is derivable to be $$\begin{align*}&\hspace {-2pc}p_{\mathrm {d}}\left ({m,\gamma,\varepsilon }\right) \\=&\int _\varepsilon ^\infty {f_{\mathcal {t}\left ({\mathbf {Y}\thinspace \vert \thinspace \mathcal {H}_{1}}\right) }\left ({x }\right)\mathrm {d}x} \\=&\frac {1}{\Gamma \left ({\kappa }\right)}\int _{\varepsilon /\theta }^\infty {\left ({\frac {x}{\theta } }\right)^{\kappa -1}\exp \left ({- \frac {x}{\theta } }\right)\mathrm {d}\frac {x}{\theta }} =\frac {\Gamma \left ({\kappa,\varepsilon /\theta }\right)}{\Gamma \left ({\kappa }\right)} \\=&\Gamma _{0}\left ({\kappa,\varepsilon /\theta }\right)=\Gamma _{0}\left ({K\xi \left ({m,\gamma }\right)\!,\frac {\varepsilon K\xi \left ({m,\gamma }\right)}{\sigma _{z}^{2}\left ({1+\gamma }\right)} }\right) \tag{13}\end{align*}$$ View Source\begin{align*}&\hspace {-2pc}p_{\mathrm {d}}\left ({m,\gamma,\varepsilon }\right) \\=&\int _\varepsilon ^\infty {f_{\mathcal {t}\left ({\mathbf {Y}\thinspace \vert \thinspace \mathcal {H}_{1}}\right) }\left ({x }\right)\mathrm {d}x} \\=&\frac {1}{\Gamma \left ({\kappa }\right)}\int _{\varepsilon /\theta }^\infty {\left ({\frac {x}{\theta } }\right)^{\kappa -1}\exp \left ({- \frac {x}{\theta } }\right)\mathrm {d}\frac {x}{\theta }} =\frac {\Gamma \left ({\kappa,\varepsilon /\theta }\right)}{\Gamma \left ({\kappa }\right)} \\=&\Gamma _{0}\left ({\kappa,\varepsilon /\theta }\right)=\Gamma _{0}\left ({K\xi \left ({m,\gamma }\right)\!,\frac {\varepsilon K\xi \left ({m,\gamma }\right)}{\sigma _{z}^{2}\left ({1+\gamma }\right)} }\right) \tag{13}\end{align*} where $\Gamma \left ({\cdot,\cdot }\right)$ denotes the upper incomplete Gamma function, $\Gamma _{0}\left ({\cdot,\cdot }\right)$ is its regularized form [13], [23], and $\varepsilon$ is the decision threshold for the detection of the result. Formula (13) indicates that $p_{\mathrm {d}}$ depends on the three parameters of $m,\gamma$ and $\varepsilon$ .

To limit the CR’s interference in the licensed system under a specified level, it is usually required that $p_{\mathrm {d}}\ge p_{\mathrm {d}0}$ , where $p_{\mathrm {d}0}$ is termed as the minimal detection probability. The substitution of eq. (13) into $p_{\mathrm {d}}\ge p_{\mathrm {d}0}$ leads to $$\begin{align*} \varepsilon\le&\varepsilon _{0}\left ({\gamma,m,p_{\mathrm {d}0} }\right) \\=&\theta \Gamma _{0}^{-1}\left ({\kappa,p_{\mathrm {d}0} }\right)\!=\!\frac {\sigma _{z}^{2}\left ({1\!+\!\gamma }\right)}{K\xi \left ({m,\gamma }\right)} \Gamma _{0}^{-1}\left ({K\xi \left ({m,\gamma }\right)\!,p_{\mathrm {d}0} }\right)\qquad \tag{14}\end{align*}$$ View Source\begin{align*} \varepsilon\le&\varepsilon _{0}\left ({\gamma,m,p_{\mathrm {d}0} }\right) \\=&\theta \Gamma _{0}^{-1}\left ({\kappa,p_{\mathrm {d}0} }\right)\!=\!\frac {\sigma _{z}^{2}\left ({1\!+\!\gamma }\right)}{K\xi \left ({m,\gamma }\right)} \Gamma _{0}^{-1}\left ({K\xi \left ({m,\gamma }\right)\!,p_{\mathrm {d}0} }\right)\qquad \tag{14}\end{align*} where $\varepsilon _{0}$ is the upper limit of $\varepsilon$ and $\Gamma _{0}^{-1}(\cdot, \cdot)$ denotes the inverse function of $\Gamma _{0}(\cdot, \cdot)$ . The above formula indicates the dependence of $\varepsilon _{0}$ on the parameters of $\gamma$ , $m$ and $p_{\mathrm {d}0}$ .

As a result of (6), the PDF of $\mathcal {t}\left ({\mathrm {Y}\thinspace \vert \thinspace \mathcal {H}_{0}}\right)$ is derived to be $$\begin{equation*} f_{\mathcal {t}\left ({\mathbf {Y}\thinspace \vert \thinspace \mathcal {H}_{0}}\right)}\left ({x }\right)=\frac {x^{K-1}K^{K}}{\Gamma \left ({K }\right)\left ({\sigma _{z}^{2} }\right)^{K}}\exp \left ({- \frac {Kx}{\sigma _{z}^{2}} }\right)\!,\quad x\ge 0. \qquad \tag{15}\end{equation*}$$ View Source\begin{equation*} f_{\mathcal {t}\left ({\mathbf {Y}\thinspace \vert \thinspace \mathcal {H}_{0}}\right)}\left ({x }\right)=\frac {x^{K-1}K^{K}}{\Gamma \left ({K }\right)\left ({\sigma _{z}^{2} }\right)^{K}}\exp \left ({- \frac {Kx}{\sigma _{z}^{2}} }\right)\!,\quad x\ge 0. \qquad \tag{15}\end{equation*} Therefore, the $p_{\mathrm {f}}$ over $\varepsilon _{0}$ becomes derivable, given by $$\begin{align*}&\hspace {-1.2pc}p_{\mathrm {f}}\left ({\tau,\sigma _{z}^{2},\varepsilon _{0} }\right) \\[3pt]=&\int _{\varepsilon _{0}}^\infty {f_{\mathcal {t}\left ({\mathbf {Y}\vert \mathcal {H}_{0}}\right)}\left ({x }\right)\mathrm {d}x} \\[3pt]=&\frac {1}{\Gamma \left ({K }\right)}\int _{K\varepsilon _{0}/\sigma _{z}^{2}}^\infty {\left ({\frac {x}{\sigma _{z}^{2}/K} }\right)^{K-1}\exp \left ({- \frac {x}{\sigma _{z}^{2}/K} }\right)\mathrm {d}\frac {x}{\sigma _{z}^{2}/K}} \\[3pt]=&\frac {\Gamma \left ({K,K\varepsilon _{0}/\sigma _{z}^{2} }\right)}{\Gamma \left ({K }\right)}=\Gamma _{0}\left ({K\left ({\tau }\right)\!,\frac {K\left ({\tau }\right)\varepsilon _{0}\left ({\gamma,m,p_{\mathrm {d}0} }\right)}{\sigma _{z}^{2}} }\right)\!. \\ \tag{16}\end{align*}$$ View Source\begin{align*}&\hspace {-1.2pc}p_{\mathrm {f}}\left ({\tau,\sigma _{z}^{2},\varepsilon _{0} }\right) \\[3pt]=&\int _{\varepsilon _{0}}^\infty {f_{\mathcal {t}\left ({\mathbf {Y}\vert \mathcal {H}_{0}}\right)}\left ({x }\right)\mathrm {d}x} \\[3pt]=&\frac {1}{\Gamma \left ({K }\right)}\int _{K\varepsilon _{0}/\sigma _{z}^{2}}^\infty {\left ({\frac {x}{\sigma _{z}^{2}/K} }\right)^{K-1}\exp \left ({- \frac {x}{\sigma _{z}^{2}/K} }\right)\mathrm {d}\frac {x}{\sigma _{z}^{2}/K}} \\[3pt]=&\frac {\Gamma \left ({K,K\varepsilon _{0}/\sigma _{z}^{2} }\right)}{\Gamma \left ({K }\right)}=\Gamma _{0}\left ({K\left ({\tau }\right)\!,\frac {K\left ({\tau }\right)\varepsilon _{0}\left ({\gamma,m,p_{\mathrm {d}0} }\right)}{\sigma _{z}^{2}} }\right)\!. \\ \tag{16}\end{align*}

Since $K=f_{\mathrm {s}}\tau$ , formula (16) implies that the four parameters of $p_{\mathrm {d}0}$ , $\tau$ , $\gamma$ and $m$ will impact the $p_{\mathrm {f}}$ along with $f_{\mathrm {s}}$ and $\sigma _{z}^{2}$ that are usually configured or considered to be fixed. The former two ($p_{\mathrm {d}0}$ and $\tau$ ) are usually programmable, while the latter two ($\gamma$ and $m$ ) are determined merely by the radio circumstance. Thus, in order to clarify such dependencies, let us rewrite (16) using a more dependence-shown formula as follows. $$\begin{equation*} p_{\mathrm {f}}\left ({p_{\mathrm {d}0},\tau,\gamma,m }\right)=\Gamma _{0}\left ({K\left ({\tau }\right)\!,\frac {K\left ({\tau }\right)\varepsilon _{0}\left ({\gamma,m,p_{\mathrm {d}0} }\right)}{\sigma _{z}^{2}} }\right)\qquad \tag{17}\end{equation*}$$ View Source\begin{equation*} p_{\mathrm {f}}\left ({p_{\mathrm {d}0},\tau,\gamma,m }\right)=\Gamma _{0}\left ({K\left ({\tau }\right)\!,\frac {K\left ({\tau }\right)\varepsilon _{0}\left ({\gamma,m,p_{\mathrm {d}0} }\right)}{\sigma _{z}^{2}} }\right)\qquad \tag{17}\end{equation*}

Based on the well-known central limit theorem (CLT) [13], it can be stated that $$\begin{align*} \mathcal {t}\left ({\mathbf {Y}\thinspace \vert \thinspace \mathcal {H}_{0}}\right)\dot {\sim }&\mathcal {N}\left ({\mu _{\mathcal {H}_{0}},\sigma _{\mathcal {H}_{0}}^{2} }\right)\!, \tag{18}\\ \mathcal {t}\left ({\mathbf {Y}\thinspace \vert \thinspace \mathcal {H}_{1}}\right)\dot {\sim }&\mathcal {N}\left ({\mu _{\mathcal {H}_{1}},\sigma _{\mathcal {H}_{1}}^{2} }\right)\!, \tag{19}\end{align*}$$ View Source\begin{align*} \mathcal {t}\left ({\mathbf {Y}\thinspace \vert \thinspace \mathcal {H}_{0}}\right)\dot {\sim }&\mathcal {N}\left ({\mu _{\mathcal {H}_{0}},\sigma _{\mathcal {H}_{0}}^{2} }\right)\!, \tag{18}\\ \mathcal {t}\left ({\mathbf {Y}\thinspace \vert \thinspace \mathcal {H}_{1}}\right)\dot {\sim }&\mathcal {N}\left ({\mu _{\mathcal {H}_{1}},\sigma _{\mathcal {H}_{1}}^{2} }\right)\!, \tag{19}\end{align*} where the symbol $\dot {\sim }$ denotes ‘approximately follows’, and $\mathcal {N}\left ({\mu,\sigma ^{2} }\right)$ as a generic symbol denotes the normal (i.e., Gaussian) distribution with $\mu$ and $\sigma ^{2}$ being its mean and variance, respectively.

The PDF formula of the normal distribution is well-known. Accordingly, an approximate decision threshold for $\mathcal {t}\left ({\mathbf {Y}\thinspace \vert \thinspace \mathcal {H}_{1}}\right)$ to meet the $p_{\mathrm {d}0}$ is easily derived, given by $$\begin{equation*} \hat {\varepsilon }_{0}=\mu _{\mathcal {H}_{1}}+\sigma _{\mathcal {H}_{1}}\mathrm {Q}^{-1}\left ({p_{\mathrm {d}0} }\right) \tag{20}\end{equation*}$$ View Source\begin{equation*} \hat {\varepsilon }_{0}=\mu _{\mathcal {H}_{1}}+\sigma _{\mathcal {H}_{1}}\mathrm {Q}^{-1}\left ({p_{\mathrm {d}0} }\right) \tag{20}\end{equation*} where $\mathrm {Q}^{-1}\left ({\cdot }\right)$ denotes the inverse of the Q function given by $\text{Q}(x)=\left ({1/\sqrt {2\pi } }\right)\int _{x}^\infty e^{- \frac {t^{2}}{2}} \mathrm {d}t$ [13].

Consequently, the second type of false alarm probability (in a sense other than that of $p_{\mathrm {f}}$ of (17)) is obtained using $\hat {\varepsilon }_{0}$ and $\mathcal {N}\left ({\mu _{\mathcal {H}_{0}},\sigma _{\mathcal {H}_{0}}^{2} }\right)$ as two approximate alternatives. It is given by $$\begin{equation*} \hat {p}_{\mathrm {f}}=\mathrm {Q}\left ({\frac {\hat {\varepsilon }_{0}-\mu _{\mathcal {H}_{0}}}{\sigma _{\mathcal {H}_{0}}} }\right)\!. \tag{21}\end{equation*}$$ View Source\begin{equation*} \hat {p}_{\mathrm {f}}=\mathrm {Q}\left ({\frac {\hat {\varepsilon }_{0}-\mu _{\mathcal {H}_{0}}}{\sigma _{\mathcal {H}_{0}}} }\right)\!. \tag{21}\end{equation*}

By replacing $\hat {\varepsilon }_{0}$ with (20) [therein $\mu _{\mathcal {H}_{1}}$ and $\sigma _{\mathcal {H}_{1}}$ are further replaced with (10.a) and (10.b), respectively], $\mu _{\mathcal {H}_{0}}$ and $\sigma _{\mathcal {H}_{0}}$ with (11.a) and (11.b), respectively, it turns into $$\begin{align*}&\hspace {-1pc}\hat {p}_{\mathrm {f}}\left ({p_{\mathrm {d}0},\tau,\gamma,m }\right) \\&=\mathrm {Q}\left ({\mathrm {Q}^{-1}\left ({p_{\mathrm {d}0} }\right)\sqrt {\gamma ^{2}/m+2\gamma + 1} +\gamma \sqrt {K\left ({\tau }\right)} }\right)\!.\qquad \tag{22}\end{align*}$$ View Source\begin{align*}&\hspace {-1pc}\hat {p}_{\mathrm {f}}\left ({p_{\mathrm {d}0},\tau,\gamma,m }\right) \\&=\mathrm {Q}\left ({\mathrm {Q}^{-1}\left ({p_{\mathrm {d}0} }\right)\sqrt {\gamma ^{2}/m+2\gamma + 1} +\gamma \sqrt {K\left ({\tau }\right)} }\right)\!.\qquad \tag{22}\end{align*}

It is still worthy to remind you that from a strict point of view, $\mathcal {t}\left ({\mathbf {Y}\thinspace \vert \thinspace \mathcal {H}_{0}}\right)$ of (5) fluctuates on the law of Gamma distribution as (6) indicates rather than the normal distribution as (18) does. Thus, the $\hat {p}_{\mathrm {f}}$ of (21) does not reflect the exact false alarm probability that actually occurs in the physical layer. Therefore, it is still necessary to present such a false alarm probability in a sense distinct from the two that were discussed, which is obtained based on (20) and (6). We denote it by $\tilde {p}_{\mathrm {f}}$ and represent it as $$\begin{equation*} \tilde {p}_{\mathrm {f}}= \frac {1}{\Gamma \left ({K }\right)}\int _{K\hat {\varepsilon }_{0}/\sigma _{z}^{2}}^\infty {\zeta ^{K-1}e^{- \zeta }} d\zeta. \tag{23}\end{equation*}$$ View Source\begin{equation*} \tilde {p}_{\mathrm {f}}= \frac {1}{\Gamma \left ({K }\right)}\int _{K\hat {\varepsilon }_{0}/\sigma _{z}^{2}}^\infty {\zeta ^{K-1}e^{- \zeta }} d\zeta. \tag{23}\end{equation*}

To clarify the major dependent variables of $\tilde {p}_{\mathrm {f}}$ , we derive it into the form of $$\begin{align*}&\hspace {-1.5pc}\tilde {p}_{\mathrm {f}}\left ({p_{\mathrm {d}0},\tau,\gamma,m }\right) \\=&\Gamma _{0}\left ({K\left ({\tau }\right)\!,\frac {K\left ({\tau }\right)\hat {\varepsilon }_{0}\left ({\mu _{\mathcal {H}_{1}}\left ({\gamma,m }\right)\!,\sigma _{\mathcal {H}_{1}}\left ({\gamma,m }\right)\!,p_{\mathrm {d}0} }\right)}{\sigma _{z}^{2}} }\right)\!. \\ \tag{24}\end{align*}$$ View Source\begin{align*}&\hspace {-1.5pc}\tilde {p}_{\mathrm {f}}\left ({p_{\mathrm {d}0},\tau,\gamma,m }\right) \\=&\Gamma _{0}\left ({K\left ({\tau }\right)\!,\frac {K\left ({\tau }\right)\hat {\varepsilon }_{0}\left ({\mu _{\mathcal {H}_{1}}\left ({\gamma,m }\right)\!,\sigma _{\mathcal {H}_{1}}\left ({\gamma,m }\right)\!,p_{\mathrm {d}0} }\right)}{\sigma _{z}^{2}} }\right)\!. \\ \tag{24}\end{align*}

Therefore, for a CR system, there exist three types of data rates corresponding to (17), (22) and (24), respectively, as follows.

1. The Exact Data Rate (EDR), which is based on an exact decision threshold and an exact PDF of $\mathcal {t}\left ({\mathbf {Y}\thinspace \vert \thinspace \mathcal {H}_{0}}\right)$ , is given by $$\begin{equation*} B\left ({\tau, \gamma }\right)=\left ({1-\frac {\tau }{T} }\right)\left ({1-p_{\mathrm {f}} }\right) \tag{25}\end{equation*}$$ View Source\begin{equation*} B\left ({\tau, \gamma }\right)=\left ({1-\frac {\tau }{T} }\right)\left ({1-p_{\mathrm {f}} }\right) \tag{25}\end{equation*} where $B$ denotes the data rate normalized to that of the non-CR case, $T$ is the period of a data frame, and $\left ({1-\tau /T }\right)$ denotes the other part of $T$ than $\tau$ for the transmission of the CR’s own data provided that the detection results indicate the absence of the licensed signal [1].

2. The CLT-resultant Physically occurring Date Rate (CPDR), which treats $\hat {\varepsilon }_{0}$ as an approximate alternative of $\varepsilon _{0}$ , is given by $$\begin{equation*} \tilde {B}\left ({\tau, \gamma }\right)=\left ({1-\frac {\tau }{T} }\right)\left ({1-\tilde {p}_{\mathrm {f}} }\right)\!. \tag{26}\end{equation*}$$ View Source\begin{equation*} \tilde {B}\left ({\tau, \gamma }\right)=\left ({1-\frac {\tau }{T} }\right)\left ({1-\tilde {p}_{\mathrm {f}} }\right)\!. \tag{26}\end{equation*}

3. The CLT-resultant imaginary date rate (CIDR) is also based on $\hat {\varepsilon }_{0}$ to detect $\mathcal {t}\left ({\mathbf {Y}\thinspace \vert \thinspace \mathcal {H}_{0}}\right)$ , the distribution of which is actually Gamma but is regarded by CR as Normal. In other words, CIDR is a type of data rate that never occurs in the physical layer. Let this data rate be expressed as $$\begin{equation*} \hat {B}\left ({\tau, \gamma }\right)=\left ({1-\frac {\tau }{T} }\right)\left ({1-\hat {p}_{\mathrm {f}} }\right)\!. \tag{27}\end{equation*}$$ View Source\begin{equation*} \hat {B}\left ({\tau, \gamma }\right)=\left ({1-\frac {\tau }{T} }\right)\left ({1-\hat {p}_{\mathrm {f}} }\right)\!. \tag{27}\end{equation*}

Based on the data rates in different senses as above, there may be two types of OSDs as follows.

1. The Exact OSD (EOSD), which depends on (25), is given by $$\begin{equation*} \tau _{0}(\gamma)= \mathop {\arg\!\max\,}_{0\leq \tau \leq T}{B\left ({\tau,\gamma }\right)}. \tag{28}\end{equation*}$$ View Source\begin{equation*} \tau _{0}(\gamma)= \mathop {\arg\!\max\,}_{0\leq \tau \leq T}{B\left ({\tau,\gamma }\right)}. \tag{28}\end{equation*}

2. The CLT-resultant imaginary OSD (CIOSD), which is based on (27), is represented as $$\begin{equation*} \hat {\tau }_{0}(\gamma)= \mathop {\arg\!\max\,}_{0\leq \tau \leq T}{\hat {B}\left ({\tau, \gamma }\right)}. \tag{29}\end{equation*}$$ View Source\begin{equation*} \hat {\tau }_{0}(\gamma)= \mathop {\arg\!\max\,}_{0\leq \tau \leq T}{\hat {B}\left ({\tau, \gamma }\right)}. \tag{29}\end{equation*}

Correspondingly, further based on the types of OSDs defined as above, two MADR-related concepts are defined as follows.

1. The Exact MADR (EMADR), which is determined by (28) and (17), is given by $$\begin{equation*} B_{0}\left ({\gamma }\right)=\left ({1-\frac {\tau _{0}(\gamma)}{T} }\right)\left ({1-p_{\mathrm {f}} }\right)\!. \tag{30}\end{equation*}$$ View Source\begin{equation*} B_{0}\left ({\gamma }\right)=\left ({1-\frac {\tau _{0}(\gamma)}{T} }\right)\left ({1-p_{\mathrm {f}} }\right)\!. \tag{30}\end{equation*}

2. The CLT-resultant MADR (CMADR), which is determined by (29) and (24), is given by $$\begin{equation*} \tilde {B}_{0}\left ({\gamma }\right)=\left ({1-\frac {\hat {\tau }_{0}(\gamma)}{T} }\right)\left ({1-\tilde {p}_{\mathrm {f}} }\right)\!. \tag{31}\end{equation*}$$ View Source\begin{equation*} \tilde {B}_{0}\left ({\gamma }\right)=\left ({1-\frac {\hat {\tau }_{0}(\gamma)}{T} }\right)\left ({1-\tilde {p}_{\mathrm {f}} }\right)\!. \tag{31}\end{equation*}

Therefore, the CLT-resultant Data Rate Loss (CDRL) that can be defined as $\Delta B_{0}\left ({\gamma }\right)=B_{0}\left ({\gamma }\right)-\tilde {B}_{0}\left ({\gamma }\right)$ . It is further expanded to be $$\begin{equation*} \Delta B_{0}\!\left ({\gamma }\right)\!=\!\left ({1\!-\!\frac {\tau _{0}(\gamma)}{T} }\right)\!\left ({1\!-\!p_{\mathrm {f}} }\right)\!-\!\left ({1\!-\!\frac {\hat {\tau }_{0}(\gamma)}{T} }\right)\!\left ({1\!-\!\tilde {p}_{\mathrm {f}} }\right) \quad \tag{32}\end{equation*}$$ View Source\begin{equation*} \Delta B_{0}\!\left ({\gamma }\right)\!=\!\left ({1\!-\!\frac {\tau _{0}(\gamma)}{T} }\right)\!\left ({1\!-\!p_{\mathrm {f}} }\right)\!-\!\left ({1\!-\!\frac {\hat {\tau }_{0}(\gamma)}{T} }\right)\!\left ({1\!-\!\tilde {p}_{\mathrm {f}} }\right) \quad \tag{32}\end{equation*}

B. Raise of Questions

Facing formula (32), three interesting questions are raised as follows.

The 1st question: Under the context of high mobility, how significant the magnitude of CDRL of (32) will be?

Under the context of high mobility, the CR data frame has to be compressed for two reasons as follows. First, the licensed signal’s rapid alternation between emergence and vanishment becomes very likely, thereby compelling the detection of its availability to be performed more frequently. To adapt to such a scenario, the data frame has to be shorter. This phenomenon will become more severe if the user’s devices shuttle rapidly through buildings. Second, the channel estimation for the CR system’s own communication also needs to occur more frequently due to the faster variation of the channel.

As a result, the sensing time slot within a data frame will be squeezed, leading to a more limited size of samples for detection. This means a larger error of optimization for the MADR due to applying the CLT. Therefore, the 1st question arises as above.

The 2nd question: For the solution of (28), is a direct formula that can replace the conventional iteration scheme applicable? If so, such a formula will be more suitable to the high mobility since its related operation saves more time.

In an effort to solve such a problem as (28) gives, one usually resorts to the iteration-based numerical approach. This is indeed a feasible scheme under a stationary or quasi-stationary context. Nevertheless, for high mobility, the availability of spectrum holes varies so rapidly that the iteration-based calculation may be incapable of keeping up with such a rapid variation. Under this scenario, raising the 2nd question becomes valuable.

The 3rd question: In the context of high mobility, is the statistical information of the OSD and MADR quickly acquirable by the CR that is detecting the licensed signal through an ergodically fading channel?

In such a case, the quick acquisition of such information becomes significantly important, since those cross-layer QoS managements (e.g., the resource allocation (RA), the call admission control (CAC) [11], [12], and the frame alignment [25]) are required to be finished within a demanding latency [7]. Thus, the 3rd question is highlighted.

For the 1st question, fortunately, our investigation in the next section will show that although a closed-form formula representing (28) is never derivable, an upper bound of it is achievable if some skill is applied. On this basis, the data rate loss due to the CLT resultant error can be estimated. The verifying simulation test in section VI indicates that such a date rate loss is still acceptable, provided that the related parameters are configured within the range regulated by IEEE 802.16e [19].

Regarding the 2nd question, our works in subsection V.B and section VI indicate that a forthright and convenient formula, based on the exponential interpolation to approximate the exact OSD will become an effective tool.

Facing the 3rd question, subsections V.C and VI will demonstrate that in an approximation sense, the probability density functions (PDF) of the OSD and MADR are achievable if applying some skills.

Proposition 2:

For the EMADR given by (30), an upper bound denoted $\mathcal {B}_{0}$ is achievable ($\mathrm {i.e.,}~B_{0}\left ({\gamma }\right)\le \mathcal {B}_{0}\left ({\gamma }\right))$ . $\mathcal {B}_{0}$ is given by (33.a), as shown at the top of the next page,

$$\begin{equation*} B_{\tau }^{\prime }\left ({\tau, \gamma }\right)=-\frac {1-p_{\mathrm {f}}}{T}-\left ({1-\frac {\tau }{T} }\right)\frac {\partial p_{\mathrm {f}}\left ({\tau, \gamma }\right)}{\partial \tau } \tag{33.b}\end{equation*}$$ View Source\begin{equation*} B_{\tau }^{\prime }\left ({\tau, \gamma }\right)=-\frac {1-p_{\mathrm {f}}}{T}-\left ({1-\frac {\tau }{T} }\right)\frac {\partial p_{\mathrm {f}}\left ({\tau, \gamma }\right)}{\partial \tau } \tag{33.b}\end{equation*}

$$\begin{align*} \frac {\partial p_{\mathrm {f}}}{\partial \tau }=\frac {\Gamma _{\tau }^{\prime }\left ({K,K\varepsilon _{0}/\sigma _{z}^{2} }\right)\Gamma \left ({K }\right)-\Gamma \left ({K,K\varepsilon _{0}/\sigma _{z}^{2} }\right)\Gamma _{\tau }^{\prime }\left ({K }\right)}{\Gamma ^{2}\left ({K }\right)} \\ \tag{33.c}\end{align*}$$ View Source\begin{align*} \frac {\partial p_{\mathrm {f}}}{\partial \tau }=\frac {\Gamma _{\tau }^{\prime }\left ({K,K\varepsilon _{0}/\sigma _{z}^{2} }\right)\Gamma \left ({K }\right)-\Gamma \left ({K,K\varepsilon _{0}/\sigma _{z}^{2} }\right)\Gamma _{\tau }^{\prime }\left ({K }\right)}{\Gamma ^{2}\left ({K }\right)} \\ \tag{33.c}\end{align*}

Meanwhile, $\acute {\tau }=\acute {\tau }^{\left ({i }\right)},\grave {\tau }=\grave {\tau }^{\left ({i }\right)}$ where $\acute {\tau }^{\left ({i }\right)}$ and $\grave {\tau }^{\left ({i }\right)}$ are the results of a set of recursive formulae¹ as follows $$\begin{align*}&{\begin{cases} \acute {\tau }^{\left ({i }\right)}=\dot {\tau }^{\left ({i }\right)} \\ \grave {\tau }^{\left ({i }\right)}=\grave {\tau }^{\left ({i-1 }\right)} \\ \end{cases}}& \mathrm {if}~B_{\tau }^{\prime }\left ({\dot {\tau }^{\left ({i }\right)},\gamma }\right)>0 \tag{34.a}\\&{\begin{cases} \acute {\tau }^{\left ({i }\right)}=\acute {\tau }^{\left ({i-1 }\right)} \\ \grave {\tau }^{\left ({i }\right)}=\dot {\tau }^{\left ({i }\right)} \\ \end{cases}}& \mathrm {if}~B_{\tau }^{\prime }\left ({\dot {\tau }^{\left ({i }\right)},\gamma }\right)<0 \tag{34.b}\\&~~\acute {\tau }^{\left ({i }\right)}=\grave {\tau }^{\left ({i }\right)}=\dot {\tau }^{\left ({i }\right)}& \mathrm {if}~B_{\tau }^{\prime }\left ({\dot {\tau }^{\left ({i }\right)},\gamma }\right)=0 \tag{34.c}\\&~~\acute {\tau }^{\left ({0}\right)}=\tau _{\mathrm {min}} \tag{34.d}\\&~~\grave {\tau }^{\left ({0}\right)}=T \tag{34.e}\end{align*}$$ View Source\begin{align*}&{\begin{cases} \acute {\tau }^{\left ({i }\right)}=\dot {\tau }^{\left ({i }\right)} \\ \grave {\tau }^{\left ({i }\right)}=\grave {\tau }^{\left ({i-1 }\right)} \\ \end{cases}}& \mathrm {if}~B_{\tau }^{\prime }\left ({\dot {\tau }^{\left ({i }\right)},\gamma }\right)>0 \tag{34.a}\\&{\begin{cases} \acute {\tau }^{\left ({i }\right)}=\acute {\tau }^{\left ({i-1 }\right)} \\ \grave {\tau }^{\left ({i }\right)}=\dot {\tau }^{\left ({i }\right)} \\ \end{cases}}& \mathrm {if}~B_{\tau }^{\prime }\left ({\dot {\tau }^{\left ({i }\right)},\gamma }\right)<0 \tag{34.b}\\&~~\acute {\tau }^{\left ({i }\right)}=\grave {\tau }^{\left ({i }\right)}=\dot {\tau }^{\left ({i }\right)}& \mathrm {if}~B_{\tau }^{\prime }\left ({\dot {\tau }^{\left ({i }\right)},\gamma }\right)=0 \tag{34.c}\\&~~\acute {\tau }^{\left ({0}\right)}=\tau _{\mathrm {min}} \tag{34.d}\\&~~\grave {\tau }^{\left ({0}\right)}=T \tag{34.e}\end{align*} where, $\dot {\tau }^{(i)}$ is given by (34,f), as shown at the top of the this page,

$$\begin{equation*} \tau _{\mathrm {min}}=\frac {\left ({\mathrm {Q}^{-1}\left ({p_{\mathrm {d}0} }\right) }\right)^{2}}{f_{\mathrm {s}}}\left ({\frac {1}{m}+\frac {2}{\gamma }+ \frac {1}{\gamma ^{2}} }\right)\!. \tag{34.g}\end{equation*}$$ View Source\begin{equation*} \tau _{\mathrm {min}}=\frac {\left ({\mathrm {Q}^{-1}\left ({p_{\mathrm {d}0} }\right) }\right)^{2}}{f_{\mathrm {s}}}\left ({\frac {1}{m}+\frac {2}{\gamma }+ \frac {1}{\gamma ^{2}} }\right)\!. \tag{34.g}\end{equation*}

Proposition 3:

The CDRL given by (32) will be arbitrarily small provided that the number of samples for detection is sufficiently large.

A. Theoretical Basis

Reference [1] has proved that $B\left ({\tau, \gamma }\right)$ of (25) (where $p_{\mathrm {f}}\left ({\tau, \gamma }\right)=\mathrm {Q}\left ({\sqrt {2\gamma +1} \mathrm {Q}^{-1}\left ({p_{\mathrm {d}0} }\right)+\gamma \sqrt {f_{\mathrm {s}}\tau } }\right))$ is concave w. r. t. $\tau$ . For the $p_{\mathrm {f}}$ of this paper in the new forms as (17), (22) and (24) give, does the concavity of the respective corresponding (25), (26) and (27) w. r. t. $\tau$ , still hold? We find it true. Therefore, another proposition is presented as follows.

It was mentioned that no solutions of $\boldsymbol{\tau }_{\mathbf {0}}\mathbf {(\gamma)}$ of (28) and $\hat {\tau }_{\mathbf {0}}\mathbf {(\gamma)}$ of (29) w. r. t. $\boldsymbol{\gamma }$ exist in explicit closed forms. However, an implicit function mapping $\boldsymbol{\gamma }$ to $\boldsymbol{\tau }_{\mathbf {0}}$ does exist [17]. Motivated by this, we consider piecewise fitting to be a good strategy for achieving an approximate direct formula of the OSD.

B. Approximate Formula Based on Exponetial Interpolation

For the sake of achieving the OSD via a direct formula instead of pure iteration, we propose a scheme that employs the exponential functions to piecewise approximate the exact OSD of (28). This direct formula is expressed in the form given by $$\begin{equation*} \tilde {\tau }_{0}^{i}\left ({\gamma }\right)=\mu _{i}+\alpha _{i}e^{\varepsilon _{i}\left ({\gamma -\beta _{i} }\right)} \tag{36}\end{equation*}$$ View Source\begin{equation*} \tilde {\tau }_{0}^{i}\left ({\gamma }\right)=\mu _{i}+\alpha _{i}e^{\varepsilon _{i}\left ({\gamma -\beta _{i} }\right)} \tag{36}\end{equation*} where $i$ is the index of the $i$ th piece, and $\mu _{i}$ , $\alpha _{i}$ , $\varepsilon _{i}$ and $\beta _{i}$ are the input parameters that can be calculated by solving a set of equations. See Appendix E for the acquisition of the above input parameters.

In Appendix E, you will observe that tiny iterations are still needed due to the need for the numerical calculations of $\tau _{0}(\gamma)$ at the starting points of all pieces. Let the $\gamma$ values (as the horizontal axis of Fig. 3 denotes) of the ends of all pieces be denoted by $\left \{{\mathcal {r}_{i}\thinspace \vert \thinspace {i\in I}}\right \}$ , where $I$ is the set of indices of all pieces. Thus, $\left \{{\mu _{i}, \alpha _{i}, \varepsilon _{i},\beta _{i} }\right \}$ will be determined by $\left \{{\mathcal {r}_{i},\tau _{0}\left ({\mathcal {r}_{i} }\right)\!,\mathrm {d}\tilde {\tau }_{0}^{i}\left ({\mathcal {r}_{i} }\right)/\mathrm {d}\gamma,\mathcal {r}_{i+1},\mathrm {d}\tilde {\tau }_{0}^{i}\left ({\mathcal {r}_{i+1} }\right)/\mathrm {d}\gamma }\right \}$ . In this sense, the scheme based on (36) falls under the category of interpolation. However, it is quite different from the popularly employed interpolation based on the Hermite spline [9], since our scheme requires the coincidences of the functions’ values merely on the starting end of all pieces ($\mathrm {i.e.,}\,\,\tilde {\tau }_{0}^{i}\left ({\mathcal {r}_{i} }\right)=\tau _{0}\left ({\mathcal {r}_{i} }\right)\!,\mathrm {\forall }i\in I)$ . Faced by the derivation of an approximate PDF of $\tau _{0}(\gamma)$ based on $\tilde {\tau }_{0}^{i}\left ({\gamma }\right)$ , the continuity of the PDF should be kept. In other words, the continuity of its 1st derivative $\mathrm {d}\tilde {\tau }_{0}^{i}\left ({\gamma }\right)/\mathrm {d}\gamma$ for all pieces as a whole must be guaranteed. This requirement is due to a fact that prior to getting the PDF of a function of a random variable, the 1st derivative of its inverse function has to be derived [13]. If the complexity or tractability of achieving such a PDF were not taken into account, undoubtedly the approaches based on polynomial interpolation (e.g., Hermite, B and rational splines) would be the better choices due to their fine performance [9]. Particularly, the cubic Hermite spline is particularly welcome since it features low order² polynomials meanwhile guarantees the curve’s smoothness. Nevertheless, unfortunately all polynomial-based interpolation schemes are not applicable here for the four reasons as follows:

1. The complexity or intractability of the inverse function of $\tilde {\tau }_{0}^{i}\left ({\gamma }\right)$ if it is expressed by a polynomial,

2. The complexity of the 1st derivative of the inverse function of $\tilde {\tau }_{0}^{i}\left ({\gamma }\right)$ even if it is tractable,

3. The complexity or intractability of further derivations of other formulae of statistic characteristics (e.g., mathematical expectation and variance), and

4. The inevitability of operations of numeric iterations or matrix’s inversions, which have superlinearly rising complexities with the increase of the number of pieces.

FIGURE 3.

Optimal sensing duration vs. the mean SNR of Nakagami-$m$ fading with various fading parameters m. ($p_{\mathrm {d0}}=0.99$ , bw = 100KHz, and $T=2$ ms).

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Consequently, the polynomial-based interpolants are not applicable to the context we are focusing on, since a highly mobile CR will surely be computational delay sensitive.

Therefore, as it is handled in Appendix E, it keeps the continuity of $\mathrm {d}\tilde {\tau }_{0}^{i}\left ({\gamma }\right)/\mathrm {d}\gamma$ at $\forall \gamma \in \left \{{\mathcal {r}_{i}\thinspace \vert \thinspace {i\in I}}\right \}$ but it permits the slight discontinuity of $\tilde {\tau }_{0}^{i}\left ({\gamma }\right)$ thereon, as Fig. 3 shows. In other words, if less continuity of $\tilde {\tau }_{0}^{i}\left ({\gamma }\right)$ can be swapped for more continuity of $\mathrm {d}\tilde {\tau }_{0}^{i}\left ({\gamma }\right)/\mathrm {d}\gamma$ , $\forall i\in I$ , it is worth being done. The reasons lie in a fact that the PDF of OSD (as (39) will show) is more sensitive to the latter than to the former, and our works for Fig. 5 have experienced the same.

FIGURE 5.

PDF of the OSD over a Nakagami-Gamma shadowed fading via-to-sense channel with different $M$ and ${p}_{\mathrm {d}0}$ . (a.1) $\tau _{0}$ ($M=1/2$ , $p_{\mathrm {d0}}=0.90$ ). (a.2) $\tau _{0}$ ($M=1/2$ , $p_{\mathrm {d0}}=0.99$ ). (b.1) $\tau _{0}$ ($M=1$ , $p_{\mathrm {d0}}=0.90$ ). (b.2) $\tau _{0}$ ($M=1$ , $p_{\mathrm {d0}}=0.99$ ). (c.1) $\tau _{0}$ ($M=3$ , $p_{\mathrm {d0}}=0.90$ ). (c.2) $\tau _{0}$ ($M=3$ , $p_{\mathrm {d0}}=0.99$ ).

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C. PDF of OSD Over a Fading Via-to-Sense Channel

As aforementioned, due to employing exponential splines $\left \{{{\tilde {\tau }_{0}^{i}(\gamma)}\thinspace \vert \thinspace {i\in I}}\right \}$ rather than polynomial-based ones, the inverse function of $\tilde {\tau }_{0}^{i}(\gamma)$ becomes smoothly derived. Therefore, based on (36), we derive it as $$\begin{equation*} \tilde {\gamma }_{i}\left ({\tau _{0} }\right)=\beta _{i}+\frac {1}{\varepsilon _{i}}\ln \left ({\frac {\tau _{0}-\mu _{i}}{\alpha _{i}} }\right)\!. \tag{37}\end{equation*}$$ View Source\begin{equation*} \tilde {\gamma }_{i}\left ({\tau _{0} }\right)=\beta _{i}+\frac {1}{\varepsilon _{i}}\ln \left ({\frac {\tau _{0}-\mu _{i}}{\alpha _{i}} }\right)\!. \tag{37}\end{equation*} Moreover, the logarithm function therein enables the extraordinarily easy derivation of its 1st derivative. The numerical result indicates that the curve of $\tau _{0}(\gamma)$ of (28) is concave-shaped as Fig. 3 shows. Thus, the PDF of $\tilde {\tau }_{0}^{i}$ can be derived via the process as follows $$\begin{align*}&\hspace {-1.2pc}f_{\tilde {\tau }_{0}^{i}}\left ({\tau }\right) \\=&\frac {\mathrm {d}F_{\tilde {\tau }_{0}^{i}}\left ({\tau }\right)}{\mathrm {d}\tau }=\frac {\mathrm {d}}{\mathrm {d}\tau }\int \limits _{-\infty }^\tau {f_{\tilde {\tau }_{0}^{i}}\left ({t }\right)\mathrm {d}t} \\=&\frac {\mathrm {d}}{\mathrm {d}\tau }\int \limits _{-\infty }^{\tilde {\gamma }_{l}\left ({\tau }\right)} {f_{\tilde {\gamma }_{l}}\left ({\gamma }\right)\mathrm {d}\gamma } +\frac {\mathrm {d}}{\mathrm {d}\tau }\int \limits _{\tilde {\gamma }_{u}\left ({\tau }\right)}^\infty {f_{\tilde {\gamma }_{u}}\left ({\gamma }\right)\mathrm {d}\gamma } \\=&\frac {\mathrm {d}F_{\tilde {\gamma }_{l}}\left ({\gamma }\right)}{\mathrm {d}\gamma }\frac {\mathrm {d}\tilde {\gamma }_{l}}{\mathrm {d}\tau }\!-\!\frac {\mathrm {d}F_{\tilde {\gamma }_{u}}\left ({\gamma }\right)}{\mathrm {d}\gamma }\frac {\mathrm {d}\tilde {\gamma }_{u}}{\mathrm {d}\tau }\!=\! \frac {f_{\gamma _{l}}\left ({\gamma }\right)}{\varepsilon _{l}\left ({\tau \!-\!\mu _{l} }\right)}\!-\!\frac {f_{\gamma _{u}}\left ({\gamma }\right)}{\varepsilon _{u}\left ({\tau \!-\!\mu _{u} }\right)} \\ \tag{38}\end{align*}$$ View Source\begin{align*}&\hspace {-1.2pc}f_{\tilde {\tau }_{0}^{i}}\left ({\tau }\right) \\=&\frac {\mathrm {d}F_{\tilde {\tau }_{0}^{i}}\left ({\tau }\right)}{\mathrm {d}\tau }=\frac {\mathrm {d}}{\mathrm {d}\tau }\int \limits _{-\infty }^\tau {f_{\tilde {\tau }_{0}^{i}}\left ({t }\right)\mathrm {d}t} \\=&\frac {\mathrm {d}}{\mathrm {d}\tau }\int \limits _{-\infty }^{\tilde {\gamma }_{l}\left ({\tau }\right)} {f_{\tilde {\gamma }_{l}}\left ({\gamma }\right)\mathrm {d}\gamma } +\frac {\mathrm {d}}{\mathrm {d}\tau }\int \limits _{\tilde {\gamma }_{u}\left ({\tau }\right)}^\infty {f_{\tilde {\gamma }_{u}}\left ({\gamma }\right)\mathrm {d}\gamma } \\=&\frac {\mathrm {d}F_{\tilde {\gamma }_{l}}\left ({\gamma }\right)}{\mathrm {d}\gamma }\frac {\mathrm {d}\tilde {\gamma }_{l}}{\mathrm {d}\tau }\!-\!\frac {\mathrm {d}F_{\tilde {\gamma }_{u}}\left ({\gamma }\right)}{\mathrm {d}\gamma }\frac {\mathrm {d}\tilde {\gamma }_{u}}{\mathrm {d}\tau }\!=\! \frac {f_{\gamma _{l}}\left ({\gamma }\right)}{\varepsilon _{l}\left ({\tau \!-\!\mu _{l} }\right)}\!-\!\frac {f_{\gamma _{u}}\left ({\gamma }\right)}{\varepsilon _{u}\left ({\tau \!-\!\mu _{u} }\right)} \\ \tag{38}\end{align*} where $F_{X}\left ({\cdot }\right)$ denotes the cumulative distribution function (CDF) of an RV $X$ . $\varepsilon _{l} \& \mu _{l}$ and $\varepsilon _{u} \& \mu _{u}$ , denote $\varepsilon _{i} \& \mu _{i}$ of (36), for which $\gamma \in \{\gamma \vert {0\le \gamma \le \gamma }_{\mathrm {p}}$ and $\gamma \in \{\gamma \vert {\gamma _{\mathrm {p}}\le \gamma \le \infty }_{}$ , respectively, where $\gamma _{\mathrm {p}}=\mathrm {arg\,max}_{0\leq \gamma \leq \infty }{\tau _{0}(\gamma)}$ .

As (4) shows, $f_{\gamma _{l}}\left ({\gamma }\right)$ and $f_{\gamma _{u}}\left ({\gamma }\right)$ within (38) follow a gamma distribution. Replacing them with (4) leads to an approximate PDF of the OSD over a Nakagami-Gamma shadowed fading via-to-sense channel. Notice that (38) is conditioned on $M$ and $\bar {\gamma }$ . On this account, we reformulate it into a conditional PDF as $$\begin{align*}&\hspace {-1pc}f_{\tilde {\tau }_{0}}\left ({\tau \thinspace \vert \thinspace {M,\bar {\gamma }}}\right) \\=&\frac {{\left ({\tilde {\gamma }_{l}\left ({\tau }\right) }\right)^{M-1}e}^{{-\tilde {\gamma }}_{l}\left ({\tau }\right)/\bar {\gamma }}}{\Gamma \left ({M }\right)\bar {\gamma }^{M}\varepsilon _{l}\left ({\tau -\mu _{l} }\right)}-\frac {{\left ({\tilde {\gamma }_{u}\left ({\tau }\right) }\right)^{M-1}e}^{{-\tilde {\gamma }}_{u}\left ({\tau }\right)/\bar {\gamma }}}{\Gamma \left ({M }\right)\bar {\gamma }^{M}\varepsilon _{u}\left ({\tau -\mu _{u} }\right)} \\=&\frac {\left [{ \beta _{l}+\frac {1}{\varepsilon _{l}}\ln \left ({\frac {\tau -\mu _{l}}{\alpha _{l}} }\right) }\right]^{M-1}\left [{ \mathrm {exp}\left ({\frac {\beta _{l}}{\bar {\gamma }} }\right)+\left ({\frac {\tau -\mu _{l}}{\alpha _{l}} }\right)^{\left ({1/\bar {\gamma }\varepsilon _{l} }\right)} }\right]}{\Gamma \left ({M }\right)\bar {\gamma }^{M}\varepsilon _{l}\left ({\tau -\mu _{l} }\right)} \\&- \frac {\left [{ \beta _{u}\!+\!\frac {1}{\varepsilon _{u}}\ln \left ({\frac {\tau -\mu _{u}}{\alpha _{u}} }\right) }\right]^{M-1}\left [{ \mathrm {exp}\left ({\frac {\beta _{u}}{\bar {\gamma }} }\right)\!+\!\left ({\frac {\tau -\mu _{u}}{\alpha _{u}} }\right)^{\left ({1/\bar {\gamma }\varepsilon _{u} }\right)} }\right]}{\Gamma \left ({M }\right)\bar {\gamma }^{M}\varepsilon _{u}\left ({\tau -\mu _{u} }\right)} \\ \tag{39.a}\end{align*}$$ View Source\begin{align*}&\hspace {-1pc}f_{\tilde {\tau }_{0}}\left ({\tau \thinspace \vert \thinspace {M,\bar {\gamma }}}\right) \\=&\frac {{\left ({\tilde {\gamma }_{l}\left ({\tau }\right) }\right)^{M-1}e}^{{-\tilde {\gamma }}_{l}\left ({\tau }\right)/\bar {\gamma }}}{\Gamma \left ({M }\right)\bar {\gamma }^{M}\varepsilon _{l}\left ({\tau -\mu _{l} }\right)}-\frac {{\left ({\tilde {\gamma }_{u}\left ({\tau }\right) }\right)^{M-1}e}^{{-\tilde {\gamma }}_{u}\left ({\tau }\right)/\bar {\gamma }}}{\Gamma \left ({M }\right)\bar {\gamma }^{M}\varepsilon _{u}\left ({\tau -\mu _{u} }\right)} \\=&\frac {\left [{ \beta _{l}+\frac {1}{\varepsilon _{l}}\ln \left ({\frac {\tau -\mu _{l}}{\alpha _{l}} }\right) }\right]^{M-1}\left [{ \mathrm {exp}\left ({\frac {\beta _{l}}{\bar {\gamma }} }\right)+\left ({\frac {\tau -\mu _{l}}{\alpha _{l}} }\right)^{\left ({1/\bar {\gamma }\varepsilon _{l} }\right)} }\right]}{\Gamma \left ({M }\right)\bar {\gamma }^{M}\varepsilon _{l}\left ({\tau -\mu _{l} }\right)} \\&- \frac {\left [{ \beta _{u}\!+\!\frac {1}{\varepsilon _{u}}\ln \left ({\frac {\tau -\mu _{u}}{\alpha _{u}} }\right) }\right]^{M-1}\left [{ \mathrm {exp}\left ({\frac {\beta _{u}}{\bar {\gamma }} }\right)\!+\!\left ({\frac {\tau -\mu _{u}}{\alpha _{u}} }\right)^{\left ({1/\bar {\gamma }\varepsilon _{u} }\right)} }\right]}{\Gamma \left ({M }\right)\bar {\gamma }^{M}\varepsilon _{u}\left ({\tau -\mu _{u} }\right)} \\ \tag{39.a}\end{align*} where $\tau$ determines which two pieces among all cover it, as Fig. 3 shows. Let the smaller and larger $i$ -valued pieces be indexed by $l$ and $u$ , respectively, they are given by $$\begin{equation*} \begin{cases} l=\mathrm {min}\left \{{i\thinspace \vert \thinspace {\tilde {\tau }_{0}^{i}=\tau,i\in I}}\right \} \\ u=\mathrm {max}\left \{{i\thinspace \vert \thinspace {\tilde {\tau }_{0}^{i}=\tau,i\in I}}\right \}\!. \\ \end{cases} \tag{39.b}\end{equation*}$$ View Source\begin{equation*} \begin{cases} l=\mathrm {min}\left \{{i\thinspace \vert \thinspace {\tilde {\tau }_{0}^{i}=\tau,i\in I}}\right \} \\ u=\mathrm {max}\left \{{i\thinspace \vert \thinspace {\tilde {\tau }_{0}^{i}=\tau,i\in I}}\right \}\!. \\ \end{cases} \tag{39.b}\end{equation*} Formula (39) is just the targeted PDF of the OSD, which may be significantly interesting to those who study the CR system’s cross-layer problems. Let us give an example. The frame alignment delay is a key contributor to the user-plane latency budget, which additionally includes the queuing, transmission and processing delays [25], [26]. Such a frame alignment can be illustrated by Fig. 1. Therein, $T$ is the frame period that is usually configured to be identical to the transmission time interval (TTI) [26]. $\tau _{0}$ and ${T-\tau }_{0}$ are the sensing and data transmission durations, respectively. $\text{a}_{k}$ and a’_k are the upper-layer data’s two possible arrival instants, while $\text{e}_{k}$ and $\text{e}_{k+1}$ are its entrance instant in the $k$ th frame and the next frame, respectively. Two random events will possibly occur for the arriving data’s entrance to the radio frame. In the first case, the data packet will arrive at the buffer output in the instant of $\text{a}_{k}$ with a time span of $t_{\mathrm {a}}^{\left ({k }\right)}, 0\le t_{\mathrm {a}}^{\left ({k }\right)}\le \tau _{0}$ to the starting instant of the frame. It will wait for a shorter duration to come into the current radio frame at $\text{e}_{k}$ . In the second case, the data packet arrives in the instant of a’_k with $t_{\mathrm {a}}^{\left ({k }\right)}$ ranging within $\tau _{0}\le t_{\mathrm {a}}^{\left ({k }\right)}\le T$ . Then, it has no chance within the current frame for its entrance, and it has to wait for a longer duration until the next frame’s entrance $\text{e}_{k+1}$ arrives.

FIGURE 1.

Constituent Time Slots of a Frame Period and the Frame Alignment Mechanism of the CR system. Therein, $T$ is the frame period, $\tau _{0}$ and ${T-\tau }_{0}$ are for sensing and data transmission, respectively; $\text{a}_{k}$ and a’_k are the upper-layer data’s two possible arrival instants while $\text{e}_{k}$ and $\text{e}_{k+1}$ are its entrance instant in the $k$ th frame and its next frame, respectively. There are two cases for the current data to enter the radio frame. In the first case, it arrives at $\text{a}_{k}$ , and enters the current frame at $\text{e}_{k}$ ; In the second case, it arrives at the instant of a’_k, then no chance for it to enter the current frame, so it has to wait till the $\text{e}_{k+1}$ arrives.

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The $t_{\mathrm {a}}^{\left ({k }\right)}$ of Fig. 1 is usually uniformly distributed within the frame period [25]. Without the loss of generality, let $t_{\mathrm {a}}$ as a RV represent $t_{\mathrm {a}}^{\left ({k }\right)}, \forall k$ . Thus, the PDF of $t_{a}$ is given by $$\begin{equation*} f_{t_{\mathrm {a}}}=1/T,\quad 0\le t_{\mathrm {a}}\le T \tag{40.a}\end{equation*}$$ View Source\begin{equation*} f_{t_{\mathrm {a}}}=1/T,\quad 0\le t_{\mathrm {a}}\le T \tag{40.a}\end{equation*} Let the frame alignment delay of the CR system be denoted as $d_{\mathrm {fa}}$ and just be the $\Delta t$ or $\Delta t^{\prime }$ in Fig. 1. Obviously, we have $d_{\mathrm {fa}}=\tau _{0}-t_{\mathrm {a}}$ with the probability of $\tau _{0}/T$ , and $d_{\mathrm {fa}}=T-t_{\mathrm {a}}+\tau _{0}$ with probability of $\left ({T-\tau _{0} }\right)/T$ , respectively. Therefore, the PDF of the frame alignment delay of a CR system denoted as $f_{d_{\mathrm {fa}}}$ is derivable based on the law of PDF of sum of RVs [13]. It is given by $$\begin{equation*} f_{d_{\mathrm {fa}}}\left ({x }\right)=\left ({f_{\tilde {\tau }_{0}}\otimes f_{\left ({-t_{\mathrm {a}} }\right)} }\right)\frac {\tau _{0}}{T}+\left ({f_{\tilde {\tau }_{0}}\otimes f_{\left ({T-t_{\mathrm {a}} }\right)} }\right)\frac {T-\tau _{0}}{T}\qquad \tag{40.b}\end{equation*}$$ View Source\begin{equation*} f_{d_{\mathrm {fa}}}\left ({x }\right)=\left ({f_{\tilde {\tau }_{0}}\otimes f_{\left ({-t_{\mathrm {a}} }\right)} }\right)\frac {\tau _{0}}{T}+\left ({f_{\tilde {\tau }_{0}}\otimes f_{\left ({T-t_{\mathrm {a}} }\right)} }\right)\frac {T-\tau _{0}}{T}\qquad \tag{40.b}\end{equation*} where $f_{\tilde {\tau }_{0}}$ , $f_{\left ({-t_{\mathrm {a}} }\right)}$ and $f_{\left ({T-t_{\mathrm {a}} }\right)}$ are given by (39) and (40.a), respectively, while the sign $\otimes$ is a convolution operator.

Accordingly, furthermore, the mean delay of the frame alignment of a CR system denoted by $\bar {d}_{\mathrm {fa}}$ is also achievable. It is given by $$\begin{equation*} \bar {d}_{\mathrm {fa}}=\int _{0}^{T} x f_{d_{\mathrm {fa}}}\left ({x }\right)\mathrm {d}x \tag{40.c}\end{equation*}$$ View Source\begin{equation*} \bar {d}_{\mathrm {fa}}=\int _{0}^{T} x f_{d_{\mathrm {fa}}}\left ({x }\right)\mathrm {d}x \tag{40.c}\end{equation*} Formulas (40.c) and (40.b) indicate that such a PDF of the OSD of (39) is of significant importance to a CR system’s frame alignment and other cross-layer operations.

Moreover, notice that formula (39) depends on $M$ and $\bar {\gamma }$ and still keeps an exponential form, which enables the particularly troubleless integral operation of its product with an exponential-formed function (e.g., (4)). Consequently, we made little effort to derive an approximate formula of mean OSD as $$\begin{align*} \mathsf {t}_{0}\left ({M,\bar {\gamma } }\right)\cong&\sum \limits _{i=1}^{N} \int _{\mathcal {r}_{i}}^{\mathcal {r}_{i+1}} {\tilde {\tau }_{0}^{i}\left ({y }\right)f_{\gamma }\left ({y\vert \bar {\gamma } }\right)\mathrm {d}y} \\=&\sum \limits _{i=1}^{N} \left [{ \mathcal {T}_{\tau }^{i}\left ({\mathcal {r}_{i+1} }\right)-\mathcal {T}_{\tau }^{i}\left ({\mathcal {r}_{i} }\right) }\right] \tag{41.a}\end{align*}$$ View Source\begin{align*} \mathsf {t}_{0}\left ({M,\bar {\gamma } }\right)\cong&\sum \limits _{i=1}^{N} \int _{\mathcal {r}_{i}}^{\mathcal {r}_{i+1}} {\tilde {\tau }_{0}^{i}\left ({y }\right)f_{\gamma }\left ({y\vert \bar {\gamma } }\right)\mathrm {d}y} \\=&\sum \limits _{i=1}^{N} \left [{ \mathcal {T}_{\tau }^{i}\left ({\mathcal {r}_{i+1} }\right)-\mathcal {T}_{\tau }^{i}\left ({\mathcal {r}_{i} }\right) }\right] \tag{41.a}\end{align*} where $f_{\gamma }\left ({y\vert \bar {\gamma } }\right)$ is given by (4) and $\mathcal {r}_{i}$ is the value of $\gamma$ at the starting end of the $i$ th piece. $\mathcal {T}_{\tau }^{i}\left ({\gamma }\right)$ is the result of $\int {\tilde {\tau }_{0}^{i}\left ({y }\right)f_{\gamma }\left ({y\vert \bar {\gamma } }\right)\mathrm {d}y}$ and is given by $$\begin{align*} \mathcal {T}_{\tau }^{i}\left ({\gamma }\right)=&-\frac {({\gamma /\bar {\gamma })}^{M}}{\Gamma \left ({M }\right)}\left ({\frac {{\alpha _{i}e}^{-\beta _{i}\varepsilon _{i}}\Gamma \left ({M,\gamma /\bar {\gamma }-\varepsilon _{i}\gamma }\right)}{\left ({\gamma /\bar {\gamma }-\varepsilon _{i}\gamma }\right)^{M}} }\right.\\&\qquad \qquad \qquad \qquad ~\left.{+\frac {\mu _{i}\Gamma \left ({M,\gamma /\bar {\gamma } }\right)}{\left ({\gamma /\bar {\gamma } }\right)^{M}} }\right)\qquad \quad \tag{41.b}\end{align*}$$ View Source\begin{align*} \mathcal {T}_{\tau }^{i}\left ({\gamma }\right)=&-\frac {({\gamma /\bar {\gamma })}^{M}}{\Gamma \left ({M }\right)}\left ({\frac {{\alpha _{i}e}^{-\beta _{i}\varepsilon _{i}}\Gamma \left ({M,\gamma /\bar {\gamma }-\varepsilon _{i}\gamma }\right)}{\left ({\gamma /\bar {\gamma }-\varepsilon _{i}\gamma }\right)^{M}} }\right.\\&\qquad \qquad \qquad \qquad ~\left.{+\frac {\mu _{i}\Gamma \left ({M,\gamma /\bar {\gamma } }\right)}{\left ({\gamma /\bar {\gamma } }\right)^{M}} }\right)\qquad \quad \tag{41.b}\end{align*}

In (41.b), $\bar {\gamma }$ and $M$ (which was explained for (4)) explicitly reflect the status of shadow fading, while $\mu _{i}$ , $\alpha _{i}$ , $\varepsilon _{i}$ and $\beta _{i}$ defined for (36) implicitly reflect the status of Nakagami-$m$ fading as Fig. 3 demonstrates. Such implicity is due to the unachievability of an explicit formula mapping $\tau _{0}$ to $m$ .

D. Ergodic-Sensing Channel Capacity

Similar to using (36) to fit (28), $B_{0}(\gamma)$ of (30) can also be approximated by piecewise exponential functions. The latter is found more easily handled, with three pieces to do it well. It is $$\begin{equation*} \tilde {B}_{0}^{j}\left ({\gamma }\right)=u_{j}+v_{j}e^{k_{j}(\gamma -w_{j})},\quad j=1,2, 3. \tag{42}\end{equation*}$$ View Source\begin{equation*} \tilde {B}_{0}^{j}\left ({\gamma }\right)=u_{j}+v_{j}e^{k_{j}(\gamma -w_{j})},\quad j=1,2, 3. \tag{42}\end{equation*} Along the same way as Appendix E addresses (36), all input parameters of (42) are acquirable.

Since formula (42) is still of an exponential form, the inverse functions of $\tilde {B}_{0}^{j}\left ({\gamma }\right)\!,j=1,2, 3$ are also quite easily derived as $$\begin{equation*} \tilde {\gamma }_{j}\left ({B_{0} }\right)=w_{j}+\frac {1}{k_{j}}\ln \left ({\frac {B_{0}-u_{j}}{\nu _{j}} }\right)\!. \tag{43}\end{equation*}$$ View Source\begin{equation*} \tilde {\gamma }_{j}\left ({B_{0} }\right)=w_{j}+\frac {1}{k_{j}}\ln \left ({\frac {B_{0}-u_{j}}{\nu _{j}} }\right)\!. \tag{43}\end{equation*}

We find that $\tilde {B}_{0}^{j}\left ({\gamma }\right)$ increases monotonically with the increase of $\gamma$ . Therefore, by directly invoking the related formula of the PDF of a function of a random variable [13], the PDF of $B_{0}(\gamma)$ can be smoothly derived still in a piecewise exponential form given by $$\begin{equation*} f_{\tilde {B}_{0}^{j}}\left ({B\thinspace \vert \thinspace {M,\bar {\gamma }}}\right) =\frac {\left ({\tilde {\gamma }_{j}\left ({B }\right) }\right)^{M-1}}{\bar {\gamma }^{M}\Gamma \left ({M }\right)\left |{ k_{i}\left ({B-u_{i} }\right) }\right |}e^{-\frac {\tilde {\gamma }_{j}\left ({B }\right)}{\bar {\gamma }}} \tag{44}\end{equation*}$$ View Source\begin{equation*} f_{\tilde {B}_{0}^{j}}\left ({B\thinspace \vert \thinspace {M,\bar {\gamma }}}\right) =\frac {\left ({\tilde {\gamma }_{j}\left ({B }\right) }\right)^{M-1}}{\bar {\gamma }^{M}\Gamma \left ({M }\right)\left |{ k_{i}\left ({B-u_{i} }\right) }\right |}e^{-\frac {\tilde {\gamma }_{j}\left ({B }\right)}{\bar {\gamma }}} \tag{44}\end{equation*} where $\tilde {\gamma }_{j}\left ({B_{0} }\right)$ is given by (43).

In some cases, the data rate performance of a CR system over a longer term must be evaluated, rendering such a definition quite necessary as follows.

So far, a set of formulae for approximately evaluating some sensing-related statistic information (i.e., the PDFs of the OSD and MADR and their statistic mean values), has been achieved. Then, one may raise such a question as follows.

Under the context of a CR or its sensed objects at high mobility, why is it so desirable to quickly acquire the OSD, the MADR, and their related statistic information? The reasons are as follows.

It can been seen from (30) and the related (17), the $B_{0}$ of MADR depends on $\gamma$ and the other three parameters of $T$ , $f_{\mathrm {s}}$ and $p_{\mathrm {d}0}$ . Obviously, the OSD depends on $\gamma$ . At high mobility, in addition to the rapid fluctuation of $\gamma$ , the latter three are also quite likely to vary quickly. Hence, they need to be timely updated to follow the rapid variation of the radio environment described as follows.

1. Since a wireless device usually moves at a varying velocity, surely its channel coherence time will vary correspondingly. Consequently, the temporal interval of channel estimation should be timely updated to adapt to the variation of its movement velocity. Therefore, $T$ (the period of a data frame) should also be swiftly adjusted for a higher efficiency of QoS management to be achieved (e.g., the maximal traffic [11], or the triple trade-off of TTI, queuing delay and spectrum efficiency [25]).

2. The bandwidth being sensed may also vary frequently since the variant licensed signals that may carry different services are ready for the CR to detect them. For example, the video service usually needs a much wider bandwidth than the audio one. Therefore, following Nyquist’s sampling theorem, $f_{\mathrm {s}}$ should also be flexible rather than fixed in order to timely adapt to the variations of sensed bandwidths.

3. Different licensed services require different QoS, making their requirements on the symbol error rate (SER) quite different [11]. For example, the audio service usually requires less SER than a text file. Therefore, as a licensed service, it requires much less $p_{\mathrm {d}0}$ of the CRs that intend to access in. Thus, to adapt to the variant licensed system with different sensitivities to the interference, $p_{\mathrm {d}0}$ should also be timely reconfigured to minimize the network latency.

Overall, we are facing the wireless environment as follows. All aforementioned parameters are varying rapidly while multiple licensed bands (e.g., the subchannels of an Orthogonal Frequency Division Multiple Access (OFDMA) system) are alternating quickly between business and idleness. Under such a context, it is of significant importance for a CR system to quickly learn the probability distribution of the OSD and the MADR of each band that certain devices intend to access. Only with awareness of the aforementioned statistical information are the relevant QoS managements (e.g., the call admission control (CAC) [11] and frame alignment [25] illustrated by Fig. 1) capable of being performed more efficiently [11], [26].

E. Complexity Analysis of EI vs. HI

For high mobility, lower complexity should be preferred to higher accuracy for an algorithm. Therefore, peer-to-peer comparisons will be done on the complexities of the above formulae, which are based on EI or HI that is supposed to be applied as EI’s alternative.

Proof:

Under hypothesis $\mathcal {H}_{1}$ , based on (2), we have $$\begin{equation*} {\vert y\left ({k }\right)\vert }^{2}=\left ({s\left ({k }\right)+z\left ({k }\right) }\right)\left ({s^{\ast }\left ({k }\right)+z^{\ast }\left ({k }\right) }\right) \tag{A.1}\end{equation*}$$ View Source\begin{equation*} {\vert y\left ({k }\right)\vert }^{2}=\left ({s\left ({k }\right)+z\left ({k }\right) }\right)\left ({s^{\ast }\left ({k }\right)+z^{\ast }\left ({k }\right) }\right) \tag{A.1}\end{equation*}

Due to the independence between $s\left ({k }\right)$ and $z\left ({k }\right)$ , the expansion of (A.1) yields $$\begin{align*} \mu _{\mathcal {H}_{1}}=E\left ({{\vert s\left ({k }\right)+z\left ({k }\right)\vert }^{2} }\right)=\sigma _{z}^{2}+\sigma _{s}^{2}=(1+\gamma)\sigma _{z}^{2} \\ \tag{A.2}\end{align*}$$ View Source\begin{align*} \mu _{\mathcal {H}_{1}}=E\left ({{\vert s\left ({k }\right)+z\left ({k }\right)\vert }^{2} }\right)=\sigma _{z}^{2}+\sigma _{s}^{2}=(1+\gamma)\sigma _{z}^{2} \\ \tag{A.2}\end{align*}

It is indicated in (3) that $\gamma \sim \mathrm {Ga}(m,\gamma /m)$ means that ${\vert s\left ({k }\right)\vert }^{2}\sim \mathrm {Ga}(m,\sigma _{S}^{2}/m)$ since $\sigma _{z}^{2}$ is assumed to be a constant here. Therefore, the mean and variance of ${\vert s\left ({k }\right)\vert }^{2}$ are derived based on a related formulae for the Gamma distribution [13] given by $$\begin{align*} E\left ({{\vert s\left ({k }\right)\vert }^{2} }\right)=&m\frac {\sigma _{S}^{2}}{m}=\sigma _{S}^{2} \tag{A.3}\\ {D(\vert s\left ({k }\right)\vert }^{2})=&m{\left({\frac {\sigma _{S}^{2}}{m}}\right)}^{2}=\frac {\sigma _{S}^{4}}{m} \tag{A.4}\end{align*}$$ View Source\begin{align*} E\left ({{\vert s\left ({k }\right)\vert }^{2} }\right)=&m\frac {\sigma _{S}^{2}}{m}=\sigma _{S}^{2} \tag{A.3}\\ {D(\vert s\left ({k }\right)\vert }^{2})=&m{\left({\frac {\sigma _{S}^{2}}{m}}\right)}^{2}=\frac {\sigma _{S}^{4}}{m} \tag{A.4}\end{align*} respectively. Thus, based on a property of variance [13], the $\mathrm {E}\left ({{\vert s\left ({n }\right)\vert }^{4} }\right)$ is accordingly derived to be $$\begin{align*} \mathrm {E}\left({{\vert s\left({k }\right)\vert }^{4} }\right)=D({\vert s\left({k }\right)\vert }^{2})+E^{2}\left ({\left |{ s\left ({k }\right) }\right |^{2} }\right)=\left({1+\frac {1}{m}}\right) \sigma _{S}^{4} \\ \tag{A.5}\end{align*}$$ View Source\begin{align*} \mathrm {E}\left({{\vert s\left({k }\right)\vert }^{4} }\right)=D({\vert s\left({k }\right)\vert }^{2})+E^{2}\left ({\left |{ s\left ({k }\right) }\right |^{2} }\right)=\left({1+\frac {1}{m}}\right) \sigma _{S}^{4} \\ \tag{A.5}\end{align*}

For any CSCG distributed noise sample $z\left ({k }\right)$ , $\mathrm {E}\left ({{\vert z\left ({k }\right)\vert }^{4} }\right)={2\sigma }_{z}^{4}$ [1]. Its and (A.5)’s substitution into (8) of [1] yields $$\begin{align*} \sigma _{\mathcal {H}_{1}}^{2}=&\frac {1}{K}[E\left ({\left |{ s\left ({k }\right) }\right |^{4} }\right)+ E\left ({\left |{ z\left ({k }\right) }\right |^{4} }\right)-\left ({\sigma _{S}^{2}-\sigma _{z}^{2} }\right)^{2}] \\=&\frac {\sigma _{z}^{4}}{K}\left [{ \frac {1}{m} \gamma ^{2}+2\gamma + 1 }\right] \tag{A.6}\end{align*}$$ View Source\begin{align*} \sigma _{\mathcal {H}_{1}}^{2}=&\frac {1}{K}[E\left ({\left |{ s\left ({k }\right) }\right |^{4} }\right)+ E\left ({\left |{ z\left ({k }\right) }\right |^{4} }\right)-\left ({\sigma _{S}^{2}-\sigma _{z}^{2} }\right)^{2}] \\=&\frac {\sigma _{z}^{4}}{K}\left [{ \frac {1}{m} \gamma ^{2}+2\gamma + 1 }\right] \tag{A.6}\end{align*}

Since $y\left ({k }\right)$ of (2) is the sum of two RVs, $s\left ({k }\right)$ and $z\left ({k }\right)$ are both CSCG distributed with $\mathrm {CN}\left ({0,\sigma _{s}^{2} }\right)$ and $\mathrm {CN}\left ({0,\sigma _{z}^{2} }\right)\!$ , respectively. Therefore, $y\sim CN\left ({0,\sigma _{S}^{2}+\sigma _{z}^{2} }\right)$ . Accordingly, $\mathrm {T}\left ({y\vert \mathcal {H}_{1} }\right)$ of (7) can be treated as the sum of multiple squared amplitudes of CSCG-distributed RVs. Therefore, $T\left ({y\vert H_{1} }\right)$ of (7) is Gamma distributed and denoted as $\mathrm {Ga}(\kappa,\theta)$ , where $\kappa$ and $\theta$ are derivable based on the property of the Gamma distribution [13] as $$\begin{align*} \theta=&\frac {\sigma _{\mathcal {H}_{1}}^{2}}{\mu _{\mathcal {H}_{1}}}=\frac {\sigma _{z}^{2}}{K}\cdot \frac {\gamma ^{2}/m +2\gamma + 1}{1+\gamma } \tag{A.7}\\ \kappa=&\frac {\mu _{\mathcal {H}_{1}}}{\theta }=\frac {K{(1+\gamma)}^{2}}{\gamma ^{2}/m +2\gamma + 1} \tag{A.8}\end{align*}$$ View Source\begin{align*} \theta=&\frac {\sigma _{\mathcal {H}_{1}}^{2}}{\mu _{\mathcal {H}_{1}}}=\frac {\sigma _{z}^{2}}{K}\cdot \frac {\gamma ^{2}/m +2\gamma + 1}{1+\gamma } \tag{A.7}\\ \kappa=&\frac {\mu _{\mathcal {H}_{1}}}{\theta }=\frac {K{(1+\gamma)}^{2}}{\gamma ^{2}/m +2\gamma + 1} \tag{A.8}\end{align*}

Proof:

First, let us prove the (33.c) to be true as follows.

The $1^{\mathrm {st}}$ derivative of $p_{\mathrm {f}}$ of (16) w.r.t. $\tau$ is given by $$\begin{align*} p_{\mathrm {f}}^{\prime }\left ({\tau }\right)=\frac {\Gamma _{\tau }^{\prime }\left ({K,K\varepsilon _{0}/\sigma _{z}^{2} }\right)\Gamma \left ({K }\right)-\Gamma \left ({K,K\varepsilon _{0}/\sigma _{z}^{2} }\right)\Gamma _{\tau }^{\prime }\left ({K }\right)}{\Gamma ^{2}\left ({K }\right)} \\ \tag{B.1}\end{align*}$$ View Source\begin{align*} p_{\mathrm {f}}^{\prime }\left ({\tau }\right)=\frac {\Gamma _{\tau }^{\prime }\left ({K,K\varepsilon _{0}/\sigma _{z}^{2} }\right)\Gamma \left ({K }\right)-\Gamma \left ({K,K\varepsilon _{0}/\sigma _{z}^{2} }\right)\Gamma _{\tau }^{\prime }\left ({K }\right)}{\Gamma ^{2}\left ({K }\right)} \\ \tag{B.1}\end{align*} where $$\begin{align*}&\hspace {-1.2pc}\Gamma _{\tau }^{\prime }\left ({K }\right) \\=&\Gamma _{K}^{\prime }\mathrm {K}_{\tau }^{\prime }=f_{\mathrm {s}}\int _{0}^\infty {\mathrm {ln}\left ({\zeta }\right)\zeta ^{K-1}e^{- \zeta }\mathrm {d}\zeta }; \tag{B.2}\\&\hspace {-1.2pc}\Gamma _{\tau }^{\prime }\left ({K,K\varepsilon _{0}/\sigma _{z}^{2} }\right) \\=&\Gamma _{K}^{\prime }\left ({K,K\varepsilon _{0}\left ({K }\right)/\sigma _{z}^{2} }\right)K_{\tau }^{\prime }\cdots \\=&f_{\mathrm {s}}\left ({\frac {\partial \Gamma \left ({K,K\varepsilon _{0}/\sigma _{z}^{2} }\right)}{\partial K}+\frac {\partial \Gamma \left ({K,K\varepsilon _{0}/\sigma _{z}^{2} }\right)}{\partial \left ({K\varepsilon _{0}/\sigma _{z}^{2} }\right)}\frac {\mathrm {d}\left ({K\varepsilon _{0}/\sigma _{z}^{2} }\right)}{\mathrm {d}K} }\right)\!. \tag{B.3.1} \end{align*}$$ View Source\begin{align*}&\hspace {-1.2pc}\Gamma _{\tau }^{\prime }\left ({K }\right) \\=&\Gamma _{K}^{\prime }\mathrm {K}_{\tau }^{\prime }=f_{\mathrm {s}}\int _{0}^\infty {\mathrm {ln}\left ({\zeta }\right)\zeta ^{K-1}e^{- \zeta }\mathrm {d}\zeta }; \tag{B.2}\\&\hspace {-1.2pc}\Gamma _{\tau }^{\prime }\left ({K,K\varepsilon _{0}/\sigma _{z}^{2} }\right) \\=&\Gamma _{K}^{\prime }\left ({K,K\varepsilon _{0}\left ({K }\right)/\sigma _{z}^{2} }\right)K_{\tau }^{\prime }\cdots \\=&f_{\mathrm {s}}\left ({\frac {\partial \Gamma \left ({K,K\varepsilon _{0}/\sigma _{z}^{2} }\right)}{\partial K}+\frac {\partial \Gamma \left ({K,K\varepsilon _{0}/\sigma _{z}^{2} }\right)}{\partial \left ({K\varepsilon _{0}/\sigma _{z}^{2} }\right)}\frac {\mathrm {d}\left ({K\varepsilon _{0}/\sigma _{z}^{2} }\right)}{\mathrm {d}K} }\right)\!. \tag{B.3.1} \end{align*} Eq. (14) leads to $K\varepsilon _{0}/\sigma _{z}^{2}=\left ({\left ({1+\gamma }\right)/\xi }\right) \Gamma _{0}^{-1}\left ({\xi K,p_{\mathrm {d}0} }\right)$ , so $$\begin{equation*} \frac {\mathrm {d}\left ({K\varepsilon _{0}/\sigma _{z}^{2} }\right)}{\mathrm {d}K}=\left ({1+\gamma }\right)\frac {\mathrm {d}\left ({\Gamma _{0}^{-1}\left ({\xi K,p_{\mathrm {d}0} }\right) }\right)}{\mathrm {d}\left ({\xi K }\right)}; \tag{B.3.2}\end{equation*}$$ View Source\begin{equation*} \frac {\mathrm {d}\left ({K\varepsilon _{0}/\sigma _{z}^{2} }\right)}{\mathrm {d}K}=\left ({1+\gamma }\right)\frac {\mathrm {d}\left ({\Gamma _{0}^{-1}\left ({\xi K,p_{\mathrm {d}0} }\right) }\right)}{\mathrm {d}\left ({\xi K }\right)}; \tag{B.3.2}\end{equation*}

Based on the formula of the derivative of inverse regularized incomplete Gamma function w.r.t. the shape parameter [23], we have $$\begin{align*}&\hspace {-1pc}\mathrm {d}\left ({\mathrm {\Gamma }_{0}^{-\mathrm {1}}\left ({\xi K\mathrm {,}p_{\mathrm {d}0} }\right) }\right)/\mathrm {d}\left ({\xi K }\right)=\cdots \\&\qquad \qquad e^{w}w^{1-\xi K}[w^{\xi K}\mathrm {\Gamma }^{2}\left ({\xi K }\right)\cdots \\&\qquad \qquad \mathrm {\cdot }{}_{2} \tilde {F}_{2}\left ({\xi K,\xi K;\xi K+1,\xi K+1;-w }\right)\cdots \\&\qquad \qquad \qquad \quad \,\,+\left ({p_{\mathrm {d}0}-1 }\right)\mathrm {\Gamma }\left ({\xi K }\right)\mathrm {log}\left ({w }\right)\cdots \\&\qquad \qquad \qquad \quad \,\,+\left ({\mathrm {\Gamma }\left ({\xi K }\right)\mathrm {-\Gamma }\left ({\xi K,w }\right) }\right)\psi \left ({\xi K }\right)]\qquad \quad \tag{B.3.3}\end{align*}$$ View Source\begin{align*}&\hspace {-1pc}\mathrm {d}\left ({\mathrm {\Gamma }_{0}^{-\mathrm {1}}\left ({\xi K\mathrm {,}p_{\mathrm {d}0} }\right) }\right)/\mathrm {d}\left ({\xi K }\right)=\cdots \\&\qquad \qquad e^{w}w^{1-\xi K}[w^{\xi K}\mathrm {\Gamma }^{2}\left ({\xi K }\right)\cdots \\&\qquad \qquad \mathrm {\cdot }{}_{2} \tilde {F}_{2}\left ({\xi K,\xi K;\xi K+1,\xi K+1;-w }\right)\cdots \\&\qquad \qquad \qquad \quad \,\,+\left ({p_{\mathrm {d}0}-1 }\right)\mathrm {\Gamma }\left ({\xi K }\right)\mathrm {log}\left ({w }\right)\cdots \\&\qquad \qquad \qquad \quad \,\,+\left ({\mathrm {\Gamma }\left ({\xi K }\right)\mathrm {-\Gamma }\left ({\xi K,w }\right) }\right)\psi \left ({\xi K }\right)]\qquad \quad \tag{B.3.3}\end{align*} where $w=\Gamma _{0}^{-1}\left ({\xi K,p_{\mathrm {d}0} }\right)$ , and ${}_{2} \tilde {F}_{2}\left ({;; }\right)$ is the hypergeometric function [24]. Based on the formula of the derivative of the incomplete Gamma function w.r.t. its shape parameter [23], we have $$\begin{align*}&\hspace {-1.8pc}\frac {\mathrm {\partial \Gamma }\left ({K,K\varepsilon _{0}\mathrm {/}\sigma _{z}^{2} }\right)}{\mathrm {\partial }K}=\cdots \\&\mathrm {\Gamma }^{2}\left ({K }\right)\left ({K\varepsilon _{0}\mathrm {/}\sigma _{z}^{2} }\right)^{K}\cdots \\&\mathrm {\cdot }{}_{2} \tilde {F}_{2}\left ({K,K;K+1,K+1;-K\varepsilon _{0}\mathrm {/}\sigma _{z}^{2} }\right)\cdots \\&-\mathrm {\Gamma }\left ({K,0,\frac {K\varepsilon _{0}}{\sigma _{z}^{2}} }\right)\mathrm {log}\left ({\frac {K\varepsilon _{0}}{\sigma _{z}^{2}} }\right)+\mathrm {\Gamma }\left ({K }\right)\psi \left ({K }\right)\qquad \tag{B.3.4}\end{align*}$$ View Source\begin{align*}&\hspace {-1.8pc}\frac {\mathrm {\partial \Gamma }\left ({K,K\varepsilon _{0}\mathrm {/}\sigma _{z}^{2} }\right)}{\mathrm {\partial }K}=\cdots \\&\mathrm {\Gamma }^{2}\left ({K }\right)\left ({K\varepsilon _{0}\mathrm {/}\sigma _{z}^{2} }\right)^{K}\cdots \\&\mathrm {\cdot }{}_{2} \tilde {F}_{2}\left ({K,K;K+1,K+1;-K\varepsilon _{0}\mathrm {/}\sigma _{z}^{2} }\right)\cdots \\&-\mathrm {\Gamma }\left ({K,0,\frac {K\varepsilon _{0}}{\sigma _{z}^{2}} }\right)\mathrm {log}\left ({\frac {K\varepsilon _{0}}{\sigma _{z}^{2}} }\right)+\mathrm {\Gamma }\left ({K }\right)\psi \left ({K }\right)\qquad \tag{B.3.4}\end{align*} where $\psi \left ({\cdot }\right)$ is the Psi function [24]. Based on that, w.r.t its lower limit of integral [23], we have $$\begin{equation*} \frac {\partial \Gamma \left ({K,K\varepsilon _{0}/\sigma _{z}^{2} }\right)}{\partial \left ({K\varepsilon _{0}/\sigma _{z}^{2} }\right)}=-\left ({\frac {K\varepsilon _{0}}{\sigma _{z}^{2}} }\right)^{K-1}e^{-\frac {K\varepsilon _{0}}{\sigma _{z}^{2}}} \tag{B.3.5}\end{equation*}$$ View Source\begin{equation*} \frac {\partial \Gamma \left ({K,K\varepsilon _{0}/\sigma _{z}^{2} }\right)}{\partial \left ({K\varepsilon _{0}/\sigma _{z}^{2} }\right)}=-\left ({\frac {K\varepsilon _{0}}{\sigma _{z}^{2}} }\right)^{K-1}e^{-\frac {K\varepsilon _{0}}{\sigma _{z}^{2}}} \tag{B.3.5}\end{equation*}

The substitutions of (B.3.2) [thereof a corresponding part replaced with (B.3.3)], (B.3.4) and (B.3.5) into (B.3.1) render it solvable. Then, the substitutions of (B.2) and (B.3.1) into (B.1) lead to a solvable $p_{\mathrm {f}}^{\prime }\left ({\tau }\right)$ in an explicit form, just as (33.c) shows.

Second, we prove that $B_{0}\left ({\gamma }\right)\le \mathcal {B}_{0}\left ({\gamma }\right)$ and $\mathcal {B}_{0}\left ({\gamma }\right)\to B_{0}\left ({\gamma }\right)$ with $i\to \infty$ where $i$ is that within (34), as follows.

Based on proposition 4, just as the context specified by [1], the $B\left ({\tau, \gamma }\right)$ of (25) is also concave if $p_{\mathrm {f}}\left ({\tau }\right)\le 0.5$ and it has a unique maximum w. r. t. $\tau$ . Since $p_{\mathrm {f}}\le \hat {p}_{\mathrm {f}}$ , $\hat {p}_{\mathrm {f}}\left ({\tau }\right)=0$ yields a solution of $\tau _{\mathrm {min}}$ that also satisfies $p_{\mathrm {f}}\left ({\tau _{\mathrm {min}} }\right)\le 0.5$ , and $\tau _{\mathrm {min}}$ is just (34.g). As shown in Fig. 7, the intersection of the two tangent lines refer to $\mathcal {B}_{0}\left ({\gamma }\right)$ of (33.a), and the solid, dash and dash-dot curves refer to the $B\left ({\tau, \gamma }\right)$ of (25), the $\tilde {B}\left ({\tau, \gamma }\right)$ of (26) and the $\hat {B}\left ({\tau, \gamma }\right)$ of (27) respectively. Consequently, the height of the intersection of the two tangent lines that contact the solid curve at $\tau _{1}$ and $\tau _{2}$ respectively is surely greater than the peak of this curve. Thus, $\mathcal {B}_{0}\left ({\gamma }\right)$ of (33.a) is justified to be an upper bound of $B_{0}\left ({\gamma }\right)$ of (30). Thus, the projection of the maximum of $\hat {B}\left ({\tau, \gamma }\right)$ on $\tilde {B}\left ({\tau, \gamma }\right)$ just represents the $\tilde {B}_{0}\left ({\gamma }\right)$ of (31). Consequently, we have that $B_{\tau }^{\prime }\left ({\acute {\tau }^{\left ({i-1 }\right)},\gamma }\right)\le B_{\tau }^{\prime }\left ({\dot {\tau }^{\left ({i }\right)},\gamma }\right)\le B_{\tau }^{\prime }\left ({\grave {\tau }^{\left ({i-1 }\right)},\gamma }\right)$ and $B_{\tau }^{\prime }\left ({\acute {\tau }^{\left ({i }\right)},\gamma }\right)\to {0}_{-}$ , $B_{\tau }^{\prime }\left ({\grave {\tau }^{\left ({i }\right)},\gamma }\right)\to {0}_{+}$ if $\mathrm {i\to \infty }$ is identical to $B_{\tau }^{\prime }\left ({\dot {\tau }^{\left ({i }\right)},\gamma }\right)\to 0$ or $\mathcal {B}_{0}\left ({\gamma }\right)=B\left ({\dot {\tau }^{\left ({i }\right)},\gamma }\right)\to B_{0}\left ({\gamma }\right)$ if $i\to \infty$ . Therefore, it is concluded.

Proof:

According to the CLT, $\left ({\mathcal {t}\left ({\mathrm {Y\vert }\mathcal {H}_{1} }\right)-\mu _{\mathcal {H}_{1}} }\right)/\sigma _{\mathcal {H}_{1}}$ will arbitrarily approach the distribution of $\mathcal {N}\left ({0,1 }\right)$ provided that the cardinality of $\mathbf {Y}$ is sufficiently large. In other words, $$\begin{equation*} \mathcal {y}_{\mathcal {H}_{1}}\!=\!\left ({\mathcal {t}\!\left ({\mathbf {Y\vert }\mathcal {H}_{1} }\right)\!-\!\mu _{\mathcal {H}_{1}} }\right)/\sigma _{\mathcal {H}_{1}}\vec {\sim }\mathcal {N}\left ({0,1 }\right)\quad \mathrm {if}~ K\to \infty \qquad \tag{C.1}\end{equation*}$$ View Source\begin{equation*} \mathcal {y}_{\mathcal {H}_{1}}\!=\!\left ({\mathcal {t}\!\left ({\mathbf {Y\vert }\mathcal {H}_{1} }\right)\!-\!\mu _{\mathcal {H}_{1}} }\right)/\sigma _{\mathcal {H}_{1}}\vec {\sim }\mathcal {N}\left ({0,1 }\right)\quad \mathrm {if}~ K\to \infty \qquad \tag{C.1}\end{equation*} where $\mu _{\mathcal {H}_{1}}$ and $\sigma _{\mathcal {H}_{1}}$ are given by (A.2) and (A.6), respectively, while the sign ‘$\vec {\sim }$ ’ denotes ‘approaches the distribution of’. The ‘$\bar {\sim }$ ’ still holds if integral operations are applied to the PDFs of both sides of (C.1), and it also holds if applied to the $\mathcal {t}\left ({\mathbf {Y\vert }\mathcal {H}_{1} }\right)$ provided so that the integral variant takes a proper transformation. Therefore, we conduct these operations to the PDF of $\mathcal {t}\left ({\mathbf {Y\vert }\mathcal {H}_{1} }\right)$ just as (13) does and to that of $\mathcal {N}\left ({0,1 }\right)$ . This results in $$\begin{equation*} \Gamma _{0}\left ({\kappa, \varepsilon _{0}/\theta }\right)\to \mathrm {Q}\left ({\left ({\hat {\varepsilon }_{0}-\mu _{\mathcal {H}_{1}} }\right)/\sigma _{\mathcal {H}_{1}} }\right)\quad \mathrm {if}~K\to \infty \qquad \tag{C.2}\end{equation*}$$ View Source\begin{equation*} \Gamma _{0}\left ({\kappa, \varepsilon _{0}/\theta }\right)\to \mathrm {Q}\left ({\left ({\hat {\varepsilon }_{0}-\mu _{\mathcal {H}_{1}} }\right)/\sigma _{\mathcal {H}_{1}} }\right)\quad \mathrm {if}~K\to \infty \qquad \tag{C.2}\end{equation*} where $\hat {\varepsilon }_{0}$ and $\varepsilon _{0}$ have been explained for (20) and (14), respectively, and $\mathrm {Q}\left ({\cdot }\right)$ denotes the Q function.

Since both $\mathrm {Q(\cdot)}$ and $\Gamma _{0}\left ({\cdot, \cdot }\right)$ are monotonic and continuous functions [13], [23], the ‘$\to$ ’ still holds if the inverse operations are applied to both sides of (C.2). This means that for a specified $p_{\mathrm {d0}}$ , we have $$\begin{align*}&\hspace {-0.3pc}{[Q}^{-1}\left ({p_{\mathrm {d0}} }\right)\sigma _{\mathcal {H}_{1}}+\mu _{\mathcal {H}_{1}}]=\hat {\varepsilon }_{0}\to \varepsilon _{0}=\theta \Gamma _{0}^{-1}\left ({\kappa, p_{\mathrm {d0}} }\right) \\[-3pt]&\!\!\!\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \mathrm {if}~K\to \infty \qquad \quad \tag{C.3}\end{align*}$$ View Source\begin{align*}&\hspace {-0.3pc}{[Q}^{-1}\left ({p_{\mathrm {d0}} }\right)\sigma _{\mathcal {H}_{1}}+\mu _{\mathcal {H}_{1}}]=\hat {\varepsilon }_{0}\to \varepsilon _{0}=\theta \Gamma _{0}^{-1}\left ({\kappa, p_{\mathrm {d0}} }\right) \\[-3pt]&\!\!\!\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \mathrm {if}~K\to \infty \qquad \quad \tag{C.3}\end{align*}

Based on (C.3), (24) and (17), we have $$\begin{equation*} \tilde {p}_{\mathrm {f}}\to p_{\mathrm {f}},\quad \mathrm {if}~K\to \infty \tag{C.4}\end{equation*}$$ View Source\begin{equation*} \tilde {p}_{\mathrm {f}}\to p_{\mathrm {f}},\quad \mathrm {if}~K\to \infty \tag{C.4}\end{equation*}

Similarly, under $\mathcal {H}_{0}$ , still based on the CLT, we have $$\begin{equation*} y_{\mathcal {H}_{0}}=\left ({\mathcal {t}\left ({\mathbf {Y\vert }\mathcal {H}_{0} }\right)-\mu _{\mathcal {H}_{0}} }\right)/\sigma _{\mathcal {H}_{0}}\vec {\sim }\mathcal {N}\left ({0,1 }\right) \tag{C.5}\end{equation*}$$ View Source\begin{equation*} y_{\mathcal {H}_{0}}=\left ({\mathcal {t}\left ({\mathbf {Y\vert }\mathcal {H}_{0} }\right)-\mu _{\mathcal {H}_{0}} }\right)/\sigma _{\mathcal {H}_{0}}\vec {\sim }\mathcal {N}\left ({0,1 }\right) \tag{C.5}\end{equation*} where $\mu _{\mathcal {H}_{0}}$ and $\sigma _{\mathcal {H}_{0}}^{2}$ can employ the related formulae of the Gamma distribution [13], [23] to be derived, given by $\mu _{\mathcal {H}_{0}}=\sigma _{z}^{2},\sigma _{\mathcal {H}_{0}}^{2}=\sigma _{z}^{4}/K$ .

Along the same way in which (C.2) is derived, conditioned on (C.3), we have $$\begin{align*} \mathrm {Q}\left ({\left ({\hat {\varepsilon }_{0}-\mu _{\mathcal {H}_{0}} }\right)/\sigma _{\mathcal {H}_{0}} }\right)\to \Gamma _{0}\left ({K,K\varepsilon _{0}/\sigma _{z}^{2} }\right) \quad \mathrm {if}~ K\to \infty \\ \tag{C.6}\end{align*}$$ View Source\begin{align*} \mathrm {Q}\left ({\left ({\hat {\varepsilon }_{0}-\mu _{\mathcal {H}_{0}} }\right)/\sigma _{\mathcal {H}_{0}} }\right)\to \Gamma _{0}\left ({K,K\varepsilon _{0}/\sigma _{z}^{2} }\right) \quad \mathrm {if}~ K\to \infty \\ \tag{C.6}\end{align*}

Based on (C.6), (21) and (17), we have $$\begin{equation*} \hat {p}_{\mathrm {f}}\to p_{\mathrm {f}} \quad \mathrm {if}~ K\to \infty \tag{C.7}\end{equation*}$$ View Source\begin{equation*} \hat {p}_{\mathrm {f}}\to p_{\mathrm {f}} \quad \mathrm {if}~ K\to \infty \tag{C.7}\end{equation*}

Consequently, based on (C.4), (C.7), (30) and (31), this proposition holds.

Proof:

Let the condition applicable to [1] labelled A describing that the received signal of the complex PSK experiences a path of line of sight (LOS) and is contaminated by CSCG distributed noise. Under condition A, [1, Formula (13)] defines its false alarm probability to be $p_{\mathrm {f}}=\mathrm {Q}\left ({\alpha +\gamma \sqrt {f_{\mathrm {s}}\tau } }\right)$ where $\alpha =\mathrm {Q}^{-1}\left ({p_{\mathrm {d0}} }\right)\sqrt {2\gamma + 1}$ as defined for (23). Let the condition of proposition 4 of this paper be labelled B. The change from condition A to B is equivalent to the replacement of $p_{\mathrm {f}}$ of [1, eq. (13)] by (22), (17) or (24) of this paper, which have distinct connotations as previously explained.

For (22), its proof is evident. Let us rewrite $\alpha =Q^{-1}\left ({p_{d0} }\right)\sqrt {\gamma ^{2}/m+2\gamma + 1}$ and substitute it into [1, eq. (13)]. All procedures of the relevant proofs will still go along in the same way as [1] does. For (17) or (24), it is not so obvious as (22)’s proof. However, the essences remain the same. Their main distinctions from (13) of [1] lie merely on the outermost operator of $\Gamma _{0}\left ({\cdot }\right)$ instead of $\text{Q}\left ({\cdot }\right)$ , and the internal parameter $\alpha$ being $\theta \Gamma _{0}^{-1}\left ({\kappa, p_{d0} }\right)\sqrt {\gamma ^{2}/m+2\gamma + 1}$ other than $Q^{-1}\left ({p_{d0} }\right)\sqrt {2\gamma + 1}$ . Despite these two differences, all corresponding proofs can still be conducted through the same approaches as [1] does.