Study of Delay Spread for Simulcast Radio Transmission in Time-Dispersive Mobile Radio Networks

Abstract

This paper analyzes and evaluates the statistical properties of the rms delay spread (delay spread) in a two-path simulcast environment and same frequency in cell repeater. Each path is subjected to multipath fading characterized by wideband Weibull distribution. A novel, unified, and accurate analytical expressions for the probability density function, the cumulative distribution function, the mean, the mean square, and the standard deviation for the delay spread have been derived. The derived expressions are then used to study the implication of the fading and scale parameters on the statistical characteristics of the delay spread. To validate the accuracy of the derived formulas, the analytical results are compared with Monte-Carlo simulation results. Full agreement has been noticed between the simulated and the analytical results over a wide range of received signals levels ratio and different values of the fading parameters.

Appendices

Appendix 1: Proof of Theorem 1

To obtain a formula for the PDF of the ratio R and under the assumption that the random variables X and Y are statistical independents, the following integral [24, (6.43)]

$$\begin{aligned} {f_R}(r) = \int \limits _0^\infty {y{f_{X}}(yr){f_{Y}}(y)dy} \end{aligned}$$

(19)

has to be expressed in closed form. After substituting (5) and (6) into (19), we get

$$\begin{aligned} {f_R}(r) = \frac{{{k_x}}}{{\lambda _x^{{k_x}}}}\frac{{{k_y}}}{{\lambda _y^{{k_y}}}}{r^{{k_x} - 1}}\int \limits _0^\infty {{y^{{k_x} + {k_y} - 1}}{e^{ - {{\left( {\frac{{yr}}{{{\lambda _x}}}} \right) }^{{k_x}}}}}{e^{ - {{\left( {\frac{y}{{{\lambda _y}}}} \right) }^{{k_y}}}}}dy}. \end{aligned}$$

(20)

Invoking identity [25, 2.9.4], namely, ${e^{ - z}} = H_{0,1}^{1,0}\left[ {z\left| {\begin{array}{c} - \\ {\left( {0,1} \right) } \end{array}} \right. } \right]$, then applying [25, Theorem 2.9], ${f_R}\left( r \right)$ can be expressed in closed form as in (7). Hence, the proof is completed.

Appendix 2: Proof of Lemma 1

The mean rms delay spread can be expressed as

$$\begin{aligned} \left\langle {{\tau _{rms}}} \right\rangle = \int _{ - \infty }^\infty {{\tau _{rms}}(r)} {f_R}\left( r \right) dr. \end{aligned}$$

(21)

Placing (3) in (21) yields

$$\begin{aligned} \left\langle {{\tau _{rms}}} \right\rangle = {\mathrm{{t}}_x}\int _{ - \infty }^\infty {\frac{{\sqrt{{r^4} + \left( {\alpha + 2} \right) {r^2} + \left( {\beta + 1} \right) } }}{{{r^2} + 1}}} {f_R}\left( r \right) dr. \end{aligned}$$

(22)

In order to derive a closed form expression for the above integral, two approximations will be carried out on both the numerator and denominator of $\tau _{rms}(R)$. Hence, the numerator of $\tau _{rms}(R)$ can be expressed as [13]

$$\begin{aligned} \tau _{rms}^{Num}(R)= & {} \sqrt{{R^4} + \left( {\alpha + 2} \right) {R^2} + \left( {\beta + 1} \right) } \nonumber \\\approx & {} \left\{ \begin{array}{ll} \frac{{{R^4} + (\alpha + 2){R^2} + (2\beta + 2 + {k_1})}}{{2\sqrt{\beta + 1 + {k_1}} }},&{} \quad 0< R< {R_1} \nonumber \\ \nonumber \\ \frac{{{R^3} + [2(\alpha + 2) + {k_2}]R + (\beta + 1){R^{ - 1}}}}{{2\sqrt{\alpha + 2 + {k_2}} }},&{} \quad {R_1}< R< {R_2}\nonumber \\ \nonumber \\ \frac{{(\alpha + 2) + (\beta + 1){R^{ - 2}} + ({k_3} + 2){R^2}}}{{2\sqrt{1 + {k_3}} }},&{} \quad {R_2} < R \end{array} \right. \\ \end{aligned}$$

(23)

Note that, an appropriate numerical algorithm has been reported in [13] which is utilized in this study in order to estimate the values of the parameters $k_i$, $i=1, 2$, and 3 and the values of $R_i$, $i=1$ and 2. The denominator can also be expanded with the appropriate Taylor series as [26]

$$\begin{aligned} \tau _{rms}^{Den}(R)= & {} {\left( {{R^2} + 1} \right) ^{ - 1}}\nonumber \\= & {} \left\{ \begin{array}{ll} \sum \limits _{i = 1}^\infty {{{( - 1)}^i}{R^{2(i - 1)}}} ,&{} \quad R < 1\\ \\ \sum \limits _{i = 1}^\infty {{{( - 1)}^i}{R^{ - 2i}}} ,&{} \quad R > 1 \end{array} \right. \end{aligned}$$

(24)

After utilizing the above approximations and splitting the integrals into appropriate partitions, $\left\langle {{\tau _{rms}}} \right\rangle$ can be expressed as

$$\begin{aligned} \left\langle {{\tau _{rms}}} \right\rangle=\, & {} {t_x}\sum \limits _{i = 1}^\infty {{{( - 1)}^i}\left\{ {{{\left( {2\sqrt{\beta + 1 + {k_1}} } \right) }^{ - 1}} } \right. } \nonumber \\&\times \left[ {\int _0^{{R_1}} {{r^{2i + 2}}{f_R}\left( r \right) dr} } \right. + (\alpha + 2)\int _0^{{R_1}} {{r^{2i}}{f_R}\left( r \right) dr} \nonumber \\&+\,(2\beta + 2 + {k_1})\left. {\int _0^{{R_1}} {{r^{2(i - 1)}}{f_R}\left( r \right) dr} } \right] \nonumber \\&+\frac{1}{{2\sqrt{\alpha + 2 + {k_2}} }}\left[ {\int _{{R_1}}^1 {{r^{2i + 1}}{f_R}\left( r \right) dr} } \right. \nonumber \\&+\left. {\int _1^{{R_2}} {{r^{ - 2i + 3}}{f_R}\left( r \right) dr} } \right] + \frac{{2(\alpha + 2) + {k_2}}}{{2\sqrt{\alpha + 2 + {k_2}} }} \nonumber \\&\times \left[ {\int _{{R_1}}^1 {{r^{2i - 1}}{f_R}\left( r \right) dr} + } \right. \left. {\int _1^{{R_2}} {{r^{ - 2i + 1}}{f_R}\left( r \right) dr} } \right] \nonumber \\&+\frac{{\beta + 1}}{{2\sqrt{\alpha + 2 + {k_2}} }}\left[ {\int _{{R_1}}^1 {{r^{2i - 3}}{f_R}\left( r \right) dr} } \right. \nonumber \\&+\left. {\int _1^{{R_2}} {{r^{ - 2i - 1}}{f_R}\left( r \right) dr} } \right] + {\left( {2\sqrt{1 + {k_3}} } \right) ^{ - 1}}\left[ {\mathop {(\alpha + 2) }} \right. \nonumber \\&\times \int _{{R_2}}^\infty {{r^{ - 2i}}{f_R}\left( r \right) dr} + (\beta + 1)\int _{{R_2}}^\infty {{r^{ - 2(i + 1)}}{f_R}\left( r \right) dr} \nonumber \\&+ \left. {\left. {({k_3} + 2)\int _{{R_2}}^\infty {{r^{ - 2(i - 1)}}{f_R}\left( r \right) dr} } \right] } \right\} . \end{aligned}$$

(25)

Placing (7) in (25), the following integrals with respect to r, namely, $\mathcal {I}_1$, $\mathcal {I}_2$, $\mathcal {I}_3$, and $\mathcal {I}_4$ appear. Utilizing [25], these integrals can be expressed in closed forms as

$$\begin{aligned} \mathcal {I}_1(m, {R_1})= & {} \int \limits _0^{{R_1}} {{r^m}{f_R}\left( r \right) dr}\nonumber \\= & {} -{k_y}{\left( {\frac{{{\lambda _x}}}{{{\lambda _y}}}} \right) ^{{k_y}}}R_1^{ - \left( {{k_y} - m} \right) } \nonumber \\&\times H_{2,2}^{2,1}\left[ {{{\left( {\frac{{{\lambda _x}}}{{{\lambda _y}}}} \right) }^{{k_y}}}R_1^{ - {k_y}}\left| \begin{array}{l} \phi _1\\ \phi _2 \end{array} \right. } \right] , \end{aligned}$$

(26)

where $\phi _1=\Big \{ \left( { - \frac{k_y}{k_x},\frac{k_y}{k_x}} \right) ,\left( {1 - {k_y} + m,{k_y}} \right) \Big \}$ and $\phi _2=\Big \{ \left( { - {k_y} + m,{k_y}} \right) ,\left( {0,1} \right) \Big \}$.

$$\begin{aligned} \mathcal {I}_2(m, {R_1}) & = \int \limits _{{R_1}}^1 {{r^m}{f_R}\left( r \right) dr} \nonumber \\ & = \mathcal {I}_1(m,1) - \mathcal {I}_1(m,{R_1}). \end{aligned}$$

(27)

$$\begin{aligned} \mathcal {I}_3(m, {R_2}) & = \int \limits _1^{{R_2}} {{r^m}{f_R}\left( r \right) dr} \nonumber \\ & = \mathcal {I}_1(m,{R_2}) - \mathcal {I}_1(m,1). \end{aligned}$$

(28)

$$\begin{aligned} \mathcal {I}_4(m, {R_2})=\, & {} \int \limits _{{R_2}}^{\infty } {{r^m}{f_R}\left( r \right) dr} \nonumber \\= & {} -{k_y}{\left( {\frac{{{\lambda _x}}}{{{\lambda _y}}}} \right) ^{{k_y}}}R_2^{ - \left( {{k_y} - m} \right) } \nonumber \\&\times H_{2,2}^{2,1}\left[ {{{\left( {\frac{{{\lambda _x}}}{{{\lambda _y}}}} \right) }^{{k_y}}}R_2^{ - {k_y}}\left| \begin{array}{l} \phi _3\\ \phi _4 \end{array} \right. } \right] , \end{aligned}$$

(29)

where $\phi _3=\Big \{ \left( {1 - {k_y} + m,{k_y}} \right) ,\left( { - \frac{{{k_y}}}{{{k_x}}},\frac{{{k_y}}}{{{k_x}}}} \right) \Big \}$ and $\phi _4=\Big \{ \left( {0,1} \right) ,\left( { - {k_y} + m,{k_y}} \right) \Big \}$. Finally, the mean value of the rms delay spread can be written as in (8). Hence, the proof is completed.

Appendix 3: Proof of Lemma 2

Utilizing the basic definition of the generalized moments in conjunction with the unified PDF in (7), that is

$$\begin{aligned} \left\langle {\tau _{rms}^2} \right\rangle= &\, {} \int _{ - \infty }^\infty {\tau _{rms}^2(r)} {f_R}\left( r \right) dr\nonumber \\=\, & {} \mathrm{{t}}_x^2\int _{ - \infty }^\infty {\frac{{{r^4} + \left( {\alpha + 2} \right) {r^2} + \left( {\beta + 1} \right) }}{{{{\left( {{r^2} + 1} \right) }^2}}}} {f_R}\left( r \right) dr\nonumber \\=\, & {} \mathrm{{t}}_x^2\int _{ - \infty }^\infty {\left[ {1 + \frac{{\alpha {r^2} + \beta }}{{{{\left( {{r^2} + 1} \right) }^2}}}} \right] } {f_R}\left( r \right) dr \end{aligned}$$

(30)

The rational expression in the above equation, namely, ${\left( {{r^2} + 1} \right) ^{ - 2}}$ can be expanded with the appropriate Taylor series as [26]

$$\begin{aligned} {\left( {{r^2} + 1} \right) ^{ - 2}} = \left\{ \begin{array}{ll} \sum \limits _{i = 1}^\infty {{{( - 1)}^{i + 1}}i{r^{2(i - 1)}}} ,&{} \quad r < 1\\ \\ \sum \limits _{i = 1}^\infty {{{( - 1)}^{i + 1}}i{r^{ -2(i + 1)}}} ,&{} \quad r > 1 \end{array} \right. \end{aligned}$$

(31)

We rearrange (31) as

$$\begin{aligned} \left\langle {\tau _{rms}^2} \right\rangle= &\, {} \mathrm{{t}}_x^2\left[ {1 + \sum \limits _{i = 1}^\infty {{{( - 1)}^{i + 1}}i} \left\{ {\alpha \left( {\int _0^1 {{r^{2i}}{f_R}\left( r \right) dr} } \right. } \right. } \right. \nonumber \\&+\left. {\int _1^\infty {{r^{ - 2i}}{f_R}\left( r \right) dr} } \right) + \beta \left( {\int _0^1 {{r^{2(i - 1)}}{f_R}\left( r \right) dr } } \right. \nonumber \\&+\left. {\left. {\left. {\int _1^\infty {{r^{ - 2(i + 1)}}{f_R}\left( r \right) dr} } \right) } \right\} } \right] \end{aligned}$$

(32)

Finally, one can easily obtain an expression for the (32) as in (9). Thus, the proof is completed.