# Exercise 1.2Z: Lognormal Fading Revisited

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We assume similar conditions as in  Exercise 1. 2  but now we summarize the purely distance-dependent path loss  $V_0$  and the mean value  $m_{\rm S}$  of the lognormal fading  (the index "S" stands for Shadowing):

$$V_{\rm 1} = V_{\rm 0} + m_{\rm S} \hspace{0.05cm}.$$

The total path loss is then given by the equation

$$V_{\rm P} = V_{\rm 1} + V_{\rm 2}(t)$$

where  $V_2(t)$  describes a  lognormal distribution  with mean value zero:

$$f_{V_{\rm S}}(V_{\rm S}) = \frac {1}{ \sqrt{2 \pi }\cdot \sigma_{\rm S}} \cdot {\rm e }^{ - { (V_{\rm S}\hspace{0.05cm}- \hspace{0.05cm}m_{\rm S})^2}/(2 \hspace{0.05cm}\cdot \hspace{0.05cm}\sigma_{\rm S}^2) }\hspace{0.05cm}.$$

The path loss model shown in the graphic is suitable for the scenario described here:

• Multiply the transmitted signal  $s(t)$  first with a constant factor  $k_1$  and further with a stochastic quantity  $z_2(t)$  with the probability density function  $\rm (PDF)$  $f_{\rm z_2}(z_2)$.
• Then the signal  $r(t)$ results at the output, whose power  $P_{\rm E}(t)$  is of course also time-dependent due to the stochastic component.
• The PDF of the lognormally distributed random variable  $z_2$  is for  $z_2 ≥ 0$:
$$f_{z_{\rm 2}}(z_{\rm 2}) = \frac {{\rm e^{- {\rm ln}^2 (z_{\rm 2}) /({2 \hspace{0.05cm}\cdot \hspace{0.05cm} C^2 \hspace{0.05cm} \cdot \hspace{0.05cm} \sigma_{\rm S}^2}) } } }{ \sqrt{2 \pi }\cdot C \cdot \sigma_{\rm S} \cdot z_2} \hspace{0.8cm}{\rm with} \hspace{0.8cm} C = \frac{{\rm ln} \hspace{0.1cm}(10)}{20\,\,{\rm dB}}\hspace{0.05cm}.$$
• For  $z_2 ≤ 0$  this PDF is equal to zero.

Notes:

$$V_{\rm 1} = 60\,{\rm dB}\hspace{0.05cm},\hspace{0.2cm} \sigma_{\rm S} = 6\,{\rm dB}\hspace{0.05cm}.$$
• The probability that a mean-free Gaussian random variable  $z$  is greater than its standard deviation  $\sigma$, is
$${\rm Pr}(z > \sigma) = {\rm Pr}(z < -\sigma) = {\rm Q}(1) \approx 0.158\hspace{0.05cm}.$$
• Also,   ${\rm Pr}(z > 2\sigma) = {\rm Pr}(z < -2\sigma) = {\rm Q}(2) \approx 0.023\hspace{0.05cm}.$
• Again for clarification:   $z_2$  is the fading coefficient in linear units, while   $V_2$  is the fading coefficient in logarithmic units.
• The following conversions apply:
$$z_2 = 10^{-V_{\rm 2}/20\,{\rm dB}}\hspace{0.05cm}, \hspace{0.2cm} V_{\rm 2} = -20\,{\rm dB} \cdot {\rm lg}\hspace{0.15cm}z_2\hspace{0.05cm}.$$

### Questions

1

How large should the constant  $k_1$  be?

 $k_1\ = \$

2

Which value range applies to the random variable  $z_2$?

 All values between  $-∞$ and $+∞$  are possible. The random size  $z_2$  is not negative. The smallest possible value is  $z_2 = 0.5$. The largest possible value is  $z_2 = 2$.

3

Calculate the PDF  $f_{\rm z2}(z_2)$  for some abscissa values.

 $f_{\rm z2}(z_2 = 0)\ = \$ $f_{\rm z2}(z_2 = 1)\ = \$ $f_{\rm z2}(z_2 = 2)\ = \$

4

Calculate the following probabilities.

 ${\rm Pr}(z_2 > 1.0)\ = \$ ${\rm Pr}(z_2 > 0.5)\ = \$ ${\rm Pr}(z_2 > 4.0)\ = \$

5

What statements are valid for the average received power  ${\rm E}\big [P_{\rm E}(t)\big]$?   Note:  $P_{\rm E}'$ is the power after multiplication by  $k_1$  (see diagram).

 ${\rm E}[P_{\rm E}(t)] = P_{\rm E}'$ ${\rm E}[P_{\rm E}(t)] < P_{\rm E}'$. ${\rm E}[P_{\rm E}(t)] > P_{\rm E}'$.

### Solution

#### Solution

(1)  The constant  $k_1$  generates the time-independent path loss  $V_1 = 60 \ \rm dB$.  From this follows:

$$k_{\rm 1} = 10^{-V_{\rm 1}/(20\hspace{0.05cm} {\rm dB})} \hspace{0.15cm} \underline{= 0.001}\hspace{0.05cm}.$$

(2)  Only the  second statement  is correct:

• For the Gaussian random variable  $V_2$  all values between  $-∞$  and  $+∞$  are (theoretically) possible.
• The transformation  $z_2 = 10^{{\it -V_2}\rm /20}$  results in only positive values for the linear random variable  $z_2$,
namely between  0  $($if  $V_2$  is positive and goes to infinity$)$  and  $+∞$  $($very large negative values  of $V_2)$.

(3)  The random value  $z_2$  can only be positive.  Therefore the PDF value  $f_{\rm z2}(z_2 = 0)\hspace{0.15cm}\underline{= 0}$.

• The PDF–value for the abscissa value  $z_2 = 1$  is obtained by inserting it into the given equation:
$$f_{z{\rm 2}}(z_{\rm 2} = 1) \hspace{-0.1cm} \ = \ \hspace{-0.1cm} \frac {{\rm e^{- {\rm ln}^2 (z_{\rm 2}=1) /({2 \hspace{0.05cm}\cdot \hspace{0.05cm} C^2 \hspace{0.05cm} \cdot \hspace{0.05cm} \sigma_{\rm S}^2}) } } }{ \sqrt{2 \pi }\cdot C \cdot \sigma_{\rm S} \cdot (z_2 = 1)} = \frac {1}{ \sqrt{2 \pi } \cdot 6\,\,{\rm dB} } \cdot \frac {20\,\,{\rm dB}}{ {\rm ln} \hspace{0.1cm}(10) } \hspace{0.15cm} \underline{\approx 0.578}\hspace{0.05cm}.$$
• The first portion is equal to the PDF–value  $f_{V2}(V_2 = 0)$.
• $C$  considers the magnitude of the derivative of the non-linear characteristic  $z_2 = g(V_2)$  for  $V_2 = 0 \ \rm dB$  or  $z_2 = 1$.
• Finally, for  $z_2 = 2$:
$$f_{z{\rm 2}}(z_{\rm 2} = 2) \hspace{-0.1cm} \ = \ \hspace{-0.1cm} \frac {f_{z{\rm 2}}(z_{\rm 2} = 1)}{ z_{\rm 2} = 2} \cdot {\rm exp } \left [ - \frac {{\rm ln}^2 (2)}{2 \cdot C^2 \cdot \sigma_{\rm S}^2} \right ]= \frac {0.578}{ 2} \cdot {\rm exp } \left [ - \frac {0.48}{0.952} \right ] \hspace{0.15cm} \underline{\approx 0.174}\hspace{0.05cm}.$$

(4)  If you take into account the relationship between  $z_2$  and  $V_2$, you get

$${\rm Pr}(z_{\rm 2} > 1) \hspace{-0.1cm} \ = \ \hspace{-0.1cm} {\rm Pr}(V_{\rm 2} < 0\,\,{\rm dB})\hspace{0.15cm} \underline{= 0.5} \hspace{0.05cm},$$
$${\rm Pr}(z_{\rm 2} > 0.5) \hspace{-0.1cm} \ = \ \hspace{-0.1cm} {\rm Pr}(V_{\rm 2} < 6\,\,{\rm dB}) = 1- {\rm Pr}(V_{\rm 2} > 6\,\,{\rm dB})= 1- {\rm Pr}(V_{\rm 2} > \sigma_{\rm S})= 1- {\rm Q}(1)\hspace{0.15cm} \underline{= 0.842} \hspace{0.05cm},$$
$${\rm Pr}(z_{\rm 2} > 4) \hspace{-0.1cm} \ = \ \hspace{-0.1cm} {\rm Pr}(V_{\rm 2} < -12\,\,{\rm dB}) = {\rm Pr}(V_{\rm 2} > +12\,\,{\rm dB}) = {\rm Pr}(V_{\rm 2} > 2 \sigma_{\rm S}) \hspace{0.05cm}.$$
• The probability that a Gaussian variable is greater than  $2 \cdot \sigma$  equals  ${\rm Q}(2)$:
$${\rm Pr}(z_{\rm 2} > 4) = {\rm Q}(2)\hspace{0.15cm} \underline{= 0.023} \hspace{0.05cm}.$$

(5)  The statement 3 is correct:

• The first statement is certainly not correct, since the mean value  $m_{\rm S}$  refers to the logarithmic received power  $($in  $\rm dBm)$.
• To clarify whether the second or the third statement is correct, we assume  $P_{\rm S} = 1 \ \rm W$, $V_1 = 60 \ \rm dB$  ⇒  $P_{\rm E}' = 1 \ {\rm µ W}$  and the following PDF for  $V_2$:
$$f_{V{\rm 2}}(V_{\rm 2}) = 0.5 \cdot \delta (V_{\rm 2}) + 0.25 \cdot \delta (V_{\rm 2}- 10\,\,{\rm dB}) + 0.25 \cdot \delta (V_{\rm 2}+ 10\,\,{\rm dB})\hspace{0.05cm}.$$
• Half of the time,  $P_{\rm E} = 1 \ \rm µ W$, while each of the following has  $25\%$ probability::
$$V_{\rm 2}= +10\,\,{\rm dB}\text{:} \hspace{0.3cm} P_{\rm E}(t) \hspace{-0.1cm} \ = \ \hspace{-0.1cm} \frac{1\,\,{\rm W}}{10^7} = 0.1\,\,{\,}{\rm µ W}\hspace{0.05cm},$$
$$V_{\rm 2}= -10\,\,{\,}{\rm dB}\text{:} \hspace{0.3cm} P_{\rm E}(t) \hspace{-0.1cm} \ = \ \hspace{-0.1cm} \frac{1\,\,{\rm W}}{10^5} = 10\,\,{\,}{\rm µ W}\hspace{0.05cm}.$$
• The mean value is then:
$${\rm E}[P_{\rm E}(t)] = 0.5 \cdot 1\,{\rm µ W}+ 0.25 \cdot 0.1\,{\rm µ W}+ 0.25 \cdot 10\,{\rm µ W}= 3.025\,{\rm µ W} > P_{\rm E}\hspace{0.05cm}' = 1\,{\rm µ W} \hspace{0.05cm}.$$
• This simple calculation with discrete probabilities instead of a continuous PDF indicates that statement 3 is correct.