Exercise 1.2: Lognormal Channel Model

PDF of lognormal fading

We consider a mobile radio cell in an urban area and a vehicle that is approximately at a fixed distance $d_0$ from the base station. For example, it moves on an arc around the base station.

Thus the total path loss can be described by the following equation: $$V_{\rm P} = V_{\rm 0} + V_{\rm S} \hspace{0.05cm}.$$

$V_0$ takes into account the distance-dependent path loss which is assumed to be constant: $V_0 = 80 \ \rm dB$ .
The loss $V_{\rm S}$ is due to shadowing caused by the lognormal–distribution with the probability density function

$$f_{V{\rm S}}(V_{\rm S}) = \frac {1}{ \sqrt{2 \pi }\cdot \sigma_{\rm S}} \cdot {\rm exp } \left [ - \frac{ (V_{\rm S}- m_{\rm S})^2}{2 \cdot \sigma_{\rm S}^2} \right ] \hspace{0.05cm}$$

is described in sufficient detail (see diagram). The following numerical values apply:

$$m_{\rm S} = 20\,\,{\rm dB}\hspace{0.05cm},\hspace{0.2cm} \sigma_{\rm S} = 10\,\,{\rm dB}\hspace{0.15cm}{\rm or }\hspace{0.15cm}\sigma_{\rm S} = 0\,\,{\rm dB}\hspace{0.15cm}{\rm (subtask\hspace{0.15cm} 2)}\hspace{0.05cm}.$$

Also make the following simple assumptions:

The transmit power is $P_{\rm S} = 10 \ \rm W$ (or $40 \ \rm dBm$).
The receive power should be at least $P_{\rm E} = 10 \ \rm pW$ (or $–80 \ \rm dBm$)

Notes:

The task belongs to the chapter Distance-dependent attenuation and shading.

You can use the following (rough) approximations for the complementary Gaussian error integral:

$${\rm Q}(1) \approx 0.16\hspace{0.05cm},\hspace{0.2cm} {\rm Q}(2) \approx 0.02\hspace{0.05cm},\hspace{0.2cm} {\rm Q}(3) \approx 10^{-3}\hspace{0.05cm}.$ * Or use the interaction module provided by $\rm LNTww$ [[Applets:Complementary_Gaussian_Error_Functions_(new_applet)|Complementary_Gaussian_Error_Functions]]. =='"`UNIQ--h-0--QINU`"'=Questionnaire== <quiz display=simple> {Would $P_{\rm E}$ without consideration of the lognormal–fading be sufficient? |type="()"} + Yes, - No. The log normal andash parameters are $m_{\rm S} = 20 \, \rm dB$ and $\sigma_{\rm S} = 0 \, \rm dB$. What percentage of the time does the system work? |type="{}"} ${\rm Pr(System \ works)} \ = \ $ { 100 3% } $\ \%$ {What is the probability with $m_{\rm S} = 20 \ \ \rm dB$ and $\sigma_{\rm S} = 10 \ \ \rm dB$? |type="{}"} ${\rm Pr(System \ works)}\ = \ $ { 98 3% } $\ \%$ {How big can $V_0$ be maximum, so that the reliability to $99.9\%$ is reached? |type="{}"} $V_0 \ = \ $ { 70 3% } $\ \ \rm dB$ </quiz ==='"`UNIQ--h-1--QINU`"' sample solution=== [[:Template:ML head]] '''(1)''' Correct is <u>YES</u>: *From the $\rm dB$–value $V_0 = 80 \ \rm dB$ follows the absolute (linear) value $K_0 = 10^8$. Thus the received power is $$P_{\rm E} = P_{\rm S}/K_0 = 10 \ {\rm W}/10^8 = 100 \ {\rm nW} > 10 \ \ \rm pW.$$ *You can also solve this problem directly with the logarithmic quantities: $10 \cdot {\rm lg}\hspace{0.15cm} \frac{P_{\rm E}}}{1\,\,{\rm mW}} = 10 \cdot {\rm lg}\hspace{0.15cm} \frac{P_{\rm S}}{1\,\,{\rm mW}} - V_0 = 40\,{\rm dBm} -80\,\,{\rm dB} = -40\,\,{\rm dBm} \hspace{0.05cm}.$$

Only the limit value $–80 \ \rm dBm$ is required.

(2) Lognormal–Fading with $\sigma_{\rm S} = 0 \ \rm dB$ is equivalent to a constant receive reading $P_{\rm E}$.

Compared to the subtask '(1) this is $m_{\rm S} = 20 \ \ \rm dB$ smaller ⇒ $P_{\rm E} = \ –60 \ \ \rm dBm$.
But it is still greater than the specified limit value ($–80 \ \rm dBm$).
It follows: The system is (almost) 100% functional. „Fast” because with a Gaussian random quantity there is always a (small) residual uncertainty.

(3) The receive power is too low (less than $–80 \ \rm dBm$) if the power loss due to the lognormal–term is $40 \ \rm dB$ or more.

The variable portion $V_{\rm S}$ must therefore not be greater than $20 \ \rm dB$.
So it follows:

$${\rm Pr}({\rm "System\hspace{0.15cm} does not work\hspace{0.15cm}"})= {\rm Q}\left ( \frac{20\,\,{\rm dB}}}{\sigma_{\rm S} = 10\,{\rm dB}\right ) = {\rm Q}(2) \approx 0.02\hspace{0.3cm} \Rightarrow \hspace{0.3cm}{\rm Pr}({\rm "System\hspace{0.15cm} works"})= 1- 0.02 \hspace{0.15cm} \underline{\approx 98\,\%}\hspace{0.05cm}.$$

loss due to lognormal fading

The graphic illustrates the result.

The probability density $f_{\rm VS}(V_{\rm S})$ of the path loss due to shadowing (Longnormal–Fading) is shown here.
Die Wahrscheinlichkeit, dass das System ausfällt, ist rot markiert:

(4) Aus der Verfügbarkeitswahrscheinlichkeit $99.9 \%$ folgt die Ausfallwahrscheinlichkeit $10^{\rm –3} \approx \ {\rm Q}(3)$.

Verringert man den entfernungsabhängigen Pfadverlust $V_0$ um $10 \ \rm dB$ auf $\underline {70 \ \rm dB}$, so kommt es erst dann zu einem Ausfall, wenn $V_{\rm S} ≥ 50 \ \rm dB$ ist.
Damit wäre genau die geforderte Zuverlässigkeit erreicht, wie die folgende Rechnung zeigt:

$${\rm Pr}({\rm "System\hspace{0.15cm}funktioniert\hspace{0.15cm}nicht"})= {\rm Q}\left ( \frac{120-70-20}{10}\right ) = {\rm Q}(3) \approx 0.001 \hspace{0.05cm}.$$

The probability that the system will fail is marked in red:

(4) From the availability probability $99.9 \%$ follows the default probability $10^{\rm –3} \approx \ {\rm Q}(3)$.

If the distance-dependent path loss $V_0$ is reduced by $10 \ \ \rm dB$ to $\underline {70 \ \rm dB}$, a failure will only occur when $V_{\rm S} ≥ 50 \ \ \rm dB$.
This would achieve exactly the required reliability, as the following calculation shows:

$${\rm Pr}({\rm "System\hspace{0.15cm} does not work\hspace{0.15cm}"})= {\rm Q}\left ( \frac{120-70-20}{10}\right ) = {\rm Q}(3) \approx 0.001 \hspace{0.05cm}.$$