# Wireless telecom system (cellular with Erlang and spectral efficiency)

We have a spectrum of 20MHz that we will use on a wireless cellular system, which requires 50kHz per channel. The system is using cluster size K = 4 and the radius of every cell is R = 1 km.

a) Find the distance between two cells that use the same kind of channels.

My ans: We can use the formula: $D = R\sqrt{3K} = 3.4 km$

b) Calculate the max number of calls that could be served simultaneously from the system per km2.

My ans: From given, $\dfrac{20000 kHz}{50 kHz} = 400$ channels, we have though K = 4, so we have 100 channel per cell. So I believe that 100 is the max calls per km2, because from given it says that cell radius is 1 km.

c) If the system becomes duplex, the users of the system during the peak hour, they do on average 1.5 calls/hour with average duration 120sec. We have Erlang-B and the GoS is 1%. Calculate the average number of calls per hour for every cell and the spectral efficiency.

My ans: Duplex: $\dfrac{20000 kHz}{100 kHz} = 200$ channels per cluster, so we have 50 channels per cell. Gos = 1% and cell got 50c, so from Erlang-b table we find that A = 37.90 or just 37.

Though $A = \lambda H$, so $\lambda$ = 37 calls / 1.5 calls per hour = 26 calls/hour.

Now this is what I tried so far but I am not sure if any of this is correct, so I just want a second opinion.

The last part, wants to find the spectral efficiency. But I really can't find how to do this. I searched a bit, and found this formula:

$$SE = \frac{A_c}{B_T\;C\;M\;S_u}$$

Though I am not sure how to use it exactly.

Check WCY Lee's book these are old AMPS calculation methods for spectral efficiency, there are some graphical methods to estimate efficiency using hexagonal geometries.

Page 186 - Mobile Communications Design Fundamentals (By William C. Y. Lee)

Also to calculate offerred traffic you have to convert call attempts/hr to attempts/second, thus

$\LARGE A_o= \frac{\frac{1.5 calls}{hr}}{\frac{3600s}{hr}} \times 120s$

$\Large A_o= 0.050 \text{ erlangs or 50 } \frac{mErlangs}{sub}$

Here is the link for the book:

Mobile Communications Design Fundamentals W CY Lee