Question

Consider a wireless communication link between a transmitter and a receiver located in free space, with finite and strictly positive capacity. If the effective areas of the transmitter and the receiver antennas, and the distance between them are all doubled, and everything else remains unchanged, the maximum capacity of the wireless link

1. increases by a factor of 2
2. decreases by a factor of 2
3. remains unchanged
4. decreases by a factor of √2

Answer 1

Correct Answer - Option 3 : remains unchanged

Concept:

The capacity of a wireless link is given by Shanon Hartley theorem as:

\(C = B~lo{g_2}\left( {1 + \frac{S}{N}} \right)\)

Channel capacity depends on \(\left( {\frac{S}{N}} \right)\), and hence on S.

S = Received power at the receiver = P_r

\(S = {P_r} = \frac{{{P_T}{G_t}{G_r}}}{{{{\left( {\frac{{4\;{\bf{\pi }}R}}{\lambda }} \right)}^2}}}\)

\(= \frac{{{P_t}\left( {\frac{{{G_\pi }}}{{{\lambda ^2}}}{A_{te}}} \right)\left( {\frac{{4\pi }}{{{\lambda ^2}}}{A_{re}}} \right)}}{{{{\left( {\frac{{4\pi R}}{\lambda }} \right)}^2}}}\)

Calculation:

It is given,

A_te' = 2A_te

A_re' = 2 A_re

R' = 2 R

\({P_{r'}} = \frac{{{P_t}\left( {\frac{{4\pi \;{A_{te}}}}{{{\lambda ^2}}}} \right)\left( {\frac{{4\pi }}{{{\lambda ^2}}}{A_{rc}}} \right)2 \times 2}}{{{{\left( {\frac{{4\pi R}}{\lambda }} \right)}^2}{{\left( 2 \right)}^2}}}\)

P_r' = P_r

S' = S

\(\frac{{S'}}{N} = \frac{S}{N}\)

⇒ C' = C, i.e.

The maximum capacity of the channel remains unchanged.