The graph in Figure $2$ contains the same information as Block $1$ Figure $5\mathrm{.}9($Block $1,$ p$.79)$ but has a logarithmic grid which makes it easier to use.

Figure $2$ Bit error graph for Question $8$

The numbers on the right of Figure $2$ are multipliers. For example, working upwards from the horizontal line representing $\backslash (10^\{-10\}\backslash ),$ the next horizontal line up represents $\backslash (2\backslash$times $10^\{-10\}\backslash )$; the next line up represents $\backslash (4\backslash$times $10^\{-10\}\backslash ),$ and so on$.$ The next line up from $\backslash (8\backslash$times $10^\{-10\}\backslash )$ has no number on the right and represents $\backslash (10\backslash$times $10^\{-10\}\backslash ),$ which is $\backslash (10^\{-9\}\backslash ),$ as shown on the vertical axis on the left.

a$.$ As a signal gets weaker $($for example, towards the edge of its range$)$ the signal$-$to$-$noise ratio decreases and the error rate increases. At a signal$-$to$-$noise ratio of $5$ dB $,$ what would the bit$-$error probability be$?$

$(1$ mark$)$

b$.$ If$,$ by moving closer to the transmitter, the signal$-$to$-$noise ratio at a receiver improves ten$-$fold, what is the new bit$-$error probability.

$(2$ marks$)$

c$.$ By what factor has the bit$-$error probability improved by this ten$-$fold improvement of signal$-$to$-$noise ratio?

$(1$ mark$)$