ABSTRACT
Since the receiver decides that any voltage above V' is a '1' and any voltage below V' is a '0' we can predict the frequency of mistakes by calculating the fraction of the plots which are the wrong side of V'. When the transmitter is trying to send V, the probability or relative frequency, p { V}, with which the received voltage is seen to be in a small interval, d V, centred at some voltage, V, will be
{-2(V - v,f} p{v} .dV = A. Exp a2 .dV ... (4.2) Since the observed voltage must always be somewhere in the range from --oo to t-oo we can say that the value of the coefficient, A, must be such that
r: A. Exp{ -Z(V a~ v,f} dV = 1 ... (4.3) i.e. the probability that the observed voltage is somewhere between~ and -too is unity. Since the total area under the distribution shape isn't allected by the choice of v, this is equivalent to saying that
When V, is being sent, the chance, C,, it will be correctly received is determined by the fraction of the distribution which lays above V'. This can be determined from integrating over the appropriate part of the curve to obtain
In a similar way, the chance Co. that V0 will be correctly received is determined by the fraction of the distribution which is below V' when V0
is being sent, i.e. we can say
and
C, = ~- [1 + Erf{V2. (V~- V')}] Co=~- [1 + Erf{V2. (V;- Vol}]
where Erf is a standard mathematical function called the Error Function. Since this iso't a pure maths book the details of this proof and the precise nature of the error function don't matter very much. It is enough for us to accept that it is just another function like sine or cos that we can look up in a book and which happens to be the right one to solve the integrals. We can now use the above expressions to see how often the receiver will pick up the correct signal level in the presence of some noise.
