Question

In a digital communication system, probability density function of the two level signal received in the receiver is: P_R(v) = P_S(v)*P_N(v) = [0.4δ(v+1) + 0.6δ(v-4)]*η(v). And , η(v) is the noise that added to the message sign as the additive Gaussian noise with a value of zero and an effective value of 3.

(* symbol means convolution process, in the solution of this problem you can use the below Q function table.) ,   η(v) = $(1/(\sigma\sqrt{2\pi }))e^{(-v^2/(2\sigma ^2)}$

A) Plot the probability density function of the signal received at the receiver.

B) Find the best decision threshold.

C) Find the probability of total error in the received signal.

Answer 1

(a) The probability density of "v" is

                           $P(v) = \int_{x = -\infty}^{\infty} \left(0.4*\delta (x+1) + 0.6*\delta (x-4) \right )*\eta(v-x)* dx$

Or

                       $P(v) =0.4*\eta(v+1) + 0.6*\eta(v-4) = 0.4*P_{1}(v) + 0.6*P_{2}(v)$

Or

                             

P() = (0.4 * e^(0+1)</202 +0.6*e-(v-4)2/202

Given that effective value of noise = 3. So sigma = 3. So we will plot P(v) from v = -15 to + 15 with steps of 0.001 in MATLAB as shown below:

sigma = 3;
v = (-15:0.001:15);
P1 = (1/(sigma*(2^0.5)))*(0.4*exp(-(v+1).^2/(2*sigma^2)));
>> P2 = (1/(sigma*(2^0.5)))*(0.4*exp(-(v-4).^2/(2*sigma^2)));
>> P = 0.4*P1 + 0.6*P2;
>> plot(v,P1)
>> hold all
>> grid on
>> plot(v,P2)
>> plot(v,P)
>> xlabel('voltage (v)')
>> ylabel('probability density function')
>> legend('P_{1}','P_{2}','P');

0.1 P1 P N- 0.08 U P 0.06 probability density function 0.04 0.02 0 -15 -10 -5 0 5 10 15 voltage (V)

(b) The probability function of 'v' is

                                                $P_{c}(v) = \int_{x= -\infty}^{v}P(x)*dx$

Or

             $P_{c}(v) = \int_{x= -\infty}^{v} \left(\frac{1}{\sigma \sqrt{2}}\right)\left[0.4*e^{-(x+1)^2/2\sigma^2} + 0.6*e^{-(x-4)^2/2\sigma^2} \right ]*dx$

Plot of Pc(v) is shown below:

Pc = cumsum(P)*0.001;
plot(v,Pc)
grid on;
xlabel('voltage(v)')
ylabel('probability function P_{c}(v)');

0.8 0.7 0.6 0.5 probability function P(V) 0.4 0.3 0.2 0.1 0 -15 -10 -5 5 10 15 0 voltage(v)

Let Vt be the threshold. Then the probabily of error is