Dump Filter Receiver | Computation of Probability of Error for Integrate block diagram

Computation of Probability of Error for Integrate and Dump Filter Receiver block diagram and working , formula :-
As discussed earlier that a noise interference may lead to wrong decision at the receiver end. As a matter of fact, the probability of error denoted by P_e is a good measure of performance of the detector.
We know that the output of the integrator is expressed as,
y(t) = x₀(t) + n₀(t)
For the positive pulse of amplitude A, x₀(t) is given as,
x₀(t) =                      for x (t)  A
Similarly, for the input pulse of amplitude – A, x₀(t) is given by,
x₀(t) =                            or   x(t) = -A
Therefore, the output y(t) may be written as
x(t) =                  or    x(t) = A                …(6.53)
Similarly,              y(t) =         or    x(t) = -A                  …(6.54)
Let us consider that x(t) = -A. Further, if noise n₀(t) is greater than , then the output y(t) would be positive according to equation (6.54). After that the receiver will decide in favour of symbol + A, which is wrong decision. This means that an error is introduced.
Similarly, let us consider that x(t) = + A. If noise n₀(t) > –  then, the output y(t) will be negative according to equation (6.53). This leads to decision in favour of –A, which is erroneous. Based on the above discussion, we can make conclusions about probability of error in the form of a Table 6.4.
Table 6.4. Probability of error in integrate and dump filter receiver

 
As shown in Table 6.4, the error will be introduced depending upon probability that n₀(t) takes a particular value. These probabilities may be obtained from PDF of n₀(t). Recall that the probability density function (PDF) of the Gaussian distributed function is expressed by standard relation as under:
EQUATION                                      …(6.55)
where, f_X(x) = the PDF of random function x.
m = the mean value and
and                          = the standard deviation.
Here, because we have to evaluate PDF for white Gaussian noise, therefore, we have
x = n₀(t)
Since this noise has zero mean value i.e., m = 0, therefore equation (6.55) may be written as,
EQUATION                                      …(6.56)
The standard deviation  is expressed as,
= [mean square value – square of mean value]^1/2
This means that
Further, we have
mean square value                    =
Also, mean value m_x = 0 for this noise.
Therefore, we have
…(6.57)
Hence equation (16.56), can be written as,
EQUATION
At this stage, it may be noted that n₀(t) is the function like ‘x’. It is a random variable and we are evaluating its PDF.
On simplifying above equation, we get
EQUATION                                      …(6.58)
This equation describes PDF of white Gaussian noise.
Figure 6.21 shows the graphical representation of this PDF.
From the property of PDF, we know that,
EQUATION                                      …(6.59)
This equation gives the probability that n₀(t) takes values greater than  .
The above integration gives the area under the curve from  onwards. It has been illustrated by shaded region in figure. Similarly, the probability that n₀(t) attains value less than  is given by area under the curve from  onwards on left side. This portion has also been shown shaded in figure 6.28.
DIAGRAM
FIGURE 6.28 Illustration of PDF of white Gaussian noise having zero mean.
Since the PDF curve is symmetric, therefore, we can write.
P                           …(6.60)
We know that the above probabilities represent error probability. Because, occurrence of – A or + A is mutually exclusive, therefore, the probability of error is given by either of the two in above equation i.e.,
P
Substituting value of P in above equation from equation (6.59), we get
EQUATION
Again, substituting value of fx[n₀(t)] from equation (6.58), in above equation, we get
EQUATION                                      …(6.61)
Now, let us put
EQUATION
So that
Thus,                                                   d[n₀(t)] =
when                                                   n0(t)  then    y
when                                                   n₀(t) =
then                                                     EQUATION
With all these substitutions, equation (6.61) takes the form
EQUATION
This equation can be rearranged as under:
EQUATION                                          …(6.62)
The integration inside brackets may be evaluated with the help of complementary error function i.e.,
EQUATION
Recall that this is a standard result and generally evaluated using numerical methods.
Thus, with the help of this definition, equation (6.62) becomes,
P_e =                                           …(6.63)
This equation describes the probability of error P_e of the integrate and dump filter receiver.
Now, since A²T = E, i.e., energy of the bit, therefore, we have
P_e =                                             …(6.64)
NOTE : It may be noted that basically erfc’ is the monotonically decreasing function. Therefore P_e falls rapidly as the ratio  increases. Hence, the maximum value of P_e is  when  is very very small. This means that even if the signal is lost entirely in the noise No even then, the probability of error P_e will be . Thus, the receiver would make incorrect decisions for half number of times.