**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) = x_{0}(t) + n_{0}(t)

For the positive pulse of amplitude A, x_{0}(t) is given as,

x_{0}(t) = for x (t) A

Similarly, for the input pulse of amplitude – A, x_{0}(t) is given by,

x_{0}(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 n_{0}(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 n_{0}(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

S. No. |
Input x(t) |
Value of n_{0}(t) for error in the output |
The probability of error P_{e} |

1. | -A | An error will be introduced if n_{0}(t)> |
In this case, the probability of error may be obtained by calculating probability that n_{0}(t) > |

2. | + A |
An error will be introduced if n_{0}(t) < |
In this case, the probability of error may be obtained by calculating the probability that, n_{0}(t) < |

As shown in Table 6.4, the error will be introduced depending upon probability that n_{0}(t) takes a particular value. These probabilities may be obtained from PDF of n_{0}(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 = n_{0}(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 n_{0}(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 n_{0}(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 n_{0}(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[n_{0}(t)] from equation (6.58), in above equation, we get

**EQUATION **…(6.61)

Now, let us put

**EQUATION**

So that

Thus, d[n_{0}(t)] =

when n0(t) then y

when n_{0}(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 A^{2}T = 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.