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702
Probability and Random Variables
Appendix B
As mentioned earlier, it is unfortunate that the integrals for Q(z) or erfc(z) cannot be evaluated
in closed form. However, for large values of z, very good closed-form approximations can
be obtained, and for small values of z, numerical integration techniques can be applied easily.
A plot of Q(z) is shown in Fig. B–7 for z 0, and a tabulation of Q(z) is given in Sec. A–10.
A relatively simple closed-form upper bound for Q(z), z 0, is
1
Q(z) 6
(B–64)
12p z e-z2>2 , z 7 0
This is also shown in Fig. B–7. It is obtained by evaluating the Q (z) integral by parts:
q
1
Q(z) = >2 q
L 12p e-l2 dl = udv = uv `
L
z
z
q
z
q
- vdu
L
z
where u = 1>(12pl) and dy = le -l2 >2 dl. Thus,
1
Q(z) = a
>2 12p l ba-e-l2 b `
q
z
q
- a-e -l2 >2 ba-
L
z
1
12p l 2 dlb
Dropping the integral, which is a positive quantity, we obtain the upper bound on Q(z), as
given by Eq. (B–64). If needed, a lower bound may also be obtained [Wozencraft and Jacobs,
1965]. A rational function approximation for Q(z) is given in Sec. (A–10), and a closed-form
approximation has an error of less than 0.27%.
For values of z 3, this upper bound has an error of less than 10% when compared with
the actual value for Q(z). That is, Q(3) = 1.35 * 10 -3 and the upper bound has a value of
1.48 * 10 -3 for z = 3. This is an error of 9.4%. For z = 4, the error is 5.6%, and for z = 5, the
error is 3.6%. In evaluating the probability of error for digital systems, as discussed in
Chapters 6, 7, and 8, the result is often found to be a Q function. Since most useful digital systems
have a probability of error of 10 -3 or less, this upper bound becomes very useful for
evaluating Q(z). At any rate, if the upper bound approximation is used, we know that the value
obtained indicates slightly poorer performance than is theoretically possible. In this sense,
this approximation will give a worst-case result.
For the case of z negative, Q(z) can be evaluated by using the identity
Q(-z) = 1 - Q(z)
(B–65)
where the Q value for positive z is used (as obtained from Fig. B–7) to compute the Q of the
negative value of z.
Example B–8 APPROXIMATION FOR Q(Z)
Write a MATLAB program to evaluate and plot Q(z) and the upper-bound approximation for
Q(z). See ExampleB_07.m for the solution. Compare these results with Fig. B–7.