Tektronix 071-0590-00 Testing for an Upper Limit on Error Rate, Reduced Test Time by Stressing

BERT Technical Articles

Testing for an Upper Limit on Error Rate

If a communication system is required to have a BER less than 10–9, the system design usually provides a couple dB of margin. This margin can make the nominal error rate on the order of 10–12, which is an average of only one error a week for fb = 1.544 Mbit/s! It is clearly impractical to measure such a low error rate accurately. But accuracy is not needed here––only some confidence that the error rate is less than 10–9. We will see that determining an upper bound on an error rate takes much less time than determining the error rate accurately.

The proposed method for determining an upper bound on an error rate is to require the system under test to be error-free for a measurement period T. The longer T is, the lower the error rate bound. Suppose you want to be sure that the actual error rate of some system is less than a specified error rate of r = 5.56 / hr. Then you must choose T so an error rate of r = 5.56 / hr or greater will have at least one error in the period T. But because of the statistical nature of the measurement, this can't be absolutely guaranteed no matter haw large you make T.

You must settle for some confidence level (say 90%) that the error rate is less than r. So choose T so r =

5.56/ hr will fail the test 90% of the time. That is, choose T so the probability of measuring n = 0 errors is only 10% when the error rate is at the limit r. The sidebar on Poisson Errors gives the probability of measuring n = 0 as

p(0) = e–rT.						(8)
Then set p(0) = 0.10 or 10% and solve for T:
T =	− ln ( 0.10)	=	2.	3	.	(9)
	r		r

For r = 5.56 / hr, this gives T = 0.414 hr. If the system is error-free for 0.414 hour (25 minutes), you are 90% confident that the error rate is less than r = 5.56 / hr.

In general, for a confidence level C that the error rate is less than r, the error-free period is given by

T =	− ln (1 − C )	.	(10)
	r

For example, for C = 0.99 and r = 5.56 / hr, then T = 0.827 hr. By doubling the test time you have increased the confidence from 90% to 99%. The tradeoff is up to you.

Reduced Test Time by Stressing

Whether the objective is to measure the error rate accurately or to determine an upper bound on the error rate, you can decrease the measurement time dramatically by stressing the system under test. The stress produces a higher error rate, and the higher error rate can be measured more quickly. Then if the error rate as a function of stress is known, you can extrapolate to the error rate the system would have when it is not stressed.

The error rate is a function of the distance S of the signal from the decision threshold compared with the level of noise. If the noise exceeds S at the decision time, there is an error. Therefore the BER is the probability the noise exceeds S. If the noise is Gaussian (or "normal distribution") with an rms value of Nrms, then the BER is given by

∞	1			−0.5 x / N 2				∞	1		−0.5 y2
BER = ∫				e			rms dx = ∫			e	dy
S	N rms	2	π					S / N rms	π
									2
	S / N rms		1			−	2
= 1 − ∫					e	0.5 y	dy = 1 − cnorm (S / N rms )
	0		2π

(11)

The second integral is the result of the substitution y = x / Nrms. The function "cnorm" is the cumulative normal distribution. For an evaluation of this function see, for example, "Probability and Statistics" in

GB1400 User Manual B-35