Agilent Measurement Journal - Issue 5 - 2008 - (Page 39)

than g with a probability given by Equation 14a. As indicated by Equation 12, this is the conﬁdence level for the upper limit. Equation 15. Equation 17. CLl = 1 – e-ν Rearranging terms, and using Equation 14b, we get: CLu = 0 P(i,ν) =0 i=0 i=0 k k 1 i -ν @ i! ν e Equation 18. As an example, let’s consider a target BER, g, of 10-8. If n is 109 then we expect to see 10 errors for k. But in the actual experiment, we observe 15 errors. CLu then works out to 0.95 or 95 percent from Equation 14a. This is obtained by using 15 for k, 109 for n and 10-8 for g. From this we can infer that Pe is higher than 10-8 with a 95 percent conﬁdence level. Its complement gives the probability that the errors are k or greater. The conﬁdence level, CLl, is then given by, Equation 16. 1 CLl = 1 – CLu = 1 – 0 i! @νie-ν i=0 k n@g = – 1n(1 – CLl ) From Equation 18, the minimum n for a given CLl can be calculated. When the conﬁdence level is speciﬁed, a lower limit can be deﬁned on the number of bits for a given target BER, g. For example, if we want to verify that a communication link has a BER of less than 10-9, and the conﬁdence level desired is 95 percent, then a minimum n of approximately 3 x 109 (or (-ln(1-0.95)/ 10-9)) bits are needed. In other words, when we receive 3 x 109 bits and there are no errors observed, then Pe is less than 10-9 with 95 percent conﬁdence. When k is non zero, the solution to Equation 16, as well as Equation 15, is difﬁcult to obtain even with Poisson approximation. Previous work in this area attempts to solve it by further approximating the Poisson distribution to a Gaussian distribution.6 This turns out to be not so efﬁcient, as it requires much higher n than would be required without the approximation. The conﬁdence level here deﬁnes the probability that Pe is less than g. From Equation 16 it is possible to ﬁnd the minimum number of bits that are required for the estimate of the probability of error as a function of the conﬁdence level, CLl. When there are no errors received in an observation window, during which n bits are analyzed or when k =0, Equation 16 reduces to: 30.0E-9 90% Conﬁdence 25.0E-9 95% Conﬁdence Measured bit error rate, k/n 20.0E-9 99% Conﬁdence 15.0E-9 10.0E-9 5.0E-9 0.0E0 0 1G 2G 3G Number of bits, n Figure 6. Upper and lower conﬁdence limits for the probability of error, Pe ; the target BER, g, is 10-8 4G 5G 5.5G Agilent Measurement Journal 39

Table of Contents Feed for the Digital Edition of Agilent Measurement Journal - Issue 5 - 2008

Agilent Measurement Journal Issue 5 Finding a New Path when Assumptions Break Down Contents Testing MIPI D-PHY Protocols Oscilloscope Firmware Update DNA Replication Joining the LTE/SAE Trial Initiative Testing Proteins Praise for DC Power Analyzer Scripting Features for AMDS Testing Hybrid PCBA Systems Improving the Gene Expression System Workflow Turning a “Good Enough” Test Strategy into One That’s Reliable, Repeatable and Traceable Understanding the Effects of Limited-Bandwidth Channels on Digital Data Signals Introducing the 3GPP LTE Downlink Validating the 26 Validating the Physical and Protocol Layers in DDR Memory Interfaces Understanding Total Jitter Measurements of Low Probabilities Using Behavioral-Model Simulation to Accurately Predict First-Order PLL Performance Creating Synchronous High-Frequency Sampling across Multiple Digitizers Overcoming the Challenges of Testing FlexRay Networks Understanding Operating Life and Repeatability of Electromechanical Switches and their Effect on Total Cost of Ownership Implementing Micro and Nano LC/MS Techniques for High Sensitivity Lipidomics Analysis

Agilent Measurement Journal - Issue 5 - 2008

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