Agilent Measurement Journal - Issue 5 - 2008 - (Page 38)

Since it is impossible to wait for an inﬁnite number of bits, it is desirable to collect as many samples as possible. If we measure a large number of errors, say 1,000, than the estimate is much more accurate. However, this poses a serious constraint on the required test time. At a 2.5-Gb/s PCI Express data rate, for example, the time required to generate 1,000 errors could easily reach 400,000 seconds, which is over 111 days! Returning to Equation 7, we observe that the value of Pe is much smaller than 1, typically in the range of 10-3 to 10-12. When Pe is small, Equation 7 can be approximated to a Poisson distribution:4 Equation 10a. 1 P(k,3) = k! @3ke-3 can also deﬁne another useful quantity, the upper limit of the conﬁdence level which deﬁnes the probability that Pe is greater than g (usually referred to as the target BER). Equation 12. CLu = Prob[Pe> g| k, n] Equation 10a gives the probability of ﬁnding k errors. Using this expression we can derive the probability of ﬁnding at most k errors when we vary the number of errors from zero to k. By summing all the probabilities we obtain the total probability of ﬁnding k or fewer errors. Equation 13. k P[errors ≤ k] = 0 P(i,μ) i–0 Equation 10b. where 3 = n@Pe This expression is much more useful for analytical purposes. In order to minimize the measurement time, we only want to measure a minimum number of bits that still guarantees a certain conﬁdence level. If we desire a higher conﬁdence level the measurement time is accordingly higher. This conﬁdence level deﬁnes a probability measure for the given estimate. Speciﬁcally, the conﬁdence level is the probability that Pe is less than a speciﬁed threshold, g, given that k errors are observed when n bits were received. Equation 11. This quantity deﬁnes the conﬁdence level for ﬁnding no greater than k errors.5 This expression is not very useful because it gives the conﬁdence level if Pe is known. If we replace Pe with g, then the resulting conﬁdence level can be used as a guideline. Equation 14a. P[errors ≤ k | g] = 0 P(i,ν), i–0 k Equation 14b. ν = n@g Let’s say we compute the conﬁdence level for a given g, n and k from Equation 14a to be high. And in the experiment, we measure a higher k. This means one of two things. Either we got lucky in picking a very unlikely event or the error mechanism is worse than anticipated. If we repeat the experiment and consistently measure higher value for g, then we conclude that the latter is indeed true. This implies that the true value, Pe, is greater CL = Prob[Pe< g| k, n] Because this conﬁdence level gives the probability that Pe is less than g, we can deﬁne it as the lower limit of conﬁdence, CLl. We 38 Agilent Measurement Journal

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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