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this is neil patwari in this segment im going to talk about the nearest neighbor approximation and the union bound is an expression that we found that provides an upper bound for the probability of symbol error for our given constellation the nearest neighbor approximation is going to give us an approximation that we can get to very quickly just by using the constellation and a few quick calculations so whats the difference the nearest neighbor approximation just looks at the minimum distance in a constellation and n-min is the number of neighbors that have that distance separating them lets do an example this is figure 6.7 from the rice book showing a 4-qam constellation symbol points are here the decision region boundaries are shown as dashed lines you can see that all of the three symbols s1 s2 and s3 are a distance of a from the origin while some other distances between neighbors so s2 and s1 are neighbors that that distance is larger than a it happens to be square root of 3 ti