Suppose you have an Analog-to-Digital Converter (or ADC) with 12 bits of precision, and the manufacturer claims it has the transfer function x = floor(4096×v/5 ) for input voltage vΣ [0, 4.9999] (where x is the digitized result you get). The resulting x can be interpreted
as representing the input voltage using the relationship v = 5×x/4096 (here, vis a random variable - don't confuse it with the original input voltage v!)
We can then create a new random variable, e = v - v, which represents the digitization error for any reading.
To simplify for you, you'll get integers n, from n = 0, 1, 2, ..., 4095, and these integers will represent voltages. (n will represent the voltage n × ρ, where ρ= 54096 ).
(a) Assume that the random variable v - v is uniformly distributed on the range [0, 5 4096 ]. What are the worst-case and the average errors if you make a single reading?
(b) Suppose we add a small amount of noise, uniformly distributed on an interval ρ, ρ], to the (non-random) input voltage v, which we'll assume happens to take a value such that the the error is the worst-case value. Approximately how many samples must we use so that the average of them approximates v to within plus or minus .0001?