Sampling Rate and Interval Partitions

In music, the sampling rate, sample rate, or sampling frequency defines the number of samples per second (or per other unit) taken from a continuous signal to make a discrete signal. For time-domain signals, the unit for sampling rate is 1/s. The inverse of the sampling frequency is the sampling period or sampling interval, which is the time between samples.

Analog Signal

Sampled Signal

In mathematics, a partition of an interval [a,b] is a finite sequence

Each [x_i,x_{i + 1}] is called a subinterval of the partition. The mesh of a partition is defined to be the length of the longest subinterval [x_i,x_{i + 1}], that is, it is max(x_{i + 1} − x_i) where . It is also called the norm of the partition.

A tagged partition of an interval is a partition of an interval together with a finite sequence of numbers subject to the conditions that for each i, . In other words, it is a partition together with a distinguished point of every subinterval. The mesh of a tagged partition is the same as that of an ordinary partition.

Suppose that together with are a tagged partition of [a,b], and that together with are another tagged partition of [a,b]. We say that and together are a refinement of together with if for each integer i with , there is an integer r(i) such that x_i = y_r(i) and such that t_i = s_j for some j with . (It is not correct to allow j to equal r(i + 1) because s_{r(i + 1)} is greater than or equal to x_{i + 1}.) Said more simply, a refinement of a tagged partition takes the starting partition and adds more tags, but does not take any away.

We can define a partial order on the set of all tagged partitions by saying that one tagged partition is bigger than another if the bigger one is a refinement of the smaller one.

There is a pretty obvious similarity between those two notions : the refinement of a tagged partition represents an increase of the sampling rate.