By Kardi Teknomo, PhD .

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Time Average

We are interested in the manipulation of average of a sequence of measurement. Let us define the time average or tag in short as the arithmetic mean of the measurement sequence up to time Time Average as

Time Average (1)

The diagram of time-average is shown in the figure below

Time Average

Diagram of time-average

Example :

We have sequence Time Average , then

Time Average , Time Average

Properties

Addition or Subtraction of Two Averages

Now we have two sequences Time Average and Time Average , where Time Average , then addition or subtraction of the two averages at the same length will produce

Time Average (2)

Proof:

Time Average QED

Addition or subtraction of two averages at the same length is equal to summation or subtraction of element of the sequences at the same position. Addition of two averages is commutative because the order of the average is not important, Time Average . However, subtraction of two averages is non-commutative Time Average .

Example :

We have sequence Time Average and Time Average , then

Time Average

Time Average

Multiplication of Averages

Multiplication of two averages is equal to summation of product permutation of the two sequences times the inverse product of their length, that is

Time Average (3)

Proof:

Time Average QED

The lengths of the two sequences are not necessarily the same. The double sum indicates product permutation, which is multiplication of each element of the first sequence to each element of the second sequence, both up to the specified length of the sequences.

Multiplication of average is commutative. The order of the average is not important. Time Average

Example :

We have sequence Time Average and Time Average , then

Time Average

Time Average

If none of elements in sequence Time Average is allowed to be zero, then division of two averages is possible:

Time Average (4)

Division of two averages is not commutative, Time Average .

Multiplication of three averages is easily extended from the two averages, that is

Time Average (5)

If we define Permutation products as

Time Average (6)

Then, we can generalize the multiplication of Time Average averages as

Time Average (7)

Distributive Law of Averages

Suppose we have averages of three sequences of the same length, then we can have distributive law of average as

Time Average (8)

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See also: Time average tutorial
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