For a discrete complex exponential
to be periodic with
period , it has to satisfy

that is, has to be a multiple of :

As is an integer, has to be a rational number (a ratio of two integers). In order for the period

to be the fundamental period, has to be the smallest integer that makes an integer, and the fundamental angular frequency is

The original signal can now be written as:

**Example 2:** Show that a discrete signal

has fundamental period

According to the discussion above, the fundamental period should satisfy

We see that for to be an integer, has to divide . But since is an integer, also has to divide . Moreover, since needs to be the smallest integer satisfying the above equation, has to be the greatest common divisor of both and , i.e., , and the fundamental period can be written as

**Example 2:** Find the fundamental period of the following discrete signal:

We first find the fundamental period for each of the two components.

- Assume the period of the first term is , then it should satisfy

where is an integer. Equating the exponents, we have

which can be solved to get . We find the smallest integer for to be an integer, the fundamental period. - Assume the period of the second term is , then it should satisfy

where is an integer. Equating the exponents, we have

which can be solved to get . We find the smallest integer for to be an integer, the fundamental period. Now the second term can be written as

Given the fundamental periods and of the two terms, the
fundamental period of their sum is easily found to be their least
common multiple

and the fundamental frequency is

Now the original signal can be written as

i.e., the two terms are the 8th and 9th harmonic of the fundamental frequency .

Similar to the continuous case, to find the fundamental frequecy of a signal containing multiple terms all expressed as a fraction multiplied by , we can rewrite these fractions in terms of the least common multiple of all the denominators.

**Example 3:**

The least common multiple of the denominators is 12, therefore

i.e., the fundamental frequency is , the fundamental period is and the three terms are the 4th, 9th and 10th harmonic of , respectively.