   # Fundamental Frequency of Discrete Signals

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.   