# Frequency aliasing

There is a special kind of aliasing, where two sinusoidal signals appear the same under sampling. A complex sinusoid is a function of the form This signal is periodic, which repeats itself at a rate of cycles per unit time. Eg. represents a 60 Hertz signal, where is measured in seconds. Two signals will get aliased at a sample interval if

Equivalently, or more simply, if . A bit of algebra shows this happens if , for some integer $N$.

Thus, two sinusoids get aliased if the difference of their frequencies is a multiple of the sampling rate .

Usually, we are interested in measuring signals with frequencies limited to some finite range. For instance, telephone sounds are typically limited to a maximum frequency of about four kilohertz. Because we use complex sinusoids in the analysis, we limit the frequencies in both negative and positive values, to a range in some interval . So, telephone sounds, you might like to measure frequencies in . The difference could be as big as 8000Hz. In order for this to not be a multiple of the sample rate, we have to choose a sample rate bigger than 8000Hz. That is, Being a little big lazy, we might choose the sample rate to be 10000Hz, and so the sampling interval is second (a tenth of a millisecond).