Negative Frequency - Sampling of Positive and Negative Frequencies and Aliasing

Sampling of Positive and Negative Frequencies and Aliasing

When a complex sinusoid is sampled at regular intervals, its frequency becomes indistinguishable from certain other frequencies, including negative ones (referred to as aliasing). The adjacent figure illustrates this effect for several cases. The red indicates 0 Hz (aka DC). Successively higher frequencies are indicated by orange, blue, purple, violet, black, and blue. Note that some frames depict "R" and "I" for the same frequency, and others depict the "I" samples of different frequencies that are aliases of each other.

For instance, the fourth frame (purple and green) compares samples of the imaginary component of the fractional frequency + with those of negative frequency, to illustrate that they are indistinguishable. Or in other words: for integer values of n, representing the sample number. The underlying waveforms are just the imaginary components of: and, where is the sample rate (samples/sec).

Likewise + is indistinguishable from . And (last plot) is indistinguishable from (first plot).