# What does it mean for a signal to alias to another?

I know the definition of this is simple and I do understand the concept of aliasing (I think) in the way that multiple signals can be aliases of each other. For example, I get it when some one says: sampling of a 7-kHz sinewave at a sample rate of 6 kHz will provide a discrete sequence of numbers whose spectrum ambiguously represents tones at 1 kHz, 7 kHz, 13 kHz, 19 kHz, etc. However, how do you determine what aliases to what? Obviously I have a terminology problem. I understand that signals can be aliases of each other (right??), but cannot see what signal aliases to what other signal.

## 1 Answer

Sampling of a signal leads to repetition in the frequency domain. Typically, the frequency domain is considered from $-\pi/2$ up to $\pi/2$ or from $0$ to $\pi$, where $\pi$ corresponds to the sampling frequency.

For a real-valued signal, the spectrum is also symmetric around $n\pi/2$ for any integer $n$. For your example where a sine wave with frequency $7$kHz is sampled at $6$ kHz, this means that the sampled signal contains the frequencies $1+6\cdot n$ kHz and the symmetric/mirrored frequencies $5+6\cdot n$ kHz, e.g., 1 kHz, 7 kHz, 13 kHz and 5 kHz, 11 kHz (and also the negative frequency parts.

You speak of 'aliasing over another signal' when the aliases overlap with other signal content. For example, when you have a signal consisting of the sine wave of 2 kHz and 7 kHz, which you sample with a rate of 6 kHz, then you observe the frequencies 2 kHz; 4 kHz and 1 kHz; 5 kHz. These frequencies do not overlap. When the original signal contains 1 kHz and 7 kHz and you sample at a rate of 6 kHz, then the '7 kHz signal aliases over the 1 kHz signal'.

Of course this principle works also with other signals, such as narrowband, low passed and high passed signals.