Aliasing is the phenomenon of when [[Sampling]] a [[Function]] $f$ which has components with higher frequency than the sampling rate $f_{s}$ / when two separate [[Signal|signals]] have the same [[Set]] of [[Sampling|samples]]. ### Phasors Given a [[Phasor]] $x(t)$ $\huge \begin{align} x(t) &= e^{i\omega_{0}t} \\ \omega_{0}&=2\pi f_{0}\\ \end{align} $ Taking [[Sampling|samples]] at a rate of $f_{s}$ [[Hertz|Hz]], $\omega_{s}=2\pi f_{s}$. The [[Period]] of $x$ is and the [[Sampling|Sampling Interval]] are: $\huge \begin{align} T_{0}&= \frac{2\pi}{\omega_{0}}=\frac{1}{f_{0}}\\ T_{s}&=\frac{2\pi}{\omega s}=\frac{1}{f_{s}} \end{align} $ $\huge $ The samples of $x(t)$ are the values $x(nT_{s}), n\in\Z$. Another phasor $y(t)= e^{i\omega_{1}t}$, can have the same samples as $x(t)$ if: $\huge \begin{align} \omega_{1} &= \pm \omega_{0} + k\omega_{s} \\ k &\in \mathbb{Z} \end{align} $ These frequencies $\omega_{1}$ are called 'aliases' of $\omega_{0}$. $\huge \begin{align} y(nT_{s}) &=e^{i\left( \pm\omega_{0} + k\omega_{s} \right) \left( nT_{s} \right) } \\ &= e^{i\left( \pm\omega_{0}nT_{s} \right) } \underbrace{ e^{ik 2\pi f_{s} n T_{s}}}_{{e^{iN{2}\pi}}=1} \\ &= e^{i\left( \pm\omega_{0}nT_{s} \right) } \\ \end{align} $ $ \huge \boxed{y(nT_{s}) = x(nT_{s})} $ The [[Set|set]] of all aliases can be denoted as: $\huge \op{Aliases}(\omega_{0}) = \left\{ \pm \omega_{0} +k\omega_{s} \mid k\in\Z \right\} $