Sampled sinusoid alias ($f_{\mathrm{alias}}={\min }_{m\in \mathbb{Z}}|f-m{f}_s|\in [0,f_s/2]$)

At uniform sampling times $t_n=n/f_s$, adding an integer multiple of the sampling rate to a sinusoid's frequency changes its complex phase by an integer multiple of $2\pi n$. The samples therefore agree. Real cosines also identify the two signs of frequency, giving the displayed representative below the Nyquist frequency. A chosen nonzero sampling-time origin changes the equivalent phase, rather than the alias frequency.