If we have a continuous time signal given by $\sin(2\pi f_0t + \phi)$ the discrete time version may be represented by sampling every $T_s$ seconds (our sampling period) Since we are sampling every $T_s$ seconds, we can replace $t$ with $nT_s$, where $n$ is the number of samples. $\sin[2\pi f_0 nT_s + \phi] = sin[2 \pi \frac{f_0}{F_s}n + \phi]$ Which may be represented as $sin[2 \pi \frac{f_0}{F_s}(n+N) + \phi]$, where $N$ is the period of the discrete time signal Now, if discrete time signal is periodic, value at $n$ must be equal to value at $n + N$, implying that $2\pi \frac{f_0}{F_s}N = 2 \pi k$ where k is an arbitrary integer If you rearrange that expression, you get $\frac{N}{k} = \frac{F_s}{f_0}$. If there are no integer values of N and k that solve this equality, the n the sampled version of the signal is non-periodic. ### Techniques for signal generation - Direct digital synthesizer (DDS) - Phase accumulator with sin() or cos() function call/table lookup system - Special cases - Sines and cosines with f = Fs/2, f = Fs/4, etc. - Digital resonator - Impulse-excited, second order, IIR filter (what does this even mean?) - Complex conjugate pole-pairs placed on unit circle - Impulse modulator - Based on scaled impulses periodically exciting FIR filter - Lots to unpack here