Sinusoid

Sinusoids are strictly defined as the trigonometric function:

$$v(t)=V_A\cos (\omega t+\phi )$$ $$v[n]=V_A\cos ({\Omega }_{0}n+\varphi )$$

where $V_A$ is the amplitude, $\omega$ is the angular frequency, and $\phi$ is the phase. From there, we define a few more terms. The cyclic frequency is defined as $f=\omega /2\pi$ in units of Hertz. The period is $T=1/f$.

For a phase shift $\phi <0$, we delay the sinusoid by $\tau =-\phi /{\omega }_0$ seconds, i.e., it’s just a time shift of the signal.

What is convenient about the sinusoidal signal is that it’s a periodic signal. A time shift also corresponds to a phase change, i.e., for some shift $t+t_0$, there’s some phase shift $\phi$. The fundamental period $T_0=2\pi /{\omega }_0=1/f_0$.

$$\sin ({\omega }_0(t+T_0))=\sin ({\omega }_{0}t+{\omega }_0{T}_0)=\sin ({\omega }_{0}t+2\pi )=\sin ({\omega }_{0}t)$$

For a discrete-time signal, if the period isn’t in the set of possible outputs we have two options. One is multiplying it out by a possible output to get something. The other is it’s not possible and it’s not periodic:

$${\Omega }_0=\frac{2\pi m}{n}$$