Periodic Function

• A function $x(t)$ is said to be periodic if there exists a nonzero number $T$ such that $x(t+T)=x(t)$ for all $t$ in the domain of $x$.
  • A nonzero constant $T$ for which this is the case is called a period of the function
  • The smallest such $T$ (if it exists) is called the fundamental period (or the period) of the function
• If $x(t)$ is periodic with period $T$, then:
  • $\forall n\in \mathbb{N},x(t+nT)=x(t)$
  • The function $G(t)=x(at)$ is periodic with period $\frac{T}{a}$ (for any nonzero constant $a$)
  • The amplitude of $x$ is $A=\frac{1}{2}(\text{max}(F)-\text{min}(F))$
  • When $x$ is a function of time $t$, then:
    • The period $T$ is the time it takes to complete one full cycle
    • The frequency of $x$ is $f=\frac{1}{T}$ (in ${\mathsf{s}}^{-1}=\mathsf{Hz}$)
    • The angular frequency of $x$ is $\omega =2\pi f$ (in $\mathsf{rad}\cdot {\mathsf{s}}^{-1}$)
  • (Fourier Series)
    • $x(t)=\frac{A_0}{2}+\sum _{n=1}^{\infty }\left[A_n\sin (2\pi n{f}_{0}t)+B_n\cos (2\pi n{f}_{0}t)\right]$
      • $f_0$ is the fundamental frequency (or fundamental harmonic) of $x(t)$
        • $n{f}_0$ are the harmonics (or overtones) of $x(t)$
      • $A_n$ and $B_n$ are the sine and cosine amplitudes of the $n$th harmonic
        • $A_n=\frac{2}{T}{\int }_0^{T}x(t)\sin (2\pi n{f}_{0}t)dt$
        • $B_n=\frac{2}{T}{\int }_0^{T}x(t)\cos (2\pi n{f}_{0}t)dt$
      • $c=\frac{2}{T}{\int }_0^{T}x(t)dt$
    • $x(t)=\sum _{n=-\infty }^{\infty }{c}_n{e}^{j2\pi n{f}_{0}t}$