# Periodic Function **A [[function]] that repeats values at regular intervals.** > [!Example] Periodic Function > A function $f$ is *periodic* if there is some non-zero constant $T$ such that > $f(t)=f(t+T)$ > for all values of $t$ within the [[domain]]. > > The constant $T$ is known as the [[Periodic Function#Period|period]] of the function. ![[PeriodicFunction.svg]] ## Period > [!Period] > The *period* of a periodic function is the time after which a periodic function repeats. > $T=\frac{1}{f}$$T=\frac{2\pi}{\omega}$ > - $f$ - *frequency*, $\text{Hz}$ > - $\omega$ - [[angular frequency]], radians per second ## Amplitude The *amplitude* of a periodic function refers to several functions of the difference between the *extreme values* of the periodic function. ![[PeriodicFunctionAmplitude.svg|400]] *A [[Sinusoid|sinusoidal]] waveform with peak (red), root mean square (orange), and peak-to-peak (yellow) amplitudes shown* ### Peak and semi-amplitude ^d3ad41 For *symmetric* periodic functions, the peak and semi-amplitudes *are the same*. > [!Peak amplitude] > The *peak amplitude* of a periodic function is the *magnitude* of the *maximum difference* from some reference level. ^c5261f > [!Example] Semi-amplitude > The *semi-amplitude* of a periodic function is *half* the value of its peak-to-peak amplitude. > > In the most common scientific usage, *amplitude* and *peak amplitude* refers to the semi-amplitude. ### Peak-to-peak amplitude > [!Example] Peak-to-peak amplitude > The *peak-to-peak amplitude* of a periodic function is the *magnitude* of the *difference* between the highest and lowest values. ### Root mean square amplitude > [!Root mean square amplitude] > The *root mean square amplitude* or RMS amplitude of a periodic function is the *square root of the mean value* of the function over time. > > For a continuous periodic function, the root mean square amplitude is > $f_{\text{RMS}}=\sqrt{\frac{1}{T}\int_{0}^{T}[f(t)]^2\;dt}$