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== Differentiation and integration == {{See also|Phasor#Differentiation and integration}} === Differentiation === [[Derivative|Differentiating]] any sinusoid with respect to time can be viewed as multiplying its amplitude by its angular frequency and advancing it by a quarter cycle: <math>\begin{align} \frac{d}{dt} [A\sin(\omega t + \varphi)] &= A \omega \cos(\omega t + \varphi) \\ &= A \omega \sin(\omega t + \varphi + \tfrac{\pi}{2}) \, . \end{align}</math> A [[differentiator]] has a [[Zeros and poles|zero]] at the origin of the [[complex frequency]] plane. The [[Gain (electronics)|gain]] of its [[frequency response]] increases at a rate of +20 [[Decibel|dB]] per [[Decade (log scale)|decade]] of frequency (for [[root-power]] quantities), the same positive slope as a 1{{Sup|st}} order [[high-pass filter]]'s [[stopband]], although a differentiator doesn't have a [[cutoff frequency]] or a flat [[passband]]. A n{{Sup|th}}-order high-pass filter approximately applies the n{{Sup|th}} time derivative of [[signals]] whose frequency band is significantly lower than the filter's cutoff frequency. === Integration === [[Integral|Integrating]] any sinusoid with respect to time can be viewed as dividing its amplitude by its angular frequency and delaying it a quarter cycle: <math>\begin{align} \int A \sin(\omega t + \varphi) dt &= -\frac{A}{\omega} \cos(\omega t + \varphi) + C\\ &= -\frac{A}{\omega} \sin(\omega t + \varphi + \tfrac{\pi}{2}) + C\\ &= \frac{A}{\omega} \sin(\omega t + \varphi - \tfrac{\pi}{2}) + C \, . \end{align}</math> The [[constant of integration]] <math>C</math> will be zero if the [[bounds of integration]] is an integer multiple of the sinusoid's period. An [[integrator]] has a [[Zeros and poles|pole]] at the origin of the complex frequency plane. The gain of its frequency response falls off at a rate of -20 dB per decade of frequency (for root-power quantities), the same negative slope as a 1{{Sup|st}} order [[low-pass filter]]'s stopband, although an integrator doesn't have a cutoff frequency or a flat passband. A n{{Sup|th}}-order low-pass filter approximately performs the n{{Sup|th}} time integral of signals whose frequency band is significantly higher than the filter's cutoff frequency.
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