Cutoff frequency

The cutoff frequency (or 3 dB cutoff, corner frequency, half-power cutoff) of a filter is the frequency at which the magnitude response drops to $1/\sqrt{2}\approx 0.707$ of its peak value — a drop of $3\text{dB}$.

For a first-order lowpass $H(j\omega )={\omega }_c/(j\omega +{\omega }_c)$:

$|H(j{\omega }_c)|=\frac{{\omega }_c}{\sqrt{{\omega }_c^2+{\omega }_c^2}}=\frac{1}{\sqrt{2}}.$

So ${\omega }_c$ is the cutoff. Same for the first-order highpass: at $\omega ={\omega }_c$, the magnitude is again $1/\sqrt{2}$.

Why $1/\sqrt{2}$ (i.e. 3 dB)

The cutoff is defined where the power has dropped to half its peak:

$\frac{|H(j{\omega }_c){|}^2}{|H(j0){|}^2}=\frac{1}{2},|H(j{\omega }_c)|=\frac{1}{\sqrt{2}}.$

In decibels, a power ratio of $1/2$ is $10{\log }_{10}(1/2)=-3\text{dB}$. Equivalently, the magnitude ratio of $1/\sqrt{2}$ is $20{\log }_{10}(1/\sqrt{2})=-3\text{dB}$. So the names “3 dB cutoff,” “half-power cutoff,” and “cutoff frequency” all refer to the same point.

In the time domain

For an RC lowpass with cutoff ${\omega }_c=1/(RC)$, the Time constant is $\tau =1/{\omega }_c=RC$. So the cutoff and the time constant are reciprocally related:

${\omega }_c=1/\tau ,f_c=\frac{1}{2\pi RC}.$

A short time constant means a high cutoff frequency (broad bandwidth, fast response). A long time constant means a low cutoff (narrow bandwidth, slow response). This is the time-bandwidth tradeoff in its simplest form.

Bode plot

On a Bode plot, the cutoff is where the asymptotic magnitude breaks from flat into a $\pm 20\text{dB/decade}$ slope (for a first-order pole or zero). The true curve crosses $-3\text{dB}$ below the asymptote at exactly the corner frequency. The phase plot is at $-45°$ (for a pole) or $+45°$ (for a zero) at the corner.

Bandwidth from cutoffs

For a lowpass, the bandwidth is simply ${\omega }_c$ (from 0 to the cutoff). For a bandpass between cutoffs ${\omega }_{ca}$ and ${\omega }_{cb}$, the bandwidth is ${\omega }_{cb}-{\omega }_{ca}$. See Filter (signal processing) for the three definitions of bandwidth (absolute, half-power, null).

Multiple cutoffs

Multi-pole filters have multiple corner frequencies, one per pole. For a Bode plot, each corner is where the slope changes by $\pm 20\text{dB/decade}$. The relationship between corner frequencies, pole locations, and the overall filter shape is what filter design is mostly about.

RC filters (Electronics I)

For the first-order RC lowpass filter and RC highpass filter, the cutoff in ordinary frequency is

$f_c=\frac{1}{2\pi RC}$

This is the point where the capacitor’s reactance equals $R$, the magnitude has fallen to $0.707$ ($-3\text{dB}$, half power), and on the Bode plot the response breaks into (or out of) a $20\text{dB/decade}$ slope. It is the single design number for a one-pole RC stage: pick $R$ and $C$ to place $f_c$ where the passband edge needs to be.