# High-pass filter

A high-pass filter is a signal-processing circuit or algorithm that passes signals with frequencies above a cutoff frequency and attenuates signals below it, with the amount of attenuation depending on the filter's design.<sup>[1](https://www.mathworks.com/discovery/high-pass-filter.html)</sup> The simplest analog example is a series resistor-capacitor (RC) circuit with the output taken across the resistor; the same two components arranged with the output across the capacitor form the low-pass filter, its mirror image.<sup>[2](https://physicslabs.colorado.edu/courses/phys-3330/lab-guides/lab3/)</sup>

| Key fact | Detail |
|---|---|
| Basic circuit | Capacitor C in series with the input, resistor R to ground, output across R; the dual of the RC low-pass filter<sup>[2](https://physicslabs.colorado.edu/courses/phys-3330/lab-guides/lab3/)</sup> |
| Transfer function | \( T = \dfrac{j\omega RC}{1 + j\omega RC} \)<sup>[2](https://physicslabs.colorado.edu/courses/phys-3330/lab-guides/lab3/)</sup> |
| Cutoff frequency | \( f_c = \dfrac{1}{2\pi RC} \), the half-power (−3 dB) point<sup>[2](https://physicslabs.colorado.edu/courses/phys-3330/lab-guides/lab3/)</sup><sup> • </sup><sup>[3](https://www.animations.physics.unsw.edu.au/jw/rcfilters.html)</sup> |
| Gain at cutoff | \( 1/\sqrt{2} \approx 0.71 \), half of maximum power; phase lead of 45°<sup>[3](https://www.animations.physics.unsw.edu.au/jw/rcfilters.html)</sup> |
| Stop-band slope | First order: +20 dB/decade (≈ 6 dB/octave) below cutoff; each added order adds 20 dB/decade<sup>[4](https://web.stanford.edu/class/archive/engr/engr40m.1178/slides/bodeplots.pdf)</sup><sup> • </sup><sup>[5](https://www.open.edu/openlearn/science-maths-technology/electronic-applications/content-section-2.5)</sup> |
| Step response | Output jumps with the input step, then decays exponentially toward 0 with time constant \( \tau = RC \)<sup>[6](https://www.sophphx.caltech.edu/Physics_5/Experiment_2.pdf)</sup> |
| Typical uses | Blocking DC while passing AC, removing low-frequency noise and trends, routing high frequencies to appropriate speakers<sup>[1](https://www.mathworks.com/discovery/high-pass-filter.html)</sup> |

## Frequency response

The RC high-pass filter is a voltage divider in which the capacitor's impedance, \( 1/(j\omega C) \), falls as frequency rises. Swapping the capacitor and resistor positions of the low-pass divider and exchanging \( Z_C \) and \( R \) in the divider equation gives the high-pass transfer function \( T = \dfrac{j\omega RC}{1 + j\omega RC} \).<sup>[2](https://physicslabs.colorado.edu/courses/phys-3330/lab-guides/lab3/)</sup> At \( \omega = \omega_o = 1/RC \) the capacitive reactance equals the resistance; the gain there is \( 1/\sqrt{2} \approx 0.71 \), the power transmitted is 50% of maximum, and the output leads the input in phase by 45°.<sup>[3](https://www.animations.physics.unsw.edu.au/jw/rcfilters.html)</sup> A reduction of power by a factor of two is a reduction of 3 dB, so a signal at \( f_o = 1/(2\pi RC) \) is attenuated by 3 dB.<sup>[3](https://www.animations.physics.unsw.edu.au/jw/rcfilters.html)</sup>

On a [Bode plot](https://www.edgepedia.org/bode-plot), a first-order high-pass filter has a sloped low-frequency asymptote rising at +20 dB/decade (each tenfold increase in frequency multiplies gain by 10) and a flat high-frequency asymptote at the passband gain; the two intersect at the corner frequency \( f_c = 1/B \) in the standard form \( V_{out}/V_{in} = Af/(1 + jBf) \).<sup>[4](https://web.stanford.edu/class/archive/engr/engr40m.1178/slides/bodeplots.pdf)</sup> The slope is independent of the gain constants A and B.<sup>[4](https://web.stanford.edu/class/archive/engr/engr40m.1178/slides/bodeplots.pdf)</sup> Higher orders steepen the stop band: a second-order high-pass filter attenuates at 40 dB/decade, a third at 60 dB/decade, and each successive order adds a further 20 dB/decade.<sup>[5](https://www.open.edu/openlearn/science-maths-technology/electronic-applications/content-section-2.5)</sup> A series [RC circuit](https://www.edgepedia.org/rc-circuit) has the same cutoff frequency whether it is configured as a low-pass or a high-pass filter.<sup>[7](https://mohamadramdhani.staff.telkomuniversity.ac.id/files/2016/08/James_W._Nilsson_Susan_Riedel_Electric_Circuits_RL-2015Book4You.pdf)</sup>

## Step and transient response

**Time-domain behavior.** For a sudden input step, the high-pass filter's output first mimics the input's sudden change and then relaxes exponentially back toward zero, because the capacitor charges and the resistor voltage falls.<sup>[6](https://www.sophphx.caltech.edu/Physics_5/Experiment_2.pdf)</sup> The relaxation follows \( a\,e^{-\omega_0 t} \) with time constant \( \tau_0 = 1/\omega_0 \), the time for the output to move a factor of \( e^{-1} \approx 0.37 \) closer to equilibrium; for the RC filter \( \tau_0 = RC \).<sup>[6](https://www.sophphx.caltech.edu/Physics_5/Experiment_2.pdf)</sup> The time constant is the reciprocal of the characteristic angular frequency; substituting a frequency in hertz would introduce an error of a factor of \( 2\pi \).<sup>[6](https://www.sophphx.caltech.edu/Physics_5/Experiment_2.pdf)</sup> Driven by a square wave whose period is many times longer than \( \tau_0 \), the output therefore consists of sharp spikes at each transition that decay back to zero, while the low-pass filter's output instead settles at the new input voltage.<sup>[6](https://www.sophphx.caltech.edu/Physics_5/Experiment_2.pdf)</sup>

## Applications

**DC blocking and AC coupling.** A high-pass filter can pass a signal into and out of a transistor amplifier stage without passing or affecting the transistor's DC bias.<sup>[3](https://www.animations.physics.unsw.edu.au/jw/rcfilters.html)</sup> In audio, AC-coupling cutoff frequencies are placed well below 20 Hz so the audible band is untouched.<sup>[8](https://wiki.analog.com/university/courses/engineering_discovery/lab_4/theory)</sup> In video, the coupling capacitor and load form a high-pass filter whose cutoff must be low enough, below about 3 Hz to 5 Hz, to pass the vertical sync interval and avoid field tilt; in a normal configuration this requires a coupling capacitor larger than 220 µF.<sup>[9](https://www.analog.com/en/resources/technical-articles/using-max4090-reconstruction-filter.html)</sup>

**Instrumentation.** The AD8232 electrocardiogram (ECG) front end applies gain and high-pass filtering simultaneously in its instrumentation amplifier, amplifying a small ECG signal by 100 while rejecting electrode offsets as large as ±300 mV; with R9 = 10 MΩ and C6 = 0.22 µF the first high-pass pole sits at 7 Hz, a cutoff 100 times higher than would typically be expected because of the amplifier's feedback architecture.<sup>[10](https://wiki.analog.com/resources/eval/ad8232-evaluation-guide/a03321a)</sup>

Applications of high-pass filters include eliminating low-frequency noise, suppressing hum in audio signals, routing higher-frequency content to the correct speakers within sound systems, and stripping low-frequency trends from time-series data so that high-frequency trends stand out.<sup>[1](https://www.mathworks.com/discovery/high-pass-filter.html)</sup>

## Active and higher-order filters

A passive RC section provides only a gentle 20 dB/decade slope, so designers cascade stages for steeper rejection, using op amps as impedance converters to avoid loading effects between stages.<sup>[11](https://www2.seas.gwu.edu/~ece121/Spring-11/filterdesign.pdf)</sup> Active topologies add gain and buffering. The Butterworth Sallen-Key high-pass filter is a second-order active filter whose Butterworth topology provides a maximally flat gain in the passband; Sallen-Key is usually preferred when a small [Q factor](https://www.edgepedia.org/q-factor), noise rejection, and a non-inverting stage gain are wanted.<sup>[12](https://www.ti.com/tool/CIRCUIT060053)</sup>

The choice of response shape trades amplitude discrimination against transient behavior. The Chebyshev response has a smaller transition region than the same-order Butterworth at the expense of ripples in the passband, while the Bessel filter is optimized for better transient response through a linear phase (constant delay) in the passband, at the cost of less amplitude discrimination.<sup>[13](https://www.analog.com/en/resources/app-notes/an-649.html)</sup>

## Digital high-pass filters

In digital signal processing a high-pass filter is implemented either as a finite impulse response (FIR) filter or an infinite impulse response (IIR) filter. FIR filters are inherently stable and can be designed with linear phase, which introduces a delay while maintaining waveform shape; IIR filters need fewer computational resources but do not have perfectly linear phase.<sup>[1](https://www.mathworks.com/discovery/high-pass-filter.html)</sup> Tools such as SciPy's `firwin` design linear-phase FIR high-pass filters by the window method, with the cutoff specified at the half-amplitude (−6 dB) point, unlike IIR design functions where the cutoff is the half-power (−3 dB) point.<sup>[14](https://docs.scipy.org/doc/scipy/reference/generated/scipy.signal.firwin.html)</sup> MATLAB's `highpass` function applies a minimum-order filter with a 60 dB stopband attenuation and compensates for the delay the filter introduces.<sup>[15](https://www.mathworks.com/help/signal/ref/highpass.html)</sup>

## References

1. [What Is a High-Pass Filter? MATLAB & Simulink, MathWorks](https://www.mathworks.com/discovery/high-pass-filter.html)
2. [Lab 3: Filters, Physics Undergraduate Labs, University of Colorado Boulder](https://physicslabs.colorado.edu/courses/phys-3330/lab-guides/lab3/)
3. [RC filters, integrators and differentiators, Physclips, University of New South Wales](https://www.animations.physics.unsw.edu.au/jw/rcfilters.html)
4. [Filters and Bode Magnitude Plots, ENGR 40M, Stanford University](https://web.stanford.edu/class/archive/engr/engr40m.1178/slides/bodeplots.pdf)
5. [Electronic applications: 2.5 Normalised first-order low-pass filters, OpenLearn, The Open University](https://www.open.edu/openlearn/science-maths-technology/electronic-applications/content-section-2.5)
6. [Experiment 2: The transient response of a simple filter, Introductory Electronics Laboratory, Caltech Physics 5](https://www.sophphx.caltech.edu/Physics_5/Experiment_2.pdf)
7. [Nilsson & Riedel, Electric Circuits (10th ed.), chapter on frequency response](https://mohamadramdhani.staff.telkomuniversity.ac.id/files/2016/08/James_W._Nilsson_Susan_Riedel_Electric_Circuits_RL-2015Book4You.pdf)
8. [Audio Amplifier with Electret Microphone, Analog Devices Wiki](https://wiki.analog.com/university/courses/engineering_discovery/lab_4/theory)
9. [Using the MAX4090 with the Reconstruction Filter, Analog Devices](https://www.analog.com/en/resources/technical-articles/using-max4090-reconstruction-filter.html)
10. [AD8232 Evaluation Board Documentation, Analog Devices Wiki](https://wiki.analog.com/resources/eval/ad8232-evaluation-guide/a03321a)
11. [Chapter 16: Active Filter Design Techniques, Texas Instruments](https://www2.seas.gwu.edu/~ece121/Spring-11/filterdesign.pdf)
12. [CIRCUIT060053: High-pass Sallen-Key filter circuit, Texas Instruments](https://www.ti.com/tool/CIRCUIT060053)
13. [AN-649: Using the Analog Devices Active Filter Design Tool, Analog Devices](https://www.analog.com/en/resources/app-notes/an-649.html)
14. [scipy.signal.firwin, SciPy v1.18.0 Manual](https://docs.scipy.org/doc/scipy/reference/generated/scipy.signal.firwin.html)
15. [highpass: Highpass-filter signals, MATLAB, MathWorks](https://www.mathworks.com/help/signal/ref/highpass.html)

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