A highpass filter is an electronic filter that passes signals with a frequency higher than a certain cutoff frequency and attenuates signals with frequencies lower than the cutoff frequency. The amount of attenuation for each frequency depends on the filter design. A highpass filter is usually modeled as a linear timeinvariant system. It is sometimes called a lowcut filter or basscut filter.^{[1]} Highpass filters have many uses, such as blocking DC from circuitry sensitive to nonzero average voltages or radio frequency devices. They can also be used in conjunction with a lowpass filter to produce a bandpass filter.
Firstorder continuoustime implementation
Figure 1: A passive, analog, firstorder highpass filter, realized by an
RC circuit
The simple firstorder electronic highpass filter shown in Figure 1 is implemented by placing an input voltage across the series combination of a capacitor and a resistor and using the voltage across the resistor as an output. The product of the resistance and capacitance (R×C) is the time constant (τ); it is inversely proportional to the cutoff frequency f_{c}, that is,

f_c = \frac{1}{2 \pi \tau} = \frac{1}{2 \pi R C},\,
where f_{c} is in hertz, τ is in seconds, R is in ohms, and C is in farads.
Figure 2: An active highpass filter
Figure 2 shows an active electronic implementation of a firstorder highpass filter using an operational amplifier. In this case, the filter has a passband gain of R_{2}/R_{1} and has a cutoff frequency of

f_c = \frac{1}{2 \pi \tau} = \frac{1}{2 \pi R_1 C},\,
Because this filter is active, it may have nonunity passband gain. That is, highfrequency signals are inverted and amplified by R_{2}/R_{1}.
Discretetime realization
Discretetime highpass filters can also be designed. Discretetime filter design is beyond the scope of this article; however, a simple example comes from the conversion of the continuoustime highpass filter above to a discretetime realization. That is, the continuoustime behavior can be discretized.
From the circuit in Figure 1 above, according to Kirchhoff's Laws and the definition of capacitance:

\begin{cases} V_{\text{out}}(t) = I(t)\, R &\text{(V)}\\ Q_c(t) = C \, \left( V_{\text{in}}(t)  V_{\text{out}}(t) \right) &\text{(Q)}\\ I(t) = \frac{\operatorname{d} Q_c}{\operatorname{d} t} &\text{(I)} \end{cases}
where Q_c(t) is the charge stored in the capacitor at time t. Substituting Equation (Q) into Equation (I) and then Equation (I) into Equation (V) gives:

V_{\text{out}}(t) = \overbrace{C \, \left( \frac{\operatorname{d} V_{\text{in}}}{\operatorname{d}t}  \frac{\operatorname{d} V_{\text{out}}}{\operatorname{d}t} \right)}^{I(t)} \, R = R C \, \left( \frac{ \operatorname{d} V_{\text{in}}}{\operatorname{d}t}  \frac{\operatorname{d} V_{\text{out}}}{\operatorname{d}t} \right)
This equation can be discretized. For simplicity, assume that samples of the input and output are taken at evenlyspaced points in time separated by \Delta_T time. Let the samples of V_{\text{in}} be represented by the sequence (x_1, x_2, \ldots, x_n), and let V_{\text{out}} be represented by the sequence (y_1, y_2, \ldots, y_n) which correspond to the same points in time. Making these substitutions:

y_i = R C \, \left( \frac{x_i  x_{i1}}{\Delta_T}  \frac{y_i  y_{i1}}{\Delta_T} \right)
And rearranging terms gives the recurrence relation

y_i = \overbrace{\frac{RC}{RC + \Delta_T} y_{i1}}^{\text{Decaying contribution from prior inputs}} + \overbrace{\frac{RC}{RC + \Delta_T} \left( x_i  x_{i1} \right)}^{\text{Contribution from change in input}}
That is, this discretetime implementation of a simple continuoustime RC highpass filter is

y_i = \alpha y_{i1} + \alpha (x_{i}  x_{i1}) \qquad \text{where} \qquad \alpha \triangleq \frac{RC}{RC + \Delta_T}
By definition, 0 \leq \alpha \leq 1. The expression for parameter \alpha yields the equivalent time constant RC in terms of the sampling period \Delta_T and \alpha:

RC = \Delta_T \left( \frac{\alpha}{1  \alpha} \right)
If \alpha = 0.5, then the RC time constant equal to the sampling period. If \alpha \ll 0.5, then RC is significantly smaller than the sampling interval, and RC \approx \alpha \Delta_T.
Algorithmic implementation
The filter recurrence relation provides a way to determine the output samples in terms of the input samples and the preceding output. The following pseudocode algorithm will simulate the effect of a highpass filter on a series of digital samples:
// Return RC highpass filter output samples, given input samples,
// time interval dt, and time constant RC
function highpass(real[0..n] x, real dt, real RC)
var real[0..n] y
var real α := RC / (RC + dt)
y[0] := x[0]
for i from 1 to n
y[i] := α * y[i1] + α * (x[i]  x[i1])
return y
The loop which calculates each of the n outputs can be refactored into the equivalent:
for i from 1 to n
y[i] := α * (y[i1] + x[i]  x[i1])
However, the earlier form shows how the parameter α changes the impact of the prior output y[i1] and current change in input (x[i]  x[i1]). In particular,

A large α implies that the output will decay very slowly but will also be strongly influenced by even small changes in input. By the relationship between parameter α and time constant RC above, a large α corresponds to a large RC and therefore a low corner frequency of the filter. Hence, this case corresponds to a highpass filter with a very narrow stop band. Because it is excited by small changes and tends to hold its prior output values for a long time, it can pass relatively low frequencies. However, a constant input (i.e., an input with (x[i]  x[i1])=0) will always decay to zero, as would be expected with a highpass filter with a large RC.

A small α implies that the output will decay quickly and will require large changes in the input (i.e., (x[i]  x[i1]) is large) to cause the output to change much. By the relationship between parameter α and time constant RC above, a small α corresponds to a small RC and therefore a high corner frequency of the filter. Hence, this case corresponds to a highpass filter with a very wide stop band. Because it requires large (i.e., fast) changes and tends to quickly forget its prior output values, it can only pass relatively high frequencies, as would be expected with a highpass filter with a small RC.
Applications
Audio
Highpass filters have many applications. They are used as part of an audio crossover to direct high frequencies to a tweeter while attenuating bass signals which could interfere with, or damage, the speaker. When such a filter is built into a loudspeaker cabinet it is normally a passive filter that also includes a lowpass filter for the woofer and so often employs both a capacitor and inductor (although very simple highpass filters for tweeters can consist of a series capacitor and nothing else). As an example, the formula above, applied to a tweeter with R=10 Ohm, will determine the capacitor value for a cutoff frequency of 5 kHz. C = \frac{1}{2 \pi f R} = \frac{1}{6.28 \times 5000 \times 10} = 3.18 \times 10^{6} , or approx 3.2 μF.
An alternative, which provides good quality sound without inductors (which are prone to parasitic coupling, are expensive, and may have significant internal resistance) is to employ biamplification with active RC filters or active digital filters with separate power amplifiers for each loudspeaker. Such lowcurrent and lowvoltage line level crossovers are called active crossovers.^{[1]}
Rumble filters are highpass filters applied to the removal of unwanted sounds near to the lower end of the audible range or below. For example, noises (e.g., footsteps, or motor noises from record players and tape decks) may be removed because they are undesired or may overload the RIAA equalization circuit of the preamp.^{[1]}
Highpass filters are also used for AC coupling at the inputs of many audio power amplifiers, for preventing the amplification of DC currents which may harm the amplifier, rob the amplifier of headroom, and generate waste heat at the loudspeakers voice coil. One amplifier, the professional audio model DC300 made by Crown International beginning in the 1960s, did not have highpass filtering at all, and could be used to amplify the DC signal of a common 9volt battery at the input to supply 18 volts DC in an emergency for mixing console power.^{[2]} However, that model's basic design has been superseded by newer designs such as the Crown MacroTech series developed in the late 1980s which included 10 Hz highpass filtering on the inputs and switchable 35 Hz highpass filtering on the outputs.^{[3]} Another example is the QSC Audio PLX amplifier series which includes an internal 5 Hz highpass filter which is applied to the inputs whenever the optional 50 and 30 Hz highpass filters are turned off.^{[4]}
A 75 Hz "low cut" filter from an input channel of a
Mackie 1402
mixing console as measured by
Smaart software. This highpass filter has a slope of 18 dB per octave.
Mixing consoles often include highpass filtering at each channel strip. Some models have fixedslope, fixedfrequency highpass filters at 80 or 100 Hz that can be engaged; other models have sweepable highpass filters, filters of fixed slope that can be set within a specified frequency range, such as from 20 to 400 Hz on the Midas Heritage 3000, or 20 to 20,000 Hz on the Yamaha M7CL digital mixing console. Veteran systems engineer and live sound mixer Bruce Main recommends that highpass filters be engaged for most mixer input sources, except for those such as kick drum, bass guitar and piano, sources which will have useful low frequency sounds. Main writes that DI unit inputs (as opposed to microphone inputs) do not need highpass filtering as they are not subject to modulation by lowfrequency stage wash—low frequency sounds coming from the subwoofers or the public address system and wrapping around to the stage. Main indicates that highpass filters are commonly used for directional microphones which have a proximity effect—a lowfrequency boost for very close sources. This low frequency boost commonly causes problems up to 200 or 300 Hz, but Main notes that he has seen microphones that benefit from a 500 Hz highpass filter setting on the console.^{[5]}
Image
Example of highpass filter applied to the right half of a photograph. Left side is unmodified, Right side is with a highpass filter applied (in this case, with a radius of 4.9)
Highpass and lowpass filters are also used in digital image processing to perform image modifications, enhancements, noise reduction, etc., using designs done in either the spatial domain or the frequency domain.^{[6]}
A highpass filter, if the imaging software does not have one, can be done by duplicating the layer, putting a gaussian blur, inverting, and then blending with the original layer using an opacity (say 50%) with the original layer.^{[7]}
The unsharp masking, or sharpening, operation used in image editing software is a highboost filter, a generalization of highpass.
See also
References

^ ^{a} ^{b} ^{c} Watkinson, John (1998). The Art of Sound Reproduction. Focal Press. pp. 268, 479.

^ Andrews, Keith; posting as ssltech (January 11, 2010). "Re: Running the board for a show this big?". Recording, Engineering & Production. ProSoundWeb. Retrieved 9 March 2010.

^ "Operation Manual: MA5002VZ". MacroTech Series. Crown Audio. 2007. Retrieved March 9, 2010.

^ "User Manual: PLX Series Amplifiers". QSC Audio. 1999. Retrieved March 9, 2010.

^ Main, Bruce (February 16, 2010). "Cut 'Em Off At The Pass: Effective Uses Of HighPass Filtering". Live Sound International (Framingham, Massachusetts: ProSoundWeb, EH Publishing).

^ Paul M. Mather (2004). Computer processing of remotely sensed images: an introduction (3rd ed.). John Wiley and Sons. p. 181.

^ "Gimp tutorial with highpass filter operation".
External links

Common Impulse Responses

ECE 209: Review of Circuits as LTI Systems, a short primer on the mathematical analysis of (electrical) LTI systems.

ECE 209: Sources of Phase Shift, an intuitive explanation of the source of phase shift in a highpass filter. Also verifies simple passive LPF transfer function by means of trigonometric identity.
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