High-pass filter

# High-pass filter

Overview
A high-pass filter is a device that passes high frequencies
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...

and attenuates
Attenuation
In physics, attenuation is the gradual loss in intensity of any kind of flux through a medium. For instance, sunlight is attenuated by dark glasses, X-rays are attenuated by lead, and light and sound are attenuated by water.In electrical engineering and telecommunications, attenuation affects the...

(i.e., reduces the amplitude of) frequencies lower than its cutoff frequency
Cutoff frequency
In physics and electrical engineering, a cutoff frequency, corner frequency, or break frequency is a boundary in a system's frequency response at which energy flowing through the system begins to be reduced rather than passing through.Typically in electronic systems such as filters and...

. A high-pass filter
Filter (signal processing)
In signal processing, a filter is a device or process that removes from a signal some unwanted component or feature. Filtering is a class of signal processing, the defining feature of filters being the complete or partial suppression of some aspect of the signal...

is usually modeled as a linear time-invariant system. It is sometimes called a low-cut filter or bass-cut filter.
Discussion
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Unanswered Questions
Encyclopedia
A high-pass filter is a device that passes high frequencies
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...

and attenuates
Attenuation
In physics, attenuation is the gradual loss in intensity of any kind of flux through a medium. For instance, sunlight is attenuated by dark glasses, X-rays are attenuated by lead, and light and sound are attenuated by water.In electrical engineering and telecommunications, attenuation affects the...

(i.e., reduces the amplitude of) frequencies lower than its cutoff frequency
Cutoff frequency
In physics and electrical engineering, a cutoff frequency, corner frequency, or break frequency is a boundary in a system's frequency response at which energy flowing through the system begins to be reduced rather than passing through.Typically in electronic systems such as filters and...

. A high-pass filter
Filter (signal processing)
In signal processing, a filter is a device or process that removes from a signal some unwanted component or feature. Filtering is a class of signal processing, the defining feature of filters being the complete or partial suppression of some aspect of the signal...

is usually modeled as a linear time-invariant system. It is sometimes called a low-cut filter or bass-cut filter.

High-pass filters have many uses, such as blocking DC from circuitry sensitive to non-zero average voltages or RF
Radio frequency
Radio frequency is a rate of oscillation in the range of about 3 kHz to 300 GHz, which corresponds to the frequency of radio waves, and the alternating currents which carry radio signals...

devices. They can also be used in conjunction with a low-pass filter
Low-pass filter
A low-pass filter is an electronic filter that passes low-frequency signals but attenuates signals with frequencies higher than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a high-cut filter, or treble cut filter...

to make a bandpass filter. The actual amount of attenuation for each frequency is a design parameter of the filter.

## First-order continuous-time implementation

The simple first-order electronic high-pass filter shown in Figure 1 is implemented by placing an input voltage across the series combination of a capacitor
Capacitor
A capacitor is a passive two-terminal electrical component used to store energy in an electric field. The forms of practical capacitors vary widely, but all contain at least two electrical conductors separated by a dielectric ; for example, one common construction consists of metal foils separated...

and a resistor
Resistor
A linear resistor is a linear, passive two-terminal electrical component that implements electrical resistance as a circuit element.The current through a resistor is in direct proportion to the voltage across the resistor's terminals. Thus, the ratio of the voltage applied across a resistor's...

and using the voltage across the resistor as an output. The product of the resistance and capacitance (R×C) is the time constant
Time constant
In physics and engineering, the time constant, usually denoted by the Greek letter \tau , is the risetime characterizing the response to a time-varying input of a first-order, linear time-invariant system.Concretely, a first-order LTI system is a system that can be modeled by a single first order...

(τ); it is inversely proportional to the cutoff frequency fc, at which the output power is half the input power. That is,

where fc is in hertz
Hertz
The hertz is the SI unit of frequency defined as the number of cycles per second of a periodic phenomenon. One of its most common uses is the description of the sine wave, particularly those used in radio and audio applications....

, τ is in second
Second
The second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock....

s, R is in ohms, and C is in farad
Farad
The farad is the SI unit of capacitance. The unit is named after the English physicist Michael Faraday.- Definition :A farad is the charge in coulombs which a capacitor will accept for the potential across it to change 1 volt. A coulomb is 1 ampere second...

s.

Figure 2 shows an active electronic implementation of a first-order high-pass filter using an operational amplifier
Operational amplifier
An operational amplifier is a DC-coupled high-gain electronic voltage amplifier with a differential input and, usually, a single-ended output...

. In this case, the filter has a passband
Passband
A passband is the range of frequencies or wavelengths that can pass through a filter without being attenuated.A bandpass filtered signal , is known as a bandpass signal, as opposed to a baseband signal....

gain of -R2/R1 and has a corner frequency of
Because this filter is active
Passivity (engineering)
Passivity is a property of engineering systems, used in a variety of engineering disciplines, but most commonly found in analog electronics and control systems...

, it may have non-unity passband gain. That is, high-frequency signals are inverted and amplified by R2/R1.

## Discrete-time realization

Discrete-time high-pass filters can also be designed. Discrete-time filter design is beyond the scope of this article; however, a simple example comes from the conversion of the continuous-time high-pass filter above to a discrete-time realization. That is, the continuous-time behavior can be discretized
Discrete signal
A discrete signal or discrete-time signal is a time series consisting of a sequence of qualities...

.

From the circuit in Figure 1 above, according to Kirchoff's Laws and the definition of capacitance
Capacitance
In electromagnetism and electronics, capacitance is the ability of a capacitor to store energy in an electric field. Capacitance is also a measure of the amount of electric potential energy stored for a given electric potential. A common form of energy storage device is a parallel-plate capacitor...

:

where is the charge stored in the capacitor at time . Substituting Equation (Q) into Equation (I) and then Equation (I) into Equation (V) gives:

This equation can be discretized. For simplicity, assume that samples of the input and output are taken at evenly-spaced points in time separated by time. Let the samples of be represented by the sequence , and let be represented by the sequence which correspond to the same points in time. Making these substitutions:

And rearranging terms gives the recurrence relation
Recurrence relation
In mathematics, a recurrence relation is an equation that recursively defines a sequence, once one or more initial terms are given: each further term of the sequence is defined as a function of the preceding terms....

That is, this discrete-time implementation of a simple continuous-time RC high-pass filter is

By definition, . The expression for parameter yields the equivalent time constant
Time constant
In physics and engineering, the time constant, usually denoted by the Greek letter \tau , is the risetime characterizing the response to a time-varying input of a first-order, linear time-invariant system.Concretely, a first-order LTI system is a system that can be modeled by a single first order...

in terms of the sampling period and :

If , then the time constant equal to the sampling period. If , then is significantly smaller than the sampling interval, and .

### 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
Pseudocode
In computer science and numerical computation, pseudocode is a compact and informal high-level description of the operating principle of a computer program or other algorithm. It uses the structural conventions of a programming language, but is intended for human reading rather than machine reading...

algorithm will simulate the effect of a high-pass filter on a series of digital samples:

// Return RC high-pass 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] := 0
for i from 1 to n
y[i] := α * y[i-1] + α * (x[i] - x[i-1])
return y

The loop which calculates each of the outputs can be refactored into the equivalent:

for i from 1 to n
y[i] := α * (y[i-1] + x[i] - x[i-1])

However, the earlier form shows how the parameter α changes the impact of the prior output y[i-1] and current change in input (x[i] - x[i-1]). 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
Time constant
In physics and engineering, the time constant, usually denoted by the Greek letter \tau , is the risetime characterizing the response to a time-varying input of a first-order, linear time-invariant system.Concretely, a first-order LTI system is a system that can be modeled by a single first order...

above, a large α corresponds to a large and therefore a low corner frequency of the filter. Hence, this case corresponds to a high-pass 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[i-1])=0) will always decay to zero, as would be expected with a high-pass filter with a large .
• A small α implies that the output will decay quickly and will require large changes in the input (i.e., (x[i] - x[i-1]) is large) to cause the output to change much. By the relationship between parameter α and time constant above, a small α corresponds to a small and therefore a high corner frequency of the filter. Hence, this case corresponds to a high-pass 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 high-pass filter with a small .

### Audio

High-pass filters have many applications. They are used as part of an audio crossover
Audio crossover
Audio crossovers are a class of electronic filter used in audio applications. Most individual loudspeaker drivers are incapable of covering the entire audio spectrum from low frequencies to high frequencies with acceptable relative volume and lack of distortion so most hi-fi speaker systems use a...

to direct high frequencies to a tweeter
Tweeter
A tweeter is a loudspeaker designed to produce high audio frequencies, typically from around 2,000 Hz to 20,000 Hz . Some tweeters can manage response up to 65 kHz...

while attenuating bass signals which could interfere with, or damage, the speaker. When such a filter is built into a loudspeaker
Loudspeaker
A loudspeaker is an electroacoustic transducer that produces sound in response to an electrical audio signal input. Non-electrical loudspeakers were developed as accessories to telephone systems, but electronic amplification by vacuum tube made loudspeakers more generally useful...

cabinet it is normally a passive filter that also includes a low-pass filter
Low-pass filter
A low-pass filter is an electronic filter that passes low-frequency signals but attenuates signals with frequencies higher than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a high-cut filter, or treble cut filter...

for the woofer
Woofer
Woofer is the term commonly used for a loudspeaker driver designed to produce low frequency sounds, typically from around 40 hertz up to about a kilohertz or higher. The name is from the onomatopoeic English word for a dog's bark, "woof"...

and so often employs both a capacitor and inductor
Inductor
An inductor is a passive two-terminal electrical component used to store energy in a magnetic field. An inductor's ability to store magnetic energy is measured by its inductance, in units of henries...

(although very simple high-pass filters for tweeters can consist of a series capacitor and nothing else). 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 bi-amplification with active RC filters
Active filter
An active filter is a type of analog electronic filter that uses an amplifier stage. Amplifiers included in a filter design can be used to improve the performance, stability and predictability of a filter. An amplifier prevents the impedance of source or load stages from affecting the...

or active digital filters with separate power amplifiers for each loudspeaker
Loudspeaker
A loudspeaker is an electroacoustic transducer that produces sound in response to an electrical audio signal input. Non-electrical loudspeakers were developed as accessories to telephone systems, but electronic amplification by vacuum tube made loudspeakers more generally useful...

. Such low-current and low-voltage line level
Line level
Line level is a term used to denote the strength of an audio signal used to transmit analog sound between audio components such as CD and DVD players, TVs, audio amplifiers, and mixing consoles, and sometimes MP3 players....

crossovers are called active crossovers.

Rumble filters are high-pass 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
RIAA equalization
RIAA equalization is a specification for the correct recording of gramophone records, established by the Recording Industry Association of America...

circuit of the preamp.

High-pass filters are also used for AC coupling at the inputs of many audio amplifier
Audio amplifier
An audio amplifier is an electronic amplifier that amplifies low-power audio signals to a level suitable for driving loudspeakers and is the final stage in a typical audio playback chain.The preceding stages in such a chain are low power audio amplifiers which perform tasks like pre-amplification,...

s, for preventing the amplification of DC currents which may harm the amplifier, rob the amplifier of headroom, and generate waste heat at the loudspeaker
Loudspeaker
A loudspeaker is an electroacoustic transducer that produces sound in response to an electrical audio signal input. Non-electrical loudspeakers were developed as accessories to telephone systems, but electronic amplification by vacuum tube made loudspeakers more generally useful...

s voice coil
Voice coil
A voice coil is the coil of wire attached to the apex of a loudspeaker cone. It provides the motive force to the cone by the reaction of a magnetic field to the current passing through it...

. One amplifier, the professional audio
Professional audio
Professional audio, also 'pro audio', refers to both an activity and a type of audio equipment. Typically it encompasses the production or reproduction of sound for an audience, by individuals who do such work as an occupation like live event support, using sound reinforcement systems designed for...

model DC300 made by Crown International
Crown International
Crown International, or Crown Audio, is a manufacturer of audio electronics, and is a subsidiary of Harman International Industries. Today the company is known primarily for its power amplifiers, but has also manufactured microphones, a line of commercial audio products as well as digital audio...

beginning in the 1960s, did not have high-pass filtering at all, and could be used to amplify the DC signal of a common 9-volt battery at the input to supply 18 volts DC in an emergency for mixing console
Mixing console
In professional audio, a mixing console, or audio mixer, also called a sound board, mixing desk, or mixer is an electronic device for combining , routing, and changing the level, timbre and/or dynamics of audio signals. A mixer can mix analog or digital signals, depending on the type of mixer...

power. However, that model's basic design has been superseded by newer designs such as the Crown Macro-Tech series developed in the late 1980s which included 10 Hz high-pass filtering on the inputs and switchable 35 Hz high-pass filtering on the outputs. Another example is the QSC Audio PLX amplifier series which includes an internal 5 Hz high-pass filter which is applied to the inputs whenever the optional 50 and 30 Hz high-pass filters are turned off.
Mixing consoles often include high-pass filtering at each channel strip
Channel strip
A channel strip is a device that allows the output of an audio device to be amplified to a line level and integrated into some other system. An audio channel strip may be a stand alone unit or one of many units built into a mixing desk. It usually includes a microphone preamp with a switchable...

. Some models have fixed-slope, fixed-frequency high-pass filters at 80 or 100 Hz that can be engaged; other models have 'sweepable HPF'—a high-pass filter of fixed slope that can be set within a specified frequency range, such as from 20 to 400 Hz on the Midas
Midas Consoles
Midas has been designing and manufacturing audio consoles since the early 1970s.Later on it became part of the Telex group. When, in January 2006, Telex Communications was acquired by the Bosch group, Midas consoles became part of the business unit "Bosch Communications Systems"...

Heritage 3000, or 20 to 20,000 Hz on the Yamaha
Yamaha
Yamaha may refer to:* Yamaha Corporation, a Japanese company with a wide range of products and services** Yamaha Motor Company, a Japanese motorized vehicle-producing company...

M7CL
M7CL
The Yamaha M7CL is a digital mixer manufactured by Yamaha Pro Audio. Two models with onboard analog input currently exist: the M7CL-32 and M7CL-48. These models have 40 - and 56 -input channels respectively, counting mono channels...

digital mixing console
Digital mixing console
In professional audio, a Digital Mixing Console , is an electronic device for combining, routing, and changing the dynamics of digital audio samples. The digital audio samples are summed to produce a combined output. A professional digital mixing console is a dedicated desk or control surface...

. Veteran systems engineer and live sound mixer Bruce Main recommends that high-pass filters be engaged for most mixer input sources, except for those such as kick drum
Bass drum
Bass drums are percussion instruments that can vary in size and are used in several musical genres. Three major types of bass drums can be distinguished. The type usually seen or heard in orchestral, ensemble or concert band music is the orchestral, or concert bass drum . It is the largest drum of...

, bass guitar
Bass guitar
The bass guitar is a stringed instrument played primarily with the fingers or thumb , or by using a pick....

and piano, sources which will have useful low frequency sounds. Main writes that DI unit
DI unit
A DI unit, DI box, Direct Box, or simply DI , is a device typically used in recording studios to connect a high-impedance, line level, unbalanced output signal to a low-impedance microphone level balanced input, usually via XLR connector...

inputs (as opposed to microphone
Microphone
A microphone is an acoustic-to-electric transducer or sensor that converts sound into an electrical signal. In 1877, Emile Berliner invented the first microphone used as a telephone voice transmitter...

inputs) do not need high-pass filtering as they are not subject to modulation by low-frequency stage wash
Stage wash (audio)
Stage wash in professional audio is unwanted sound entering a microphone on stage during a concert. Stage wash can come from the main public address system, from monitor loudspeakers, from instrument amplifiers such as for guitars and keyboards, and from loud instruments such as drums...

—low frequency sounds coming from the subwoofer
Subwoofer
A subwoofer is a woofer, or a complete loudspeaker, which is dedicated to the reproduction of low-pitched audio frequencies known as the "bass". The typical frequency range for a subwoofer is about 20–200 Hz for consumer products, below 100 Hz for professional live sound, and below...

s or the public address
Public address
A public address system is an electronic amplification system with a mixer, amplifier and loudspeakers, used to reinforce a sound source, e.g., a person giving a speech, a DJ playing prerecorded music, and distributing the sound throughout a venue or building.Simple PA systems are often used in...

system and wrapping around to the stage. Main indicates that high-pass filters are commonly used for directional microphones which have a proximity effect
Proximity effect (audio)
The proximity effect in audio is an increase in bass or low frequency response when a sound source is close to a microphone.-Technical explanation:...

—a low-frequency 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 HPF setting on the console.

### Image

High-pass and low-pass filters are also used in digital image processing
Image processing
In electrical engineering and computer science, image processing is any form of signal processing for which the input is an image, such as a photograph or video frame; the output of image processing may be either an image or, a set of characteristics or parameters related to the image...

to perform image modifications, enhancements, noise reduction, etc., using designs done in either the spatial domain or the frequency domain
Frequency domain
In electronics, control systems engineering, and statistics, frequency domain is a term used to describe the domain for analysis of mathematical functions or signals with respect to frequency, rather than time....

.

A high-pass 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.

The unsharp masking
Unsharp masking
Unsharp masking is an image manipulation technique, often available in digital image processing software.The "unsharp" of the name derives from the fact that the technique uses a blurred, or "unsharp," positive to create a "mask" of the original image...

, or sharpening, operation used in image editing software is a high-boost filter, a generalization of high-pass.

## See also

• DSL filter
DSL filter
A DSL filter is an analog low-pass filter installed between analog devices and a plain old telephone service telephone line, in order to prevent interference between such devices and a digital subscriber line service operating on the same line...

• Band-stop filter
Band-stop filter
In signal processing, a band-stop filter or band-rejection filter is a filter that passes most frequencies unaltered, but attenuates those in a specific range to very low levels. It is the opposite of a band-pass filter...

• Band-pass filter
Band-pass filter
A band-pass filter is a device that passes frequencies within a certain range and rejects frequencies outside that range.Optical band-pass filters are of common usage....

• Bias tee
Bias tee
A bias tee is a three port network used for setting the DC bias point of some electronic components without disturbing other components. The bias tee is a diplexer. The low frequency port is used to set the bias; the high frequency port passes the radio frequency signals but blocks the biasing...

• Low-pass filter
Low-pass filter
A low-pass filter is an electronic filter that passes low-frequency signals but attenuates signals with frequencies higher than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a high-cut filter, or treble cut filter...

## External links

• Common Impulse Responses
• ECE 209: Review of Circuits as LTI Systems – Short primer on the mathematical analysis of (electrical) LTI systems.
• ECE 209: Sources of Phase Shift – Gives an intuitive explanation of the source of phase shift in a high-pass filter. Also verifies simple passive LPF transfer function
Transfer function
A transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a linear time-invariant system. With optical imaging devices, for example, it is the Fourier transform of the point spread function i.e...

by means of trigonometric identity.