Ahigh-pass filter (HPF) is anelectronic filter that passessignals with afrequency higher than a certaincutoff frequency andattenuates signals with frequencies lower than the cutoff frequency. The amount ofattenuation for each frequency depends on the filter design. A high-passfilter is usually modeled as alinear time-invariant system. It is sometimes called alow-cut filter orbass-cut filter in the context of audio engineering.[1] High-pass filters have many uses, such as blocking DC from circuitry sensitive to non-zero average voltages orradio frequency devices. They can also be used in conjunction with alow-pass filter to produce aband-pass filter.
In the optical domain filters are often characterised by wavelength rather than frequency.High-pass andlow-pass have the opposite meanings, with a "high-pass" filter (more commonly "short-pass") passing onlyshorter wavelengths (higher frequencies), and vice versa for "low-pass" (more commonly "long-pass").
In electronics, afilter is atwo-portelectronic circuit which removesfrequency components from asignal (time-varying voltage or current) applied to its input port. A high-pass filter attenuates frequency components below a certain frequency, called its cutoff frequency, allowing higher frequency components to pass through. This contrasts with alow-pass filter, which attenuates frequencies higher than a certain frequency, and abandpass filter, which allows a certain band of frequencies through and attenuates frequencies both higher and lower than the band.
Inoptics a high pass filter is a transparent or translucent window of colored material that allows light longer than a certainwavelength to pass through and attenuates light of shorter wavelengths. Since light is often measured not by frequency but bywavelength, which is inversely related to frequency, a high pass optical filter, which attenuates light frequencies below a cutoff frequency, is often called a short-pass filter; it attenuates longer wavelengths.
A resistor and either a capacitor or an inductor can be configured as a first-order high-pass filter. The simple first-order capacitive high-pass filter shown in Figure 1 is implemented by placing an input voltage across the series combination of acapacitor and aresistor and using the voltage across the resistor as an output. Thetransfer function of thislinear time-invariant system is:
The product of the resistance and capacitance (R×C) is thetime constant (τ); it is inversely proportional to the cutoff frequencyfc, that is,
wherefc is inhertz,τ is inseconds,R is inohms, andC is infarads. At the cutoff frequency, the filter'sfrequency response reaches -3dB referenced to the gain at an infinite frequency.
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Figure 2 shows an active electronic implementation of a first-order high-pass filter using anoperational amplifier. The transfer function of this linear time-invariant system is:
In this case, the filter has apassband gain of −R2/R1 and has a cutoff frequency of
Because this filter isactive, it may havenon-unity passband gain. That is, high-frequency signals are inverted and amplified byR2/R1.
All of these first-order high-pass filters are calleddifferentiators, because they performdifferentiation for signals whosefrequency band is well below the filter's cutoff frequency.
| Linear analog electronic filters |
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Simple filters |
Filters of higher order have steeper slope in the stopband, such that the slope of nth-order filters equals 20n dB per decade. Higher order filters can be achieved simply by cascading these first order filters. Whileimpedance matching and loading must be taken into account when chaining passive filters, active filters can be easily chained because the signal is restored by the output of the op amp at each stage. Variousfilter topologies andnetwork synthesis filters for higher orders exist, which ease design.
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 bediscretized.
From the circuit in Figure 1 above, according toKirchhoff's Laws and the definition ofcapacitance:
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 therecurrence relation
That is, this discrete-time implementation of a simple continuous-time RC high-pass filter is
By definition,. The expression for parameter yields the equivalenttime constant in terms of the sampling period and:
Recalling that
then and are related by:
and
If, then the time constant equal to the sampling period. If, then is significantly smaller than the sampling interval, and.
The filter recurrence relation provides a way to determine the output samples in terms of the input samples and the preceding output. The followingpseudocode algorithm will simulate the effect of a high-pass filter on a series of digital samples, assuming equally spaced samples:
// Return RC high-pass filter output samples, given input samples,// time intervaldt, and time constantRCfunction highpass(real[1..n] x,real dt,real RC)varreal[1..n] yvarreal α := RC / (RC + dt) y[1] := x[1]for ifrom 2to n y[i] := α × y[i−1] + α × (x[i] − x[i−1])return y
The loop which calculates each of the outputs can berefactored into the equivalent:
for ifrom 2to 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 outputy[i-1] and currentchange in input(x[i] - x[i-1]). In particular,
High-pass filters have many applications. They are used as part of anaudio crossover to direct high frequencies to atweeter while attenuating bass signals which could interfere with, or damage, the speaker. When such a filter is built into aloudspeaker cabinet it is normally apassive filter that also includes alow-pass filter for thewoofer and so often employs both a capacitor andinductor (although very simple high-pass filters for tweeters can consist of a series capacitor and nothing else).As an example, theformula above, applied to a tweeter with a resistance of 10 Ω, will determine the capacitor value for a cut-off frequency of 5 kHz., 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 employbi-amplification withactive RC filters or active digital filters with separate power amplifiers for eachloudspeaker. Such low-current and low-voltageline level crossovers are calledactive crossovers.[1]
Rumble filters are high-pass filters applied to the removal of unwanted sounds near to the lower end of theaudible range or below. For example, noises (e.g., footsteps, or motor noises fromrecord players andtape decks) may be removed because they are undesired or may overload theRIAA equalization circuit of thepreamp.[1]
High-pass filters are also used forAC coupling at the inputs of manyaudio power amplifiers, for preventing the amplification of DC currents which may harm the amplifier, rob the amplifier of headroom, and generate waste heat at theloudspeakersvoice coil. One amplifier, theprofessional audio model DC300 made byCrown International 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 formixing console power.[2] 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.[3] Another example is theQSC 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.[4]
Mixing consoles often include high-pass filtering at eachchannel strip. Some models have fixed-slope, fixed-frequency high-pass filters at 80 or 100 Hz that can be engaged; other models have sweepable high-pass filters, filters of fixed slope that can be set within a specified frequency range, such as from 20 to 400 Hz on theMidas Heritage 3000, or 20 to 20,000 Hz on theYamahaM7CLdigital mixing console. 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 askick drum,bass guitar and piano, sources which will have useful low-frequency sounds. Main writes thatDI unit inputs (as opposed tomicrophone inputs) do not need high-pass filtering as they are not subject to modulation by low-frequencystage wash—low frequency sounds coming from thesubwoofers or thepublic address system and wrapping around to the stage. Main indicates that high-pass filters are commonly used for directional microphones which have aproximity effect—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 high-pass filter setting on the console.[5]
High-pass and low-pass filters are also used in digitalimage processing to perform image modifications, enhancements, noise reduction, etc., using designs done in either thespatial domain or thefrequency domain.[6] Theunsharp masking, or sharpening, operation used in image editing software is a high-boost filter, a generalization of high-pass.
