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High-pass filter
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A high-pass filter is a filter that passes high frequencies well, but attenuates (reduces the amplitude of) frequencies lower than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a low-cut filter; the terms bass-cut filter or rumble filter are also used in audio applications. A high-pass filter is the opposite of a low-pass filter, and a band-pass filter is a combination of a high-pass and a low-pass.
It is useful as a filter to block any unwanted low frequency components of a complex signal while passing the higher frequencies.
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Encyclopedia
A high-pass filter is a filter that passes high frequencies well, but attenuates (reduces the amplitude of) frequencies lower than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a low-cut filter; the terms bass-cut filter or rumble filter are also used in audio applications. A high-pass filter is the opposite of a low-pass filter, and a band-pass filter is a combination of a high-pass and a low-pass.
It is useful as a filter to block any unwanted low frequency components of a complex signal while passing the higher frequencies. The meanings of 'low' and 'high' frequencies are relative to the cutoff frequency chosen by the filter designer.
Implementation
The simplest electronic high-pass filter consists of a capacitor in series with the signal path in conjunction with a resistor in parallel with the signal path. The resistance times the capacitance (R×C) is the time constant (τ); it is inversely proportional to the cutoff frequency, at which the output power is half the input (-3 dB):
Where f is in hertz, τ is in seconds, R is in ohms, and C is in farads.
Discrete-time realization
The effect of a high-pass filter can be simulated on a computer by analyzing its behavior in the time domain, and then discretizing the model.
From the circuit diagram above, according to Kirchoff's Laws and the definition of capacitance:
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
That is, this discrete-time implementation of a simple RC high-pass filter is
By definition, . The expression for parameter yields the equivalent time constant 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 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] := x[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 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 .
Applications
Such a filter could be used as part of an audio crossover to direct high frequencies to a tweeter while blocking 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 low-pass filter for the woofer and so often employs both a capacitor and inductor (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 with separate power amplifiers for each loudspeaker making an active crossover.
Rumble filters are high-pass filters applied to the removal of unwanted sounds below or near to the lower end of the audible range. 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.
High-pass and low-pass filters are also used in digital image processing to perform transformations in the spatial frequency domain.
High-pass filters are also used for AC coupling at the input and output of amplifiers.
See also
External links
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- — Short primer on the mathematical analysis of (electrical) LTI systems.
- — Gives an intuitive explanation of the source of phase shift in a high-pass filter. Also verifies simple passive LPF transfer function by means of trigonometric identity.
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