- High-pass filter
A

**high-pass filter**is a filter that passes high frequencies well, but attenuates (reduces the amplitude of) frequencies lower than thecutoff 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 alow-pass filter , and aband-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. Of course, 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 aresistor in parallel with the signal path. The resistance times the capacitance (R×C) is thetime constant (τ); it is inversely proportional to the cutoff frequency, at which the output power is half the input (−3 dB)::$f\; =\{1\; over\; 2\; pi\; au\}\; =\; \{1\; over\; 2\; pi\; R\; C\}$

Where "f" is in

hertz , "τ" is insecond s, "R" is in ohms, and "C" is infarad s.**Digital simulation**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 ofcapacitance ::$V\_\{out\}(t)\; =\; I(t)\; R$:$Q\_c(t)\; =\; C\; left\; [V\_\{in\}(t)\; -\; V\_\{out\}(t)\; ight]$:$I(t)\; =\; left(\; frac\{\; d\; Q\_c\}\{\; d\; t\}\; ight)$

Taking the time derivative of the second equation, $I(t)\; =\; C\; left\; [\; frac\{dV\_\{in\{dt\}\; -\; frac\{dV\_\{out\{dt\}\; ight]$. Combining this with the first equation:

:$V\_\{out\}(t)\; =\; C\; left\; [\; frac\{dV\_\{in\{dt\}\; -\; frac\{dV\_\{out\{dt\}\; ight]\; R\; =\; R\; C\; left\; [\; frac\{dV\_\{in\{dt\}\; -\; frac\{dV\_\{out\{dt\}\; ight]$

Now we may discretize the equation. Let us represent $V\_\{in\}$ by a series of samples $x\_\{1...n\}$. We will likewise represent $V\_\{out\}$ by a series of sample $y\_\{1...n\}$ at thesame points in time. For simplicity we assume that the samples are taken at evenly-spaced points in time separated by $Delta\; t$. Making these substitutions:

:$y\_\{i\}\; =\; R\; C\; frac\{x\_\{i\}\; -\; x\_\{i-1\}\; -\; y\_\{i\}\; +\; y\_\{i-1\{Delta\; t\}$

And rearranging terms:

:$y\_\{i\}\; =\; left(\; frac\{RC\}\{RC\; +\; Delta\; t\}\; ight)\; left(\; y\_\{i-1\}\; +\; x\_\{i\}\; -\; x\_\{i-1\}\; ight)$

or more succinctly,

:$y\_n\; =\; alpha\; (\; y\_\{n-1\}\; +\; x\_\{n\}\; -\; x\_\{n-1\})$ :where $alpha\; =\; frac\{RC\}\{RC\; +\; Delta\; t\}$

This gives us a way to determine the output samples in terms of the input samples and the preceding output. The following 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" alpha := RC / (RC + dt) y [0] := x [0]**for**i**from**1**to**n y [i] := alpha*(y [i-1] + x [i] - x [i-1] )**return**y**Applications**Such a filter could be used to direct high frequencies to a

tweeter speaker while blocking bass signals which could interfere with or damage the speaker. A low-pass filter could be realized by using an inductor instead of a capacitor, to simultaneously direct low frequencies to thewoofer . "Seeaudio crossover ". However, inductors are prone to parasitic coupling; hence the RClow-pass filter .High-pass and low-pass filters are also used in digital

image processing to perform transformations in the spatialfrequency domain .Most high-pass filters have zero gain (-"inf" dB) at DC. Such a high-pass filter with very low cutoff frequency can be used to block DC from a signal that is undesired in that signal (and pass nearly everything else). These are sometimes called

DC blocking filters .**ee also***

DSL filter

*Band-stop filter

*Bias tee

*Low-pass filter **External links*** [

*http://www.dspguide.com/ch7/1.htm Common Impulse Responses*]

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