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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. The meanings of 'low' and 'high' frequencies are relative to the cutoff frequency chosen by the filter designer.

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[edit] Implementation

A passive, analog, first-order high-pass filter, realized by an RC circuit

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):

f ={1 \over 2 \pi \tau} = {1 \over 2 \pi R C}

Where f is in hertz, τ is in seconds, R is in ohms, and C is in farads.

[edit] Discrete-time realization

For another method of conversion from continuous- to discrete-time, see Bilinear transform.

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:

\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 Qc(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 evenly-spaced points in time separated by ΔT time. Let the samples of Vin be represented by the sequence (x_1, x_2, \ldots, x_n), and let Vout 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_{i-1}}{\Delta_T} - \frac{y_i - y_{i-1}}{\Delta_T} \right)

And rearranging terms gives the recurrence relation

y_i = \overbrace{\frac{RC}{RC + \Delta_T} y_{i-1}}^{\text{Decaying contribution from prior inputs}} + \overbrace{\frac{RC}{RC + \Delta_T} \left( x_i - x_{i-1} \right)}^{\text{Contribution from change in input}}

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

y_i = \alpha y_{i-1} + \alpha (x_{i} - x_{i-1}) \qquad \text{where} \qquad \alpha \triangleq \frac{RC}{RC + \Delta_T}

By definition, 0 \leq \alpha \leq 1. The expression for parameter α yields the equivalent time constant RC in terms of the sampling period ΔT and α:

RC = \Delta_T \left( \frac{\alpha}{1 - \alpha} \right)

If α = 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.

[edit] 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 n 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 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 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 RC.
  • 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 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 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 RC.

[edit] 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.[citation needed]

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.[citation needed]

High-pass and low-pass filters are also used in digital image processing to perform transformations in the spatial frequency domain.[citation needed]

High-pass filters are also used for AC coupling at the input and output of amplifiers.[citation needed]

[edit] See also

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