High-pass filter

A high-pass filter is an LTI filter that passes high frequencies well but attenuates (i.e., reduces the amplitude of) frequencies lower than the cutoff frequency. The actual amount of attenuation for each frequency is a design parameter of the filter. It is sometimes called a low-cut filter; the terms bass-cut filter or rumble filter are also used in audio applications.^{[citation needed]}

Contents [hide] 1 First-order continuous-time implementation 2 Discrete-time realization 2.1 Algorithmic implementation 3 Applications 4 See also 5 References 6 External links

[edit] First-order continuous-time implementation

Figure 1: A passive, analog, first-order high-pass filter, realized by an RC circuit

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 and a resistor and using the voltage across the resistor as an output. The product of the resistance and capacitance (R×C) is the time constant (τ); it is inversely proportional to the cutoff frequency f_c, at which the output power is half (−3 dB) the input power. That is,

where f_c is in Hertz, τ is in seconds, R is in Ohms, and C is in Farads.

Figure 2: An active high-pass filter

Figure 2 shows an active electronic implementation of a first-order high-pass filter using an operational amplifier. In this case, the filter has a passband gain of -R₂/R₁ and has a corner frequency of

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

[edit] Discrete-time realization

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

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.

From the circuit in Figure 1 above, according to Kirchoff's Laws and the definition of capacitance:

where Q_c(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:

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 Δ_T time. Let the samples of V_in be represented by the sequence , and let V_out 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 continuous-time RC high-pass filter is

By definition, . The expression for parameter α yields the equivalent time constant RC in terms of the sampling period Δ_T and α:

If α = 0.5, then the RC time constant equal to the sampling period. If , then RC is significantly smaller than the sampling interval, and .

[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]}