A highpass filter, or HPF, is an LTI filter that passes high frequencies well but attenuates (i.e., reduces the amplitude of) frequencies lower than the filter's cutoff frequency. The actual amount of attenuation for each frequency is a design parameter of the filter. It is sometimes called a lowcut filter or basscut filter.^{[1]}
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Firstorder continuoustime implementation
The simple firstorder electronic highpass 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 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 shows an active electronic implementation of a firstorder highpass filter using an operational amplifier. In this case, the filter has a passband gain of R_{2}/R_{1} and has a corner frequency of
Because this filter is active, it may have nonunity passband gain. That is, highfrequency signals are inverted and amplified by R_{2}/R_{1}.
Discretetime realization
Discretetime highpass filters can also be designed. Discretetime filter design is beyond the scope of this article; however, a simple example comes from the conversion of the continuoustime highpass filter above to a discretetime realization. That is, the continuoustime 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:
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