A high-pass 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 low-cut filter or bass-cut filter.
First-order continuous-time implementation
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 fc, at which the output power is half the input power. That is,
where fc 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 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.
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 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:
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