# High-Pass Filter

"A high-pass filter is an electronic filter that allows high-frequency signals to pass freely while attenuating signals with frequencies lower than the cut-off frequency. [...] High-pass filters have many uses, such as blocking DC from circuitry sensitive to non-zero average voltages or RF devices. They can also be used in conjunction with a low-pass filter to make a bandpass filter."[1]

#### Passive high-pass filter

A passive electronic high-pass filter can be created using a resistor and capacitor voltage divider as shown in Figure 1. The impedance of a capacitor is $\frac{1}{\omega C}$, where $\omega$ is the frequency and $C$ is the capacitance. The higher the frequency or capacitance the lower the "resistance" to oscillating current. The capacitor acts like a short for high frequencies and acts like an open circuit for low frequencies. High frequency signals can pass through the capacitor relatively unimpeded while lower frequencies are resisted. This has the effect of suppressing low frequencies from the output signal.

*Figure 1: A passive electronic high-pass filter built from a resistor-capacitor voltage divider.*

The frequency at which the amplitudes begin to attenuate is called the cut-off frequency. It is the frequency at which the signal has attenuated by 3dB as shown in Figure 2. All lower signal frequencies are attenuated at a rate of 20 dB/decade which means a drop in signal strength by 20 dB for each order of magnitude greater frequency than the cut off frequency. For example, if the cut-off frequency is 1kHz, a frequency of 100Hz would attenuate by 20dB and a frequency of 10Hz is attenuated by 40dB.

*Figure 2: The amplitude suppressing effect of a high-pass filter. This figure is a derivative made with permission.*

The cut-off frequency for a passive RC filter can be determined using the following equation:

$$f_c = {1\over{2\pi RC}} $$

where $f_c$ is the cut-off frequency, $R$ is the resistor' resistance and $C$ is the capacitor's capacitance. A low resistance resistor may be desired in order to not overload the source signal.

#### Active high-pass filter

Another high-pass filter type is an active filter which can be built using an op-amp with a few resistors and capacitors as shown in Figure 3. The cut-off frequency for this type of filter is:

$$f_c = \frac{1}{2\pi R_1C}$$

The gain for this type of filter is $-\frac{R_2}{R_1}$ and the attenuation rate after the cut-off frequency is 20dB per decade similarly to the passive low-pass filter.

The op-amp is in a non-inverting amplifier configuration. The output signal is an inverted and amplified version of the input signal. It provides negative feedback to the the input signal. The capacitor acts as a short for high frequencies so high frequencies are allowed to pass through to the input while the low frequencies experience a higher impedance.

*Figure 3: An active high-pass filter built using an op-amp.*