Butterworth High-Pass Filter Frequency Response

The Butterworth high-pass filter is a type of signal processing filter designed to have a flat frequency response in the passband. Its primary feature is the smooth and monotonic transition from passband to stopband, making it useful in applications requiring minimal signal distortion within the passband. The key parameters for designing a Butterworth high-pass filter are the cutoff frequency and the order of the filter.

Historical Background

The Butterworth filter was introduced by the British engineer Stephen Butterworth in his 1930 paper, "On the Theory of Filter Amplifiers." He aimed to create a filter with as flat a frequency response as possible within its passband.

Calculation Formula

The frequency response \( H(\omega) \) of a Butterworth high-pass filter is given by:

\[ H(\omega) = \sqrt{\frac{1}{1 + \left(\frac{\omega_c}{\omega}\right)^{2N}}} \]

Where:

• \( \omega \) is the angular frequency of the input signal.
• \( \omega_c \) is the angular cutoff frequency.
• \( N \) is the order of the filter.

Example Calculation

For a Butterworth high-pass filter with a cutoff frequency of 100 Hz and an order of 2, the frequency response at different frequencies can be calculated as follows:

1. Cutoff frequency \( f_c = 100 \) Hz, thus \( \omega_c = 2 \pi \times 100 \) radians/second.
2. At \( f = 50 \) Hz, \( \omega = 2 \pi \times 50 \): \[ H(50) = \sqrt{\frac{1}{1 + \left(\frac{2 \pi \times 100}{2 \pi \times 50}\right)^{4}}} = \sqrt{\frac{1}{1 + 16}} = 0.2425 \]
3. At \( f = 200 \) Hz, \( \omega = 2 \pi \times 200 \): \[ H(200) = \sqrt{\frac{1}{1 + \left(\frac{2 \pi \times 100}{2 \pi \times 200}\right)^{4}}} = \sqrt{\frac{1}{1 + 0.0625}} = 0.9844 \]

Importance and Usage Scenarios

Butterworth high-pass filters are widely used in audio processing, communications, and control systems where a smooth and flat passband is desired. They are essential in removing low-frequency noise and improving signal quality.

Common FAQs

1. What is a high-pass filter?

   • A high-pass filter allows signals with frequencies higher than a certain cutoff frequency to pass through while attenuating signals with frequencies lower than the cutoff frequency.

2. What is the order of a Butterworth filter?

   • The order of a Butterworth filter determines the steepness of the filter's response near the cutoff frequency. Higher-order filters have a sharper transition between the passband and the stopband.

3. Why use a Butterworth filter?

   • Butterworth filters are used for their maximally flat response in the passband, meaning they do not have ripples in the passband and provide a smooth frequency response.

By using this calculator, users can determine the frequency response of a Butterworth high-pass filter, aiding in the design and analysis of electronic and signal processing systems.