What is a filter?
Think of a filter as a “black box” with an input to which a signal is applied, and an output where the filtered signal is measured. As the frequency of the signal applied to the input is varied, the amplitude of the output varies. If the input to the filter is a sine wave, the output will be as well. Other wave shapes, square waves for instance, will become distorted when filtered. Filters do not act on the amplitude of the input signal, only its frequency. If you vary the amplitude of the input signal, the amplitude of its output varies with it, regardless of the frequency of the signal. The following figure illustrates the concept.
In figure above, the input signal to the filter is a constant amplitude sine wave whose frequency increases as time goes on. The filter modifies the input signal by decreasing the amplitude of the output at higher frequencies. The higher the frequency of the input signal, the greater the attenuation of the output waveform.
In general, filters are categorized by which part of the frequency range they modify as shown in the following figure.
The plots shown above depict the output voltage on the vertical axis versus the frequency of the signal on the horizontal axis. If the filter allows low frequency signals to pass through unaltered and attenuates high frequency signals, the filter is called a lowpass filter. If the situations is reversed and high frequencies pass through unaltered and low frequencies are attenuated, the filter is called a highpass filters. Likewise, bandpass filters pass the midrange frequencies and attenuate the others, while bandstop filters act in the opposite way.
The regions of a particular filter shape are given names based on the how the signal is changing with frequency. This is illustrated in the following figure.
The passband region of a filter is given by specifying the frequencies in which the filter’s output is between its maximum value and -3dB (≈0.707) of the maximum value. In the case of a lowpass filter, the filters passband is given by a single frequency value which is also referred to the filters bandwidth.
The stopband begins at an amplitude that is dependent on the design of the filter or its design requirement (there is no fixed value for this), and extends to all output values lower in amplitude than the stopband threshold.
The transition band, sometimes called the filter skirt, occupies the region between the passband and the stopband. The same definitions apply for all filter types. In the case of a bandpass filter, there are two stopbands, and in the case of a bandstop filter, there are two passbands. For these filter types (bandpass and bandstop), the filter is usually described by the center frequency of the band and the bandwidth. The bandwidth is the value, in hertz, between the -3dB values.
Passive and Active
Filters can be made from passive components only (resistors, capacitors, and inductors) or passive components in combination with active ones such as op-amps and transistors.
In general, the steepness of the transition band and how much attenuation is present in the stop band is based on the number of reactive components (capacitors and inductors) used in the filter. For instance, a second order filter uses two reactive components while a fourth order filter uses four.
This website contains solvers for first order passive (RC) filters and second order and higher active filters. For active filters, an option is provided for the specific implementation. These implementations (Butterworth, Chebyshev, Bessel) trade off steepness in the skirt for a flat response in the pass band. The following figure illustrates this.
This implementation is characterized by a steep filter slope to the stop band. This comes at the expense of ripple in the pass band and a nonlinear phase response. In response to a step input (think square wave) there is an appreciable amount of overshoot. The greater the ripple, the steeper the cutoff slope.
This implementation is characterized by a maximally flat pass band with a reasonably steep filter slope to the stop band. Like the Chebyshev, this filter has a nonlinear phase response. In response to a step input, there is small amount of overshoot.
This implementation is characterized by pass band response that is similar to the same order RC passive filter with an improved filter slope to the stop band. This filter has maximally linear phase response or time delay. In response to a step input, there is no overshoot.