Low Pass Filter

What is a Low Pass Filter?
A low-pass filter is an electronic filtering device that allows signals below the cutoff frequency to pass, but not signals above the cutoff frequency.

Basic Information on Low Pass Filters

For different filters, the strength of the signal at each frequency is different. When used in audio applications, it is sometimes called a high-frequency clipping filter, or a high-frequency cancellation filter.
The low-pass filter concept takes many different forms, including electronic circuits (such as hiss filters used in audio equipment), digital algorithms to smooth data, acoustic barriers, image blurring, and more. Both tools provide a smoothed version of the signal by removing short-term fluctuations and preserving long-term developments.
The role of low-pass filters in signal processing is equivalent to that of moving averages in other fields such as finance;
There are many types of low-pass filters, among which the most common ones are Butterworth filters and Chebyshev filters.

Butterworth filter:
The Butterworth filter is a design classification of the filter, which uses the Butterworth transfer function, and has various filter types such as high-pass, low-pass, band-pass, and band-stop.
The Butterworth filter has stable amplitude-frequency characteristics inside and outside the passband, but has a long transition band, which is easy to cause distortion in the transition band.

Chebyshev filter:
The Chebyshev filter is a design classification of the filter, which uses the Chebyshev transfer function, and also has a variety of filter types such as high-pass, low-pass, band-pass, high-resistance, and band-rejection.
Compared with the Butterworth filter, the transition band of the Chebyshev filter is very narrow, but the internal amplitude-frequency characteristics are very unstable.

Classification of Low Pass Filters

The most common topology for high-pass and low-pass filters is the Sallen Key.

high pass filter
It requires only one op amp (Figures 1a and 1b). Multi-channel (channel) filters are often used as bandpass filters (Figure 1c), and they require only one op amp. Figures 2 and 3 show the topology of the biquad filter section. Each structure implements a complete general purpose filter transfer function. The circuit shown in Figure 2 uses three op amps, and the purpose of using the central op amp is only to make the overall feedback path negative. The same filter with switched capacitors requires only two op amps (Figure 3).
And the purpose of using the central op amp is only to make the overall feedback path negative feedback

low pass filter
A low-pass filter allows signals from dc up to a certain cutoff frequency (fCUTOFF) to pass. Setting the high-pass and band-pass coefficients of the second-order transfer function of the general-purpose filter to zero yields a second-order low-pass filter transfer formula:
For frequencies above f0, the signal falls at the rate squared by the frequency. At frequency f0, the damping value attenuates the output signal. You can cascade multiple such filter sections to get a higher order (steeper roll-off) filter. Assume that the design calls for a fourth-order Bessel low-pass filter with a cutoff frequency of 10kHz. According to Reference 1, the roll-off frequencies of each part are 16.13 and 18.19 kHz, the damping values are 1.775 and 0.821, and the high-pass, band-pass and low-pass coefficients of the two filter partitions are 0, 0 and 1, respectively. You can use these two filter sections with the above parameters to achieve the required filter. The cutoff frequency is the frequency point at which the output signal is attenuated by 3 dB.

Low Pass Filter Example

A solid barrier is a low-pass filter for sound waves. When playing music in another room, it was easy to hear the music's bass, but the highs were mostly filtered out. Similarly, very loud music in one car sounds like a bass beat to someone in another car, because the enclosed car (and air space) acts as a low-pass filter, attenuating the all the high notes.
Electronic low-pass filters are used to drive subwoofers and other types of amplifiers and block the high-pitched beats they can't effectively transmit.
Radio transmitters use low-pass filters to block harmonic emissions that could cause interference with other communications.
DSL splitters use low-pass and high-pass filters to separate DSL and POTS signals that share twisted-pair cables.
Low-pass filters also play an important role in the sound processing of electronic music synthesized by analog synthesizers such as the Roland Corporation.
Ideal and Real Filters An ideal low-pass filter completely rejects all frequencies above the cutoff frequency and passes signals below the cutoff frequency unaffected. The actual transition area no longer exists either. An ideal low-pass filter can be obtained mathematically (theoretically) by multiplying the signal by a rectangular function in the frequency domain. As a method with the same effect, it can also be obtained by convolution with the sinc function in the time domain.
However, such a filter is not achievable for actual real signals, because the sinc function is a function extending to infinity, so such a filter needs to predict the future and needs to have the past in order to perform the convolution all data. This is achievable for pre-recorded digital signals (with zero padding at the end of the signal such that the resulting filtered error is smaller than the quantization error) or infinitely recurring signals.
Real filters in real-time applications approximate ideal filters by delaying signals by a small amount of time so that they can "see" a small portion of the future, as evidenced by phase shifting. The higher the approximation accuracy, the longer the delay required.
The Nyquist-Shannon sampling theorem describes how to reconstruct a continuous signal from digital signal samples using a perfect low-pass filter and the Nyquist-Shannon interpolation formula. Actual DACs use approximation filters.

electronic low pass filter

There are many, many different types of filter circuits with different frequency responses. The frequency response of a filter is usually represented by a Bode plot.
For example, a first-order filter cuts the signal strength in half (approximately -6dB) when the frequency is doubled (increases the octave). A first-order filter magnitude bode plot is a horizontal line below the cutoff frequency and a sloped line above the cutoff frequency. There is also a "knee curve" at the boundary between the two that transitions smoothly between the two straight regions.
The second-order filter can have a higher effect on cutting high-frequency signals. The Bode plot of this type of filter is similar to that of a first-order filter, except that it has a faster roll-off rate. For example, a second-order Butterworth filter (which is a critically attenuating RLC circuit with no spikes) attenuates the signal strength by a quarter of its original strength (-12dB per octave) as the frequency doubles. Other 2nd order filters may have an initial roll-off speed that depends on their Q factor, but the final speed is -12dB per octave.
The third and higher order filters are similar. In summary, the roll-off rate of the last nth order filter is 6ndB per octave.
For any Butterworth filter, if you extend the horizontal line to the right and the sloped line to the left (the asymptote of the function), they will intersect at the "cutoff frequency". The frequency response of a first order filter at the cutoff frequency is below the horizontal line - 3dB. Different types of filters - Butterworth, Chebyshev, etc. - have "knee curves" of different shapes. Many 2nd order filters are designed to have "peaks" or resonances to get the cutoff frequency The frequency response at is above the horizontal line.
The meaning of 'low' and 'high' - eg cutoff frequency - depends on the characteristics of the filter. (The term "low-pass filter" simply refers to the shape of the filter response. A high-pass filter can be designed with a lower cutoff frequency than any low-pass filter. The different frequency responses are what differentiate them.) Filters can be designed for any desired frequency range - down to microwave frequencies (over 1000 MHz) and beyond.
In many cases, a simple gain or suppress amplifier (see Op Amp) is converted into a low-pass filter by adding capacitor C. This dampens the frequency response at high frequencies and avoids oscillations within the amplifier. For example, an audio amplifier can be made as a low-pass filter with a cutoff frequency of 100kHz to reduce gain at frequencies that would cause oscillation. Since the human ear can hear audio up to about 20kHz, the frequency of interest falls well within the passband, so the amplifier behaves exactly like the audio of interest.