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Introduction
A filter is a frequencyselective device that makes certain frequency components of a signal pass and greatly attenuates other frequency components. In the test device, the interference noise can be filtered or the spectrum analysis can be carried out by using the frequency selection function of the filter. The contents of this article belong to the scope of analog filtering. This paper will mainly introduces the principle, type, mathematical model, main parameters, RC filter design of analog filter (continuous timeinvariant system). Although digital filtering technology has been widely used, analog filtering is still widely used in automatic detection, automatic control and electronic measurement instruments.
Filter Characteristics: RC and RL Circuits
Catalog
Ⅰ What is Filter
A filter is a frequencyselective device that makes certain frequency components of a signal pass and greatly attenuates other frequency components. In the test device, the interference noise can be filtered or the spectrum analysis can be carried out by using the frequency selection function of the filter.In a broad sense, any channel of information transmission (medium) can be regarded as a filter. Because the response characteristic of any device is a function of the excitation frequency, these transfer characteristics can be described in terms of frequencydomain functions. Therefore, any part of the test system, such as the mechanical system, electrical network, instrumentation and even the connecting wires and so on, will transform and process the signals that pass through in a certain frequency range according to their frequency domain characteristics.
According to the nature of the filter signal processing, filters are divided into analog filters and digital filters.
(a) General Frequency Filter; (b) High Pass Filter; (c) Low Pass Filter; (d) Band Pass Filtr
II Classification of Filter
2.1 Classification Based on Filter Frequency Selection

Lowpass Filter
From 0 to f2 , the amplitude  frequency characteristic is flat , which can make the frequency components below f2 in the signal almost unattenuated , and the frequency components above f2 are greatly attenuated .

Highpass Filter
In contrast to lowpass filtering, the amplitudefrequency characteristic is flat from the frequency f _ 1 ~ ∞, which makes the frequency component above f _ 1 pass through almost without attenuation, while the frequency component below f _ 1 will be greatly attenuated.

Bandpass Filter
The passband of bandpass filter is between f1 and f2 . It allows the frequency components in the signal higher than f1 and lower than f2 pass without attenuation while other components are attenuated .

Band Stop Filter
In contrast to bandpass filtering, the stop band of band stop filter is between frequency f1 ~ f2. It attenuates the frequency components above f1 and below f2 in the signal and passes the rest of the frequency components almost without attenuation.
 Note
Lowpass filters and highpass filters are the two most basic forms of filters. Other filters can be divided into these two types of filters. For example, the series connection of a lowpass filter and a highpass filter is a bandpass filter; The parallel connection of a low pass filter and a high pass filter is a bandstop filter.
Series Connection of A Lowpass Filter and A Highpass Filter
Parallel Connection of A Low Pass Filter and A High Pass Filter
2.2 Classification Based on " Best Approximation Properties " Standard

Butterworth Filter
The requirement of Butterworth filter will be raised from the amplitudefrequency characteristic without regard of the phasefrequency characteristic. Butterworth filter has the characteristic of maximum flat amplitude.The amplitude frequency response expression is :
In is the order of the filter; wc is the cutoff frequency of the filter,
When w=wc, H(wc)2=1/2, so wc corresponds to 3db point of the filter.
The Butterworth lowpass filter uses the Butterworth function as the transfer function H (s) of the filter and approximates the ideal rectangular shape of the filter in the form of the highest order Taylor series.

Chebyshev Filter
The approximation requirement of Chebyshev filter is also raised from the amplitudefrequency characteristic.The amplitude frequency response expression is :
ε is the coefficient of fluctuation that determines the size of the passband ripple, 0 <ε <1. The ripple is due to the presence of reactance elements in the actual filter network; wc is the passband cutoff frequency and Tn is the norder Chebyshev polynomial.
Compared with the Butterworth approximation property, although this characteristic is fluctuating within the passband, it attenuates more steeply for the same value of n after entering the stopband and is much closer to the ideal case. The smaller the value of ε, the smaller the fluctuation will be in the passband. At the same time, the decibel value of the attenuation at the cutoff frequency will also be smaller , but the attenuation characteristic after entering the stopband will slow down. Compared with Butterworth filters, there is ripple in the Chebyshev filter passband , and the transitional band of Chebyshev filter is light steep straight; therefore, when there is not allowed ripple in the passband, Butterworth filter is much more desirable.
Phase response from the point of view, Butterworth type is better than the Chebyshev type, through the above two comparison can be seen, the former phase frequency response is closer to a straight line.
From the response of phase frequency, Butterworth type is better than that of Chebyshev type, through comparison of the two figures below, it can be seen that the phase frequency response of the former one is closer to straight line.

Bessel filter
Only meet the phase frequency characteristics and do not care about the amplitudefrequency characteristic. Bessel filter, is also known as the most flattime delay or delay filter. The phase shift is proportional to the frequency, that is, a linear relationship. However, due to its poor amplitudefrequency characteristic, its application is often limited.
2.3 Classification Based on the Nature of the Filter Components

Passive filter (R, L, C)

Active filter (including op amp)
III Ideal Filter
The ideal filter is a filter that can make the amplitude and phase of the signal in the passband not distorted, and the frequency components in the stopband all decay to zero. There is a clear dividing line between passband and stopband. That is to say, the amplitudefrequency characteristic of the ideal filter in the passband should be constant, the slope of the phasefrequency characteristic is a constant value, and the amplitudefrequency characteristic outside the passband should be zero.
The ideal lowpass filter frequency response function is:
The amplitudefrequency and phasefrequency characteristic curve is as follows:
By analyzing the frequency characteristic of the upper expression, we can know that The pulse response function H (T) of the filter in the time domain is a sinc function, the graph is shown below. The waveform of the impulse response h (t) extends infinitely along the left and right of the abscissa. It can be seen from the figure that the filter has responded before the unit pulse is input to the filter at t = 0, ie at t <0 . Obviously, this is a noncausal relationship that can not be physically realized. This shows that the amplitudefrequency characteristic of rightangle sharpness at the cutoff frequency, or the ideal filter described by the rectangular window function in the frequency domain, is not possible. The actual filter's frequency domain graph will not be completely cut off at a certain frequency, but will gradually attenuate and extend to ∞.
IV Actual Filter
The basic parameters of the actual filter
The ideal filter does not exist. In the graph of actual filter amplitudefrequency characteristics, there should be no strict boundaries between passband and stopband. There is a transition band between the pass and stop bands. The frequency component in the transition zone will not be completely suppressed, and it will only be attenuated in varying degrees. Of course, it is believed that the narrower the transition band the better, that is, the faster and more the frequency components out of the passband decay the better. Therefore, when designing an actual filter, it is always possible to approximate the ideal filter by various methods as much as possible.
Shown in the figure: the ideal bandpass (dashed line) and the actual (solid line) bandpass filter amplitude frequency characteristic.
As can be seen from the figure, the ideal filter characteristics only need to use the cutoff frequency description, and the actual filter curve without significant turning point, the two cutoff frequency amplitudefrequency characteristics are not constant, so it requires more parameters to describe .

Ripple amplitude d
In a certain frequency range, the amplitudefrequency characteristic of the actual filter may fluctuate. The smaller the amplitude d is compared with the average A0 of the amplitudefrequency characteristic, the better, and generally it should be far less than 3dB.

Cutoff frequency fc
The frequency corresponding to the amplitudefrequency characteristic value equal to 0.707A0 is called the cutoff frequency of the filter. Taking A0 as a reference value, 0.707A0 corresponds to a 3dB point, that is, attenuates 3dB with respect to A0. If the signal amplitude squared signal power, the corresponding point is exactly halfpower point.

Bandwidth B and quality factor Q
The frequency range between the upper and lower cutoff frequencies is called filter bandwidth or 3dB bandwidth in Hz. The bandwidth determines the ability of the filter to separate the adjacent frequency components in the signal frequency resolution. In Electrotechnics, Q is usually used to represent the quality factor of the resonant circuit.
In the second order oscillation link, the Q value is equal to the amplitude gain coefficient of the resonance point, Q = ½ξ (ξ  damping rate). For the bandpass filter, the ratio of the center frequency f0 () to the bandwidth B is usually referred to as the quality factor Q of the filter. For example, a center frequency of 500Hz filter, if the 3dB bandwidth of 10Hz, the Q value of 50. The larger the Q value, the higher the filter frequency resolution.
For example, if a filter with a center frequency of 500Hz with a bandwidth of 3dB is 10 Hz, the Q value is 50. The greater the Q is, the higher the frequency resolution of the filter is.

Octave selective W
Outside the two cutoff frequencies, the actual filter has a transitional band. The slope of the amplitudefrequency curve of this transition band indicates how fast the amplitudefrequency characteristic decays, which determines the filter's ability to reject the bandwidthfrequency components. It is usually characterized by octave selectivity. The socalled octave selectivity refers to the attenuation value of the amplitudefrequency characteristic between the upper cutoff frequency fc2 and 2fc2 or between the lower cutoff frequency fc1 and fc1/2, that is, the attenuation when the frequency changes by one octave
or
Octave attenuation is expressed in dB / oct (octave, octave). Obviously, the faster the decay (ie, the larger the W value), the better the selectivity of the filter. The attenuation rate that is far away from the cutoff frequency can also be expressed in terms of 10fold frequency decay, ie. [dB / 10 oct].

Filter factor (or rectangular coefficient) λ
The filter factor is another representation of filter selectivity, which is a measure of filter selectivity using the ratio of the 60dB bandwidth to the 3dB bandwidth of the filter's amplitudefrequency characteristics, denoted as λ, ie,
Ideal filter λ= 1, common filter λ= 15, obviously, the closer to 1, the better the filter selectivity will be.
V RC Passive Filter
In the test system, RC filters are commonly used . Because in this area, the signal frequency is relatively low. The RC filter has the advantages of simple circuit, strong antiinterference, better lowfrequency performance and a selection of standard resistance capacitance element. so the most commonly used filter in the field of engineering testing is RC filter.
5.1 Firstorder RC Lowpass Filter
RC lowpass filter circuit and the amplitude and phase frequency characteristics is shown below
Suppose the input voltage of the filter is ex, the output voltage is ey, the differential equation of the circuit is
This is a typical firstorder system. Let λ =RC, called time constant, take Laplace transform on the Upper Formula
or
The amplitude frequency and the phase frequency characteristic formula:
Analysis shows that when f is small, A (f) = 1, the signal passes without attenuation; when f is large, A (f) = 0, the signal is completely blocked and can not pass. Lowpass filter upload stop frequency:
5.2 Firstorder RC Highpass Filter
RC highpass filter circuit and its amplitude and phase frequency characteristics is shown below
Suppose the input voltage of the filter is ex, the output voltage is ey, the differential equation of the circuit is
Similarly, Let λ =RC, called time constant, take Laplace transform on the Upper Formula
or
The amplitude frequency and the phase frequency characteristic formula:
Analysis shows that when f is small, A (f) = 0, the signal is completely blocked and can not pass; when f is large, A (f) = 1, the signal is not attenuated.
5.3 RC Bandpass Filter
Bandpass filter can be seen as the lowpass filter and highpass filter in series, the circuit and its amplitudefrequency, phasefrequency characteristics is shown below
The amplitude frequency and the phase frequency characteristic formula:
In the formula, H1 (s) is the transfer function of the highpass filter and H2 (s) is the transfer function of the lowpass filter.
At this moment, the extremely low and extremely high frequency components are completely blocked and can not pass; only the frequency components of the signal located in the frequency passband can pass through.
Lower cutoff frequency:
Upper cut off frequency:
It should be noticed that when two stages of high and low pass are connected in series, the interaction between the two stages should be eliminated because the latter stage becomes the "load" of the previous stage and the former stage is the signal source internal resistance of the latter stage. In fact, In fact , the two  stage common emitter follower is isolated by an operational amplifier. So the actual bandpass filter is often active. The active filter consists of an RC tuning network and an op amp. Operational amplifier can not only play the role of isolation, but also play an amplifying role of the signal amplitude.
VI Application of Analog Filters
Analog filters are a common type of conversion device used in test systems or specialized instrumentation. Such as bandpass filters used as frequencyselective devices in spectrum analyzers; lowpass filters used as antialiasing in digital signal analysis systems; highpass filters used as elimination of lowfrequency interference noise in acoustic emission detector; band stop filter used as eddy current vibration meter trap.
The bandpass filter that is used in the spectrum analyzer can be divided into two types according to the numerical relationship between the center frequency and the bandwidth:
One is the bandwidth B does not change depending on the center frequency, known as constant bandwidth bandpass filter. As shown in the figure, the bandwidth is the same when the center frequency is at any frequency band.
The other one is the ratio of bandwidth B to the center frequency is constant, known as constant bandwidth ratio bandpass filter. As shown in the figure, the higher the center frequency, the bandwidth is wider.
Normally, in order to make the filter have good frequency resolution in any band, you can use a constant bandwidth bandpass filter (such as the radio frequency). The narrower the selected bandwidth is, the higher the frequency resolution is, but at this time, the number of filters required is large to cover the entire frequency range to be detected. Therefore, in many cases, the constant bandwidth bandpass filter does not necessarily have to be a fixed center frequency, but instead uses a reference signal that varies the center frequency of the filter by the frequency of the reference signal. In the process of signal spectrum analysis, the reference signal is supplied by a frequencyswept signal generator. This variable center frequency, constant bandwidth, bandpass filter is used in correlation filtering and in scantracking filtering.
The constant bandwidth ratio bandpass filter is used in the octave spectrum analyzer, which is a filter bank with different center frequencies, so that the individual bandpass filters can be combined to cover the entire frequency range of the signal to be analyzed. Its center frequency and bandwidth are configured according to certain rules.
If the lower cutoff frequency of any of the bandpass filters is fc1, the upper cutoff frequency is fc2, the relationship between fc1 and fc2: fc1=2nfc1
In the formula, n value is called the octave range, if n = 1, it is called the octave filter; n = 1/3, then called 1/3 octave filter. The center frequency f0 of the filter is taken as the geometric mean, ie:
According to the above two formulas, we can obtain:
Filter bandwidth:
If the Q value of the filter's quality factor is expressed, then:
So in octave filter, if n = l, then Q = 1.41; if n = 1/3, then Q = 4.38; if n = 1/5, then Q = 7.2. The smaller the octave n value is, the larger the Q value will be, which indicating that the higher the filter resolution. According to the above relationship, the center frequency f0 and the bandwidth B value of the common octave filter can be determined.
In order to make the frequency components of the signal to be analyzed without lost, the center frequency of the bandpass filter bank is octave and the bandwidth is contiguous at the same time. The usual practice is to match the cutoff frequency of one 3dB on the previous filter with the lower cutoff frequency of the later one of the filter, which is shown in the figure. Such a set of filters will cover the entire frequency range, which is known as "abutting type".
The following figure shows the adjacent octave filter, the box number represents the center frequency of each bandpass filter. By the analysis of signal input, input, output band switch sequence connected with each filter, if the signal has a bandpass filter pass band frequency components, then we can show that the recording instrument observed on the frequency components.
After learning about filters, if you are still interested, then you can watch this video or check the Classification of Electronic Filters.
Filter Design
In short, filters can be built in a number of different technologies. The same transfer function can be realised in several different ways, that is the mathematical properties of the filter are the same but the physical properties are quite different. Often the components in different technologies are directly analogous to each other and fulfill the same role in their respective filters.
In Telephony, filtering is useful for stacking together many modulated 0–4kHz voice channels into a single widebandwidth signal (think fiber, not copper) transmitting, and then decomposing the signal back into separate 0–4kHz voice channels again.
In Audio Engineering, filtering (specifically highpass filtering) recorded tracks of instruments is useful, so the bass frequencies from a guitar amplifier would not obscure or overshadow the bass frequencies coming from a bass amplifier.
Filter technology is so different in virous applications, such as important measures to prevent electromagnetic interference, it is also a new field for many people.
Ⅶ Recommended Reading
Common Applications of Filter
What Is A Low Pass Filter Circuit?
What is a High Pass Filter?
Classification of Electronic Filters
Four Typical Electronic Filters (Signal Processing)
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2 comments
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