Basic knowledge of filters

1. Function and type of filter
1. Function: The filter is a frequency-selective circuit or a processing system that has the function of filtering out noise and separating various signals.
2. Type:
According to the form of processed signals: analog filter and digital filter;
According to function: low-pass, high-pass, band-pass, band-stop
; According to circuit composition: LC passive, RC passive, passive filter composed of special components, RC active filter;
According to the order of the differential equation of the transfer function: first order, second order, high order

2. Transfer function and frequency characteristics of analog filters
(I) Transfer function of analog filters
The characteristics of analog filter circuits can be described by transfer functions. The transfer function is the ratio of the output to the Laplace transform of the input signal voltage or current.
After analysis, after any number of mutually isolated linear networks are cascaded, the total transfer function is equal to the product of the transfer functions of each network. In this way, any complex filter network can be composed of a number of simple first-order and second-order filter circuits cascaded.

(II) Frequency characteristics of analog filters
The transfer function H(s) of an analog filter expresses the transfer relationship between the input and output of the filter. If the input signal Ui of the filter is a unit signal with an angular frequency of w, the output of the filter Uo(jw)=H(jw) expresses the relationship between the output signal and the frequency change under the condition of unit signal input, which is called the frequency characteristic function of the filter, or simply the frequency characteristic. The
frequency characteristic H(jw) is a complex function, whose amplitude A(w) is called the amplitude-frequency characteristic, and its amplitude angle ∮(w) represents the change in the phase of the output signal relative to the phase of the input signal, which is called the phase-frequency characteristic.

(III) Main characteristic indicators of filters
1. Characteristic frequency:
① Passband cutoff frequency fp=wp/(2) is the frequency of the boundary point between the passband and the transition band, at which the signal gain drops to an artificially specified lower limit.
② Stopband cutoff frequency fr=wr/(2) is the frequency of the boundary point between the stopband and the transition band, at which the signal attenuation (the inverse of the gain) drops to an artificially specified lower limit.
③ Turn frequency fc=wc/(2) is the frequency when the signal power attenuates to 1/2 (about 3dB). In many cases, fc is often used as the passband or stopband cutoff frequency.
④ Natural frequency f0=w0/(2) is the resonant frequency of the filter when there is no loss in the circuit. Complex circuits often have multiple natural frequencies.

2. Gain and attenuation
The gain of the filter in the passband is not a constant.
① For low-pass filters, the passband gain Kp generally refers to the gain when w=0; high-pass refers to the gain when w→∞; and bandpass refers to the gain at the center frequency.
② For band-stop filters, the stopband attenuation should be given, and the attenuation is defined as the inverse of the gain.
③ The passband gain change △Kp refers to the maximum change in gain at each point in the passband. If △Kp is in dB, it refers to the change in the gain dB value.

3. Damping coefficient and quality factor
The damping coefficient is a measure of the damping effect of the filter on the signal with a angular frequency of w0, and is an indicator of energy loss in the filter.
The reciprocal of the damping coefficient is called the quality factor, which is an important indicator for evaluating the frequency selection characteristics of bandpass and bandstop filters, Q= w0/△w. In the formula, △w is the 3dB bandwidth of the bandpass or bandstop filter, and w0 is the center frequency. In many cases, the center frequency is equal to the natural frequency.
4. Sensitivity
The filter circuit is composed of many components, and changes in the parameter values ​​of each component will affect the performance of the filter. The sensitivity of a certain performance indicator y of the filter to the change of a certain component parameter x is recorded as Sxy, and is defined as: Sxy=(dy/y)/(dx/x).
This sensitivity is not the same as the sensitivity of the measuring instrument or circuit system. The smaller the sensitivity, the stronger the circuit fault tolerance and the higher the stability.

5. Group delay function
When the amplitude-frequency characteristics of the filter meet the design requirements, in order to ensure that the output signal distortion does not exceed the allowable range, certain requirements should also be put forward for its phase-frequency characteristics ∮(w). In filter design, the group delay function d∮(w)/dw is often used to evaluate the degree of phase distortion of the signal after filtering. The closer the group delay function d∮(w)/dw is to a constant, the smaller the signal phase distortion.