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A filter is a frequency-selective device that allows certain frequency components in a signal to pass while greatly attenuating other frequency components. Using this frequency selection function of the filter, interference noise can be filtered out or spectrum analysis can be performed.
In other words, any device or system that can pass a specific frequency component in a signal and greatly attenuate or suppress other frequency components is called a filter.
filter concept
Filtering is an important concept in signal processing. The function of the filter circuit is to reduce the AC component of the pulsating DC voltage as much as possible, retain its DC component, reduce the ripple coefficient of the output voltage, and make the waveform smoother.
Generally speaking, filtering is divided into classical filtering and modern filtering.
Classical filtering is an engineering concept based on Fourier analysis and transformation. According to advanced mathematical theory, any signal that meets certain conditions can be regarded as a superposition of infinite sine waves.
In other words, the engineering signal is a linear superposition of sine waves of different frequencies, and the sine waves of different frequencies that make up the signal are called frequency components or harmonic components of the signal. A circuit that only allows signal components within a certain frequency range to pass normally while preventing the passage of other frequency components is called a classical filter or filter circuit.
In classical filtering and modern filtering, the filter model is actually the same (hardware filters have not actually developed much), but modern filtering also adds many concepts of digital filtering.
The principle of filter circuit
When the current flowing through the inductor changes, the induced electromotive force generated in the inductor coil will prevent the current from changing. When the current passing through the inductance coil increases, the self-inductive electromotive force generated by the inductance coil is opposite to the current direction, preventing the increase of the current, and at the same time converting a part of the electric energy into a magnetic field that can be stored in the inductance; when the current passing through the inductance coil decreases, The self-induced electromotive force is in the same direction as the current, preventing the reduction of the current, and at the same time releasing the stored energy to compensate for the reduction of the current.
Therefore, after inductive filtering, not only the pulsation of load current and voltage is reduced, the waveform becomes smooth, and the conduction angle of the rectifier diode is increased.
When the inductance coil remains unchanged, the smaller the load resistance, the smaller the AC component of the output voltage. Only when RL>>ωL can obtain better filtering effect. The larger the L, the better the filtering effect.
The role of the filter
1. Separate the useful signal from the noise to improve the anti-interference and signal-to-noise ratio of the signal;
2. Filter out uninteresting frequency components to improve analysis accuracy;
3. Separate single frequency components from complex frequency components.
Ideal filter vs. real filter
ideal filter
The amplitude and phase of the signal in the passband are not distorted, and the frequency components in the noise are attenuated to zero, and there is a clear dividing line between the passband and the stopband.
For example, the frequency response function of an ideal low-pass filter is:
actual filter
An ideal filter does not exist, and in the amplitude-frequency characteristic diagram of an actual filter, there should be no strict boundaries between the passband and the stopband. There is a transition band between passband and stopband. The frequency components in the transition band will not be completely suppressed, but will only be attenuated to varying degrees.
Of course, it is hoped that the narrower the transition band, the better, that is, the faster and more attenuated frequency components outside the passband, the better. Therefore, when designing the actual filter, always try to approximate the ideal filter through various methods.
As can be seen from the amplitude-frequency characteristic diagrams of the ideal band-pass filter and the actual band-pass filter above, the characteristics of the ideal filter only need to be described by the cut-off frequency, while the characteristic curve of the actual filter has no obvious turning point. The amplitude-frequency characteristic between the two cut-off frequencies It is also not constant, so more parameters are needed to describe it.
1. Ripple amplitude d
In a certain frequency range, the amplitude-frequency characteristics of the actual filter may change in ripples. Compared with the average value A0 of the amplitude-frequency characteristics, the smaller the fluctuation range d, the better, generally it should be much less than -3dB.
2. Cutoff frequency fc
The frequency corresponding to the amplitude-frequency characteristic value equal to 0.707A0 is called the cut-off frequency of the filter. Taking A0 as a reference value, 0.707A0 corresponds to the -3dB point, which is a 3dB attenuation relative to A0. If the signal power is represented by the square of the amplitude of the signal, the corresponding point is exactly the half-power point.
3. Bandwidth B and quality factor Q value
The frequency range between the upper and lower cutoff frequencies is called the filter bandwidth, or -3dB bandwidth, in Hz. Bandwidth determines the ability of a filter to separate adjacent frequency components in a signal—frequency resolution. In electrical engineering, Q is usually used to represent the quality factor of a resonant tank.
In the second-order oscillation link, the Q value is equivalent to the amplitude gain coefficient of the resonance point, Q=1/2ξ (ξ - damping rate). For a bandpass filter, the ratio of the center frequency f0 ( ) to the bandwidth B is usually called the quality factor Q of the filter. For example, a filter whose center frequency is 500Hz, if its -3dB bandwidth is 10Hz, its Q value is 50. The larger the Q value, the higher the frequency resolution of the filter.
4. Octave selectivity W
Outside the two cutoff frequencies, the actual filter has a transition band. The slope of the amplitude-frequency curve of this transition band indicates the attenuation of the amplitude-frequency characteristic, which determines the filter's ability to attenuate the frequency components outside the bandwidth.
It is usually characterized by octave selectivity. The so-called octave selectivity refers to the attenuation value of the amplitude-frequency characteristic between the upper cut-off frequency fc2 and 2fc2, or between the lower cut-off frequency fc1 and fc1/2, that is, the attenuation or times when the frequency changes by one octave. The frequency range attenuation is expressed in dB/oct (octave, octave).
Obviously, the faster the attenuation (that is, the larger the W value), the better the filter selectivity. For the attenuation rate far from the cutoff frequency, it can also be expressed by the 10-octave attenuation number. That is [dB/10oct].
5. Filter factor (or rectangle factor)
Filter factor is another way of expressing filter selectivity, which uses the ratio of -60dB bandwidth to -3dB bandwidth of filter amplitude-frequency characteristic to measure filter selectivity. Ideal filter = 1, common filter = 1-5, obviously, the closer it is to 1, the better the filter selectivity.
Classification of filters
Classification according to the frequency selection function of the filter
low pass filter
From 0 to f2 frequency, the amplitude-frequency characteristic is flat, which can make the frequency components lower than f2 in the signal pass through almost without attenuation, while the frequency components higher than f2 are greatly attenuated.
high pass filter
Contrary to low-pass filtering, the amplitude-frequency characteristic is flat from the frequency f1 to ∞. It allows the frequency components of the signal above f1 to pass through with little attenuation, while the frequency components below f1 will be greatly attenuated.
bandpass filter
Its passband is between f1 and f2. It allows the frequency components of the signal above f1 and below f2 to pass through without attenuation, while other components are attenuated.
band stop filter
Contrary to bandpass filtering, the stopband is between frequencies f1-f2. It makes the frequency components higher than f1 and lower than f2 in the signal attenuated, and the signal of the remaining frequency components passes almost unattenuated.
Low-pass filter and high-pass filter are the two most basic forms of filter, other filters can be decomposed into these two types of filters, for example: the series connection of low-pass filter and high-pass filter is band-pass The parallel connection of the low-pass filter and the high-pass filter is a band-stop filter.
Low-pass filter and high-pass filter in series
Parallel connection of low-pass filter and high-pass filter
Classification according to the "Best Approximate Properties" criterion
Butterworth filter
The requirements are made from the amplitude-frequency characteristics without considering the phase-frequency characteristics. The Butterworth filter has a maximum flat amplitude characteristic, and its amplitude-frequency response is expressed as:
Chebyshev filter
The Chebyshev filter also proposes approximation requirements from the aspect of amplitude-frequency characteristics, and its amplitude-frequency response expression is:
ε is a coefficient that determines the size of the passband ripple, and the ripple is caused by the fact that the actual filter network contains reactive elements; Tn is the first type of Chebyshev polynomial.
Compared with the Butterworth approximation characteristic, although this characteristic fluctuates in the passband, it decays more steeply after entering the stopband for the same value of n, which is closer to the ideal situation. The smaller the value of ε, the smaller the fluctuation of the pass band, and the smaller the decibel value of the cut-off frequency point attenuation, but the attenuation characteristic changes slowly after entering the stop band.
The Chebyshev filter is compared with the Butterworth filter. The passband of the Chebyshev filter is rippled, and the transition band is light and steep. Therefore, when ripples in the passband are not allowed, the Butterworth type It is preferable; from the perspective of phase-frequency response, the Butterworth type is better than the Chebyshev type. It can be seen from the comparison of the above two figures that the phase-frequency response of the former is closer to a straight line.
Bessel filter
The Bessel filter is also called the most flat delay or constant delay filter. The phase shift is proportional to the frequency, which is a linear relationship. However, its application is often limited due to its poor amplitude-frequency characteristics.
According to the components used, it is divided into passive and active filters
passive filter
Passive filters are filters composed only of passive components, which are constructed using the principle that the reactance of capacitive and inductive components changes with frequency.
The advantages of this type of filter are: the circuit is relatively simple, no DC power supply is required, and the reliability is high;
The disadvantage is: the signal in the passband has energy loss, the load effect is relatively obvious, and electromagnetic induction is easily caused when the inductance element is used. When the inductance L is large, the volume and weight of the filter are relatively large, which is not applicable in the low frequency domain.
Active filter
Active filters consist of passive components and active devices.
The advantages of this type of filter are: the signal in the passband not only has no energy loss, but also can be amplified, the load effect is not obvious, and the mutual influence is very small when multiple stages are connected. filter, and the filter is small in size, light in weight, and does not require magnetic shielding;
The disadvantage is that the passband range is limited by the bandwidth of the active device, and it needs a DC power supply. The reliability is not as high as that of a passive filter, and it is not suitable for high voltage, high frequency, and high power applications.
Divided into analog filter and digital filter according to the signal processed
Fundamentals of Digital Filters
The signal processing procedure for importing the digital filter is shown in Fig. Among them analog signal (continuous signal)
Sampling must be done using the sampling theorem. The input signal is subjected to analog low-pass filtering, that is, an anti-aliasing filter to remove high-frequency components in the input signal. The smoothed analog signal is reused for sampling. In addition, after the DA conversion, the analog signal should be smoothed by a smoothing filter, which can be completed by an analog low-pass filter.
In addition, the digital equalizer used in digital communication can also be regarded as a digital filter, but when the digital equalizer is used for digital signal processing directly, the AD converter and DA converter in the figure are no longer needed. .
The so-called digital filter is to transform the input sequence into an output sequence through a certain operation. As shown in FIG. Its time-domain input-output relationship is
If the Fourier transform of x(n) , y(n) exists, the frequency domain relationship of input and output is
Assuming that |X(ejw)| and |H(ejw)| are shown in (a) and (b) in the figure, the formula |Y(ejw)| is shown in figure (c).
In this way, the result of x(n) passing through the system h(n) is that the frequency part of |w|>wc is no longer contained in the output y(n), and the components of |w|<wc are passed through without distortion. Therefore, by designing |H(ejw)| with different shapes, different filtering effects can be obtained.
Main features of digital filters:
1. The digital filter is less sensitive to the external environment and has higher reliability.
2. Digital filters can achieve functions that cannot be achieved by analog filters, such as accurate linear phase and multi-rate processing.
3. As long as the word length of the digital filter is increased, the signal processing of arbitrary precision can be realized.
4. The realization of digital filter is more flexible, and can store signals at the same time.
5. The frequency domain width of the digitally processed signal is limited by the sampling rate.
The main difference between digital filters and analog filters
1. Digital filters are used in discrete systems, analog filters are used in continuous time systems, and can also be used in discrete time systems, such as SC (switched capacitor) filters.
2. A digital filter is an algorithm or device composed of digital multipliers, adders and delay units. The function of the digital filter is to perform arithmetic processing on the digital code of the input discrete signal to achieve the purpose of changing the signal spectrum.
The digital filter can be realized by computer software or by large-scale integrated digital hardware. There are active and passive analog filters. Active filters are mainly composed of op amps, or spanning op amps, and resistors and capacitors. Passive filters are mainly composed of R, L, and C. Analog filters have issues such as voltage drift, temperature drift, and noise, but digital filters do not have these issues, so they can achieve high stability and accuracy.
3. In terms of implementation methods, analog filters are generally built with analog devices such as capacitors and inductors, and digital filters can be implemented through software or digital chips. It is very troublesome to replace capacitors and inductors when analog filter parameters are changed. When the parameters of the digital filter are changed, sometimes it is only necessary to modify the coefficients (such as software implementation).
4. From the perspective of technical indicators, it is very difficult for an analog filter to reach -60dB, while a digital filter can easily achieve this indicator.
5. The biggest difference between analog filters and digital filters is that digital filters are inverted with respect to the Fs/2 frequency, that is, symmetrical; while analog filters are not. Therefore, a large number of interpolation filters are selected in the DAC, and the image frequency is placed at a far frequency point, and then an analog filter such as a sound meter is used in the radio frequency section to filter out the image frequency. So digital and analog filters are indispensable.
6. The expressions of analog filters and digital filters are different: analog filters are represented by H(S), while digital filters are represented by H(Z). The analog filter is mainly based on the approximation of the amplitude-frequency characteristic, while the digital filter can achieve phase matching.