Digital filter (IV) -- converting analog filter into digital filter

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Digital filter ( One )–IIR And FIR The basic structure and MATLAB Realization
Digital filter ( Two )– Minimum phase delay system and all pass system
Digital filter ( 3、 ... and )– Analog filter design

1. Mapping method

The purpose of mapping is to convert from analog filter to digital filter , This process is from the known analog filter system function $H_a(s)$ System functions mapped to digital filters $H(z)$, Therefore, from analog filter to digital filter is from s Convert plane to z Plane . This mapping needs to meet two requirements ：

• $H(z)$ The frequency response of should be able to imitate $H_a(s)$ Frequency response of ,s The imaginary axis of the plane is mapped to z The unit circle of the plane .
• Causal stable $H_a(s)$ Can be mapped to causal instability $H(z)$, namely s The left half plane of the plane $Re[s]<0$ To map to z Inside the plane unit circle $|z|<1$

There are two general transformation methods ： Impulse response invariance method and bilinear transformation method

2. Impulse response invariant method

2.1 Transformation steps

Impulse response invariable method is to make the unit impulse response of digital filter $h(n)$ Simulate the impulse response of analog filter $h_a(t)$, in other words , We will $h_a(t)$ Sampling at equal intervals , bring $h(n)$ Exactly equal to $h_a(t)$ Of T Interval sampling value , namely ：
$$h(n)=h_a(t){|}_{t=nT}$$
Assume $h(n)<->H(z),h_a(t)<->H_a(s)$, The process of digitizing the analog filter is ：
$$H_a(s)->h_a(t)->h(n)->H(z)$$
This process is also called time domain sampling 、 The process of periodic extension in frequency domain .

The steps of designing digital low-pass filter using impulse response invariance method are as follows ：

• First step
  According to the given index of digital low-pass filter $w_p$, $w_{st}$, ${\delta }_p$, ${\delta }_s$;
• The second step
  Choose the right one T value , Solve the simulation index ${\Omega }_p=\frac{w_p}{T}$, ${\Omega }_{st}=\frac{w_{st}}{T}$
• The third step
  According to the index $w_p$, $w_{st}$, ${\delta }_p$, ${\delta }_s$, Design analog filter , And the system function $H_a(s)$
• Step four
  Set the system function $H_a(s)$ Carry out partial fractional expansion , Unfold into （ Look up the table for Factorization ）
  $$H_a(s)=\sum _{k=1}^N\frac{A_k}{s-s_k}$$
• Step five
  According to the impulse response unchanged method , The system function of the digital filter is
  $$H(z)=\sum _{k=1}^N\frac{T{A}_k}{1-e^{s_{k}T{z}^{-1}}}$$
  Sampling interval T The value of does not affect the design of the filter , For the sake of calculation , Usually take 1 Mostly .

Here are two examples to illustrate ：

• example 1
  
  
• example 1
  
  

2.2 Advantages and disadvantages of impulse response unchanged method

Advantages of impulse response unchanged method ：

• The time domain approximation of impulse response invariable method is good ;
• Analog frequency $\Omega$ And digital frequency $w$ There is a linear mapping relationship between $w=\Omega T$

Disadvantages of impulse response unchanged method ：

• The filter designed by impulse response invariable method will have the aliasing effect of frequency response
• Impulse response invariant method Only applicable to band limited analog filters （ For example, the attenuation characteristics are very good Low pass or bandpass filter ）, And the faster the high-frequency attenuation , The smaller the aliasing effect ; And for qualcomm 、 Band stop filter , It is inconvenient to use this method for design .

3. bilinear transformation

3.1 Transformation steps

Bilinear transformation method first adopts the method of nonlinear frequency compression , Compress the frequency azimuth on this frequency axis to $[-\frac{\pi }{T},\frac{\pi }{T}]$, Through $z=e^{sT}$ take s The plane is mapped to z Plane , such s Flat and z The plane establishes a one-to-one single value relationship , The multi value transformation is eliminated , Thus, the phenomenon of spectrum aliasing is eliminated , As shown in the figure below ：

The expression of bilinear transformation relation is ：
$$s=\frac{2}{T}\frac{1-z^{-1}}{1+z^{-1}}$$

The steps of designing analog low-pass filter by bilinear transformation method are ：

• First step
  According to the given index of digital low-pass filter $w_p$, $w_{st}$, ${\delta }_p$, ${\delta }_s$;
• The second step
  Through pre distortion , Determine simulation indicators ：$\Omega =\frac{2}{T}tan(\frac{w}{2})$(T The value of is generally 2)
• The third step
  According to the index $w_p$, $w_{st}$, ${\delta }_p$, ${\delta }_s$, Design analog filter , And get the system function $H_a(s)$
• Step four
  According to bilinear transformation , The system function of the digital filter is
  $$H(z)=H(s){|}_{s=\frac{2}{T}\frac{1-z^{-1}}{1+z^{-1}}}$$

It is worth noting that , During the transformation , Sampling interval T Usually take 2, Will be offset by calculation , Therefore, its value does not affect the design .

Now let's experience the design of digital filters by bilinear transformation through two examples ：

• example 1
  
  
• example 2
  
  

3.2 Advantages and disadvantages of bilinear transformation

advantage ：
It eliminates the aliasing effect of impulse response invariable method , Various types of filters can be designed .
shortcoming ：
There are serious nonlinear frequency transformations ：
$$\Omega =\frac{2}{T}tan(\frac{w}{2})$$
For filters with piecewise constants , After bilinear transformation , The filter with piecewise constant amplitude frequency characteristic is still obtained , But the position of the critical frequency point of each segment edge will produce distortion , This frequency distortion , It can be corrected by pre distortion of frequency .
Pre distortion refers to the distortion of the critical analog frequency realization , After transformation, it can be mapped to the required digital frequency ; The expression of pre distortion is ：
$${\Omega }_p=\frac{2}{T}tan(\frac{w_p}{2})$$
When designing digital filter with bilinear transformation method, pre distortion operation must be carried out .