Digital filter (III) -- Design of analog 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

Preface

IIR Filtering is also known as infinite unit impulse response filter , The basic structure and MATLAB For implementation, please refer to the first blog in this column ： Digital filter ( One )–IIR And FIR The basic structure and MATLAB Realization . IIR The design steps of the filter mainly consist of three major steps shown in the figure below . First, we need to design an analog prototype filter , Then the filter is digitized into the corresponding digital low-pass filter , Then the conversion in the memory digital domain is called the frequency selection filter we want to get （ Lowpass 、 qualcomm 、 Bandpass 、 Band stop ）.

Let's talk about it again IIR Advantages and disadvantages of filter ：

• IIR Advantages of filter

  • Analog filters can be used to design , There are a lot of charts for the design of analog filters , Convenient and simple .

• IIR Disadvantages of filter

  • The phase is nonlinear
  • The shock response is infinite , So we can't do fast convolution
  • There are stability problems

  This blog mainly talks about how to design and implement common analog filters , Butterworth filter and Chebyshev Ⅰ Type a filter .

1. Analog filter system function

The system function of analog filter is commonly used $H_a(s)$ Express , The response of the analog filter can be $H_a(j\Omega )$ Express , Therefore, the expression of the amplitude square function of the analog filter is ：

For real filters , Zero pole at s The plane is Pair In the form of , As shown in the figure below ：

Let's take a look s Flat and z The correspondence of planes ：

s In plane $\sigma <0$ The area of corresponds to z Unit circle on plane , Why discuss this relationship ？ Because filters must be causal stable systems , Its system function is z The poles of the plane are inside the unit circle , So it corresponds to s Plane , The poles should be distributed in the left half plane . If the minimum phase delay system is required , Then the zeros of the system function should also be distributed in s The left half plane of the plane .

2 Butterworth filter

2.1 Design indicators

First, let's understand the shape of Butterworth filter , As shown in the figure below ：

As can be seen from the above figure , Butterworth analog filter has the characteristic of maximum flat amplitude , The stopband changes monotonically . Order N The bigger it is , The curve trembles , The better the filter characteristics .

The design of Butterworth filter needs to consider some indicators ( The indicators correspond to the following figure )：

• N: The order of the filter
• ${\Omega }_p$： Passband cut-off frequency
• ${\Omega }_c$： Filter cutoff frequency , also called 3dB bandwidth , Its corresponding assignment is $\frac{1}{\sqrt{2}}$
• ${\Omega }_{s}t$: Stopband cut-off frequency
• ${\delta }_p$: Passband attenuation ( corrugated )（dB）
• ${\delta }_s$: Stopband attenuation ( corrugated )（dB）
  

2.2 Design process

• First step ： Determine the filter order according to the index requirements N( Rounding up ) And filter cut-off frequency

• The second step ： Check the table according to the order , Get the normalized system function $H_{a}N(s)$:
  
  The normalized expression of the system function is ：

• The third step ： De normalize the system function , Get the final system function ：
  

Now let's experience the process of designing Butterworth filter through an example ：

3. Chebyshev Ⅰ Type a filter

3.1 Design indicators

Chebyshev analog filters are divided into Ⅰ The type and Ⅱ Type two , Let's look at the response curves of oneortwo filters .

• Chebyshev Ⅰ type
  The amplitude characteristic is equiripple in the passband , Monotonic descent in stopband ：
  
  For the order of the filter N, The passband ripple of the filter is different when it is odd and even ,N In an odd number of ,$\Omega =0$ When the amplitude is 1,N For even when ,$\Omega =0$ When the amplitude is $\frac{1}{\sqrt{1+{\epsilon }^2}}$
• Chebyshev Ⅱ type
  The amplitude characteristic is equiripple in the stopband , It is stable in the passband ：
  
  This blog post focuses on Chebyshev Ⅰ Type a filter .
  Design Bischoff Ⅰ The following indexes need to be considered for type a filter ：
• $N$： The order of the filter
• ${\Omega }_c$： Passband width
• ${\delta }_p$: Passband attenuation ( corrugated )（dB）
• ${\delta }_s$: Stopband attenuation ( corrugated )（dB）

3.2 Design process

• First step ： Determine the filter order according to the index requirements
  The order of the filter can be determined by the passband 、 Stopband attenuation determination ：
  
  Where hyperbolic cosine function $arcch(x)=ln(x+\sqrt{x^2-1})$

• The second step ： Check the table according to the order and passband ripple
  At this time, the cut-off frequency is normalized ：
  
  The normalized system function of analog filter can be determined by looking up the table ：
  
  Coefficient of $d_0$ Can pass $H_a(j0)=H_{aN}(s){|}_{s=0}$ determine , When the order N In an odd number of , $H_{aN}(s){|}_{s=0}=1$, When the order N For even when ,$H_{aN}(s){|}_{s=0}=\frac{1}{\sqrt{1+{\epsilon }^2}}$, The parameter $\epsilon$ You can use the formula ：${\epsilon }^2=10^{0.1{\delta }_p}-1$ To make sure .

• The third step ： De normalize the system function , Get the final system function
  
  Now let's experience the design of Chebyshev through an example Ⅰ Type filter process ：