Digital filter (V) -- design IIR filter

lead

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
Digital filter ( Four )– Analog filter into digital filter

Preface

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 ）.

We've done that Analog filter design , Digitalization of analog filter , What we need to do in this section is IIR The last step of filter design , Digital frequency selection filter .

1. Digital low pass to digital low pass

set up ${\theta }_c$ Is the cut-off frequency of a given digital low-pass filter ,$w_c$ Is the cut-off frequency of the desired low-pass filter , That is, we need to use ${\theta }_c$ Demapping $w_c$, As shown in the figure below ：

The mapping relationship from digital low-pass to digital low-pass is ：

take ${\theta }_c$ and $w_c$ Bring in the above formula to get ：

The above formula can be solved ：

Here is an example to illustrate how to convert ：

2. Digital low pass to digital high pass

set up ${\theta }_c$ Is the cut-off frequency of a given digital low-pass filter ,$w_c$ Is the cut-off frequency of the desired high pass filter , That is, we need to use ${\theta }_c$ Demapping $w_c$, As shown in the figure below ：

The mapping relationship from digital low pass to digital high pass is ：

take ${\theta }_c$ and $w_c$ Bring in the above formula to get ：

The above formula can be solved ：

3. Digital low pass to digital bandpass

set up ${\theta }_c$ Is the cut-off frequency of a given digital low-pass filter ,$w_1$,$w_2$ For the desired band-pass filter 、 Lower cut-off frequency , That is, we need to use ${\theta }_c$ Demapping $w_c$, As shown in the figure below ：

The mapping relationship from digital lowpass to digital bandpass is ：

take ${\theta }_c$ and $w_1$,$w_2$ Bring in the above formula to solve ：

among ：

4. Digital low pass to digital band stop

set up ${\theta }_c$ Is the cut-off frequency of a given digital low-pass filter ,$w_1$,$w_2$ For the desired band stop filter 、 Lower cut-off frequency , That is, we need to use ${\theta }_c$ Demapping $w_c$, As shown in the figure below ：

The mapping relationship from digital low-pass to digital band stop is ：

take ${\theta }_c$ and $w_1$,$w_2$ Bring in the above formula to solve ：

among

5. Design IIR Filter steps

Combined with the previous blogs , Design IIR The steps of the filter can be shown as ：