IIR数字滤波器的设计_专栏

基于FPGA IIR数字滤波器的设计

IIR（Infinite Impulse Response）无线脉冲响应滤波器。
系统传递函数为：

系统的差分方程可写为：

IIR优缺点:
1)在相同的幅频条件下，滤波器阶数比FIR滤波器低。
2)IIR滤波器占用的硬件资源比较少（相比FIR滤波器）。
3)不具备严格的线性相位特性。

1 IIR数字滤波器的基本结构及类型

图1 直接I型

图2 直接II型

图3 级联型

图4 并联型
2 设计目标

采用matlab buffer函数设计一个IIR滤波器低通滤波器，通带截止频率为1khz，输入信号为1khz+3khz sin波形，经过IIR滤波器后输出为1KHZ sin波，其他不做要求。（本文只对IIR设计思想进行验证不做性能要求）。

3 matlab的设计验证

Matlab源码：

%参数定义

<pre code="" consolas="" courier="" inconsolata="" menlo="" monaco="" mono="" monospace="" pro="" rgb="" sans="" sans-serif="" sc="" source="" style="box-sizing: border-box;font-family: " yahei="">```
FS <span style="box-sizing: border-box;color: rgb(166, 127, 89);font-variant-ligatures: normal !important;font-variant-numeric: normal !important;font-variant-east-asian: normal !important;font-stretch: normal !important;line-height: normal !important;">=</span><span style="box-sizing: border-box;color: rgb(152, 104, 1);font-variant-ligatures: normal !important;font-variant-numeric: normal !important;font-variant-east-asian: normal !important;font-stretch: normal !important;line-height: normal !important;">44100</span><span style="box-sizing: border-box;color: rgb(153, 153, 153);font-variant-ligatures: normal !important;font-variant-numeric: normal !important;font-variant-east-asian: normal !important;font-stretch: normal !important;line-height: normal !important;">;</span> <span style="box-sizing: border-box;color: rgb(166, 127, 89);font-variant-ligatures: normal !important;font-variant-numeric: normal !important;font-variant-east-asian: normal !important;font-stretch: normal !important;line-height: normal !important;">%</span>Sample rate Frequncy<br></br>fc <span style="box-sizing: border-box;color: rgb(166, 127, 89);font-variant-ligatures: normal !important;font-variant-numeric: normal !important;font-variant-east-asian: normal !important;font-stretch: normal !important;line-height: normal !important;">=</span> <span style="box-sizing: border-box;color: rgb(152, 104, 1);font-variant-ligatures: normal !important;font-variant-numeric: normal !important;font-variant-east-asian: normal !important;font-stretch: normal !important;line-height: normal !important;">1000</span><span style="box-sizing: border-box;color: rgb(153, 153, 153);font-variant-ligatures: normal !important;font-variant-numeric: normal !important;font-variant-east-asian: normal !important;font-stretch: normal !important;line-height: normal !important;">;</span> <span style="box-sizing: border-box;color: rgb(166, 127, 89);font-variant-ligatures: normal !important;font-variant-numeric: normal !important;font-variant-east-asian: normal !important;font-stretch: normal !important;line-height: normal !important;">%</span><span style="box-sizing: border-box;color: rgb(152, 104, 1);font-variant-ligatures: normal !important;font-variant-numeric: normal !important;font-variant-east-asian: normal !important;font-stretch: normal !important;line-height: normal !important;">1</span>khz<br></br>fe <span style="box-sizing: border-box;color: rgb(166, 127, 89);font-variant-ligatures: normal !important;font-variant-numeric: normal !important;font-variant-east-asian: normal !important;font-stretch: normal !important;line-height: normal !important;">=</span> <span style="box-sizing: border-box;color: rgb(152, 104, 1);font-variant-ligatures: normal !important;font-variant-numeric: normal !important;font-variant-east-asian: normal !important;font-stretch: normal !important;line-height: normal !important;">3000</span><span style="box-sizing: border-box;color: rgb(153, 153, 153);font-variant-ligatures: normal !important;font-variant-numeric: normal !important;font-variant-east-asian: normal !important;font-stretch: normal !important;line-height: normal !important;">;</span> <span style="box-sizing: border-box;color: rgb(166, 127, 89);font-variant-ligatures: normal !important;font-variant-numeric: normal !important;font-variant-east-asian: normal !important;font-stretch: normal !important;line-height: normal !important;">%</span>外部输入信号  <span style="box-sizing: border-box;color: rgb(152, 104, 1);font-variant-ligatures: normal !important;font-variant-numeric: normal !important;font-variant-east-asian: normal !important;font-stretch: normal !important;line-height: normal !important;">3</span>khz<br></br>N <span style="box-sizing: border-box;color: rgb(166, 127, 89);font-variant-ligatures: normal !important;font-variant-numeric: normal !important;font-variant-east-asian: normal !important;font-stretch: normal !important;line-height: normal !important;">=</span> <span style="box-sizing: border-box;color: rgb(152, 104, 1);font-variant-ligatures: normal !important;font-variant-numeric: normal !important;font-variant-east-asian: normal !important;font-stretch: normal !important;line-height: normal !important;">1024</span><span style="box-sizing: border-box;color: rgb(153, 153, 153);font-variant-ligatures: normal !important;font-variant-numeric: normal !important;font-variant-east-asian: normal !important;font-stretch: normal !important;line-height: normal !important;">;</span><br></br>Q <span style="box-sizing: border-box;color: rgb(166, 127, 89);font-variant-ligatures: normal !important;font-variant-numeric: normal !important;font-variant-east-asian: normal !important;font-stretch: normal !important;line-height: normal !important;">=</span><span style="box-sizing: border-box;color: rgb(152, 104, 1);font-variant-ligatures: normal !important;font-variant-numeric: normal !important;font-variant-east-asian: normal !important;font-stretch: normal !important;line-height: normal !important;">16</span><span style="box-sizing: border-box;color: rgb(153, 153, 153);font-variant-ligatures: normal !important;font-variant-numeric: normal !important;font-variant-east-asian: normal !important;font-stretch: normal !important;line-height: normal !important;">;</span><br></br>


%波形产生

```
sin_osc =sin(t*fc);

sin_e =sin(t*fe);

sin_add = sin_osc+sin_e;


```
```

%IIR 滤波器系数（低通滤波器）

```
```
[b a] = butter(3,fc/(FS/2),'low');

 

%滤波（混频后）

y = filter(b,a,sin_add);

 

f_osc =fft(sin_osc,N);

f_osc=20*log(abs(f_osc))/log(10);        %换算成dBW单位

ft=[0:(FS/N):FS/2];              %转换横坐标以Hz为单位

f_osc=f_osc(1:length(ft));


```
```

%滤波器系数量化

```
```
Mab =max(max(abs(a),abs(b)));

%16bit 量化

Qb = round((b/Mab)*(2^(Q-1)-1));

Qa = round((a/Mab)*(2^(Q-1)-1));

%%%%

Qm =floor(log2(Mab/a(1)));

if Qmlog2(Mab/a(1))

    Qm = Qm +1;

end

Qm = 2^Qm;

Qb1=round(b/Qm*(2^(Q-1)-1));

Qa1=round(a/Qm*(2^(Q-1)-1));


```
```

%绘图  
%时域波形图

```
```
figure(1),

hold on

subplot(221),plot(t(1:128),sin_osc(1:128),'-');

legend('sin 1khz');title('sin 1khz');

subplot(222),plot(t(1:128),sin_e(1:128),'-');

legend('sin 3khz');title('sin 3khz');

subplot(223),plot(t(1:128),sin_add(1:128),'-');

legend('sin 1khz add 3khz');title('sin 1khz add 3khz');

subplot(224),plot(t(1:128),y(1:128),'-');

legend('LPF 结果');title('LPF 结果');

grid;

hold off


```
```

%频域波形

```
```
figure(2),

hold on

subplot(221);plot(ft,f_osc);

xlabel('频率(Hz)','fontsize',8); ylabel('功率(dBW)','fontsize',8);

title('信号频谱图 2KHZ','fontsize',8);legend('sinosc');

subplot(222);plot(ft,f_e);

xlabel('频率(Hz)','fontsize',8); ylabel('功率(dBW)','fontsize',8);

title('信号频谱图3KHZ','fontsize',8);legend('sine');

subplot(223);plot(ft,f_add);

xlabel('频率(Hz)','fontsize',8); ylabel('功率(dBW)','fontsize',8);

title('信号频谱图2KHZ 和 3KHZ','fontsize',8);legend('sin add');

subplot(224);plot(ft,y_f);

xlabel('频率(Hz)','fontsize',8); ylabel('功率(dBW)','fontsize',8);

title('信号频谱图滤波后','fontsize',8);legend('LPF结果');

hold off


```
```

%幅频响应

```
```
figure(3);

subplot(211);stem(Fb);

title('Fb单位抽样响应','fontsize',8);

subplot(212);plot(f,mag);

xlabel('频率(Hz)','fontsize',8);

ylabel('幅度(dB)','fontsize',8);

title('freqz()幅频响应','fontsize',8);


```
```

![](https://ebaina.oss-cn-hangzhou.aliyuncs.com/wechat-official-crawl/2022-02/164480041494098.jpg)

图5 matlab时域波形如图5所示，（5,1）波形加（5,2）波形得到（5,3）波形，经过IIR滤波器后得到（5,4）时域波形。

![](https://ebaina.oss-cn-hangzhou.aliyuncs.com/wechat-official-crawl/2022-02/164480041411783.jpg)

图6 matlab频域分析如图6所示，（6,3）与（6,4）相比3khz大概被削弱25DB左右。

![](https://ebaina.oss-cn-hangzhou.aliyuncs.com/wechat-official-crawl/2022-02/164480041434395.jpg)  
图7 IIR幅频响应分析

如图7所示，通过幅频响应（7,2）可知3khz所在位置大概被削弱25DB。

4 FPGA设计验证
----------

FPGA IIR.v设计源码：

```
```
`timescale 1ps/1ps

module iir(

       input               mclk,//45.1584MHZ

      input               reset_n,

    input  signed[31:0] pcm_in,

     output signed[31:0] pcm_out

    );

       

localparam LAST_CYCLE = 1023;

reg [9:0] i;



wire signed [15:0] b1,b2,b3,b4;

wire signed [15:0] a2,a3,a4;



wire signed [31:0] xn;

reg signed [31:0] xn1,xn2,xn3;

reg signed [31:0] yn,yn1,yn2,yn3;

reg signed [47:0] r_x1;

reg signed [47:0] r_x2;

reg signed [47:0] r_y;

reg signed [47:0] r_s;

reg signed [47:0] r_s1;



//coffe b

assign b1 = 3;

assign b2 = 8;

assign b3 = 8;

assign b4 = 3;



//coffe a

assign a2 = -22243;

assign a3 = 20231;

assign a4 = -6159;



assign xn = pcm_in;

assign pcm_out = yn;





always @(posedge mclk or negedge reset_n) begin

  if(reset_n == 1'b0) begin

    i  0;

 xn1  0;

 xn2  0;

 xn3  0;

 

   yn   0;

 yn1  0;

 yn2  0;

 yn3  0;

 

   r_x1  0;

    r_x2  0;

    r_y   0;

    r_s   0;

    r_s1  0;

  end

  else begin

    i i+1;

    if(i==1) begin

     r_x1  b1*(xn + xn3);

   r_x2  b2*(xn1 + xn2);//Zero(n)

    r_y   a2*yn1+a3*yn2+a4*yn3;//Pole(n)

      $display("r_x1 = %d,r_x2 = %d,r_y = %d",r_x1,r_x2,r_y);

 end

    if(i==2) begin

      r_s  r_x1 +r_x2 - r_y;//8192y(n) = Zero(n)-Pole(n)

      $display("%d",r_s);

  end

    if(i==3) r_s1  (r_s>>13);

 if(i==4) yn  r_s1[31:0];

    if(i==5) begin //pipeline

     xn1  xn;

     xn2  xn1;

    xn3  xn2;

    

   yn1  yn;

     yn2  yn1;

    yn3  yn2;

  end

  end

end


```
```

Endmodule

实验结果：  
Modelsim波

![](https://ebaina.oss-cn-hangzhou.aliyuncs.com/wechat-official-crawl/2022-02/164480041567959.jpg)

图8 modelsim 时域波形由图8可知，pcm\_out1(1khz + 3khz)经过IIR滤波器后滤除了3khz，设计成功。  
Matlab结果分析：  
![](https://ebaina.oss-cn-hangzhou.aliyuncs.com/wechat-official-crawl/2022-02/16448004156152.jpg)

图9 FPGA输入波形matlab时域分析![](https://ebaina.oss-cn-hangzhou.aliyuncs.com/wechat-official-crawl/2022-02/16448004163937.jpg)

图10 FPGA结果数据分析由图10的频域分析结果可知3khz大概被削弱25db左右，设计成功。

感谢阅读，别走！点赞、关注、转发后再走吧

![](https://ebaina.oss-cn-hangzhou.aliyuncs.com/wechat-official-crawl/2022-02/16448004166370.jpg)

转载：全栈芯片工程师