书接上回

上一篇讲述了一阶IIR滤波器的一些特点，并设计了一个高通的IIR。实际应用中往往需要动态调整滤波器的参数满足不同场合，不同平台的需要，本章基于上一节的高通滤波器来继续给出一个通用的灵活的高通IIR：$$H(s)=\frac{s}{s+{\Omega }_c}$$
数字滤波器的截止频率计算公式：$${\omega }_c=\frac{2\pi f}{f_s}$$

开始推算

数字和模拟频率的转换关系：$${\Omega }_c=\frac{{\omega }_c}{T}$$
直接带入传递函数得到：
$$H(s)=\frac{s}{s+\frac{{\omega }_c}{T}}$$
带入双线性变换公式：$$s=\frac{2}{T}\frac{1-z^{-1}}{1+z^{-1}}和{\omega }_c=\frac{2\pi f}{f_s}$$
我们得到
$$H(s)=\frac{\frac{2}{T}\frac{1-z^{-1}}{1+z^{-1}}}{\frac{2}{T}\frac{1-z^{-1}}{1+z^{-1}}+\frac{\frac{2\pi f}{f_s}}{T}}=\frac{1-z^{-1}}{(\pi \frac{f_c}{f_s}+1)+(\pi \frac{f_c}{f_s}-1)z^{-1}}=\frac{Y(Z)}{X(Z)}$$
根据这个得出时域差分方程$$y(n)=\frac{1}{\pi \frac{f_c}{f_s}+1}(x(n)-x(n-1))-\frac{\pi \frac{f_c}{f_s}-1}{\pi \frac{f_c}{f_s}+1}y(n-1)$$
同理得出预畸变(双线性变换对频率作预畸变处理)：$${\Omega }_c=\frac{2}{T}\tan \frac{{\omega }_c}{2}$$的差分方程$$y(n)=\frac{1}{tan(\frac{\pi f_c}{f_s})+1}(x(n)-x(n-1))-\frac{tan(\frac{\pi f_c}{f_s})-1}{tan(\frac{\pi f_c}{f_s})+1}y(n-1)$$
为何提出这两种形式呢？ 主要是因为在低频段两者基本可以互换，而前者能省掉tan计算，对于一些比较弱鸡的嵌入式系统，三角计算是不支持的。
下边给了一个例子，在48k采样率的情况下，1000Hz以内的截止频率，两者的系数计算结果基本可以互换。

PRE_WARP: temp_alpha1=-0.986995 temp_beta1=0.993497 freqz: 100, srate:48000
PRE_WARP: TRANSTO INT: temp_alpha1=-16170 temp_beta1=16277 freqz: 100, srate:48000
NORMAL: alpha1=-0.986995 beta1=0.993498 freqz: 100, srate:48000
NORMAL TRANSTO INT: a0=-16170 a1=16277 freqz: 100, srate:48000


PRE_WARP: temp_alpha1=-0.974157 temp_beta1=0.987078 freqz: 200, srate:48000
PRE_WARP: TRANSTO INT: temp_alpha1=-15960 temp_beta1=16172 freqz: 200, srate:48000
NORMAL: alpha1=-0.974158 beta1=0.987079 freqz: 200, srate:48000
NORMAL TRANSTO INT: a0=-15960 a1=16172 freqz: 200, srate:48000



PRE_WARP: temp_alpha1=-0.961481 temp_beta1=0.980741 freqz: 300, srate:48000
PRE_WARP: TRANSTO INT: temp_alpha1=-15752 temp_beta1=16068 freqz: 300, srate:48000
NORMAL: alpha1=-0.961486 beta1=0.980743 freqz: 300, srate:48000
NORMAL TRANSTO INT: a0=-15752 a1=16068 freqz: 300, srate:48000


PRE_WARP: temp_alpha1=-0.948965 temp_beta1=0.974482 freqz: 400, srate:48000
PRE_WARP: TRANSTO INT: temp_alpha1=-15547 temp_beta1=15965 freqz: 400, srate:48000
NORMAL: alpha1=-0.948976 beta1=0.974488 freqz: 400, srate:48000
NORMAL TRANSTO INT: a0=-15548 a1=15966 freqz: 400, srate:48000



PRE_WARP: temp_alpha1=-0.936602 temp_beta1=0.968301 freqz: 500, srate:48000
PRE_WARP: TRANSTO INT: temp_alpha1=-15345 temp_beta1=15864 freqz: 500, srate:48000
NORMAL: alpha1=-0.936624 beta1=0.968312 freqz: 500, srate:48000
NORMAL TRANSTO INT: a0=-15345 a1=15864 freqz: 500, srate:48000


PRE_WARP: temp_alpha1=-0.924390 temp_beta1=0.962195 freqz: 600, srate:48000
PRE_WARP: TRANSTO INT: temp_alpha1=-15145 temp_beta1=15764 freqz: 600, srate:48000
NORMAL: alpha1=-0.924428 beta1=0.962214 freqz: 600, srate:48000
NORMAL TRANSTO INT: a0=-15145 a1=15764 freqz: 600, srate:48000


PRE_WARP: temp_alpha1=-0.912326 temp_beta1=0.956163 freqz: 700, srate:48000
PRE_WARP: TRANSTO INT: temp_alpha1=-14947 temp_beta1=15665 freqz: 700, srate:48000
NORMAL: alpha1=-0.912384 beta1=0.956192 freqz: 700, srate:48000
NORMAL TRANSTO INT: a0=-14948 a1=15666 freqz: 700, srate:48000


PRE_WARP: temp_alpha1=-0.900404 temp_beta1=0.950202 freqz: 800, srate:48000
PRE_WARP: TRANSTO INT: temp_alpha1=-14752 temp_beta1=15568 freqz: 800, srate:48000
NORMAL: alpha1=-0.900491 beta1=0.950245 freqz: 800, srate:48000
NORMAL TRANSTO INT: a0=-14753 a1=15568 freqz: 800, srate:48000


PRE_WARP: temp_alpha1=-0.888622 temp_beta1=0.944311 freqz: 900, srate:48000
PRE_WARP: TRANSTO INT: temp_alpha1=-14559 temp_beta1=15471 freqz: 900, srate:48000
NORMAL: alpha1=-0.888744 beta1=0.944372 freqz: 900, srate:48000
NORMAL TRANSTO INT: a0=-14561 a1=15472 freqz: 900, srate:48000


PRE_WARP: temp_alpha1=-0.876976 temp_beta1=0.938488 freqz: 1000, srate:48000
PRE_WARP: TRANSTO INT: temp_alpha1=-14368 temp_beta1=15376 freqz: 1000, srate:48000
NORMAL: alpha1=-0.877141 beta1=0.938571 freqz: 1000, srate:48000
NORMAL TRANSTO INT: a0=-14371 a1=15377 freqz: 1000, srate:48000



PRE_WARP: temp_alpha1=-0.865464 temp_beta1=0.932732 freqz: 1100, srate:48000
PRE_WARP: TRANSTO INT: temp_alpha1=-14179 temp_beta1=15281 freqz: 1100, srate:48000
NORMAL: alpha1=-0.865681 beta1=0.932840 freqz: 1100, srate:48000
NORMAL TRANSTO INT: a0=-14183 a1=15283 freqz: 1100, srate:48000



PRE_WARP: temp_alpha1=-0.854081 temp_beta1=0.927040 freqz: 1200, srate:48000
PRE_WARP: TRANSTO INT: temp_alpha1=-13993 temp_beta1=15188 freqz: 1200, srate:48000
NORMAL: alpha1=-0.854359 beta1=0.927179 freqz: 1200, srate:48000
NORMAL TRANSTO INT: a0=-13997 a1=15190 freqz: 1200, srate:48000


PRE_WARP: temp_alpha1=-0.842824 temp_beta1=0.921412 freqz: 1300, srate:48000
PRE_WARP: TRANSTO INT: temp_alpha1=-13808 temp_beta1=15096 freqz: 1300, srate:48000
NORMAL: alpha1=-0.843174 beta1=0.921587 freqz: 1300, srate:48000
NORMAL TRANSTO INT: a0=-13814 a1=15099 freqz: 1300, srate:48000



PRE_WARP: temp_alpha1=-0.831691 temp_beta1=0.915846 freqz: 1400, srate:48000
PRE_WARP: TRANSTO INT: temp_alpha1=-13626 temp_beta1=15005 freqz: 1400, srate:48000
NORMAL: alpha1=-0.832123 beta1=0.916061 freqz: 1400, srate:48000
NORMAL TRANSTO INT: a0=-13633 a1=15008 freqz: 1400, srate:48000


PRE_WARP: temp_alpha1=-0.820679 temp_beta1=0.910339 freqz: 1500, srate:48000
PRE_WARP: TRANSTO INT: temp_alpha1=-13446 temp_beta1=14915 freqz: 1500, srate:48000
NORMAL: alpha1=-0.821204 beta1=0.910602 freqz: 1500, srate:48000
NORMAL TRANSTO INT: a0=-13454 a1=14919 freqz: 1500, srate:48000



PRE_WARP: temp_alpha1=-0.809784 temp_beta1=0.904892 freqz: 1600, srate:48000
PRE_WARP: TRANSTO INT: temp_alpha1=-13267 temp_beta1=14825 freqz: 1600, srate:48000
NORMAL: alpha1=-0.810414 beta1=0.905207 freqz: 1600, srate:48000
NORMAL TRANSTO INT: a0=-13277 a1=14830 freqz: 1600, srate:48000


PRE_WARP: temp_alpha1=-0.799004 temp_beta1=0.899502 freqz: 1700, srate:48000
PRE_WARP: TRANSTO INT: temp_alpha1=-13090 temp_beta1=14737 freqz: 1700, srate:48000
NORMAL: alpha1=-0.799751 beta1=0.899876 freqz: 1700, srate:48000
NORMAL TRANSTO INT: a0=-13103 a1=14743 freqz: 1700, srate:48000


PRE_WARP: temp_alpha1=-0.788336 temp_beta1=0.894168 freqz: 1800, srate:48000
PRE_WARP: TRANSTO INT: temp_alpha1=-12916 temp_beta1=14650 freqz: 1800, srate:48000
NORMAL: alpha1=-0.789213 beta1=0.894607 freqz: 1800, srate:48000
NORMAL TRANSTO INT: a0=-12930 a1=14657 freqz: 1800, srate:48000



PRE_WARP: temp_alpha1=-0.777778 temp_beta1=0.888889 freqz: 1900, srate:48000
PRE_WARP: TRANSTO INT: temp_alpha1=-12743 temp_beta1=14563 freqz: 1900, srate:48000
NORMAL: alpha1=-0.778798 beta1=0.889399 freqz: 1900, srate:48000
NORMAL TRANSTO INT: a0=-12759 a1=14571 freqz: 1900, srate:48000


PRE_WARP: temp_alpha1=-0.767327 temp_beta1=0.883663 freqz: 2000, srate:48000
PRE_WARP: TRANSTO INT: temp_alpha1=-12571 temp_beta1=14477 freqz: 2000, srate:48000
NORMAL: alpha1=-0.768503 beta1=0.884252 freqz: 2000, srate:48000
NORMAL TRANSTO INT: a0=-12591 a1=14487 freqz: 2000, srate:48000

浮点转定点

低端的数字信号处理器大都是定点处理器，所以真要这个滤波器活蹦乱掉的跑起来，还得做一个浮点转定点的工作。听着很高大上，其实很简单，就是系数转换为Qxx的整形格式（以2的整数次幂放大），然后一顿加乘计算后，再等量缩减。细心的读者可以发现脚本例子中“TRANSTO INT”一行的系数算出了一个巨大的整数值，这个值就是浮点转定点的系数。实在觉得费尽，可以找一些现成的代码，下面的代码就是一个Q15的转换函数。

int To_Q15_Fixed_Point(double Data_Float)
{
    unsigned int  Data_Q15_Format = 0;

    if(Data_Float == 1) return (unsigned short)0x7FFF;
    else if(Data_Float == -1) return (unsigned short)0x8000;
    else if(Data_Float < 0)
    {
        Data_Float = -Data_Float;
        Data_Q15_Format |= (unsigned short)0x8000;
    }

    Data_Q15_Format |= (short)(Data_Float*0x8000);

    if(Data_Q15_Format&0x8000)
    {
        Data_Q15_Format &= 0x7FFF;
        Data_Q15_Format = ((~Data_Q15_Format) + 0x0001);
    }

    return (int)Data_Q15_Format;
}

最后

有了系数和差分方程，写函数这件事就是最简单的了。在上一篇的基础上，我们彻底解放了这个一阶iir的高通滤波器，使用的时候需要评估采用哪种系数计算方法更满足要求。

最后

以上就是年轻乐曲为你收集整理的通用的一阶IIR数字高通滤波器的实现书接上回开始推算浮点转定点最后的全部内容，希望文章能够帮你解决通用的一阶IIR数字高通滤波器的实现书接上回开始推算浮点转定点最后所遇到的程序开发问题。

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