Dynamic range compression （RMS Limit）

Overview

• 背景 ： 人类的听感动态范围[能承受的最大响度和能感受的最安静声音响度的范围可达100万:1（即106倍）] 达120dB。扩声系统声音重放的动态范围由于受电子设备的限制，远比人耳的动态范围小很多。最低声音的响度受系统中不相关噪声的限制，使小的声音信号淹没在噪声中而无法听到；最大声音的响度受信号削波失真，而且信号削波后产生大量谐波，损坏扬声器单元（尤其是高音扬声器）。为此，在控制信号动态范围中，音量动态平衡器得到了广泛的应用。简单地说，它是一种限制音频信号动态范围的电子装置，把安静的小信号变得更响，把高幅度的尖峰信号变得更小些，并且不产生信号削波，保护扬声器和功放免受冲击和损坏。
• 动态范围压缩特点：(1) 限制了音乐信号中极大的峰值信号，保护扬声器系统和功放系统免受损坏；(2) 可获得更大的声音增益，因为压低了信号峰值，可使节目信号中的其他幅度较小的信号得到充分的提升；(3) 使音频中的小信号不会落到调音台的噪声中去，提高了节目信号的信噪比；(4) 如果这些装置中的四个基本参数（门槛电平、压缩比、动作时间和释放时间） 调整得不适当时，会出现“泵浦声”、“呼吸声”或根本没法对信号的动态范围进行控制。

Limiting

• 不同的参数压缩的理论效果图如下
  
• 实际调试的效果图如下
  
  

Algorithm

• 一阶滞后滤波算法公式：S(n) = αS(n-1) + (1 - α)G；

• 压缩时间根据α的大小变化，α最小可以设置成1，这样压缩时间为0；

• 原理图示如下：

• 

• Root Mean Square $RMS=\sqrt[2]{\frac{1}{n}{x}_1^2+x_2^2+...x_n^2}$
  This caculation is used to get a mean value over a buffer of 512 frames. A frame is
  defined as one float per channel of sound. That can later be used to get a smoother
  compression.

• Gain $X_{dB}=20\cdot {\log }_{10}(RMS)$
  Using the RMS result, it can be converted to the logarithmic plane,it have soft knee.
  $$X_{sc}=\{\begin{array}{rcl}{X}_{dB}& & X_{dB}<(T-\frac{W}{2})\\ X_{dB}+\frac{(\frac{1}{R}-1)(X_{dB}-T+\frac{W}{2}{)}^2}{2W}& & (T-\frac{W}{2})\le X_{dB}<(T+\frac{W}{2})\\ T+\frac{X_{dB}-T}{R}& & X_{dB}>(T+\frac{W}{2})\end{array}$$
  Where $T$ = Threshold, $R$ = Ratio and $W$ = KneeWidth. This is the main part of
  the equations, where the gain is computed according to the set values of $T$,$R$ and
  W.
  $$gc=X_{sc}+X_{dB}$$The computed gain and the original signal is added to each other.

• Smoothing Gain
  $$g_s=\{\begin{array}{rcl}{\alpha }_A\cdot g_s[n-1]+(1-{\alpha }_A)\cdot g_c& & g_c[n]<g_s[n-1]\\ {\alpha }_R\cdot g_s[n-1]+(1-{\alpha }_R)\cdot g_c& & g_c[n]\ge g_s[n-1]\end{array}$$
  $${\alpha }_A=exp(\frac{-log(9)}{F_s\cdot T_A})$$
  $${\alpha }_R=exp(\frac{-log(9)}{F_s\cdot T_R})$$
  Where $T_A$ = AttackTime Period, $T_R$ = ReleaseTime Period and $F_s$ = Sample
  Frequency. The smoothing function is there to give a natural transition from no compression to the calculated compression. Mathworks (2017)

Program

• 程序中算法使用不含soft knee，其变形公式如下
  $$X_{sc}=\{\begin{array}{rcl}{X}_{dB}& & X_{dB}<(T-\frac{W}{2})\\ T+\frac{X_{dB}-T}{R}& & X_{dB}\ge (T+\frac{W}{2})\end{array}$$

#if 0
void vRMSLimExec(float* input, LIM_Type* pLim)
{
    int i;
    volatile float aA = 0.0;
    for(i = 0; i < MG_DEF_BLK_SZ; i++)
    {
        if(pLim->Pro.fGc <= pLim->Pro.fGs)
        {
            aA = pLim->Set.aA;    /* compress time*/
            pLim->Pro.Cnt = 0;
        }
        else
        {
            aA = 1.0;    /*hold time*/
            pLim->Pro.Cnt++;
            if(pLim->Pro.Cnt >= pLim->Pro.HdCnt)
            {
                pLim->Pro.Cnt = pLim->Pro.HdCnt;
                aA = pLim->Set.aR;    /*release time*/
            }
        }
        pLim->Pro.fGs = aA * pLim->Pro.fGs + (1 - aA) * pLim->Pro.fGc;    /* one order lag algrithm*/
        input[i] = pLim->Pro.fGs * input[i];
        input[i] = pLim->Pro.fGm * input[i];
    }
    *pLim->reduc = (1.0 / pLim->Pro.fGs);
    *pLim->reduc = ((*pLim->reduc) > 1.0) ? (*pLim->reduc) : 1.0;
}
#endif
#if 1
void vRMSLimExec(float* input, LIM_Type* pLim)
{
    fGc = pLim->Pro.fGc;
    fGm = pLim->Pro.fGm;
    aA = pLim->Set.aA;
    aR = pLim->Set.aR;
    HoldTime = pLim->Pro.HdCnt;
    HoldCnt = &pLim->Pro.Cnt;
    RmsLimExec(input, &pLim->Pro.fGs);
    *pLim->reduc = (1.0 / pLim->Pro.fGs);
    *pLim->reduc = ((*pLim->reduc) > 1.0) ? (*pLim->reduc) : 1.0;
}
#endif

/**
  * @brief  receive RMS Limiter param
  * @param
  * @retval None
  */
void cmp_RMSLim_param(LIM_Type* pLim, int ChType)
{
    ENV_RMS_LIMITER_T* pCfgLim = (ENV_RMS_LIMITER_T*)(pLim->CfgLim);

    SetMaxMinValue_Float(&pCfgLim->fRatio, 1.2, 24.9);
    SetMaxMinValue_Float(&pCfgLim->fGain, -12, 12);
    SetMaxMinValue_Int(&pCfgLim->nAttackTime, 1, 10000);
    SetMaxMinValue_Int(&pCfgLim->nReleaseTime, 1, 10000);
    SetMaxMinValue_Int(&pCfgLim->nHoldTime, 1, 10000);
    if(ChType == OUTPUT_CHANNEL)
    {
        pLim->HwGain = fOutGainHW;
        SetMaxMinValue_Float(&pCfgLim->fThreshold, 12, 42);
    }
    if(ChType == INPUT_CHANNEL)
    {
        pLim->HwGain = fInGainHW;
        SetMaxMinValue_Float(&pCfgLim->fThreshold, -24, 24);
    }
    pLim->Set.fVth = pCfgLim->fThreshold + pLim->HwGain;
    pLim->Set.aA = (Factor / (pCfgLim->nAttackTime * pLim->Set.Fs));
    pLim->Set.aA = exp(pLim->Set.aA);
    pLim->Set.aR = (Factor / (pCfgLim->nReleaseTime * pLim->Set.Fs));
    pLim->Set.aR = exp(pLim->Set.aR);
    pLim->Set.fW = 2;
    pLim->Set.fRatio = pCfgLim->fRatio;
    pLim->Set.fM = pCfgLim->fGain;
    pLim->Pro.HdCnt = pCfgLim->nHoldTime * pLim->Set.Fs;
}

/**
  * @brief  get RMS limit phase status
  * @param  length
  * @retval None
  */
void gPhaseStatus(LIM_Type* pLim, float* pRMS)
{
    pLim->Pro.fInRMSdBu = 20 * log10(*pRMS) + dBuVrefCoef;
    pLim->Pro.fInRMSdBu = (pLim->Pro.fInRMSdBu > (-140)) ? pLim->Pro.fInRMSdBu : (-140);
    //pLim->Pro.fSCdBu = pLim->Pro.fInRMSdBu;
    #if 1
    /*no soft knee*/
    if(pLim->Pro.fInRMSdBu <= (pLim->Set.fVth))
    {
        pLim->Pro.fSCdBu = pLim->Pro.fInRMSdBu;
    }
    else
    {
        pLim->Pro.fSCdBu = ((pLim->Pro.fInRMSdBu - pLim->Set.fVth) / pLim->Set.fRatio);
        pLim->Pro.fSCdBu = pLim->Set.fVth + pLim->Pro.fSCdBu;
    }
    #endif

    #if 0
    /*have soft knee*/
    if(pLim->Pro.fInRMSdBu < (pLim->Set.fVth - (pLim->Set.fW / 2)))
    {
        pLim->Pro.fSCdBu = pLim->Pro.fInRMSdBu;
    }
    if((pLim->Pro.fInRMSdBu >= (pLim->Set.fVth - (pLim->Set.fW / 2))) &&
        (pLim->Pro.fInRMSdBu <= (pLim->Set.fVth + (pLim->Set.fW / 2))))
    {
        pLim->Pro.fSCdBu = (pLim->Pro.fInRMSdBu - pLim->Set.fVth + pLim->Set.fW / 2);
        pLim->Pro.fSCdBu = pLim->Pro.fSCdBu * pLim->Pro.fSCdBu * ((1 / pLim->Set.fRatio) - 1) / (2 * pLim->Set.fW) + pLim->Pro.fInRMSdBu;
    }
    if(pLim->Pro.fInRMSdBu > (pLim->Set.fVth + (pLim->Set.fW / 2)))
    {
        pLim->Pro.fSCdBu = pLim->Set.fVth + ((pLim->Pro.fInRMSdBu - pLim->Set.fVth) / pLim->Set.fRatio);
    }
    #endif
    pLim->Pro.fGc = pLim->Pro.fSCdBu - pLim->Pro.fInRMSdBu;
    pLim->Pro.fGc = gain_computer(pLim->Pro.fGc, 20);    /* dBu convert to physical unit*/
    pLim->Pro.fGm = gain_computer(pLim->Set.fM, 20);
}