See more about filtering here:

See filtering summary:LINK

Low pass filter design AN2874- application notes:LINK

Audio-EQ Coockbock:LINK

Second order recursive linear filter, containing two poles and two zeros, so called "biQuad" form LINK

main.cpp

            #include "signal_gen.h"
            #include < cmath >
            #define PI 3.14159265358979323846

            struct Coeff
            {
                    double a1,a2;
                    double b0,b1,b2;        
                  
            };
          

            // second order butterworth filter 
            Coeff LPF_butter(double fc, double fs)
            {  Coeff LPF; 
            
                const double ita =1.0/ tan(M_PI*fc/fs);
                const double q=sqrt(2.0);
                
                LPF.b0 =  1.0 / (1.0 + q*ita + ita*ita);
                LPF.b1 =  2*LPF.b0;
                LPF.b2 =  LPF.b0;
            
                LPF.a1 = -2.0 * (ita*ita - 1.0) * LPF.b0;
                LPF.a2 = (1.0 - q*ita + ita*ita) * LPF.b0;           
            
               return LPF; 
            }
          

            struct DEQ
            {
              
                // set up the x[n-1], x[n-2] ... 
                double x1,x2;
                double y0,y1, y2;

                double b0,b1,b2;
                double a1,a2; 

                double process(const Coeff& fltCoeff,  double x0)
                {
                    // read in the coefficients 
                    b0 = fltCoeff.b0;
                    b1 = fltCoeff.b1;
                    b2 = fltCoeff.b2;
                        
                    a1 = fltCoeff.a1;
                    a2 = fltCoeff.a2;
                
                    y0 = b0*x0 + b1*x1 + b2*x2 - a1*y1 -a2*y2;  
                    x2 = x1;   
                    x1 = x0;    
                    y2 = y1;  
                    y1 = y0;              
                    return y0;
                }
                
            };
          

              
              int main()
              {
                  
                  std::cout << "|||||||||||||||||||||||||||||||||||||||||||||||||||||| FILTER prototype 2. \n" ; 

                        
                      double fs = 10e3;   // 10kHz sampling frequency 
                      DEQ sys;            // difference equations 
                      double fc = 20;     // cutoff frequency ... 

                      Coeff lpf = LPF_butter(fc,fs);

                      cout << "\n  ";
                      cout << lpf.b0 << endl << lpf.b1 << endl << lpf.b2 << endl << lpf.a1 << endl << lpf.a2 << endl;   


                      std::cout << "|||||||||||||||||||||||||||||||||||||||||||||||||||||| Generate signal Pt2. \n" ; 
                      
                      //|*************************************************************************************************
                      //| SIGNAL GENERATION 
                      //| fs, f, Amp, phi, t_max
                      signal_gen signal; signal.sinus(fs,5,5,0,1);  
                      signal_gen overTone; overTone.sinus(fs,70,2,0,1);

                      vector x = signal.getSignal();
                      vector x1 = overTone.getSignal();
                      
                      vector signal_sum;      
                      for (size_t i = 0; i < x.size(); i++)  { signal_sum.push_back(x[i] + x1[i] ); }


                      std::cout << "|||||||||||||||||||||||||||||||||||||||||||||||||||||| Filter and Write signals . \n" ; 
                      //| filter output ... 
                      //|
                      vector yout;
                      for(size_t i = 0; i < signal_sum.size(); i++) {yout.push_back(sys.process(lpf, signal_sum[i]));}

          

Filtering process outcome

Sine + Overtone (green) --> IIR --> Sine (black)

Sine + Overtone + Random (first) --->|IIR|---> Sine (second fig.)

Original v.s. Cleaned signal(third on fig.)