Digital filters are used to apply a mathematical operation to a sampled signal. One of the common formulations is the "direct form II transposed" which can represent both infinite impulse response (IIR) and finite impulse response (FIR) filters, as well as being more numerically stable than other forms. [1]

Task

Filter a signal using an order 3 low-pass Butterworth filter. The coefficients for the filter are a=[1.00000000, -2.77555756e-16, 3.33333333e-01, -1.85037171e-17] and b = [0.16666667, 0.5, 0.5, 0.16666667]

The signal that needs filtering is the following vector: [-0.917843918645, 0.141984778794, 1.20536903482, 0.190286794412, -0.662370894973, -1.00700480494, -0.404707073677 ,0.800482325044, 0.743500089861, 1.01090520172, 0.741527555207, 0.277841675195, 0.400833448236, -0.2085993586, -0.172842103641, -0.134316096293, 0.0259303398477, 0.490105989562, 0.549391221511, 0.9047198589]

fcn direct_form_II_transposed_filter(b,a,signal){
   out:=List.createLong(signal.len(),0.0);  // vector of zeros
   foreach i in (signal.len()){
      tmp:=0.0;
      foreach j in (b.len()){ if(i-j >=0) tmp += b[j]*signal[i-j] }
      foreach j in (a.len()){ if(i-j >=0) tmp -= a[j]*out[i-j]    }
      out[i] = tmp/a[0];
   }
   out
}