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]

with Ada.Text_IO;
 
procedure Apply_Filter is
 
   generic
      type Num is digits <>;
      type Num_Array is array (Natural range <>) of Num;
      A : Num_Array;
      B : Num_Array;
   package Direct_Form_II_Transposed is
      pragma Assert (A'First = 0 and B'First = 0);
      pragma Assert (A'Last = B'Last);
      pragma Assert (A (0) = 1.000);
 
      function Filter (X : in Num) return Num;
   end Direct_Form_II_Transposed;
 
   package body Direct_Form_II_Transposed
   is
      W : Num_Array (A'Range) := (others => 0.0);
 
      function Filter (X : in Num) return Num is
         Y : constant Num := X * B (0) + W (0);
      begin
         --  Calculate delay line for next sample
         for I in 1 .. W'Last loop
            W (I - 1) := X * B (I) - Y * A (I) + W (I);
         end loop;
         return Y;
      end Filter;
 
   end Direct_Form_II_Transposed;
 
   type Coeff_Array is array (Natural range <>) of Float;
 
   package Butterworth is
      new Direct_Form_II_Transposed (Float, Coeff_Array,
                                     A => (1.000000000000, -2.77555756e-16,
                                           3.33333333e-01, -1.85037171e-17),
                                     B => (0.16666667, 0.50000000,
                                           0.50000000, 0.16666667));
 
   subtype Signal_Array is Coeff_Array;
 
   X_Signal : constant Signal_Array :=
     (-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);
 
   package Float_IO is
      new Ada.Text_IO.Float_IO (Float);
   Y : Float;
begin
   for Sample of X_Signal loop
      Y := Butterworth.Filter (Sample);
      Float_IO.Put (Y, Exp => 0, Aft => 6);
      Ada.Text_IO.New_Line;
   end loop;
end Apply_Filter;