user
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]

See also

[Wikipedia on Butterworth filters]
assistant
(formerly Perl 6)

**Works with**: Rakudo version 2016.11

**Translation of**: zkl

```
sub TDF-II-filter ( @signal, @a, @b ) {
    my @out = 0 xx @signal;
    for ^@signal -> $i {
        my $this;
        $this += @b[$_] * @signal[$i-$_] if $i-$_ >= 0 for ^@b;
        $this -= @a[$_] *    @out[$i-$_] if $i-$_ >= 0 for ^@a;
        @out[$i] = $this / @a[0];
    }
    @out
}

my @signal = [
    -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
];
my @a = [ 1.00000000, -2.77555756e-16, 3.33333333e-01, -1.85037171e-17 ];
my @b = [ 0.16666667,  0.5,            0.5,             0.16666667     ];

say TDF-II-filter(@signal, @a, @b)».fmt("% 0.8f")
    Z~ flat (', ' xx 4, ",\n") xx *;

```

Output:

```
(-0.15297399,  -0.43525783,  -0.13604340,   0.69750333,   0.65644469,
 -0.43548245,  -1.08923946,  -0.53767655,   0.51704999,   1.05224975,
  0.96185430,   0.69569009,   0.42435630,   0.19626223,  -0.02783512,
 -0.21172192,  -0.17474556,   0.06925841,   0.38544587,   0.65177084,
)
```