# Apply a digital filter (direct form II transposed)

Apply a digital filter (direct form II transposed)
You are encouraged to solve this task according to the task description, using any language you may know.

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]

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]

## 11l

Translation of: Nim
```F apply_filter(a, b, signal)
V result = [0.0] * signal.len
L(i) 0 .< signal.len
V tmp = 0.0
L(j) 0 .< min(i + 1, b.len)
tmp += b[j] * signal[i - j]
L(j) 1 .< min(i + 1, a.len)
tmp -= a[j] * result[i - j]
tmp /= a[0]
result[i] = tmp
R result

V a = [1.00000000, -2.77555756e-16, 3.33333333e-01, -1.85037171e-17]
V b = [0.16666667, 0.5, 0.5, 0.16666667]

V 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]

V result = apply_filter(a, b, signal)
L(r) result
print(‘#2.8’.format(r), end' ‘’)
print(I (L.index + 1) % 5 != 0 {‘, ’} E "\n", end' ‘’)```
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
```

```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
Y : Float;
begin
for Sample of X_Signal loop
Y := Butterworth.Filter (Sample);
Float_IO.Put (Y, Exp => 0, Aft => 6);
end loop;
end Apply_Filter;
```

## ALGOL 68

Translation of: C++
... via Yabasic

... with the "j" loops transformed to not needlessly iterate beyond i.
The default lower bound in Algol 68 arrays is 1, so the loops/subscripts have been adjusted accordingly.

```BEGIN # apply a digital filter #
# the lower bounds of a, b, signal and result must all be equal #
PROC filter = ( []REAL a, b, signal, REF[]REAL result )VOID:
IF LWB a /= LWB b OR LWB a /= LWB signal OR LWB a /= LWB result THEN
print( ( "Array lower bounds must be equal for filter", newline ) );
stop
ELSE
FOR i FROM LWB result TO UPB result DO result[ i ] := 0 OD;
FOR i FROM LWB signal TO UPB signal DO
REAL tmp := 0;
FOR j FROM LWB b TO IF i > UPB b THEN UPB b ELSE i FI DO
tmp +:= b[ j ] * signal[ LWB signal + ( i - j ) ]
OD;
FOR j FROM LWB a + 1 TO IF i > UPB a THEN UPB a ELSE i FI  DO
tmp -:= a[ j ] * result[ LWB result + ( i - j ) ]
OD;
result[ i ] := tmp / a[ LWB a ]
OD
FI # filter # ;
[  4 ]REAL a := []REAL( 1, -2.77555756e-16, 3.33333333e-01, -1.85037171e-17 );
[  4 ]REAL b := []REAL( 0.16666667, 0.5, 0.5, 0.16666667 );
[ 20 ]REAL signal
:= []REAL( -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
);
[ 20 ]REAL result;
filter( a, b, signal, result );
FOR i FROM LWB result TO UPB result DO
print( ( " ", fixed( result[ i ], -9, 6 ) ) );
IF i MOD 5 /= 0 THEN print( ( ", " ) ) ELSE print( ( newline ) ) FI
OD
END```
Output:
``` -0.152974,  -0.435258,  -0.136043,   0.697503,   0.656445
-0.435482,  -1.089239,  -0.537677,   0.517050,   1.052250
0.961854,   0.695690,   0.424356,   0.196262,  -0.027835
-0.211722,  -0.174746,   0.069258,   0.385446,   0.651771
```

## AppleScript

Translation of: Julia
— except that j starts from 2 in the second inner repeat, there being no point in fetching and performing math with the zero about to be overwritten. This change in turn allows the result list to be populated on the fly instead of being pre-populated with zeros.
```on min(a, b)
if (b < a) then return b
return a
end min

on DF2TFilter(a, b, sig)
set aCount to (count a)
set bCount to (count b)
set sigCount to (count sig)
set rst to {}

repeat with i from 1 to sigCount
set tmp to 0
set iPlus1 to i + 1
repeat with j from 1 to min(i, bCount)
set tmp to tmp + (item j of b) * (item (iPlus1 - j) of sig)
end repeat
repeat with j from 2 to min(i, aCount)
set tmp to tmp - (item j of a) * (item (iPlus1 - j) of rst)
end repeat
set end of rst to tmp / (beginning of a)
end repeat

return rst
end DF2TFilter

local acoef, bcoef, signal
set acoef to {1.0, -2.77555756E-16, 0.333333333, -1.85037171E-17}
set bcoef to {0.16666667, 0.5, 0.5, 0.16666667}
set signal to {-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, ¬
0.025930339848, 0.490105989562, 0.549391221511, 0.9047198589}
DF2TFilter(acoef, bcoef, signal)
```
Output:
```{-0.1529739895, -0.43525782905, -0.136043396988, 0.697503326548, 0.656444692469, -0.435482453256, -1.089239461153, -0.537676549563, 0.517049992313, 1.052249747155, 0.961854300374, 0.69569009401, 0.424356295096, 0.196262231822, -0.027835124463, -0.21172191545, -0.174745562223, 0.069258408901, 0.385445874308, 0.651770838819}
```

## C

Given the number of values a coefficient or signal vector can have and the number of digits, this implementation reads data from a file and prints it to the console if no output file is specified or writes to the specified output file. Usage printed on incorrect invocation.

```#include<stdlib.h>
#include<string.h>
#include<stdio.h>

#define MAX_LEN 1000

typedef struct{
float* values;
int size;
}vector;

vector extractVector(char* str){
vector coeff;
int i=0,count = 1;
char* token;

while(str[i]!=00){
if(str[i++]==' ')
count++;
}

coeff.values = (float*)malloc(count*sizeof(float));
coeff.size = count;

token = strtok(str," ");

i = 0;

while(token!=NULL){
coeff.values[i++] = atof(token);
token = strtok(NULL," ");
}

return coeff;
}

vector processSignalFile(char* fileName){
int i,j;
float sum;
char str[MAX_LEN];
vector coeff1,coeff2,signal,filteredSignal;

FILE* fp = fopen(fileName,"r");

fgets(str,MAX_LEN,fp);
coeff1 = extractVector(str);

fgets(str,MAX_LEN,fp);
coeff2 = extractVector(str);

fgets(str,MAX_LEN,fp);
signal = extractVector(str);

fclose(fp);

filteredSignal.values = (float*)calloc(signal.size,sizeof(float));
filteredSignal.size = signal.size;

for(i=0;i<signal.size;i++){
sum = 0;

for(j=0;j<coeff2.size;j++){
if(i-j>=0)
sum += coeff2.values[j]*signal.values[i-j];
}

for(j=0;j<coeff1.size;j++){
if(i-j>=0)
sum -= coeff1.values[j]*filteredSignal.values[i-j];
}

sum /= coeff1.values[0];
filteredSignal.values[i] = sum;
}

return filteredSignal;
}

void printVector(vector v, char* outputFile){
int i;

if(outputFile==NULL){
printf("[");
for(i=0;i<v.size;i++)
printf("%.12f, ",v.values[i]);
printf("\b\b]");
}

else{
FILE* fp = fopen(outputFile,"w");
for(i=0;i<v.size-1;i++)
fprintf(fp,"%.12f, ",v.values[i]);
fprintf(fp,"%.12f",v.values[i]);
fclose(fp);
}

}

int main(int argC,char* argV[])
{
char *str;
if(argC<2||argC>3)
printf("Usage : %s <name of signal data file and optional output file.>",argV[0]);
else{
if(argC!=2){
str = (char*)malloc((strlen(argV[2]) + strlen(str) + 1)*sizeof(char));
strcpy(str,"written to ");
}
printf("Filtered signal %s",(argC==2)?"is:\n":strcat(str,argV[2]));
printVector(processSignalFile(argV[1]),argV[2]);
}
return 0;
}
```

Input file, 3 lines containing first ( a ) and second ( b ) coefficient followed by the signal, all values should be separated by a single space:

```1.00000000 -2.77555756e-16 3.33333333e-01 -1.85037171e-17
0.16666667 0.5 0.5 0.16666667
-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
```

Invocation and output for writing to file :

```C:\rosettaCode>filterSignal.exe signalData.txt signalOut1.txt
Filtered signal written to signalOut1.txt
```

Output file :

```-0.152973994613, -0.435257852077, -0.136043429375, 0.697503268719, 0.656444668770, -0.435482472181, -1.089239478111, -0.537676513195, 0.517050027847, 1.052249789238, 0.961854279041, 0.695690035820, 0.424356281757, 0.196262255311, -0.027835110202, -0.211721926928, -0.174745559692, 0.069258414209, 0.385445863008, 0.651770770550
```

## C#

Translation of: Java
```using System;

namespace ApplyDigitalFilter {
class Program {
private static double[] Filter(double[] a, double[] b, double[] signal) {
double[] result = new double[signal.Length];
for (int i = 0; i < signal.Length; ++i) {
double tmp = 0.0;
for (int j = 0; j < b.Length; ++j) {
if (i - j < 0) continue;
tmp += b[j] * signal[i - j];
}
for (int j = 1; j < a.Length; ++j) {
if (i - j < 0) continue;
tmp -= a[j] * result[i - j];
}
tmp /= a[0];
result[i] = tmp;
}
return result;
}

static void Main(string[] args) {
double[] a = new double[] { 1.00000000, -2.77555756e-16, 3.33333333e-01, -1.85037171e-17 };
double[] b = new double[] { 0.16666667, 0.5, 0.5, 0.16666667 };

double[] signal = new double[] {
-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
};

double[] result = Filter(a, b, signal);
for (int i = 0; i < result.Length; ++i) {
Console.Write("{0,11:F8}", result[i]);
Console.Write((i + 1) % 5 != 0 ? ", " : "\n");
}
}
}
}
```
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```

## C++

This uses the C++11 method of initializing vectors. In g++, use the -std=c++0x compiler switch.

```#include <vector>
#include <iostream>
using namespace std;

void Filter(const vector<float> &b, const vector<float> &a, const vector<float> &in, vector<float> &out)
{

out.resize(0);
out.resize(in.size());

for(int i=0; i < in.size(); i++)
{
float tmp = 0.;
int j=0;
out[i] = 0.f;
for(j=0; j < b.size(); j++)
{
if(i - j < 0) continue;
tmp += b[j] * in[i-j];
}

for(j=1; j < a.size(); j++)
{
if(i - j < 0) continue;
tmp -= a[j]*out[i-j];
}

tmp /= a[0];
out[i] = tmp;
}
}

int main()
{
vector<float> sig = {-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};

//Constants for a Butterworth filter (order 3, low pass)
vector<float> a = {1.00000000, -2.77555756e-16, 3.33333333e-01, -1.85037171e-17};
vector<float> b = {0.16666667, 0.5, 0.5, 0.16666667};

vector<float> result;
Filter(b, a, sig, result);

for(size_t i=0;i<result.size();i++)
cout << result[i] << ",";
cout << endl;

return 0;
}
```
Output:
`-0.152974,-0.435258,-0.136043,0.697503,0.656445,-0.435483,-1.08924,-0.537677,0.51705,1.05225,0.961854,0.69569,0.424356,0.196262,-0.0278351,-0.211722,-0.174746,0.0692584,0.385446,0.651771,`

## Common Lisp

Translation of: zkl
```(defparameter a #(1.00000000L0 -2.77555756L-16 3.33333333L-01 -1.85037171L-17))
(defparameter b #(0.16666667L0 0.50000000L0 0.50000000L0 0.16666667L0))
(defparameter s #(-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))

(loop with out = (make-array (length s) :initial-element 0.0D0)
for i below (length s)
do (setf (svref out i)
(/ (- (loop for j below (length b)
when (>= i j) sum (* (svref b j) (svref s (- i j))))
(loop for j below (length a)
when (>= i j) sum (* (svref a j) (svref out (- i j)))))
(svref a 0)))
(format t "~%~16,8F" (svref out i)))
```
Output:
```     -0.15297399
-0.43525784
-0.13604341
0.69750331
0.65644468
-0.43548247
-1.08923949
-0.53767657
0.51705000
1.05224976
0.96185428
0.69569007
0.42435630
0.19626225
-0.02783512
-0.21172192
-0.17474557
0.06925841
0.38544587
0.65177083
```

## D

Translation of: Kotlin
```import std.stdio;

alias T = real;
alias AT = T[];

AT filter(const AT a, const AT b, const AT signal) {
AT result = new T[signal.length];

foreach (int i; 0..signal.length) {
T tmp = 0.0;
foreach (int j; 0..b.length) {
if (i-j<0) continue;
tmp += b[j] * signal[i-j];
}
foreach (int j; 1..a.length) {
if (i-j<0) continue;
tmp -= a[j] * result[i-j];
}
tmp /= a[0];
result[i] = tmp;
}

return result;
}

void main() {
AT a = [1.00000000, -2.77555756e-16, 3.33333333e-01, -1.85037171e-17];
AT b = [0.16666667, 0.5, 0.5, 0.16666667];

AT 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
];

AT result = filter(a,b,signal);
foreach (i; 0..result.length) {
writef("% .8f", result[i]);
if ((i+1)%5 != 0) {
write(", ");
} else {
writeln;
}
}
}
```
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,  0.06925840,  0.38544587,  0.65177084```

## FreeBASIC

Translation of: Yabasic
```Sub Filtro(a() As Double, b() As Double, senal() As Double, resultado() As Double)
Dim As Integer j, k
Dim As Double tmp

For j = 0 To Ubound(senal)
tmp = 0
For k = 0 To Ubound(b)
If (j-k < 0) Then Continue For
tmp = tmp + b(k) * senal(j-k)
Next k
For k = 0 To Ubound(a)
If (j-k < 0) Then Continue For
tmp = tmp - a(k) * resultado(j-k)
Next k
tmp /= a(0)
Next j
End Sub

Dim Shared As Double a(4), b(4), senal(20), resultado(20)
Dim As Integer i
For i = 0 To 3 : Read a(i) : Next i
For i = 0 To 3 : Read b(i) : Next i
For i = 0 To 19 : Read senal(i) : Next i

For i = 0 To 19
If (i+1) Mod 5 <> 0 Then
Print ", ";
Else
Print
End If
Next i

'' a()
Data 1, -2.77555756e-16, 3.33333333e-01, -1.85037171e-17
'' b()
Data 0.16666667, 0.5, 0.5, 0.16666667
'' senal()
Data -0.917843918645, 0.141984778794, 1.20536903482, 0.190286794412
Data -0.662370894973, -1.00700480494, -0.404707073677, 0.800482325044
Data 0.743500089861, 1.01090520172, 0.741527555207, 0.277841675195
Data 0.400833448236, -0.2085993586, -0.172842103641, -0.134316096293
Data 0.0259303398477, 0.490105989562, 0.549391221511, 0.9047198589

Sleep
```
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,  0.06925840,  0.38544587,  0.65177084
```

## Go

```package main

import "fmt"

type filter struct {
b, a []float64
}

func (f filter) filter(in []float64) []float64 {
out := make([]float64, len(in))
s := 1. / f.a[0]
for i := range in {
tmp := 0.
b := f.b
if i+1 < len(b) {
b = b[:i+1]
}
for j, bj := range b {
tmp += bj * in[i-j]
}
a := f.a[1:]
if i < len(a) {
a = a[:i]
}
for j, aj := range a {
tmp -= aj * out[i-j-1]
}
out[i] = tmp * s
}
return out
}

//Constants for a Butterworth filter (order 3, low pass)
var bwf = filter{
a: []float64{1.00000000, -2.77555756e-16, 3.33333333e-01, -1.85037171e-17},
b: []float64{0.16666667, 0.5, 0.5, 0.16666667},
}

var sig = []float64{
-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,
}

func main() {
for _, v := range bwf.filter(sig) {
fmt.Printf("%9.6f\n", v)
}
}
```
Output:
```-0.152974
-0.435258
-0.136043
0.697503
0.656445
-0.435482
-1.089239
-0.537677
0.517050
1.052250
0.961854
0.695690
0.424356
0.196262
-0.027835
-0.211722
-0.174746
0.069258
0.385446
0.651771
```

## Groovy

Translation of: Java
```class DigitalFilter {
private static double[] filter(double[] a, double[] b, double[] signal) {
double[] result = new double[signal.length]
for (int i = 0; i < signal.length; ++i) {
double tmp = 0.0
for (int j = 0; j < b.length; ++j) {
if (i - j < 0) continue
tmp += b[j] * signal[i - j]
}
for (int j = 1; j < a.length; ++j) {
if (i - j < 0) continue
tmp -= a[j] * result[i - j]
}
tmp /= a[0]
result[i] = tmp
}
return result
}

static void main(String[] args) {
double[] a = [1.00000000, -2.77555756e-16, 3.33333333e-01, -1.85037171e-17] as double[]
double[] b = [0.16666667, 0.5, 0.5, 0.16666667] as double[]
double[] 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
] as double[]

double[] result = filter(a, b, signal)
for (int i = 0; i < result.length; ++i) {
printf("% .8f", result[i])
print((i + 1) % 5 != 0 ? ", " : "\n")
}
}
}
```
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```

The solution is based not on the explicit loops, as in strict imperative languages, but on lazy recursive trick known as "tying a knot".

```import Data.List (tails)

-- lazy convolution of a list by given kernel
conv :: Num a => [a] -> [a] -> [a]
conv ker = map (dot (reverse ker)) . tails  . pad
where
pad v = replicate (length ker - 1) 0 ++ v
dot v = sum . zipWith (*) v

-- The lazy digital filter
dFilter :: [Double] -> [Double] -> [Double] -> [Double]
dFilter (a0:a) b s = tail res
where
res = (/ a0) <\$> 0 : zipWith (-) (conv b s) (conv a res)
```

### Examples

Demonstration of convolution:

```λ> take 10 \$ conv [1,10,100,1000] [1..]
[1,12,123,1234,2345,3456,4567,5678,6789,7900]
```

```λ> let a = [1, -2.77555756e-16, 3.33333333e-01, -1.85037171e-17]
λ> let b = [0.16666667, 0.5, 0.5, 0.16666667]
λ> let s = [-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]
λ> dFilter a b s
[-0.15297398950031305,-0.4352578290502175,-0.13604339698849033,0.6975033265479628,
0.6564446924690288,-0.4354824532561055,-1.089239461152929,-0.5376765495627545,
0.517049992313214,1.0522497471553531,0.961854300373645,0.6956900940096052,
0.4243562950955321,0.19626223182178906,-2.7835124463393313e-2,-0.21172191545011776,
-0.17474556222276072,6.925840890119485e-2,0.3854458743074388,0.6517708388193053,
0.6802579154588558,0.326668188810626,-7.596599209379973e-2,-0.10888939616131928]
```

The last line is redundant and appears due to the finiteness of a signal stream. The digital filter is able to handle infinite lists (as streams):

```λ> take 10 \$ dFilter a b \$ cycle [1,-1]
[0.16666667,0.33333333000000004,0.11111111338888897,-0.11111110988888885,-3.703703775925934e-2,3.70370365925926e-2,1.2345679240740749e-2,-1.2345678851851824e-2,-4.1152264094650535e-3,4.115226279835409e-3]```

## J

There's probably a nicer way to do this:

```Butter=: {{
t=. (#n) +/ .*&(|.n)\(}.n*0),y
A=.|.}.m
for_i.}.i.#y do.
t=. t i}~ (i{t) - (i{.t) +/ .* (-i){.A
end.
t%{.m
}}

sig=: ". rplc&('-_') {{)n
-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
}}-.LF

a=: 1.00000000 _2.77555756e_16 3.33333333e_01 _1.85037171e_17
b=: 0.16666667 0.5 0.5 0.16666667

4 5\$ a Butter b sig
_0.152974 _0.435258 _0.136043 0.697503   0.656445
_0.435482  _1.08924 _0.537677  0.51705    1.05225
0.961854   0.69569  0.424356 0.196262 _0.0278351
_0.211722 _0.174746 0.0692584 0.385446   0.651771
```

## Java

Translation of: Kotlin
```public class DigitalFilter {
private static double[] filter(double[] a, double[] b, double[] signal) {
double[] result = new double[signal.length];
for (int i = 0; i < signal.length; ++i) {
double tmp = 0.0;
for (int j = 0; j < b.length; ++j) {
if (i - j < 0) continue;
tmp += b[j] * signal[i - j];
}
for (int j = 1; j < a.length; ++j) {
if (i - j < 0) continue;
tmp -= a[j] * result[i - j];
}
tmp /= a[0];
result[i] = tmp;
}
return result;
}

public static void main(String[] args) {
double[] a = new double[]{1.00000000, -2.77555756e-16, 3.33333333e-01, -1.85037171e-17};
double[] b = new double[]{0.16666667, 0.5, 0.5, 0.16666667};

double[] signal = new double[]{
-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
};

double[] result = filter(a, b, signal);
for (int i = 0; i < result.length; ++i) {
System.out.printf("% .8f", result[i]);
System.out.print((i + 1) % 5 != 0 ? ", " : "\n");
}
}
}
```
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```

## Julia

Translation of: zkl
```function DF2TFilter(a::Vector, b::Vector, sig::Vector)
rst = zeros(sig)
for i in eachindex(sig)
tmp =  sum(b[j] * sig[i-j+1] for j in 1:min(i, length(b)))
tmp -= sum(a[j] * rst[i-j+1] for j in 1:min(i, length(a)))
rst[i] = tmp / a[1]
end
return rst
end

acoef = [1.00000000, -2.77555756e-16, 3.33333333e-01, -1.85037171e-17]
bcoef = [0.16666667, 0.5, 0.5, 0.16666667]
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]
@show DF2TFilter(acoef, bcoef, signal)
```
Output:
`DF2TFilter(acoef, bcoef, signal) = [-0.152974, -0.435258, -0.136043, 0.697503, 0.656445, -0.435482, -1.08924, -0.537677, 0.51705, 1.05225, 0.961854, 0.69569, 0.424356, 0.196262, -0.0278351, -0.211722, -0.174746, 0.0692584, 0.385446, 0.651771]`

## Kotlin

Translation of: C++
```// version 1.1.3

fun filter(a: DoubleArray, b: DoubleArray, signal: DoubleArray): DoubleArray {
val result = DoubleArray(signal.size)
for (i in 0 until signal.size) {
var tmp = 0.0
for (j in 0 until b.size) {
if (i - j < 0) continue
tmp += b[j] * signal[i - j]
}
for (j in 1 until a.size) {
if (i - j < 0) continue
tmp -= a[j] * result[i - j]
}
tmp /= a[0]
result[i] = tmp
}
return result
}

fun main(args: Array<String>) {
val a = doubleArrayOf(1.00000000, -2.77555756e-16, 3.33333333e-01, -1.85037171e-17)
val b = doubleArrayOf(0.16666667, 0.5, 0.5, 0.16666667)

val signal = doubleArrayOf(
-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
)

val result = filter(a, b, signal)
for (i in 0 until result.size) {
print("% .8f".format(result[i]))
print(if ((i + 1) % 5 != 0) ", " else "\n")
}
}
```
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
```

## Lua

Translation of: C++
```function filter(b,a,input)
local out = {}
for i=1,table.getn(input) do
local tmp = 0
local j = 0
out[i] = 0

for j=1,table.getn(b) do
if i - j < 0 then
--continue
else
tmp = tmp + b[j] * input[i - j + 1]
end
end

for j=2,table.getn(a) do
if i - j < 0 then
--continue
else
tmp = tmp - a[j] * out[i - j + 1]
end
end

tmp = tmp / a[1]
out[i] = tmp
end
return out
end

function main()
local sig = {
-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
}

--Constants for a Butterworth filter (order 3, low pass)
local a = {1.00000000, -2.77555756e-16, 3.33333333e-01, -1.85037171e-17}
local b = {0.16666667, 0.5, 0.5, 0.16666667}

local result = filter(b,a,sig)
for i=1,table.getn(result) do
io.write(result[i] .. ", ")
end
print()

return nil
end

main()
```
Output:
`-0.15297398950031, -0.43525782905022, -0.13604339698849, 0.69750332654796, 0.65644469246903, -0.43548245325611, -1.0892394611529, -0.53767654956275, 0.51704999231321, 1.0522497471554, 0.96185430037364, 0.6956900940096, 0.42435629509553, 0.19626223182179, -0.027835124463393, -0.21172191545012, -0.17474556222276, 0.069258408901195, 0.38544587430744, 0.65177083881931,`

## Mathematica/Wolfram Language

```b = {0.16666667, 0.5, 0.5, 0.16666667};
a = {1.00000000, -2.77555756*^-16, 3.33333333*^-01, -1.85037171*^-17};
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};
RecurrenceFilter[{a, b}, signal]
```
Output:
`{-0.152974,-0.435258,-0.136043,0.697503,0.656445,-0.435482,-1.08924,-0.537677,0.51705,1.05225,0.961854,0.69569,0.424356,0.196262,-0.0278351,-0.211722,-0.174746,0.0692584,0.385446,0.651771}`

## MATLAB

MATLAB is commonly used for filter design and implementation. To implement this filter, and display the original signal and the filtered result:

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

out = filter(b,a,signal)

figure
subplot(1,2,1)
stem(0:19, signal)
xlabel('n')
title('Original Signal')

subplot(1,2,2)
stem(0:19, out)
xlabel('n')
title('Filtered Signal')
```
Output:
```out =

Columns 1 through 10

-0.1530   -0.4353   -0.1360    0.6975    0.6564   -0.4355   -1.0892   -0.5377    0.5170    1.0522

Columns 11 through 20

0.9619    0.6957    0.4244    0.1963   -0.0278   -0.2117   -0.1747    0.0693    0.3854    0.6518
```

## Nim

Translation of: Kotlin
```import strformat

func filter(a, b, signal: openArray[float]): seq[float] =

result.setLen(signal.len)

for i in 0..signal.high:
var tmp = 0.0
for j in 0..min(i, b.high):
tmp += b[j] * signal[i - j]
for j in 1..min(i, a.high):
tmp -= a[j] * result[i - j]
tmp /= a[0]
result[i] = tmp

#———————————————————————————————————————————————————————————————————————————————————————————————————

let a = [1.00000000, -2.77555756e-16, 3.33333333e-01, -1.85037171e-17]
let b = [0.16666667, 0.5, 0.5, 0.16666667]

let 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]

let result = filter(a, b, signal)
for i in 0..result.high:
stdout.write fmt"{result[i]: .8f}"
stdout.write if (i + 1) mod 5 != 0: ", " else: "\n"
```
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```

## Objeck

Translation of: Java
```class DigitalFilter {
function : Main(args : String[]) ~ Nil {
a := [1.00000000, -2.77555756e-16, 3.33333333e-01, -1.85037171e-17];
b := [0.16666667, 0.5, 0.5, 0.16666667];
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];

result := Filter(a, b, signal);
each(i : result) {
System.IO.Console->Print(result[i])->Print(((i + 1) % 5 <> 0) ? ",\t" : "\n");
};
}

function : Filter(a : Float[], b : Float[], signal : Float[]) ~ Float[] {
result := Float->New[signal->Size()];

each(i : signal) {
tmp := 0.0;

each(j : b) {
if(i-j >= 0) {
tmp += b[j] * signal[i - j];
};
};

each(j : a) {
if(i-j >= 0) {
tmp -= a[j] * result[i - j];
};
};

tmp /= a[0];
result[i] := tmp;
};

return result;
}
}```
Output:
```-0.152974,      -0.435258,      -0.136043,      0.697503,       0.656445
-0.435482,      -1.08924,       -0.537677,      0.51705,        1.05225
0.961854,       0.69569,        0.424356,       0.196262,       -0.0278351
-0.211722,      -0.174746,      0.0692584,      0.385446,       0.651771
```

## ooRexx

```/* REXX */
a=.array~of(1.00000000, -2.77555756e-16, 3.33333333e-01, -1.85037171e-17)
b=.array~of(0.16666667, 0.5, 0.5, 0.16666667)
s=.array~of(-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)

ret=.array~new(s~items)~~fill(0) /* create array and fill with zeroes */

Call filter a,b,s,ret
Do i=1 To ret~items
Say format(i,2) format(ret[i],2,12)
End
Exit

::Routine filter
Use Arg a,b,s,ret
Do i=1 To s~items
temp=0
Do j=1 To b~items
if i-j>=0 Then
temp=temp+b[j]*s[i-j+1]
End
Do j=1 To a~items
if i-j>=0 Then Do
u=i-j+1
temp=temp-a[j]*ret[u]
End
End
ret[i]=temp/a[1]
End
Return

::OPTIONS digits 24      /* Numeric Digits 24, everywhere */
```
output:
``` 1 -0.152973989500
2 -0.435257829050
3 -0.136043396988
4  0.697503326548
5  0.656444692469
6 -0.435482453256
7 -1.089239461153
8 -0.537676549563
9  0.517049992313
10  1.052249747155
11  0.961854300374
12  0.695690094010
13  0.424356295096
14  0.196262231822
15 -0.027835124463
16 -0.211721915450
17 -0.174745562223
18  0.069258408901
19  0.385445874307
20  0.651770838819```

## Perl

Translation of: Raku
```use strict;
use List::AllUtils 'natatime';

sub TDF_II_filter {
our(@signal,@a,@b);
local(*signal,*a,*b) = (shift, shift, shift);
my @out = (0) x \$#signal;
for my \$i (0..@signal-1) {
my \$this;
map { \$this += \$b[\$_] * \$signal[\$i-\$_] if \$i-\$_ >= 0 } 0..@b;
map { \$this -= \$a[\$_] *    \$out[\$i-\$_] if \$i-\$_ >= 0 } 0..@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     );

my @filtered = TDF_II_filter(\@signal, \@a, \@b);
my \$iter = natatime 5, @filtered;
while( my @values = \$iter->() ) {
printf(' %10.6f' x 5 . "\n", @values);
}
```
Output:
```  -0.152974  -0.435258  -0.136043   0.697503   0.656445
-0.435482  -1.089239  -0.537677   0.517050   1.052250
0.961854   0.695690   0.424356   0.196262  -0.027835
-0.211722  -0.174746   0.069258   0.385446   0.651771```

## Phix

Translation of: Julia

Note however that the a[j]* starts from index 2, unlike Julia/C/Raku/Rust/Sidef/zkl, but the same as C++/C#/D/Java/Kotlin - and it does not seem to make any difference...

```with javascript_semantics
function direct_form_II_transposed_filter(sequence a, b, signal)
sequence result = repeat(0,length(signal))
for i=1 to length(signal) do
atom tmp = 0
for j=1 to min(i,length(b)) do tmp += b[j]*signal[i-j+1] end for
for j=2 to min(i,length(a)) do tmp -= a[j]*result[i-j+1] end for
result[i] = tmp/a[1]
end for
return result
end function

constant acoef = {1.00000000, -2.77555756e-16, 3.33333333e-01, -1.85037171e-17},
bcoef = {0.16666667, 0.5, 0.5, 0.16666667},
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}

pp(direct_form_II_transposed_filter(acoef, bcoef, signal),{pp_FltFmt,"%9.6f",pp_Maxlen,110})
```
Output:
```{-0.152974,-0.435258,-0.136043, 0.697503, 0.656445,-0.435482,-1.089239,-0.537677, 0.517050, 1.052250,
0.961854, 0.695690, 0.424356, 0.196262,-0.027835,-0.211722,-0.174746, 0.069258, 0.385446, 0.651771}
```

## Phixmonti

Translation of: Phix
```include ..\Utilitys.pmt

( 1.00000000  -2.77555756e-16  3.33333333e-01  -1.85037171e-17 ) var a
( 0.16666667   0.5             0.5              0.16666667 ) var b
( -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 )

len dup 0 swap repeat >ps

for var i
0 >ps
b len i min for var j
j get rot i j - 1 + get rot * ps> + >ps swap
endfor
drop
a len i min for var j
j get ps> tps i j - 1 + get nip rot * - >ps
endfor
drop
ps> a 1 get nip / ps> swap i set >ps
endfor
drop ps> ?```
Output:
```[-0.152973989500313, -0.435257829050217, -0.13604339698849, 0.697503326547963, 0.656444692469029, -0.435482453256106, -1.089239461152929, -0.537676549562755, 0.517049992313214, 1.052249747155353, 0.961854300373645, 0.695690094009605, 0.424356295095532, 0.196262231821789, -0.0278351244633933, -0.211721915450118, -0.174745562222761, 0.0692584089011949, 0.385445874307439, 0.651770838819305]

=== Press any key to exit ===```

## Python

```#!/bin/python
from __future__ import print_function
from scipy import signal
import matplotlib.pyplot as plt

if __name__=="__main__":
sig = [-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]

#Create an order 3 lowpass butterworth filter
#Generated using b, a = signal.butter(3, 0.5)
a = [1.00000000, -2.77555756e-16, 3.33333333e-01, -1.85037171e-17]
b = [0.16666667, 0.5, 0.5, 0.16666667]

#Apply the filter to signal
filt = signal.lfilter(b, a, sig)
print (filt)

plt.plot(sig, 'b')
plt.plot(filt, 'r--')
plt.show()
```
Output:
```[-0.15297399 -0.43525783 -0.1360434   0.69750333  0.65644469 -0.43548245
-1.08923946 -0.53767655  0.51704999  1.05224975  0.9618543   0.69569009
0.4243563   0.19626223 -0.02783512 -0.21172192 -0.17474556  0.06925841
0.38544587  0.65177084]```

## Racket

Translation of: C
Strangely, C was more informative than Common Lisp in helping figure out what was going on here.
```#lang racket

(define a (vector 1.00000000E0 -2.77555756E-16 3.33333333E-01 -1.85037171E-17))
(define b (vector 0.16666667E0 0.50000000E0 0.50000000E0 0.16666667E0))
(define s (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))

(define (filter-signal-direct-form-ii-transposed coeff1 coeff2 signal)
(define signal-size (vector-length signal))
(define filtered-signal (make-vector signal-size 0))
(for ((i signal-size))
(vector-set! filtered-signal
i
(/ (for/fold ((s (for/fold ((s 0)) ((j (vector-length coeff2)) #:when (>= i j))
(+ s (* (vector-ref coeff2 j) (vector-ref signal (- i j)))))))
((j (vector-length coeff1)) #:when (>= i j))
(- s (* (vector-ref coeff1 j) (vector-ref filtered-signal (- i j)))))
(vector-ref coeff1 0))))
filtered-signal)

(filter-signal-direct-form-ii-transposed a b s)
```
Output:
```'#(-0.15297398950031305
-0.4352578290502175
-0.13604339698849033
0.6975033265479628
0.6564446924690288
-0.4354824532561056
-1.0892394611529292
-0.5376765495627545
0.5170499923132141
1.0522497471553531
0.9618543003736449
0.6956900940096049
0.42435629509553213
0.19626223182178917
-0.027835124463393326
-0.21172191545011781
-0.17474556222276072
0.06925840890119488
0.3854458743074388
0.6517708388193052)```

## Raku

(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,
)```

## REXX

### version 1

Translation of: Julia
```/*REXX pgm filters a signal with a order3 lowpass Butterworth, direct form II transposed*/
@a= '1           -2.77555756e-16  3.33333333e-1  -1.85037171e-17'  /*filter coefficients*/
@b=  0.16666667   0.5             0.5             0.16666667       /*  "          "     */
@s= '-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  '
\$.=0;            N=words(@s);    w=length(n);   numeric digits 24  /* [↑]  signal vector*/
do i=1  for N;              #=0           /*process each of the vector elements. */
do j=1  for words(@b); if i-j >= 0  then #= # + word(@b, j) * word(@s, i-j+1);  end
do k=1  for words(@a); _= i -k +1;  if i-k >= 0  then #= # - word(@a, k) * \$._; end
\$.i= # / word(@a ,1);         call tell
end   /*i*/                                 /* [↑]  only show using ½ the dec. digs*/
exit                                             /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
tell: numeric digits digits()%2;  say right(i, w)   " "   left('', \$.i>=0)\$.i /1;   return
```
output:
``` 1   -0.1529739895
2   -0.43525782905
3   -0.136043396988
4    0.697503326548
5    0.656444692469
6   -0.435482453256
7   -1.08923946115
8   -0.537676549563
9    0.517049992313
10    1.05224974716
11    0.961854300374
12    0.69569009401
13    0.424356295096
14    0.196262231822
15   -0.0278351244634
16   -0.21172191545
17   -0.174745562223
18    0.0692584089012
19    0.385445874307
20    0.651770838819
```

### version 2

Translation of: Julia
```/* REXX */
Numeric Digits 24
acoef = '1.00000000, -2.77555756e-16, 3.33333333e-01, -1.85037171e-17'
bcoef = '0.16666667, 0.5, 0.5, 0.16666667'
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'

Do i=1 By 1 While acoef>'';  Parse Var acoef a.i . ',' acoef;   End; a.0=i-1
Do i=1 By 1 While bcoef>'';  Parse Var bcoef b.i . ',' bcoef;   End; b.0=i-1
Do i=1 By 1 While signal>''; Parse Var signal s.i . ',' signal; End; s.0=i-1

ret.=0
Do i=1 To s.0
temp=0.0
Do j=1 To b.0
if i-j>=0 Then Do
u=i-j+1
temp=temp+b.j*s.u
End
End
Do j=1 To a.0
if i-j>=0 Then Do
u=i-j+1
temp=temp-a.j*ret.u
End
End
ret.i=temp/a.1
Say format(i,2) format(ret.i,2,12)
End
```
output:
``` 1 -0.152973989500
2 -0.435257829050
3 -0.136043396988
4  0.697503326548
5  0.656444692469
6 -0.435482453256
7 -1.089239461153
8 -0.537676549563
9  0.517049992313
10  1.052249747155
11  0.961854300374
12  0.695690094010
13  0.424356295096
14  0.196262231822
15 -0.027835124463
16 -0.211721915450
17 -0.174745562223
18  0.069258408901
19  0.385445874307
20  0.651770838819
```

## RPL

We use here useful list handling functions that are available from HP48G or newer models.

RPL code Comment
```  ≪
ROT REVLIST ROT REVLIST → signal a b
≪ { } 1 signal SIZE FOR j
signal j b SIZE - 1 + j SUB
WHILE DUP SIZE b SIZE < REPEAT 0 SWAP + END
b * ∑LIST
OVER j a SIZE - 1 + j 1 - SUB 0 +
WHILE DUP SIZE a SIZE < REPEAT 0 SWAP + END
a * ∑LIST -
a DUP SIZE GET / +
NEXT
≫ ≫ 'FILTR' STO
```
```FILTR ( {a} {b} {signal} → {filtered} )
Reverse a and b
For j = 1 to last signal item
extract what to multiply to b
prepend 0's if necessary
multiply by b and sum
extract what to multiply by a except a[0]
prepend 0's if necessary
multiply by a, sum and substract
divide by a[0] which is the last item once a reversed

```

Figures have been rounded to 3 digits after the decimal point to ease the 100% manual data transfer.

```{ 1 -2.778E-16 0.3333 -1.851E-17 }
{ 0.1667 0.5 0.5 0.1667}
{ -0.9178 0.1420  1.2054  0.1903 -0.6624
-1.007 -0.4047  0.8005  0.7435  1.0109
0.7415 0.2778  0.4008 -0.2086 -0.17281    -
-0.1343 0.02593 0.4901  0.5494  0.90472 }
FILTR 3 RND
```

Output:

``` 1: { -.153 -.435 -.136 .697 .656
-.436 -1.089 -.538 .517 1.052
.962 .696 .425 .196 .028
-.212 -.175 .069 .386 .652 }
```

## Ruby

Translation of: C#
```def filter(a,b,signal)
result = Array.new(signal.length(), 0.0)
for i in 0..signal.length()-1 do
tmp = 0.0
for j in 0 .. b.length()-1 do
if i - j < 0 then next end
tmp += b[j] * signal[i - j]
end
for j in 1 .. a.length()-1 do
if i - j < 0 then next end
tmp -= a[j] * result[i - j]
end
tmp /= a[0]
result[i] = tmp
end
return result
end

def main
a = [1.00000000, -2.77555756e-16, 3.33333333e-01, -1.85037171e-17]
b = [0.16666667, 0.5, 0.5, 0.16666667]
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
]

result = filter(a,b,signal)
for i in 0 .. result.length() - 1 do
print "%11.8f" % [result[i]]
if (i + 1) % 5 == 0 then
print "\n"
else
print ", "
end
end
end

main()
```
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```

## Rust

Translation of: Java
```use std::cmp::Ordering;

struct IIRFilter<'f>(&'f [f32], &'f [f32]);

impl<'f> IIRFilter<'f> {
pub fn with_coefficients(a: &'f [f32], b: &'f [f32]) -> IIRFilter<'f> {
IIRFilter(a, b)
}

// Performs the calculation as an iterator chain.
pub fn apply<I: Iterator<Item = &'f f32> + 'f>(
&self,
samples: I,
) -> impl Iterator<Item = f32> + 'f {
// Name some things for readability
let a_coeff = self.0;
let b_coeff = self.1;

let mut prev_results = Vec::<f32>::new();
let mut prev_samples = Vec::<f32>::new();

// The actual calculation, done one number at a time
samples.enumerate() // (i, sample[i])
.map(move |(i, sample)| { // for each sample, apply this function
prev_samples.push(*sample);
prev_results.push(0f32); // the initial version of the previous result

let sum_b: f32 = b_coeff.iter() // for each coefficient in b
.enumerate() // (j, b_coeff[j])
.map(|(j, c)| { // calculate the weight of the coefficient
if i >= j {
(*c) * prev_samples[i-j]
} else {
0f32
}
})
.sum(); // add them all together

let sum_a: f32 = a_coeff.iter() // for each coefficient in a
.enumerate() // (j, a_coeff[j])
.map(|(j, c)| { // calculate the weight of the coefficient
if i >= j {
(*c) * prev_results[i-j]
} else {
0f32
}
})
.sum(); // add them all together

// perform the final calculation
let result = (sum_b - sum_a) / a_coeff[0];

// update the previous result for the next iteration
prev_results[i] = result;

// return the current result in this iteration
result
}
)
}
}

fn main() {
let a: &[f32] = &[1.00000000, -2.77555756e-16, 3.33333333e-01, -1.85037171e-17];
let b: &[f32] = &[0.16666667, 0.5, 0.5, 0.16666667];

let samples: Vec<f32> = vec![
-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,
];

for (i, result) in IIRFilter::with_coefficients(a, b)
.apply(samples.iter())
.enumerate()
{
print!("{:.8}", result);
if (i + 1) % 5 != 0 {
print!(", ");
} else {
println!();
}
}
println!();
}
```
output:
```-0.15297399, -0.43525785, -0.13604343, 0.69750333, 0.65644467
-0.43548250, -1.08923948, -0.53767651, 0.51705003, 1.05224979
0.96185434, 0.69568992, 0.42435625, 0.19626230, -0.02783510
-0.21172196, -0.17474557, 0.06925842, 0.38544586, 0.65177077
```

## Scala

Output:
See it yourself by running in your browser either by ScalaFiddle (ES aka JavaScript, non JVM) or Scastie (remote JVM).
Works with: Scala version 2.13
```object ButterworthFilter extends App {
private def filter(a: Vector[Double],
b: Vector[Double],
signal: Vector[Double]): Vector[Double] = {

@scala.annotation.tailrec
def outer(i: Int, acc: Vector[Double]): Vector[Double] = {
if (i >= signal.length) acc
else {
@scala.annotation.tailrec
def inner0(j: Int, tmp: Double): Double = if (j >= b.length) tmp
else if ((i - j) >= 0) inner0(j + 1, tmp + b(j) * signal(i - j)) else inner0(j + 1, tmp)

@scala.annotation.tailrec
def inner1(j: Int, tmp: Double): Double = if (j >= a.length) tmp
else if (i - j >= 0) inner1(j + 1, tmp - a(j) * acc(i - j)) else inner1(j + 1, tmp)

outer(i + 1, acc :+ inner1(1, inner0(0, 0D)) / a(0))
}
}

outer(0, Vector())
}

filter(Vector[Double](1, -2.77555756e-16, 3.33333333e-01, -1.85037171e-17),
Vector[Double](0.16666667, 0.5, 0.5, 0.16666667),
Vector[Double](
-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)
).grouped(5)
.map(_.map(x => f"\$x% .8f"))
.foreach(line => println(line.mkString(" ")))

}
```

## Sidef

Translation of: Raku
```func TDF_II_filter(signal, a, b) {
var out = [0]*signal.len
for i in ^signal {
var this = 0
for j in ^b { i-j >= 0 && (this += b[j]*signal[i-j]) }
for j in ^a { i-j >= 0 && (this -= a[j]*   out[i-j]) }
out[i] = this/a[0]
}
return out
}

var 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
]

var a = [1.00000000, -2.77555756e-16, 3.33333333e-01, -1.85037171e-17]
var b = [0.16666667,  0.5,            0.5,             0.16666667    ]
var f = TDF_II_filter(signal, a, b)

say "["
say f.map { "% 0.8f" % _ }.slices(5).map{.join(', ')}.join(",\n")
say "]"
```
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
]```

## Visual Basic .NET

Translation of: C#
```Module Module1

Function Filter(a As Double(), b As Double(), signal As Double()) As Double()
Dim result(signal.Length - 1) As Double
For i = 1 To signal.Length
Dim tmp = 0.0
For j = 1 To b.Length
If i - j < 0 Then
Continue For
End If
tmp += b(j - 1) * signal(i - j)
Next
For j = 2 To a.Length
If i - j < 0 Then
Continue For
End If
tmp -= a(j - 1) * result(i - j)
Next
tmp /= a(0)
result(i - 1) = tmp
Next
Return result
End Function

Sub Main()
Dim a() As Double = {1.0, -0.000000000000000277555756, 0.333333333, -1.85037171E-17}
Dim b() As Double = {0.16666667, 0.5, 0.5, 0.16666667}

Dim signal() As Double = {
-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
}

Dim result = Filter(a, b, signal)
For i = 1 To result.Length
Console.Write("{0,11:F8}", result(i - 1))
Console.Write(If(i Mod 5 <> 0, ", ", vbNewLine))
Next
End Sub

End Module
```
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```

## V (Vlang)

Translation of: Go
```struct Filter {
b []f64
a []f64
}

fn (f Filter) filter(inp []f64) []f64 {
mut out := []f64{len: inp.len}
s := 1.0 / f.a[0]
for i in 0..inp.len {
mut tmp := 0.0
mut b := f.b
if i+1 < b.len {
b = b[..i+1]
}
for j, bj in b {
tmp += bj * inp[i-j]
}
mut a := f.a[1..]
if i < a.len {
a = a[..i]
}
for j, aj in a {
tmp -= aj * out[i-j-1]
}
out[i] = tmp * s
}
return out
}

//Constants for a Butterworth Filter (order 3, low pass)
const bwf = Filter{
a: [f64(1.00000000), -2.77555756e-16, 3.33333333e-01, -1.85037171e-17],
b: [f64(0.16666667), 0.5, 0.5, 0.16666667],
}

const sig = [
f64(-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,
]

fn main() {
for v in bwf.filter(sig) {
println("\${v:9.6}")
}
}```
Output:
```-0.152974
-0.435258
-0.136043
0.697503
0.656445
-0.435482
-1.089239
-0.537677
0.517050
1.052250
0.961854
0.695690
0.424356
0.196262
-0.027835
-0.211722
-0.174746
0.069258
0.385446
0.651771
```

## Wren

Translation of: Kotlin
Library: Wren-fmt
```import "/fmt" for Fmt

var filter = Fn.new { |a, b, signal|
var result = List.filled(signal.count, 0)
for (i in 0...signal.count) {
var tmp = 0
for (j in 0...b.count) {
if (i - j < 0) continue
tmp = tmp + b[j] * signal[i - j]
}
for (j in 1...a.count) {
if (i - j < 0) continue
tmp = tmp - a[j] * result[i - j]
}
tmp = tmp / a[0]
result[i] = tmp
}
return result
}

var a = [1.00000000, -2.77555756e-16, 3.33333333e-01, -1.85037171e-17]
var b = [0.16666667, 0.5, 0.5, 0.16666667]

var 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
]

var result = filter.call(a, b, signal)
for (i in 0...result.count) {
Fmt.write("\$11.8f", result[i])
System.write(((i + 1) % 5 != 0) ? ", " : "\n")
}
```
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
```

## XPL0

```real A, B, Signal, Temp, Result(20);
int  I, J;
[A:= [1.00000000, -2.77555756e-16, 3.33333333e-01, -1.85037171e-17];
B:= [0.16666667, 0.5, 0.5, 0.16666667];
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 ];
Format(2, 8);
for I:= 0 to 20-1 do
[Temp:= 0.;
for J:= 0 to 4-1 do
if I-J >= 0 then
Temp:= Temp + B(J)*Signal(I-J);
for J:= 1 to 4-1 do
if I-J >= 0 then
Temp:= Temp - A(J)*Result(I-J);
Result(I):= Temp / A(0);
RlOut(0, Result(I));
Text(0, if rem(I/5) = 4 then "^m^j" else ", ");
];
]```
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
```

## Yabasic

Translation of: D
```sub filter(a(), b(), signal(), result())
local i, j, tmp

for i = 0 to arraysize(signal(), 1)
tmp = 0
for j = 0 to arraysize(b(), 1)
if (i-j<0) continue
tmp = tmp + b(j) * signal(i-j)
next
for j = 0 to arraysize(a(), 1)
if (i-j<0) continue
tmp = tmp - a(j) * result(i-j)
next
tmp = tmp / a(0)
result(i) = tmp
next
end sub

dim a(4), b(4), signal(20), result(20)

// a()
data 1, -2.77555756e-16, 3.33333333e-01, -1.85037171e-17
// b()
data 0.16666667, 0.5, 0.5, 0.16666667
// signal()
data -0.917843918645, 0.141984778794, 1.20536903482, 0.190286794412
data -0.662370894973, -1.00700480494, -0.404707073677, 0.800482325044
data 0.743500089861, 1.01090520172, 0.741527555207, 0.277841675195
data 0.400833448236, -0.2085993586, -0.172842103641, -0.134316096293
data 0.0259303398477, 0.490105989562, 0.549391221511, 0.9047198589

for i = 0 to 3 : read a(i) : next
for i = 0 to 3 : read b(i) : next
for i = 0 to 19 : read signal(i) : next

filter(a(),b(),signal(),result())

for i = 0 to 19
print result(i) using "%11.8f";
if mod(i+1, 5) <> 0 then
print ", ";
else
print
end if
next```

## zkl

Translation of: C++
```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
}```
```signal:=T(-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 );
a:=T(1.0, -2.77555756e-16, 3.33333333e-01, -1.85037171e-17 );
b:=T(0.16666667, 0.5, 0.5, 0.16666667 );
result:=direct_form_II_transposed_filter(b,a,signal);
println(result);```
Output:
```L(-0.152974,-0.435258,-0.136043,  0.697503, 0.656445,-0.435482,
-1.08924, -0.537677, 0.51705,   1.05225,  0.961854, 0.69569,
0.424356, 0.196262,-0.0278351,-0.211722,-0.174746, 0.0692584,
0.385446, 0.651771)
```

References