⚠️ Warning: This is a draft ⚠️

This means it might contain formatting issues, incorrect code, conceptual problems, or other severe issues.

If you want to help to improve and eventually enable this page, please fork RosettaGit's repository and open a merge request on GitHub.

[[Category:Digital signal processing]] {{task}} 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. [https://ccrma.stanford.edu/~jos/fp/Transposed_Direct_Forms.html]

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]

## 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++

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;
}
```

{{out}}

```-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,
```

## C#

{{trans|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");
}
}
}
}
```

{{out}}

```-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
```

## Common Lisp

{{trans|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)))
```

{{out}}

```     -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

{{trans|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;
}
}
}
```

{{out}}

```-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)
}
}
```

{{out}}

```
-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

```

## Java

{{trans|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");
}
}
}
```

{{out}}

```-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

{{trans|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

{{trans|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")
}
}
```

{{out}}

```
-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

```

## 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')

```

{{out}}

```
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

```

## Objeck

{{trans|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 */

```

{{out|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

{{trans|Perl 6}}

```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);
}

```

{{out}}

```  -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
```

## Perl 6

{{works with|Rakudo|2016.11}} {{trans|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 *;
```

{{out}}

```(-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,
)
```

## Phix

{{trans|Julia}} Note however that the a[j]* starts from index 2, unlike Julia/C/Perl6/Rust/Sidef/zkl, but the same as C++/C#/D/Java/Kotlin - and it does not seem to make any difference...

```function direct_form_II_transposed_filter(sequence a, b, signal)
sequence result = repeat(0,length(signal))
for i=1 to length(in) 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})
```

{{out}}

```
{-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}

```

## 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()
```

{{out}}

```[-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]
```

## REXX

### version 1

{{trans|Julia}}

```/*REXX pgm filters a signal with a order3 lowpass Butterworth, direct form II transposed*/
numeric digits 24                                                  /*use 20 decimal digs*/
@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)                    /* [↑]  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
```

{{out|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

{{trans|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
```

{{out|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

```

## Rust

{{trans|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!();
}
```

{{out|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

{{Out}}See it yourself by running in your browser either by [https://scalafiddle.io/sf/0D4zyWF/0 ScalaFiddle (ES aka JavaScript, non JVM)] or [https://scastie.scala-lang.org/mUPbvXTQTXijp2DV1piQgQ Scastie (remote JVM)]. {{libheader|Scala Tail recursion}} {{libheader|Scala Digital Signal Processing}} {{libheader|ScalaFiddle qualified}} {{libheader|Scastie qualified}} {{works with|Scala|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

{{trans|Perl 6}}

```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 "]"
```

{{out}}

```
[
-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

{{trans|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

{{trans|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);
```

{{out}}

```
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'''