Fourier series expansion from data points

Question

Bartosz Kurylek 2021년 10월 21일

0
링크

이 질문에 대한 바로 가기 링크

https://kr.mathworks.com/matlabcentral/answers/1568758-fourier-series-expansion-from-data-points

답변: Neelanshu 2024년 5월 9일

Hi all,

I was trying to find Fourer approximation for any function data set. I managed to write algorythm that gives me exact shape that i wanted, but for some reason when i exceed ~320 itarations graph starts to scale up. I spent too much time looking through code and maybe anyone else had simmilar problem.

Here are functions I used to find Foureir coefficients and final function:

Thank you for any help!

My code:

%Cleaning comand window

close all

clear

clc

%Number of iterations

it = 500;

%Data points input

%loop just to create dropdown

for data_input = 1:1

t = [0

0.017952

0.035904

0.053856

0.071808

0.08976

0.107712

0.125664

0.143616

0.161568

0.17952

0.197472

0.215423

0.233375

0.251327

0.269279

0.287231

0.305183

0.323135

0.341087

0.359039

0.376991

0.394943

0.412895

0.430847

0.448799

0.466751

0.484703

0.502655

0.520607

0.538559

0.556511

0.574463

0.592415

0.610367

0.628319

0.64627

0.664222

0.682174

0.700126

0.718078

0.73603

0.753982

0.771934

0.789886

0.807838

0.82579

0.843742

0.861694

0.879646

0.897598

0.91555

0.933502

0.951454

0.969406

0.987358

1.00531

1.023262

1.041214

1.059166

1.077117

1.095069

1.113021

1.130973

1.148925

1.166877

1.184829

1.202781

1.220733

1.238685

1.256637

1.274589

1.292541

1.310493

1.328445

1.346397

1.364349

1.382301

1.400253

1.418205

1.436157

1.454109

1.472061

1.490013

1.507964

1.525916

1.543868

1.56182

1.579772

1.597724

1.615676

1.633628

1.65158

1.669532

1.687484

1.705436

1.723388

1.74134

1.759292

1.777244

1.795196

1.813148

1.8311

1.849052

1.867004

1.884956

1.902908

1.92086

1.938811

1.956763

1.974715

1.992667

2.010619

2.028571

2.046523

2.064475

2.082427

2.100379

2.118331

2.136283

2.154235

2.172187

2.190139

2.208091

2.226043

2.243995

2.261947

2.279899

2.297851

2.315803

2.333755

2.351706

2.369658

2.38761

2.405562

2.423514

2.441466

2.459418

2.47737

2.495322

2.513274

2.531226

2.549178

2.56713

2.585082

2.603034

2.620986

2.638938

2.65689

2.674842

2.692794

2.710746

2.728698

2.74665

2.764602

2.782553

2.800505

2.818457

2.836409

2.854361

2.872313

2.890265

2.908217

2.926169

2.944121

2.962073

2.980025

2.997977

3.015929

3.033881

3.051833

3.069785

3.087737

3.105689

3.123641

3.141593

3.159545

3.177497

3.195449

3.2134

3.231352

3.249304

3.267256

3.285208

3.30316

3.321112

3.339064

3.357016

3.374968

3.39292

3.410872

3.428824

3.446776

3.464728

3.48268

3.500632

3.518584

3.536536

3.554488

3.57244

3.590392

3.608344

3.626296

3.644247

3.662199

3.680151

3.698103

3.716055

3.734007

3.751959

3.769911

3.787863

3.805815

3.823767

3.841719

3.859671

3.877623

3.895575

3.913527

3.931479

3.949431

3.967383

3.985335

4.003287

4.021239

4.039191

4.057143

4.075094

4.093046

4.110998

4.12895

4.146902

4.164854

4.182806

4.200758

4.21871

4.236662

4.254614

4.272566

4.290518

4.30847

4.326422

4.344374

4.362326

4.380278

4.39823

4.416182

4.434134

4.452086

4.470038

4.48799

4.505941

4.523893

4.541845

4.559797

4.577749

4.595701

4.613653

4.631605

4.649557

4.667509

4.685461

4.703413

4.721365

4.739317

4.757269

4.775221

4.793173

4.811125

4.829077

4.847029

4.864981

4.882933

4.900885

4.918836

4.936788

4.95474

4.972692

4.990644

5.008596

5.026548

5.0445

5.062452

5.080404

5.098356

5.116308

5.13426

5.152212

5.170164

5.188116

5.206068

5.22402

5.241972

5.259924

5.277876

5.295828

5.31378

5.331732

5.349683

5.367635

5.385587

5.403539

5.421491

5.439443

5.457395

5.475347

5.493299

5.511251

5.529203

5.547155

5.565107

5.583059

5.601011

5.618963

5.636915

5.654867

5.672819

5.690771

5.708723

5.726675

5.744627

5.762579

5.78053

5.798482

5.816434

5.834386

5.852338

5.87029

5.888242

5.906194

5.924146

5.942098

5.96005

5.978002

5.995954

6.013906

6.031858

6.04981

6.067762

6.085714

6.103666

6.121618

6.13957

6.157522

6.175474

6.193426

6.211377

6.229329

6.247281

6.265233

6.283185];

f = [0

0

1.345656

1.46359

1.853533

1.908695

1.859239

1.731795

1.611959

0.938595

0.234796

1.739403

1.904891

1.581524

0.969029

0.390773

0.858758

1.022642

1.117587

1.19447

1.259128

1.315828

1.36746

1.414924

1.45791

1.496073

1.530751

1.565343

1.594598

1.608704

1.614128

1.613152

1.606161

1.585375

1.55787

1.52766

1.493909

1.452975

1.401511

1.344671

1.285349

1.220261

1.146139

1.059762

1.970569

2.228344

2.195105

2.098704

2.002302

1.905901

1.809499

1.713098

1.616696

1.520295

1.423893

1.327491

1.23109

1.134688

1.038287

0.941885

0.845484

0.749082

0.652681

0.556279

0.459878

0.363476

0.267075

0.203542

0.737289

0.648332

0.860617

1.126093

0.604427

0.10148

0.116437

0.138592

0.154547

0.167611

0.174579

0.178716

0.18133

0.182603

0.183212

0.183552

0.183756

0.183878

0.183948

0.183981

0.263889

0.327583

0.354472

0.625727

0.674006

0.700521

0.719297

0.731877

0.740955

0.748438

0.760127

0.782385

0.799697

0.780789

0.76188

0.750682

0.745784

0.739718

0.730804

0.715644

0.689119

0.639306

0.237436

0.132665

0

0];

end

%Ploting graphs

f1 = figure;

title('Skyline graph from datapoints')

hold on

plot(t,f,"Color",[0,0,0])

scatter(t,f,'red')

hold off

%Period and array size

T = 2*pi;

m = size(t);

m = m(1);

m1 = m;

%Fourier 0 coefficient

a0 = (2/m)*sum(f,"all");

%Finding array for sin and cosine Fourier coefficiants for all iterations

for i = 1:it

san = 0;

sbn = 0;

j=0;

for j = 1:m

san = san + (f(j)*cos((2*pi/T)*i*t(j)));

sbn = sbn + (f(j)*sin((2*pi/T)*i*t(j)));

an(i) = (2/m)*san;

bn(i) = (2/m)*sbn;

end

%Final an and bn arrays

fan = an;

fbn = bn;

%Finding function array values

for ifn = 1:m

sfn = 0;

for jfn = 1:it

sfn = sfn + (fan(jfn)*cos(jfn*t(ifn))+fbn(jfn)*sin(jfn*t(ifn)));

fn(ifn) = (a0/2) + sfn;

end

fnn = fn;

%Data for plotting graphs

t = t(1:m1);

f = f(1:m1);

fnn = fnn(1:m1);

fn = (1:m1);

f2 = figure;

title('Best fourier approximation over graph from data points')

hold on

plot(t,fnn)

plot(t,f)

hold off

%Finidning quality of function

vi = fnn - f;

vi2 = vi.^2;

vi = sum(vi2,'all');

RMS = sqrt((vi)/m1)

RMS = 51.0386

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Answer 1

Neelanshu 2024년 5월 9일

0
링크

이 답변에 대한 바로 가기 링크

https://kr.mathworks.com/matlabcentral/answers/1568758-fourier-series-expansion-from-data-points#answer_1454981

MATLAB Online에서 열기

Hi Bartosz,

The functions you used for computation of the fourier coefficient is erroneous. The formula for discrete time fourier coefficients is as follows :

Furthermore, the functions defined in the code for the coefficients of the fourier series are erroneous. f(1) corresponds to the coefficient a0, not coefficient a(1) and therefore, the index of either the function or the sinusoidals needs to be changed appropriately as shown in the code snippet below:

m = size(t);
m = m(1);
m1 = m;
T = m;
%Finding array for sin and cosine Fourier coefficiants for all iterations
    for i = 1:m
    
        san = 0;
        sbn = 0;
        j=0;
        
        for j = 1:m
            
                san = san + (f(j)*cos((2*pi/T)*(i-1)*(j-1)));
                sbn = sbn - (f(j)*sin((2*pi/T)*(i-1)*(j-1)));
                
                an(i) = (1/m)*san;
                bn(i) = (1/m)*sbn; 
                
        end
    end
    
%Final an and bn arrays 
    fan = an;
    fbn = bn;
%Finding function array values
    for ifn = 1:m
    sfn = 0; 
        for jfn = 1:m
           
            sfn = sfn + (fan(jfn)*cos((jfn-1)*(2*pi/T)*(ifn-1))-fbn(jfn)*sin((jfn-1)*(2*pi/T)*(ifn-1)));
            fn(ifn) =  sfn;
            
        end
        
    end
   
fnn = fn;
%Code for RMSE
vi = fnn - f';
vi2 = vi.^2;
vi = sum(vi2,'all');
RMS = sqrt((vi)/m1)