Representation of Fourier series depends on tabulation points - c++

Question

Well, I had task to create function that does Fourier series with some mathematical function, so I found all the formulas, but the main problem is when I change count of point on some interval to draw those series I have very strange artifact:
This is Fourier series of sin(x) on interavl (-3.14; 314) with 100 point for tabulation
And this is same function with same interval but with 100000 points for tabulation
Code for Fourier series coeficients:
void fourieSeriesDecompose(std::function<double(double)> func, double period, long int iterations, double *&aParams, double *&bParams){
aParams = new double[iterations];
aParams[0] = integrateRiemans(func, 0, period, 1000);
for(int i = 1; i < iterations; i++){
auto sineFunc = [&](double x) -> double { return 2 * (func(x) * cos((2 * x * i * M_PI) / period)); };
aParams[i] = integrateRiemans(sineFunc, -period / 2, period / 2, 1000) / period;
}
bParams = new double[iterations];
for(int i = 1; i < iterations; i++){
auto sineFunc = [&](double x) -> double { return 2 * (func(x) * sin(2 * (x * (i + 1) * M_PI) / period)); };
bParams[i] = integrateRiemans(sineFunc, -period / 2, period / 2, 1000) / period;
}
}
This code I use to reproduce function using found coeficients:
double fourieSeriesCompose(double x, double period, long iterations, double *aParams, double *bParams){
double y = aParams[0];
for(int i = 1; i < iterations; i++){
y += sqrt(aParams[i] * aParams[i] + bParams[i] * bParams[i]) * cos((2 * i * x * M_PI) / period - atan(bParams[i] / aParams[i]));
}
return y;
}
And the runner code
double period = M_PI * 2;
auto startFunc = [](double x) -> double{ return sin(x); };
fourieSeriesDecompose(*startFunc, period, 1000, aCoeficients, bCoeficients);
auto readyFunc = [&](double x) -> double{ return fourieSeriesCompose(x, period, 1000, aCoeficients, bCoeficients); };
tabulateFunc(readyFunc);
scaleFunc();
//Draw methods after this

Answer 1

see:
How to compute Discrete Fourier Transform?
So if I deciphered it correctly the aParams,bParams represent the real and imaginary part of the result then the angles in sin and cos must be the same but you have different! You got this:
auto sineFunc = [&](double x) -> double { return 2*(func(x)*cos((2* x* i *M_PI)/period));
auto sineFunc = [&](double x) -> double { return 2*(func(x)*sin( 2*(x*(i+1)*M_PI)/period));
as you can see its not the same angle. Also what is period? You got iterations! if it is period of the function you want to transform then it should be applied to it and not to the kernel ... Also integrateRiemans does what? its the nested for loop to integrate the furrier transform? Btw. hope that func is real domain otherwise the integration/sumation needs both real and imaginary part not just one ...
So what you should do is:
create (cplx) table of the func(x) data on the interval you want with iterations samples
so for loop where x = x0+i*(x1-x0)/(iterations-1) and x0,x1 is the range you want the func to sample. Lets call it f[i]
for (i=0;i<iteration;i++) f[i]=func(x0+i*(x1-x0)/(iterations-1));
furrier transform it
something like this:
for (i=0;i<iteration;i++) a[i]=b[i]=0;
for (j=0;j<iteration;j++)
for (i=0;i<iteration;i++)
{
a[j]+=f[i]*cos(-2.0*M_PI*i*j/iterations);
b[j]+=f[i]*sin(-2.0*M_PI*i*j/iterations);
}
now a[],b[] should hold your slow DFT result ... beware integer rounding ... depending on compiler you might need to cast some stuff to double to avoid integer rounding.