math - Keeping exact wave form in memory -

let's have program calculates value of sine wave @ time t. sine wave of form sin(f*t + phi). amplitude 1.

if have 1 sin term fine. can calculate value @ time t.

but, @ runtime, wave form becomes modified when combines other waves. sin(f1 * t + phi1) + sin(f2 * t + phi2) + sin(f3 * t + phi3) + ...

the simplest solution have table columns phi , f, iterate on rows, , sum results. me feels once reach thousands of rows, computation become slow.

is there different way of doing this? combining sines 1 statement/formula?

if have fourier series (i.e. f_i = i f f) can use clenshaw recurrence relation faster computing sines (but might less accurate).

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in case can consider sequence:

f_k = exp( ( k f t + phi_k) ) , imaginary unit. 

notice im(f_k) = sin( k f t + phi_k ), sequence.

also

f_k = exp( ( k f t + phi_k) ) = exp( k f t ) exp( phi_k ) 

hence have a_k = exp(i phi_k). can precompute these values , store them in array. simplicity on assume a_0 = 0.

now, exp( (k + 1) f t) = exp(i k f t) * exp(i f t), alpha_k = exp(i f t) , beta_k = 0.

you can apply recurrence formula, in c++ can this:

complex<double> clenshaw_fourier(double f, double t, const vector< complex<double> > & ) {     const complex<double> alpha = exp(f * t * i);      complex<double> b = 0;      (int k = a.size() - 1; k >0; -- k )         b = a[k] + alpha * b;      return a[0] + alpha * b; } 

assuming a[k] == exp( phi_k ).

the real part of answer sum of cos(k f t + phi_k), while imaginary part sum of sin(k f t + phi_k).

as can see uses addition , multiplications, except exp(f * t * i) computed once.