使用IIR滤波器生成粉红噪声

Question

I implement the following code from `'Proceedings of the IEEE'', N. Jeremy Kasdin (page 825) pdf . But I don't understand theses lines because I don't have Numerical Recipes book :

/* perform the discrete Fourier transform */  
realft (hfa,n_pts, 1); 
realft (wfa,n_pts, 1);

wfa[1]=wfa[1]*hfa[1]; 
wfa[2]=wfa[2]*hfa[2]; 

for(i=3;i<=nn;i+=2) { 
wr=wfa[i]; 
wi=wfa[i+1]; 
wfa[il=wr*hfa[i]-wi*hfa[i+1]; 
wfa[i+l]=wr*hfa[i+1]+wi*hfa[i];
}

Can someone give me some directions ?

Answer 1

The realft function in NR does the following. You feed it an array of N real numbers. (N has to be a power of 2.) The discrete Fourier transform of this consists of N complex numbers that obey a conjugate-symmetry relation: F(k) and F(Nk) are conjugate. In particular, F(0) and F(N/2) are real. So realft returns N real numbers, as follows: F(0), F(N/2), real part of F(1), imaginary part of F(1), real part of F(2), ..., imaginary part of F(N/2-1).

NR was originally all Fortran, and (at least in older editions) uses 1-based indexing instead of 0-based. Even in C. That's why the code starts by operating on element 1 and runs up to nn inclusively rather than exclusively.

So, you've taken the FT of hfa and wfa , in place. The rest of the code is simply computing the elementwise product of the results -- the first two lines are simple real multiplications, and the rest is multiplying complex numbers.

I'll guess that after that there's another call to realft with -1 as the last argument (meaning to do the inverse operation). So the whole thing is: FT of hfa and of wfa ; multiply them elementwise; inverse FT. In other words, a convolution.