CMSIS DSP库中的复杂点积

Question

Recently I've been checking out the CMSIS DSP complex math functions library and I've seen something I cannot fully comprehend, thus my first post on SO.

What I'm unable to grasp is how the he11 can the complex dot product function yeild a proper result? The function may be found here: Complex Dot Product

As far as I'm concerned the part

for(n=0; n<numSamples; n++) {  
   realResult += pSrcA[(2*n)+0]*pSrcB[(2*n)+0] - pSrcA[(2*n)+1]*pSrcB[(2*n)+1];  
   imagResult += pSrcA[(2*n)+0]*pSrcB[(2*n)+1] + pSrcA[(2*n)+1]*pSrcB[(2*n)+0];  
}  

is A-okay, but how's that:

/* CReal = A[0]* B[0] + A[2]* B[2] + A[4]* B[4] + .....+ A[numSamples-2]* B[numSamples-2] */
real_sum += (*pSrcA++) * (*pSrcB++);
/* CImag = A[1]* B[1] + A[3]* B[3] + A[5]* B[5] + .....+ A[numSamples-1]* B[numSamples-1] */
imag_sum += (*pSrcA++) * (*pSrcB++);

supposed to work, since it misses the product of real*imag parts of the samples?

It might - and most probably is - a really dumb question, but somehow I simply cannot see it working.

Answer 1

That looks simply wrong, and the implementation doesn't match the description.

Suppose we have z = x + i*y and w = u + i*v with x, y, u, v real. Then

z*w = (x + i*y)*(u + i*v) = (x*u - y*v) + i*(x*v + y*u)

and

z*conjugate(w) = (x + i*y)*(u - i*v) = (x*u + y*v) + i*(y*u - x*v)

So with the loop

while(blkCnt > 0u)
{
  /* CReal = A[0]* B[0] + A[2]* B[2] + A[4]* B[4] + .....+ A[numSamples-2]* B[numSamples-2] */
  real_sum += (*pSrcA++) * (*pSrcB++);
  /* CImag = A[1]* B[1] + A[3]* B[3] + A[5]* B[5] + .....+ A[numSamples-1]* B[numSamples-1] */
  imag_sum += (*pSrcA++) * (*pSrcB++);
  /* Decrement the loop counter */
  blkCnt--;
}

you will get real_sum + imag_sum = Real part of hermitian inner product finally.

Neither real_sum nor imag_sum is in any simple way related to the real/imaginary part of the inner product nor the bilinear product.