Advanced DSP

Complex Vectors

An N component complex valued vector u has N components, each one being a complex number. This can be thought of equivalently as a 2N component real vector, where the 2N real components are u_r and u_i, such that u=u_r+iu_i. Complex vector addition, energy and real scaling become the natural addition, energy and real scaling in this “realized” domain.

Let us see what happens when u is scaled by a complex number c=c_r+ic_i. We will get the vector cu=c_r(u_r+iu_i)+c_i(iu_r-u_i)=c_ru1+c_iu2. In the Euclidean sense, the dot product of u₁ and u₂ is zero – in the 2N dimensional realized space, u₁ and u₂ are orthogonal.

Now suppose we want to find the best complex scale c of u that fits another complex vector x the best. Since realized u₁ and u₂ are orthogonal, this can be achieved by projecting realized x on both of them to get the real and imaginary components of c This gives c=‹x,u›/‹u,u›, where ‹x,u› is defined as the componentwise multiplication and addition (mad product!) of x and u* (the componentwise conjugate of u).

The componentwise multiplication of two Fourier spirals gives a third Fourier spiral – of the sum frequency. Thus, the mad product of two Fourier spirals is zero unless these two spirals are conjugate spirals. Thus the dot product of two Fourier spirals is zero unless they are the same spiral. This proves that the Fourier transform is complex-orthogonal. Since we use the requisite number of spirals, the Fourier transform is obviously complete.

With an almost-trivial application of the Least Squares Method we found a fitting formula for complex vectors which is a generalization of the Dot Product.