Complex Representation?

Question

Why we use complex representation for periodic wave?

Exp(-jwt)=cos(wt)-jsin(wt)

Is it because it looks short and we just need the real part? The imaginary part is what for?

I understand the real part (x) means the amplitude of the signal. Then what is the physical meaning of the imaginary part (y)?

Answer 1

The imaginary part is the y-coordinate for the tip of a rotating vector that rotates counterclockwise about the origin at w rad/s.

Answer 2

No, you need both parts. The 'imaginary' component is the 'y value' for the tip of a phasor (rotating vector) that rotates counterclockwise about the origin at w rad/s.

But it's also a very useful form when you start doing things such as LaPlace and Fourier integrals.

Also note, the complex frequency is *not* just jw. It is actually (jw + ß) where ß is the 'Naperian' frequency and represents a time exponential decay in amplitude of that particular sinewave. This gets *real* convenient when you're using LaPlace to solve for impulse excitation and damped oscillation(s) in things such as filters, servo loops, etc. ☺


Doug