It is standard practice in signal processing to employ complex numbers whenever possible. One of the main reasons for doing this is that it enables us to represent the important sine and cosine functions in terms of complex exponential functions and to replace trigonometric identities with the somewhat simpler rules for the manipulation of exponents.

The complex numbers are the points in the x, y-plane: the complex number z = ( a, b) is identified with the point in the plane having a = Re( z), the real part of z, for its x-coordinate and b = Im( z), the imaginary part of z, for its y-coordinate. We call ( a, b) the rectangular form of the complex number z. The conjugate of the complex number z is z = ( a, ? b). We can also represent z in its polar form: let the magnitude of z be and the phase angle of z, denoted ?( z), be the angle in [0 , 2 ?) with cos ?( z) = a/ z. Then the polar form for z is

Any complex number z =...