A.3  Manipulating Complex Numbers and Rational Functions

The complex-conjugate of a sum of two complex numbers $p$ and $q$ is the sum of the conjugate of these numbers:

Assuming that $p=a+\mathit{bj}$ and $q=c+\mathit{dj}$, one can also write this result as:

where $a,b,c,d\in ℝ$. This is also valid for the conjugate of a difference, product, or quotient of two numbers, which is the difference, product, or quotient, respectively, of their individual conjugates.

This is useful when manipulating a rational system function $H(z)$ to obtain $H^{\ast }(z)$, as required in Eq. (4.59). For instance, suppose $H(z)=(z-2e^{j4})(z-3+j5)∕(z+6e^{-j4})$, then $H^{\ast }(z)=(z^{\ast }-2e^{-j4})(z^{\ast }-3-j5)∕(z^{\ast }+6e^{j4})$.