Manipulating Complex Exponentials

In Fourier and Z transforms it is common to encounter expressions such as $1 − e^{−j𝜃}$ . In some cases it is convenient to rewrite them as

1 − e^{−j𝜃} = \frac{e^{j𝜃∕2}}{e^{j𝜃∕2}} (1 − e^{−j𝜃}) = \frac{e^{j𝜃∕2} − e^{−j𝜃∕2}}{e^{j𝜃∕2}} = 2j e^{−j𝜃∕2} sin (𝜃∕2).

(A.14)

Similarly, one can write

1 + e^{−j𝜃} = \frac{e^{j𝜃∕2}}{e^{j𝜃∕2}} (1 + e^{−j𝜃}) = \frac{e^{j𝜃∕2} + e^{−j𝜃∕2}}{e^{j𝜃∕2}} = 2 e^{−j𝜃∕2} cos (𝜃∕2).

(A.15)