A.4  Manipulating complex exponentials

In Fourier and Z transforms it is common to encounter expressions such as 1 ej𝜃. In some cases it is convenient to rewrite them as

1 ej𝜃 = ej𝜃2 ej𝜃2(1 ej𝜃) = ej𝜃2 ej𝜃2 ej𝜃2 = 2jej𝜃2 sin (𝜃2).
(A.14)

Similarly, one can write

1 + ej𝜃 = ej𝜃2 ej𝜃2(1 + ej𝜃) = ej𝜃2 + ej𝜃2 ej𝜃2 = 2ej𝜃2 cos (𝜃2).
(A.15)