Show that 1 exp ja 2 sin a2 exp ja pi2 we will use this p

Show that 1 - exp (ja) = 2 sin (a/2) exp (j[a - pi]/2) we will use this property in the part of the course that covers Fourier transforms.

Solution

Write 1-exp(ja) = exp(ja/2)(exp(-ja/2) - exp(ja/2)).

Recall exp(jx) = cos(x) + jsin(x) so that exp(ja/2) = cos(a/2) + jsin(a/2) and exp(-ja/2) = cos(a/2) - jsin(a/2). Then (exp(-ja/2) - exp(ja/2)) = -2jsin(a/2).

Now recall that -j = cos(-pi/2) + j sin(-pi/2) = exp(-pi/2 j) ==> -2jsin(a/2) = 2sin(a/2)exp(-pi/2 j).

So 1-exp(ja) = 2sin(a/2)exp(-pi/2 j)exp(ja/2) = 2sin(a/2)exp(j(a-pi)/2)