Digital Signal Processing Reference
In-Depth Information
Equation (D.15) gives a procedure for conversion from the exponential form
of a complex number to the rectangular form. We now develop a procedure for
converting from the rectangular form to the exponential form. From (D.15), with
s = a + ib,
a
2
+ b
2
= A
2
cos
2
u + A
2
sin
2
u = A
2
,
and thus,
A = (a
2
+ b
2
)
1/2
= ([Re(s)]
2
+ [Im(s)]
2
)
1/2
.
(D.16)
In addition,
b
a
=
A sin u
A cos
u
= tan u,
Im
b
A
a
Re
Figure D.3
Rectangular form and
exponential form.
or
a
= tan
-1
Im(s)
u = tan
-1
b
Re(s)
.
(D.17)
The relationship of the rectangular form to the exponential form is illustrated in the
complex plane in Figure D.3 and is quadrant dependent.
