Introduction to the complex-number system - Continuous and Discrete Time Signals and Systems - page 802

where r represents the magnitude or length of the vector r obtained by mapping
the complex number x onto the Cartesian plane. The length r and angle θ
associated with vector r are obtained from Eqs (B.14) and (B.15) with r x = a
and r y = b . We demonstrate the conversion of a complex number from one
representation to another with a series of examples.

Example B.2
Converting rectangular format into polar format Consider a complex num-
ber x = 2 + j4. Clearly, x is represented in the rectangular format. To derive its
equivalent polar format, we map the complex number into the Cartesian plane
and calculate the parameters r and θ . Using Eqs (B.14) and (B.15), we obtain

√

r = 2 2 + 4 2 = 20

and

− 1 (4 / 2) = 0 . 35 π radians .

θ = tan

= √ 20e j0 . 35 π . The polar representation of x = 2 + j4 is x

Example B.3
Converting polar format into rectangular format

Consider a complex num-

4e j π/ 3 . The rectangular representation of x is derived ber in the polar format x =

using Eq. (B.13) as

π
3

a = r x = 4 cos = 2

and

√

π
3

b = r y = 4 sin = 2 3 .

√

4e j π/ 3 is x

The rectangular representation of x = = 2 + j2 3.

In terms of polar representations, the basic arithmetic operations between two
complex numbers x 1

= r 1 e j θ 1 = r 2 e j θ 2 and x 2 are defined as follows.

B.4.1 Addition

Addition of two complex numbers in polar format:

= r 1 e j θ 1 + r 2 e j θ 2 x 1 + x 2 = ( r 1 cos θ 1 + j r 1 sin θ 1 ) + ( r 2 cos θ 2 + j r 2 sin θ 2 )

= ( r 1 cos θ 1 + r 2 cos θ 2 ) + j ( r 1 sin θ 1 + r 2 sin θ 2 )

+ r 2 cos θ 2 ) 2 + ( r 1 sin θ 1 + r 2 sin θ 2 ) 2

= ( r 1 cos θ 1

r 1 sin θ 1 + r 2 sin θ 2

− 1

exp j tan

r 1 cos θ 1 + r 2 cos θ 2

r 1 + r 2 = + 2 r 1 r 2 cos( θ 1 − θ 2 )

r 1 sin θ 1 + r 2 sin θ 2

− 1

exp j tan . (B.18)

r 1 cos θ 1 + r 2 cos θ 2