Signals and Systems - Electrical Engineering

0.4 Complex or Real? 21

FIGURE 0.11
(a) Representation of a complex numberzby a
vector (b) addition of complex numberszandv;
(c) integer powers ofj; and (d) complex conjugate.

(x, y)

θ

|z|

x

(a)

(c) (d)

y

z+v

(b)

v z

− 1 =j^2 , j^6 , ...

1 =j^0 , j^4 , ...

j=j^1 , j^5 , ...

−j=j^3 , j^7 , ...

(x, y)

(x, −y)

θ

−θ

|z|

|z|

Using their polar representations requires a geometric interpretation: the addition of vectors (see

Figure 0.11). On the other hand, the multiplication ofzandvis easily done using their polar

forms as

zv=|z|ej∠z|v|ej∠v=|z||v|ej(∠z+∠v)

but it requires more operations if done in the rectangular form—that is,

zv=(x+jy)(p+jq)=(xp−yq)+j(xq+yp)

It is even more difficult to obtain a geometric interpretation. Such an interpretation will be seen

later on. Addition and subtraction as well as multiplication and division can thus be done more

efficiently by choosing the rectangular and the polar representations, respectively. Moreover, the polar

representation is also useful when finding powers of complex numbers. For the complex variable

z=|z|e∠z, we have that

zn=|z|nejn∠z

forninteger or rational. For instance, ifn=10, thenz^10 =|z|^10 ej^10 ∠z, and ifn= 3 /2, thenz1.5=

(

√

|z|)^3 ej1.5∠z. The powers ofjare of special interest. Given thatj=

√

−1 then, we have

jn=(− 1 )n/^2 =

{

(− 1 )m n= 2 m, neven

(− 1 )mj n= 2 m+1, nodd