Signals and Systems - Electrical Engineering

22 C H A P T E R 0: From the Ground Up!

so thatj^0 =1,j^1 =j,j^2 =−1,j^3 =−j, and so on. Lettingj= 1 ejπ/^2 , we can see that the increasing

powers ofjn= 1 ejnπ/^2 are vectors with angles of 0 whenn=0,π/2 whenn=1,πwhenn=2,

and 3π/2 whenn=3. The angles repeat for the next four values, the four after that, and so on. See

Figure 0.11.

One operation possible with complex numbers that is not possible with real numbers iscomplex

conjugation. Given a complex numberz=x+jy=|z|ej∠zits complex conjugate isz∗=x−jy=

|z|e−j∠z—that is, we negate the imaginary part ofzor reflect its angle. This operation gives that

(i) z+z∗= 2 x or Re[z]=0.5[z+z∗]

(ii) z−z∗= 2 jy or Im[z]=0.5[z−z∗]

(iii) zz∗=|z|^2 or |z|=

√

zz∗

(iv)

z

z∗

=ej^2 ∠z or ∠z=−j0.5[log(z)−log(z∗)] (0.17)

The complex conjugation provides a different approach to the division of complex numbers in rect-

angular form. This is done by making the denominator a positive real number by multiplying both

numerator and denominator by the complex conjugate of the denominator. For instance,

z=

1 +j 1

3 +j 4

=

( 1 +j 1 )( 3 −j 4 )

( 3 +j 4 )( 3 −j 4 )

=

7 −j

9 + 16

=

7 −j

25

Finally, the conversion of complex numbers from rectangular to polar needs to be done with care,

especially when computing the angles. For instance,z= 1 +jhas a vector representing in the first

quadrant of the complex plane, and its magnitude is|z|=

√

2 while the tangent of its angleθis

tan(θ)=1 orθ=π/4 radians. Ifz=− 1 +j, the vector representing it is now in the second quadrant

with the same magnitude as before, but its angle is now

θ=π−tan−^1 ( 1 )= 3 π/ 4

That is, we find the angle with respect to the negative real axis and subtract it fromπ. Likewise, if

z=− 1 −j, the magnitude does not change but the phase is now

θ=π+tan−^1 ( 1 )= 5 π/ 4

which can also be expressed as− 3 π/4. Finally, whenz= 1 −j, the angle is−π/4 and the magnitude

remains the same. The conversion from polar to rectangular form is much easier. Indeed, given a

complex number in polar formz=|z|ejθits real part isx=|z|cos(θ)(i.e., the projection of the vector

corresponding tozonto the real axis) and the imaginary part isy=|z|sin(θ), so thatz=x+jy. For

instance,z=

√

2 e^3 π/^4 can be written as

z=

√

2 cos( 3 π/ 4 )+j

√

2 sin( 3 π/ 4 )=− 1 +j