Signals and Systems - Electrical Engineering

Problems 59

0.14. Euler’s identity and trigonometric identities
Use Euler’s identity to find an expression forcos(α)cos(β), and from the relation between cosines and sines
obtain an expression forsin(α)sin(β).

0.15. Algebra of complex numbers
(a) The complex conjugate ofz=x+jyisz∗=x−jy. Using these rectangular representations, show that

zz∗=x^2 +y^2

1

z

=

z∗

zz∗

(b) Show that it is easier to find the above results by using the polar representationz=|z|ejθofzwhere

|z|=

√

x^2 +y^2

is the magnitude ofzand

θ=tan−^1

(y

x

)

is the angle or phase ofz. Thus, whenever we are multiplying or dividing complex numbers the polar

form is more appropriate.

(c) Whenever we are adding or subtracting complex numbers the rectangular representation is more

appropriate. Show that for two complex numbersz=x+jyandw=v+jq; then,

(z+w)∗=z∗+w∗

On the other hand, when showing that(zw)∗=z∗w∗the polar form is more appropriate.

(d) If the above conclusions still do not convince you, consider then the case of multiplying two complex

numbers:

z=rcos(θ)+jrsin(θ)

w=ρcos(φ)+jρsin(φ)

Find the polar forms ofzandwand then findzwby using the rectangular and then the polar forms

and decide which is easier. As a bonus you should get the trigonometric identities forcos(θ+φ)and

sin(θ+φ). What are they?

0.16. Vectors and complex numbers
Using the vectorial representation of complex numbers it is possible to get some interesting inequalities:
(a) Is it true that for a complex numberz=x+jy:

|x|≤|z|?

Show it geometrically by representingzas a vector.

(b) The so-calledtriangle inequalitysays that for any complex (or real) numberszandvwe have that

|z+v|≤|z|+|v|

Show a geometric example that verifies this.

0.17. Complex functions of time—MATLAB
Consider the complex functionx(t)=( 1 +jt)^2 for−∞<t<∞.
(a) Find the real and the imaginary parts ofx(t)and carefully plot them with MATLAB. Try to make
MATLAB plotx(t)directly. What do you get? Does MATLAB warn you? Does it make sense?