Signals and Systems - Electrical Engineering

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

The relation between the complex exponentials and the sinusoidal functions is of great importance

in signals and systems analysis. Using Euler’s identity the cosine can be expressed as

cos(θ)=Re[ejθ]=

ejθ+e−jθ

2

(0.19)

while the sine is given by

sin(θ)=Im[ejθ]=

ejθ−e−jθ

2 j

(0.20)

Indeed, we have

ejθ=cos(θ)+jsin(θ)

e−jθ=cos(θ)−jsin(θ)

Adding them we get the above expression for the cosine, and subtracting the second from the first we

get the given expression for the sine. The variableθis in radians, or in the corresponding angle in

degrees (recall that 2πradians equals 360 degrees).

These relations can be used to define the hyperbolic sinusoids as

cos(jα)=

e−α+eα

2

=cosh(α) (0.21)

jsin(jα)=

e−α−eα

2

=−sinh(α) (0.22)

from which the other hyperbolic functions are defined. Also, we obtain the following expression for

the real-valued exponential:

e−α=cosh(α)−sinh(α) (0.23)

Euler’s identity can also be used to find different trigonometric identities. For instance,

cos^2 (θ)=

[

ejθ+e−jθ

2

] 2

=

1

4

[2+ej^2 θ+e−j^2 θ]=

1

2

+

1

2

cos( 2 θ)

sin^2 (θ)= 1 −cos^2 (θ)=

1

2

−

1

2

cos( 2 θ)

sin(θ)cos(θ)=

ejθ−e−jθ

2 j

ejθ+e−jθ

2

=

ej^2 θ−e−j^2 θ

4 j

=

1

2

sin( 2 θ)

0.4.3 Phasors and Sinusoidal Steady State

A sinusoidx(t)is a periodic signal represented by

x(t)=Acos( 0 t+ψ) −∞<t<∞ (0.24)

whereAis the amplitude, 0 = 2 πf 0 is the frequency in rad/sec, andψis the phase in radians. The

signalx(t)is defined for all values oft, and it repeats periodically with a periodT 0 = 1 /f 0 (sec), so