Signals and Systems - Electrical Engineering

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

(b)Compute the derivativey(t)=dx(t)/dtand plot its real and imaginary parts. How do these relate to the

real and the imaginary parts ofx(t)?

(c)Compute the integral

∫^1

0

x(t)dt

(d)Would the following statement be true (remember∗indicates complex conjugate)?





∫^1

0

x(t)dt





∗

=

∫^1

0

x∗(t)dt

0.18. Euler’s equation and orthogonality of sinusoids

Euler’s equation,

ejθ=cos(θ)+jsin(θ)

is very useful not only in obtaining the rectangular and polar forms of complex numbers, but in many other

respects as we will explore in this problem.

(a) Carefully plotx[n]=ejπnfor−∞<n<∞. Is this a real or a complex signal?

(b)Suppose you want to find the trigonometric identity corresponding to

sin(α)sin(β)

Use Euler’s equation to express the sines in terms of exponentials, multiply the resulting exponentials,

and use Euler’s equation to regroup the expression in terms of sinusoids.

(c)As we will see later on, two periodic signalsx(t)andy(t)of periodT 0 are said to be orthogonal if the

integral over a periodT 0 is

∫

T 0

x(t)y(t)dt= 0

For instance, considerx(t)=cos(πt)andy(t)=sin(πt). Check first that these functions repeat every

T 0 = 2 (i.e., show thatx(t+ 2 )=x(t)and thaty(t+ 2 )=y(t)). Thus,T 0 = 2 can be seen as their period.

Then use the representation of a cosine in terms of complex exponentials,

cos(θt)=

ejθ+e−jθ

2

to express the integrand in terms of exponentials and calculate the integral.

0.19. Euler’s equation and trigonometric expressions

Obtain using Euler’s equation an expression forsin(θ)in terms of exponentials and then

(a) Use it to obtain the trigonometric identity forsin^2 (θ).

(b)Compute the integral

∫^1

0

sin^2 ( 2 πt)dt