Signals and Systems - Electrical Engineering

Problems 61

0.20. De Moivre’s theorem for roots
Consider the calculation of roots of an equation,

zN=α

whereN≥ 1 is an integer andα=|α|ejφa nonzero complex number.

(a) First verify that there are exactlyNroots of this equation and that they are given by

zk=rejθk

wherer=|α|^1 /Nandθk=(φ+ 2 πk)/Nfork=0, 1,...,N− 1.

(b)Use the above result to find the roots of the following equations:

z^2 = 1

z^2 =− 1

z^3 = 1

z^3 =− 1

and plot them in a polar plane (i.e., indicating their magnitude and phase).

(c)Explain how the roots are distributed around a circle of radiusrin the complex polar plane.

0.21. Natural log of complex numbers
Suppose you want to find the log of a complex numberz=|z|ejθ. Its logarithm can be found to be

log(z)=log(|z|ejθ)=log(|z|)+log(ejθ)=log(|z|)+jθ

Ifzis negative it can be written asz=|z|ejπand we can findlog(z)by using the above derivation. The log

of any complex number can be obtained this way also.

(a) Justify each one of the steps in the above equation.

(b) Find

log(− 2 )

log( 1 +j 1 )

log( 2 ejπ/^4 )

0.22. Hyperbolic sinusoids—MATLAB
In filter design you will be asked to use hyperbolic functions. In this problem we relate these functions to
sinusoids and obtain a definition of these functions so that we can actually plot them.
(a) Consider computing the cosine of an imaginary number—that is, use

cos(x)=

ejx+e−jx

2

Letx=jθand findcos(x). The resulting function is called the hyperbolic cosine or

cos(jθ)=cosh(θ)

(b) Consider then the computation of the hyperbolic sinesinh(θ); how would you do it? Carefully plot it

as a function ofθ.