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# WRITTEN HOMEWORK ASSIGNMENT

KEVIN YOKOYAMA

Abstract. Digital signal processing is concerned with processing digital sig-

nals. Digital sound and images are manipulated using the methods of DSP.

1. Complex Numbers

The basic manipulations of complex numbers are reviewed in this section. The

algebraic rules for combining complex numbers are reviewed.

1.1. **Introduction. **A complex number system is an extension of the real
number

system. Complex numbers are necessary to solve equations such as

The symbol i is introduced to stand for
, so the previous equation (1.1) has

two solutions z =± i . More generally, complex numbers are needed to solve for

the two roots of a quadratic equation

which, according to the quadratic formula, has the two solutions:

1.2.** Notation for Complex Numbers**. Several
different mathematical nota-

tions can be used to represent complex numbers. The two basic types are polar

form and rectangular form. Converting between them quickly and easily is an

important skill.

1.2.1. Rectangular Form. In rectangular form, the
following notations define the

same complex number z:

The ordered pair (x, y) can be interpreted as a point in
the two-dimensional

plane. A complex number can also be drawn as a vector whose tail is located at

the origin and whose head is at the point (x, y), in which case x is the
horizontal

coordinate of the vector and y the vertical coordinate. The complex number z =

2 +5i is represented by the point (2, 5), which lies in the first quadrant.
Likewise,

z = 4− 3i lies in the fourth quadrant at the location (4, 3).

1.2.2. Polar Form. In polar form, the vector is defined by
its length (r) and its

direction ( θ ). The following descriptive notation is often used:

1.2.3. Conversion: Rectangular and Polar. Both the polar
and rectangular forms

are commonly used to represent complex numbers. Given z = r∠θ , the x
and y

coordinates are given by the following equations

Therefore, a valid formula for z is

**Theorem 1.1.** The sum of two or more cosine signals
each having the same fre-

quency, but having different amplitudes and phase shifts, can be expressed as a

single equivalent cosine signal

proof

This completes the proof.

1.3. **Sound Demo.** Run the MATLAB sound demo by
typing xpsound at the

MATLAB prompt. To generate a tone in MATLAB type in the following code:

1.4. **Phasor Demo. **The process of phasor addition
can be accomplished easily

using MATLAB. The function zprint was used to create the following output.

2. The Mandelbrot Set

2.1. **Introduction.** The Mandelbrot set is a subset of the parameter plane
for

iteration of the complex quadratic function .
Here the parameter

c is complex. The Mandelbrot set consists of those c values for which the orbit
of

0 is bounded.