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Lecture 1: Complex Numbers

Reading assignments

Ch. 2
Appendix A: Complex Numbers

Quick Exercise -- Find the Roots

Find the roots of the following polynomials
a) ?
b) ?

The first polynomial has two real roots (one repeated twice). The second polynomial has two imaginary roots.
→ Recall the fundamental theorem of algebra

Complex Numbers

Complex numbers are a powerful way to represent signals

A complex number can be represented in the form

Where

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Note

Some polynomials have no real roots (e.g. ), but all order-N polynomials have N (possibly complex, possible repeated) roots.

An aside on j, the imaginary unit

Recall that this pattern repeats as power is incremented by 1

Operations with Complex Numbers

Complex Addition = Vector Addition

Complex addition can be performed the same way as vector addition
Add the real and imaginary components separately

Exercise

Add to

Solution
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Complex Multiply = Vector Rotation

Multiplication/division scales and rotates vectors
When dividing, magnitude is divided and phase is subtracted

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Powers

Involves repeated multiplication
Rotating by equal angle size

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Bottom line

Cartesian: add, subtract
Polar: multiply, divide, power

Euler’s Formula and Euler’s Identity

Consider: What is the most beautiful equation?

Google search yields the answer
This equation is called Euler’s Identity
Operations with two irrational numbers, resulting in 0

Polynomial Approximations

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Additional proof using power-series expansions. Note for ECE2026, we prefer to use the notation

{\displaystyle {\begin{aligned}e^{ix}&=1+ix+{\frac {(ix)^{2}}{2!}}+{\frac {(ix)^{3}}{3!}}+{\frac {(ix)^{4}}{4!}}+{\frac {(ix)^{5}}{5!}}+{\frac {(ix)^{6}}{6!}}+{\frac {(ix)^{7}}{7!}}+{\frac {(ix)^{8}}{8!}}+\cdots \\[8pt]&=1+ix-{\frac {x^{2}}{2!}}-{\frac {ix^{3}}{3!}}+{\frac {x^{4}}{4!}}+{\frac {ix^{5}}{5!}}-{\frac {x^{6}}{6!}}-{\frac {ix^{7}}{7!}}+{\frac {x^{8}}{8!}}+\cdots \\[8pt]&=\left(1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+{\frac {x^{8}}{8!}}-\cdots \right)+i\left(x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots \right)\\[8pt]&=\cos x+i\sin x,\end{aligned}}}

Euler’s Formula: An Unexpected Triangle

The figure below depicts:

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Graphical representation of a vector normalized to magnitude of 1. Applying trig to find components.

Euler’s formula

Generalized form with a general magnitude

Change of coordinates
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Changing to and from rectangular and polar forms. Note you need to be mindful of which quadrant the vector is in when using arctan

Polar and Rectangular Coordinates -- Exercise

Put these common values in polar and rectangular form.
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Review unit circle if needed

Phase Ambiguity

Due to periodicity, values of phase that are apart are equivalent. For ECE2026, phase will be constrained to

Additionally, make magnitude a positive real number by adding/subtracting from the phase

Complex conjugate (z*)

Definition: Complex Conjugate

To obtain the complex conjugate of a vector , change the sign of all ‘s

In other words, we change the sign of the imaginary component

Rectangular form

Polar form

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Complex conjugation reflects the vector about the real axis

Aside on cos() and sin()

Note how cosine is an even function. We can observe that the real component (obtained through cosine) is not affected when we find the complex conjugate.

Uses of Conjugation

When dividing two complex numbers in rectangular form, we may make the denominator real
- Multiply both the numerator and denominator by the complex conjugate of the denominator
- Recall:

Using Conjugates: Inverse Euler⭐⭐⭐

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We can see the connection between sinusoids and complex numbers.

Important

In other words…

yields
yields

Magnitude and Magnitude Squared ⭐⭐⭐

Magnitude squared

Magnitude of complex exponential is one

Roots of Unity

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Recall power operations (repeated multiplication – scaling and rotating)

How many solutions to

According to the funamental theorem of algebra, N solutions!

Note: As angles are periodic, we account for that using

We can solve for r to be 1, as the magnitiude of the complex number is 1
We set to solve for

Obtaining the result ⭐

Roots of Unity for N=6

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N equally spaced vectors on unit circle
What happens if we add them?
- Looks like the answer is zero for the case of N=6

Sum of Roots of Unity

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Write as geometric sum
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Roots of Unity – Example Problem

Find values of for which:

Hint: Let . We now have the equation to solve.

Roots of Unity – General Form of Solution

important ⭐⭐⭐

- Therefore,
- and

We now have the equation

Hooray we have both magnitude and phase

Example

Solve for , given

Answer: r = 1,

Note how contains an initial angle/changes when k = 0. This as a result rotates all solutions.

Next Lecture: 05.18 Lecture - ECE2026

Lecture 1: Complex Numbers

Quick Exercise -- Find the Roots

Complex Numbers

Operations with Complex Numbers

Complex Addition = Vector Addition

Exercise

Complex Multiply = Vector Rotation

Powers

Euler’s Formula and Euler’s Identity

Polynomial Approximations

Euler’s Formula: An Unexpected Triangle

Polar and Rectangular Coordinates -- Exercise

Complex conjugate (z*)

Uses of Conjugation

Using Conjugates: Inverse Euler⭐⭐⭐

Magnitude and Magnitude Squared ⭐⭐⭐

Roots of Unity

Roots of Unity for N=6

Sum of Roots of Unity

Roots of Unity – Example Problem

Roots of Unity – General Form of Solution

Graph

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