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Trigonometric Fourier Series. Outline Introduction Visualization Theoretical Concepts Qualitative Analysis Example Class Exercise. Introduction. What is Fourier Series? Representation of a periodic function with a weighted, infinite sum of sinusoids. Why Fourier Series?

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trigonometric fourier series
Trigonometric Fourier Series
  • Outline
    • Introduction
    • Visualization
    • Theoretical Concepts
    • Qualitative Analysis
    • Example
    • Class Exercise
introduction
Introduction
  • What is Fourier Series?
    • Representation of a periodic function with a weighted, infinite sum of sinusoids.
  • Why Fourier Series?
    • Any arbitrary periodic signal, can be approximated by using some of the computed weights
    • These weights are generally easier to manipulate and analyze than the original signal
periodic function
Periodic Function
  • What is a periodic Function?
    • A function which remains unchanged when time-shifted by one period
      • f(t) = f(t + To) for all values of t
  • What is To

To

To

properties of a periodic function 1
Properties of a periodic function 1
  • A periodic function must be everlasting
    • From –∞ to ∞
  • Why?
  • Periodic or Aperiodic?
properties of a periodic function
Properties of a periodic function
  • You only need one period of the signal to generate the entire signal
    • Why?
  • A periodic signal cam be expressed as a sum of sinusoids of frequency F0 = 1/T0 and all its harmonics
visualization
Visualization

Can you represent this simple function using sinusoids?

Single sinusoid representation

visualization7

amplitude

New amplitude

amplitude

Fundamental frequency

2nd Harmonic

4th Harmonic

Visualization

To obtain the exact signal, an infinite number

of sinusoids are required

theoretical concepts

Period

Cosine terms

Sine terms

Theoretical Concepts
dc offset

A

-2

-1

0

1

2

A

-A

-1

0

1

2

-2

DC Offset

What is the difference between these two functions?

Average Value = 0

Average Value ?

dc offset11
DC Offset

If the function has a DC value:

qualitative analysis
Qualitative Analysis
  • Is it possible to have an idea of what your solution should be before actually computing it?

For Sure

properties dc value

-A

-1

1

2

A

A

-1

0

1

2

-2

Properties – DC Value
  • If the function has no DC value, then a0 = ?

DC?

DC?

properties symmetry

f(-t) = f(t)

A

0

π

π/2

3π/2

-A

f(-t) = -f(t)

A

0

π

π/2

3π/2

A

Properties – Symmetry
  • Even function
  • Odd function
properties symmetry15

Even

Even

Odd

Even

=

=

Odd

Even

Properties – Symmetry
  • Note that the integral over a period of an odd function is?

If f(t) is even:

X

X

properties symmetry16

Odd

Odd

Even

Odd

=

=

Odd

Even

Properties – Symmetry
  • Note that the integral over a period of an odd function is zero.

If f(t) is odd:

X

X

properties symmetry17
Properties – Symmetry
  • If the function has:
    • even symmetry: only the cosine and associated coefficientsexist
    • odd symmetry: only the sine and associated coefficientsexist
    • even and odd: both terms exist
properties symmetry18

-A

-1

1

2

A

Properties – Symmetry
  • If the function is half-wave symmetric, then only odd harmonics exist

Half wave symmetry: f(t-T0/2) = -f(t)

properties discontinuities

-A

-1

1

2

A

A

-1

0

1

2

-2

Properties – Discontinuities
  • If the function has
    • Discontinuities: the coefficients will be proportional to 1/n
    • No discontinuities: the coefficients will be proportional to 1/n2
  • Rationale:

Which function has discontinuities?

Which is closer to a sinusoid?

example

-A

-1

1

2

A

Example
  • Without any calculations, predict the general form of the Fourier series of:

DC?

Half wave

symmetry?

Yes, only odd harmonics

No, a0 = 0;

Discontinuities?

Symmetry?

No, falls of as

1/n2

Even, bn = 0;

Prediction an 1/n2 for n = 1, 3, 5, …;

example21

zero for n even

Example
  • Now perform the calculation
example22

DC?

Half wave

symmetry?

Yes, only odd harmonics

No, a0 = 0;

Discontinuities?

Symmetry?

No, falls of as

1/n2

Even, bn = 0;

Example
  • Now compare your calculated answer with your predicted form
class exercise

A

-1

0

1

2

-2

Class exercise
  • Discuss the general form of the solution of the function below and write it down
  • Compute the Fourier series representation of the function
  • With your partners, compare your calculations with your predictions and comment on your solution