Chapter Contents

• Fourier Series

• Helpful Revision

• 1. Overview of Fourier Series
• 2. Full Range Fourier Series
• 3. Fourier Series of Even and Odd Functions
• Fourier Series Graph Interactive
• 4. Fourier Series of Half Range Functions
• 5. Harmonic Analysis
• 6. Line Spectrum
• 7. Fast Fourier Transform
• Fourier Series Problem Solver

• Comments, Questions?

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Helpful Revision for Fourier Series

This page contains some background information that will help you to better understand this chapter on Fourier Series.

You have seen most of this before, but I have included it here to give you some help before getting into the heavy stuff.

Properties of Sine and Cosine Functions

These properties can simplify the integrations that we will perform later in this chapter.

The Cosine Function

The function `f(x) = cos\ x` is an even function. That is, it is symmetrical about the vertical axis.

We have: `cos(-x) = cos(x)`, and

`int_(-pi)^picos\ theta\ d theta=0`

The yellow "negative" portions of the graph when added to the green "positive" portion cancel each other out. They add to zero.

The Sine Function

The function `f(x) = sin\ x` is an odd function. That is, it is symmetrical about the origin.

We have: `sin(-x) = -sin(x)`, and

`int_(-pi)^pi sin\ theta\ d theta=0`

Once again, the yellow negative portion is the same size as the green positive portion, so the sum is 0.

Multiples of π for Sine and Cosine Curves

Consider the function `y = sin\ x`.

From the graph (or using our calculator), we can observe that:

`sin(nπ) = 0`	for n = 0, 1, 2, 3, ... (in fact, all integers)
`sin{:((2n-1)pi)/2:}=(-1)^(n+1`	for n = 0, 1, 2, 3, ... (in fact, all integers)

---

Next, we consider the curve y = cos x

`cos(2nπ) = 1`	for n = 0, 1, 2, 3, ... (in fact, all integers)
`cos[(2n − 1)π] = −1`	for n = 0, 1, 2, 3, ... (in fact, all integers)
`cos(nπ) = (−1)^n`	for n = 0, 1, 2, 3, ... (in fact, all integers)

Periodic Functions

A function `f(t)` is said to be periodic with period p if

`f(t + p) = f(t)`

for all values of t and if `p > 0`.

The period of the function `f(t)` is the interval between two successive repetitions.

Examples of Periodic Functions

1a. `f(t) = sin\ t`.

For `f(t) = sin\ t`, we have: `f(t) = f(t + 2π)`. The period is 2π.

1b. Saw tooth waveform, period `= 2`:

For this function, we have:

`f(t) = 3t` (for −1 ≤ t < 1)

`f(t) = f(t + 2)` [This expression indicates it is periodic with period `2`.]

1c. Parabolic, period `= 2`.

For this function, we have:

`f(t) = t^2` (for `0 ≤ t < 2`)

`f(t) = f(t + 2)` [Indicating it is periodic with period 2.]

1d. Square wave, period = 4.

For this function, we have:

`f(t) = -3` for `-1 ≤ t < 1` and `3` for `1 ≤ t < 3`

`f(t) = f(t + 4)` [The period is 4.]

NOTE: In this example, the period `p = 4`. We can write this as `2L = 4`.

In the diagram we are thinking of one cycle starting at ` −2` and finishing at `2`. For convenience when integrating later, we let `L = 2` and the cycle goes from `-L` to `L`.

Continuity

If a graph of a function has no sudden jumps or breaks, it is called a continuous function.

Examples:

• sine functions
• cosine functions
• exponential functions
• parabolic functions

Finite discontinuity - a function makes a finite jump at some point or points in the interval.

Examples:

• Square wave function
• Saw tooth functions

Split Functions

Much of the behaviour of current, charge and voltage in an AC circuit can be described using split functions.

Examples of Split Functions

Sketch the following functions:

2a. `f(t)={ {: (-t,if -pi<=t < 0),(t,if 0 <=t < pi) :}`

Answer

2b. `f(t)={ {:(t,if 0 <=t < pi), (t-pi, if pi <= t <2pi):}`

Answer

2c. `f(t)={ {: ((t+pi)^2,if -pi <=t <0),((t-pi)^2,if 0 <= t < pi) :}`

Answer

2d. `f(t)={(t+pi, if -pi <= t < -pi/2),(-1, if -pi/2 <= t < pi/2),(-t+pi, if pi/2 <= t < pi) :}`

Answer

Summation Notation

It is important to understand summation notation when dealing with Fourier series.

Examples

Expand the following and simplify where possible:

3a. `sum_(n=1)^3n/(n+1)`

Answer

3b. `sum_(n=1)^5(2n-1)`

Answer

3c. `sum_(n=1)^5n^2a_n`

Answer

3d. `sum_(n=1)^4(npit)/L`

Answer

Some Useful Integrals

The next 2 integrals are obtained from integration by parts and can be found in the Table of Common Integrals. We use them quite a bit in this Fourier Series chapter.

`int t\ sin\ nt\ dt=1/n^2(sin\ nt-nt\ cos\ nt)`

`int t\ cos\ nt\ dt=1/n^2(cos\ nt+nt\ sin\ nt)`