Fourier Series: BIOE40004 Lecture 4

• This lecture is on Fourier series
• It is lecture 4 in Spring term 2025
• Courses: BIOE40004 – Mathematics 1 and BIOE40005 – Mathematics and Engineering 1 Delta
• The lecturer is Dr Amy Howard
• Engage with GTAs during study groups to get help
• Office hours are Tuesdays at 12pm, RSM 4.37
• Email to contact : [email protected]
• Solutions to study group questions are available Monday evenings
• Solutions to questions in lecture are available Tuesday evenings
• Practice exam-style questions can be found in SG and via BlackBoard

Formula Sheet

• Formulas from the delta course are available on the formula sheet
• Taylor series, Fourier series, Fourier transform and inverse Fourier transform, Parseval's identity, trig identities, and integral calculus topics are included
• This is available on Blackboard and in exams

Remainder Term of Taylor Series

• The remainder term, 𝑅𝑛 , describes the error from approximating a function using a Taylor series about point 𝑎 with a finite number of terms
• 𝑓 𝑥 = ෍ ℎ𝑘/𝑘! 𝑓𝑘(𝑎) + 𝑅𝑛 ℎ from n=0 to infinity
• 𝑅𝑛 ℎ = 𝑓𝑛+1(𝑐) ∗ ℎ𝑛+1/(𝑛+1)! where 𝑎 ≤ 𝑐 ≤ 𝑥
• 𝑅𝑛 ℎ = 𝑓𝑛+1(𝑎 + 𝜃ℎ) ∗ ℎ𝑛+1/(𝑛+1)! where 0 ≤ 𝜃 ≤ 1
• ℎ = 𝑥 − 𝑎
• The (n + 1)th function derivative is not evaluated at 𝑥 = 𝑎, but at an unknown point, 𝑐, between a and x (𝑎 ≤ 𝑐 ≤ 𝑥)
• The max value of the remainder term can be found if the range of possible points is known

Learning Objectives

• After this course, students should be able to:
• Describe periodic, odd and even functions
• Approximate periodic functions using the Fourier series, using odd/even behavior to simplify the calculation of Fourier coefficients
• Find the odd/even periodic expansions of a function and compute its Fourier sine/cosine series representation -Comment on the convergence of the Fourier series expansion
• Assert the conservation of energy using the Parseval identity

Text Book Reference

• See K.A. Stroud Engineering Mathematics

Periodic Functions

• A periodic function repeats itself at regular intervals
• 𝑓 𝑥 = 𝑓(𝑥 + 𝑘𝑇) when a function is periodic with period T, where k is any integer
• The function integrated over period T will always be the same, regardless of the starting point 𝑥0
• ∫0𝑇 𝑓 𝑥 𝑑𝑥 = ∫𝑥0 𝑥0+𝑇 𝑓 𝑥 𝑑𝑥, ∀𝑥0 ∈ [0, 𝑇]

Odd and Even Functions

• A function is odd when 𝑓 −𝑥 = −𝑓(𝑥)
• It is identical when rotated 180° about the origin
• A function is even when 𝑓 −𝑥 = 𝑓(𝑥)
• It is symmetric about the y-axis
• Odd functions on [-L,L] integrate to 0 ∫−𝐿 𝐿 𝑓 𝑥 𝑑𝑥 = 0
• Even functions on [-L,L] integrate to twice the integral on on [0,L] ∫−𝐿 𝐿 𝑓 𝑥 𝑑𝑥 = 2 ∫0 𝐿 𝑓 𝑥 𝑑𝑥
• Even × Even = Even
• Odd × Odd = Even
• Even × Odd = Odd
• ∫−𝐿 𝐿 𝐸𝑣𝑒𝑛 x 𝑂𝑑𝑑 𝑑𝑥 = 0
• ∫−𝐿 𝐿 𝐸𝑣𝑒𝑛 x 𝐸𝑣𝑒𝑛 𝑑𝑥 = 2 ∫0 𝐿 𝑓 𝑥 𝑑𝑥
• ∫−𝐿 𝐿 𝑂𝑑𝑑 x 𝐸𝑣𝑒𝑛 𝑑𝑥 = 0
• ∫−𝐿 𝐿 𝑂𝑑𝑑 x 𝑂𝑑𝑑 𝑑𝑥 = 2 ∫0 𝐿 𝑓 𝑥 𝑑𝑥

Fourier Series Analysis

• A Fourier series expansion decomposes a periodic function into sine and cosine sums
• Simplifies analysis by breaking down complex signals into their fundamental frequency components

Square Wave

• Square wave - periodic wave

Other waves

• Approximations of square wave shown with 4/π sin 𝑥

• Approximations of square wave shown with 4/π (sin 𝑥 + 1/3 sin 3𝑥)

• Approximations of square wave shown with 4/π (sin 𝑥 + 1/3 sin 3𝑥)

• Approximations of square wave shown with 4/π (sin 𝑥 + 1/3 sin 3𝑥 + 1/5 sin 5𝑥)

• Approximations of square wave shown with 4/π ∑ sin nx / n, from n=1,3,5... up to 19

• The lowest frequency (n=1) is called the fundamental or first harmonic

Fourier expansion

• Less good at discontinuities, converges to midpoint
• There is also Gibbs ringing at sharp transitions (over/undershooting doesn't go away)
• An odd function implies only sine terms
• An even function would imply only cosine terms
• Otherwise there is a mixture

Fourier Series Definition

• Function 𝑓(𝑥) in the interval (−𝐿, 𝐿) where 𝐿 > 0, that can be defined outside the interval as well and has a periodic nature with period 2L
• Then its Fourier series expansion is 𝑓 𝑥 = 𝑎0/2 + ෍ 𝑎𝑛 cos 𝑛𝜋𝑥/𝐿 + 𝑏𝑛 sin 𝑛𝜋𝑥/𝐿, from n=1 to infinity
• The numbers a0, an and bn are called Fourier coefficients
• 𝑎0 = 1/𝐿 ∫−𝐿 𝐿 𝑓 𝑥 𝑑𝑥
• 𝑎𝑛 = 1/𝐿 ∫−𝐿 𝐿 𝑓 𝑥 cos 𝑛𝜋𝑥/𝐿 𝑑𝑥 𝑛 = 1,2,3, …
• 𝑏𝑛 = 1/𝐿 ∫−𝐿 𝐿 𝑓 𝑥 sin 𝑛𝜋𝑥/𝐿 𝑑𝑥 𝑛 = 1,2,3, …
• The function 𝑓(𝑥) and its derivatives are piecewise continuous in the interval [−𝐿, 𝐿]

Orthogonality

• Fourier coefficients come from integrating both sides over period –L,L
• ∫−𝐿 𝐿 𝑓 𝑥 𝑑𝑥 = ∫−𝐿 𝐿 𝑎0/2 + ∑ 𝑎𝑛 cos 𝑛𝜋𝑥/𝐿 + 𝑏𝑛 sin 𝑛𝜋𝑥/𝐿 𝑑𝑥, from n=0 to infinity
• Integrals of sine and cosine over one full period are zero: ∫−𝐿 𝐿 𝑓 𝑥 𝑑𝑥 = ∫−𝐿 𝐿 𝑎0/2 𝑑𝑥 = 2𝐿/2 =𝐿
• Multiply both sides by cos 𝑚𝜋𝑥/𝐿 and integrate to find a𝑛
• ∫−𝐿 𝐿 𝑓 𝑥 cos 𝑚𝜋𝑥/𝐿 𝑑𝑥 = ∫−𝐿 𝐿 𝑎0/2 + ∑ 𝑎𝑛 cos 𝑛𝜋𝑥/𝐿 + 𝑏𝑛 sin 𝑛𝜋𝑥/𝐿 cos 𝑚𝜋𝑥/𝐿 𝑑𝑥 from n=1 to infinity
• Orthogonality properties of sin and cosine: ∫−𝐿 𝐿sin 𝑛𝜋𝑥/𝐿 cos 𝑚𝜋𝑥/𝐿 𝑑𝑥 = 0, for all 𝑚, 𝑛 ∫−𝐿 𝐿 cos 𝑛𝜋𝑥/𝐿 cos 𝑚𝜋𝑥/𝐿 𝑑𝑥 = 𝐿 when 𝑚 = 𝑛 and 0 otherwise
• Only the m=n cosine term survives, and the recovered identity is: ∫−𝐿 𝐿 𝑓 𝑥 cos 𝑚𝜋𝑥/𝐿 𝑑𝑥 = 𝑎𝑚 𝐿 and 𝑎𝑛 = 1/𝐿 ∫−𝐿 𝐿 𝑓 𝑥 cos𝑛𝜋𝑥/𝐿 𝑑𝑥 𝑛 = 1,2,3, ...
• To find 𝑏𝑛 , multiply both sides by sin 𝑚𝜋𝑥/𝐿 and integrate
• ∫−𝐿 𝐿 𝑓 𝑥 sin 𝑚𝜋𝑥/𝐿 𝑑𝑥 = ∫−𝐿 𝐿 𝑎0/2 + ∑ 𝑎𝑛 cos 𝑛𝜋𝑥/𝐿 + 𝑏𝑛 sin 𝑛𝜋𝑥/𝐿 sin 𝑚𝜋𝑥/𝐿 𝑑𝑥 from n=1 to infinity
• ∫−𝐿 𝐿 sin 𝑛𝜋𝑥/𝐿 cos 𝑚𝜋𝑥/𝐿 𝑑𝑥 = 0, for all 𝑚, 𝑛
• ∫−𝐿 𝐿 sin 𝑛𝜋𝑥/𝐿 sin 𝑚𝜋𝑥/𝐿 𝑑𝑥 = 𝐿 when 𝑚 = 𝑛 and 0 otherwise
• Only the m=n cosine term survives, and the recovered identity is: ∫−𝐿 𝐿 𝑓 𝑥 sin 𝑚𝜋𝑥/𝐿 𝑑𝑥 = 𝑏𝑚 𝐿 and 𝑏𝑛 = 1/𝐿 ∫−𝐿 𝐿 𝑓 𝑥 sin 𝑛𝜋𝑥/𝐿 𝑑𝑥 𝑛 = 1,2,3, ...

Piecewise Continuous Definition

• A function is continuous on the interval except for a finite number of discontinuities where the one-sided limits exist and are not infinite

Summary of Methodology – 1 to find the Fourier series expansion of a function.

• Roughly plot the function in the interval of interest. -Check if the function fand its first derivative are piecewise continuous in the interval of interest.
• Calculate the Fourier coefficients.
• Form the Fourier series expansion. -Check and exclude plausible points of the interval in which the expansion is not defined (e.g. dividing by 0).

Fourier Series on a different interval

• For 𝑓=𝑓 𝑥 where 0 ≤ 𝑥 ≤ 2𝐿, the period is 2L
• 𝑎0 = 1/𝐿 ∫0 2𝐿 𝑓 𝑥 𝑑𝑥
• 𝑎𝑛 = 1/𝐿 ∫0 2𝐿 𝑓 𝑥 cos 𝑛𝜋𝑥/𝐿 𝑑𝑥
• 𝑏𝑛 = 1/𝐿 ∫0 2𝐿 𝑓(𝑥) sin 𝑛𝜋𝑥/𝐿 𝑑𝑥

Periodic Expansion Definition

• The periodic expansion 𝐹 𝑥 of a function 𝑓(𝑥) defined in the interval (−𝐿, 𝐿), 𝐿 > 0, is defined as: 𝐹 𝑥 =𝑓 𝑥 , −𝐿 < 𝑥 < 𝐿 𝐹 𝑥 + 2𝐿 = 𝐹 𝑥 , ∀𝑥 ∈ ℜ

Odd and Even Expansion Definitions

• Consider a function 𝑓(𝑥), defined in the interval (0, 𝐿), 𝐿 > 0.
• Even Expansion: 𝑓𝑒𝑣𝑒𝑛(𝑥) = 𝑓(𝑥), 0≤𝑥L
• Odd Expansion: 𝑓𝑜𝑑𝑑(𝑥) = 𝑓(𝑥), 0≤𝑥L

Sine & Cosine Series Definitions

• 𝑓 𝑥 = 𝑎0/2 + ∑ 𝑎𝑛 cos𝑛𝜋𝑥/𝐿 + 𝑏𝑛 sin 𝑛𝜋𝑥/𝐿 from n=1 to infinity
• The Fourier series expansion of an even function 𝑓 (𝑥) contains only cosine terms and is called the Fourier cosine series expansion of 𝑓(x)
• The Fourier series expansion of an odd function 𝑓 (𝑥) contains only sine terms and is called the Fourier sine series expansion of 𝑓(x) and an=0

Summary of Methodology – 2

In order to find the sine or cosine Fourier series expansion of a function defined in the interval (0, L):

• Expand the function over the interval (-L, L) such that the function is even / odd for cosine / sine expansion, respectively.
• Expand the new, even / odd function periodically in R.
• Calculate the Fourier coefficients. -Form the cosine / sine Fourier series expansion.
• Check and exclude plausible points of the interval in which the expansion is not defined (e.g. division by 0 for some n).

Convergence of Fourier Series

• If a function 𝑓 (𝑥) has a Fourier series expansion of the form 𝑓(x) = 𝑎0/2 + ∑ 𝑎𝑛 cos (𝑛𝜋𝑥/𝐿) + 𝑏𝑛 sin (𝑛𝜋𝑥/𝐿) where 𝑎𝑛 and 𝑏𝑛 are the Fourier coefficients of 𝑓 (𝑥) in (−𝐿, 𝐿),
• Then the Fourier series converges to the original function 𝑓(𝑥) at points of continuity and the average of the two limits at points of discontinuity, 𝑥0.
• [𝑓(𝑥) + 𝑓(𝑥)], ∀𝑥 ∈ ℜ
• Series converges to midpoint for discontinuity
• Odd finction means 𝑆𝑖𝑛𝑒 series

Parseval Identity

• Asserts the total energy (or power) of the signal is conserved between the spatial and frequency domain.
• The integral of the function squared = the sum of the squared Fourier coefficients
• Can figure out the sum of series ∫−𝐿 𝐿(𝑓(𝑥))2𝑑𝑥 = 𝑎0/2+∑(𝑎𝑛2 +𝑏𝑛2) from n=1 to infinity