EEE 3396 Chapter Notes - Chapter 9: Fourier-Transform Infrared Spectroscopy, List Of Trigonometric Identities, Square Wave

50 views11 pages
Fourier Series & Fourier Transforms
nicholas.harrison@imperial.ac.uk
19th October 2003
Synopsis
Lecture 1 :
Review of trigonometric identities
Fourier Series
Analysing the square wave
Lecture 2:
The Fourier Transform
Transforms of some common functions
Lecture 3:
Applications in chemistry
FTIR
Crystallography
Bibliography
1. The Chemistry Maths Book (Chapter 15), Erich Steiner, OUP, 1996.
1
Unlock document

This preview shows pages 1-3 of the document.
Unlock all 11 pages and 3 million more documents.

Already have an account? Log in
Introduction
Chemistry often involves the measurement of properties which are the aggregate
of many fundamental processes. A variety of techniques have been developed for
extracting information about these underlying processes. Fourier analysis is one
of the most important and is very widely used - eg: in crystallography, X-ray
adsorbtion spectroscopy, NMR, vibrational spectroscopy (FTIR) etc.. As it involves
decomposition of functions into partial waves it also appears in many quantum
mechanical calculations.
A Little Trigonometry
You will need to be able to manipulate sin() and cos() in order to understand
Fourier analysis - a good understanding of the UK's A-level Pure Maths syllabus is
sucient. Here is a brief reminder of some important properties.
Units angles are typically measured in radians: 0360ois equivalent to 02π
radians
Cos and sin curves look like this:
2
Unlock document

This preview shows pages 1-3 of the document.
Unlock all 11 pages and 3 million more documents.

Already have an account? Log in
Both sin(x) and cos(x) are periodic on the interval 2πand integrate to 0 over a full
period, ie:
Z+π
π
cos (x)dx =Z+π
π
sin (x)dx = 0
Wavelength
It should be clear that sin(2x) repeats on the interval 0πand sin(3x) on the
interval 02π/3etc. In general sin(nx) and cos(nx) repeat on the interval 0
2π/n. The repeat distance is the wavelength λand so in general, λ= 2π/n.
The discrete family of functions sin(nx), cos(nx) are all said to be commensurate
with the period 2π- that is, they all have wavelengths which divide exactly into 2π.
The function sin(kx) for some real number k has an arbitrary wavelength λ= 2π/k.
kis usually referred to as the wavevector.
Note: A simple Mathermatica notebook, trig_1.nb, is provided with the course
and can be used to play with sin and cos functions.
Fourier Series
The idea of a Fourier series is that any (reasonable) function, f(x), that is peri-
odic on the interval 2π(ie: f(x+ 2πn) = f(x)for all n) can be decomposed into
contributions from sin(nx) and cos(nx).
The general Fourier series may be written as:
f(x) = a0
2+a1cos (x) + a2cos (2x) + a3cos (3x) + . . . +ancos (nx)
+b1sin (x) + b2sin (2x) + b3sin (3x) + . . . +bnsin (nx)(1)
Note:
1. cos (nx)and sin (nx)are periodic on the interval 2πfor any integer n.
2. The anand bncoecients measure the strength of contribution from each
harmonic.
Orthogonality
The functions cos (nx)and sin (nx)can be used in this way because they satisfy the
following orthogonality conditions:
3
Unlock document

This preview shows pages 1-3 of the document.
Unlock all 11 pages and 3 million more documents.

Already have an account? Log in

Document Summary

T r : tr tr t t s, r r r s, s t sq r . T r : r r r s r , r s r s s t s. S r t s tr s r t s tr s t s t s. Ts s r t s r r s 0 360o s q t t 0 2 r s. T s s r r t t r 2 t r t t r . R cos (x) dx = z + . T s r t t s r ts t t r 0 s t . T r 0 2 /3 t r s s r t t t r 0 . 2 /n r t st s t t s r = 2 /n . T t r 2 t t s t t s t t 2 .

Get access

Grade+20% off
$8 USD/m$10 USD/m
Billed $96 USD annually
Grade+
Homework Help
Study Guides
Textbook Solutions
Class Notes
Textbook Notes
Booster Class
40 Verified Answers
Class+
$8 USD/m
Billed $96 USD annually
Class+
Homework Help
Study Guides
Textbook Solutions
Class Notes
Textbook Notes
Booster Class
30 Verified Answers

Related Documents