EEE 3396 Chapter Notes - Chapter 9: Fourier-Transform Infrared Spectroscopy, List Of Trigonometric Identities, Square Wave
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.
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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
sucient. Here is a brief reminder of some important properties.
Units angles are typically measured in radians: 0−360ois equivalent to 0−2π
radians
Cos and sin curves look like this:
2
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 0→πand sin(3x) on the
interval 0→2π/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 bncoecients 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:
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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 .