Applications of Fourier Analysis (F32SMS):Lecture Notes Set 1A Synoptic Module∗

Philip Moriarty

School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, UK

(Dated: January 27, 2007)

Fourier analysis impacts on practically every area of physics. A knowledge of Fourier seriesand Fourier transforms thus forms an essential component of your training as a physicist. Aparticular goal of this module is to impress upon you the remarkable ubiquity of Fourier analysis.Although mathematics - and, in particular, the solution of many integrals - will form a sizeablecomponent of the module, a key objective is to teach you to look beyond the mathematics anddevelop an intuition for the relationships between waveforms/signals in the time domain andtheir counterparts in the frequency domain. (Or, alternatively, between real space and reciprocalspace structures).In the Applications of Fourier Analysis lecture course we will thus focus on thepractical uses of Fourier methods across a range of scientific disciplines.

NB This set of notes covers lectures 1 - 4. Another set of notes covering theremainder of the course will be handed out (and will be available from the modulewebsite) on March 5.

Contents

I. Module Overview 1A. What Will We Cover And Why Is It So Important? 1B. Module Organisation And What Is Expected Of You 2

1. Lectures 22. Problems Classes 23. Assessment 34. Recommended Textbooks and Websites 3

II. A Gentle Introduction to Fourier Analysis 3A. Overview 3B. The Basic Premise 4C. Fourier Series 4D. Amplitude Spectra 5E. The Gibbs Phenomenon 6F. Dirichlet conditions 6G. Complex Fourier series 7

III. Fourier Transforms and Conjugate Variables 7A. From Fourier Series to Fourier Transforms 8B. Conjugate Variables, Reciprocity, and the Uncertainty

Principle 91. Real Space and Reciprocal (Inverse) Space 102. Position-momentum transformations in quantum

mechanics and the Uncertainty Principle 10C. The Dirac delta function 11D. Magnitude, Phase, and Power Spectra 12

1. The Importance of Phase 122. Fourier transforms, power spectra and symmetry 12

E. Parseval’s Theorem: Energy and Power 13F. Important Properties of Fourier Transforms and the

Dirac delta function 13

IV. Measurement, Impulse Response, and Convolution 13

∗module website: www.nottingham.ac.uk/~ppzpjm/F32SMS

A. Impulse response 13B. Convolution and Impulse Response 14C. The Convolution Theorem and Fourier Transforms 15

1. Convolution of δ(x− x0) with a function f(x) 152. The frequency convolution theorem 15

D. Impulse Response and Audio Signals 15

V. Reciprocal space, Diffraction, and Fourier Optics 15A. Fraunhofer diffraction for single- and double slit

experiments 16B. The diffraction grating 16C. 2D Apertures and 2D Reciprocal Space 16D. Reciprocal Space, Spatial Frequencies, and Image

Processing 17

I. MODULE OVERVIEW

Vibrations and waves are at the core of a vast amountof physics (and play a central role in very many otherdisciplines including chemistry, engineering, and materi-als science). Indeed, it is difficult to identify an area ofphysics where the influence of vibrations and waves is notfelt: acoustics, electronics, image processing and analy-sis, fluid mechanics, optics, quantum mechanics, spec-troscopy, transmission and reception of electromagneticsignals - the list is almost endless. An understanding ofeach of these topics relies fundamentally on the analysisof the structure and propagation of waves.

A. What Will We Cover And Why Is It So Important?

Jean Baptiste Joseph Fourier (see Fig.1) made the re-markable discovery that virtually all periodic functions

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FIG. 1 Jean Baptiste Joseph Fourier (1768 -1830) .

can be constructed from a sum of harmonic (i.e. sine orcosine) terms of different amplitudes and phases. How-ever - and as also shown by Fourier - we are not limitedto analyzing periodic functions in terms of their harmon-ics: even aperiodic functions (e.g. spikes, steps, noise,etc...) may be decomposed into their constituent fre-quency components. Plotting the amplitudes (and/orphases) of these harmonic components versus frequencyproduces a spectrum, a frequency domain representationof the signal that is entirely equivalent to the time domainrepresentation with which you are most familiar. Fur-thermore, we are not restricted to time-frequency trans-formations: both the theory of diffraction and the pro-cessing of digital images, for example, rest on Fourieranalysis of waves in terms of spatial frequencies.

A key objective of this synoptic module is to highlighthow the mathematics of Fourier analysis may be appliedacross very many areas (which at first may seem ratherunrelated) without presenting a substantial amount ofnew physics. Given the ubiquity of Fourier analysis, how-ever, occasionally we will need to introduce a topic thatmay not have been covered in the modules you have stud-ied to date. In such cases, we will ensure that the newtopic is discussed in terms of physics that you have pre-viously encountered.

B. Module Organisation And What Is Expected Of You

Contact DetailsPhilip Moriarty, philip.moriarty@nottinghamOffice: B403, Tel.: Ext. 15156

Please first contact me by e-mail to organise anappointment if you would like to discuss any aspect ofthe module.

FIG. 2 Electron waves on the surface of a copper crystal.Fourier analysis underlies many aspects of our understandingof electrons in solids.

1. Lectures

A style of lecturing similar to that used for the 1st yearThermal and Kinetic Physics module will be employed.Printed lecture notes will be handed out and are avail-able from the module website. You will gain significantlymore from this module if you read the lecture notes be-fore the lecture. Throughout the module, the materialcovered will be presented as computer-based slides andthe blackboard will be used in parallel. You are not ex-pected to slavishly copy down everything from the slidesdisplayed during the lectures. However, you are ad-vised to make your own notes (to complementthose in this handout) as each lecture progresses.

The information in the lectures will largely be given inthe printed lecture notes. However, the lectures will notsimply mirror the printed lecture notes you’ll be givenand you will need to attend the lectures to ensure thatyou get the ’complete picture’. There are two specific andkey areas where the lectures will strengthen and comple-ment the printed notes:

1. The lectures will feature computer-based and ’realworld’ demonstrations of Fourier analysis

2. The notes feature a number of questions in eachsection, the answers to which will be explored inthe lectures.

2. Problems Classes

Eight of the ten weekly lectures are accompanied by a2 hour problems class in B23. A timetable for the prob-lems classes is given below. All 2nd year students havebeen sent an e-mail detailing the problems class groupto which they have been assigned. This information isavailable on the Applications of Fourier Analysis web-site. Numerical answers and, subsequently, worked solu-tions for the problems will also be posted on the website.

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3. Assessment

This module will be assessed via two open book classtests which will take place at 09:00 in B1 (i.e. the regularlecture slot) on Feb. 26th (worth 40 % of the module) andin the week beginning May 14th (worth 60%).

4. Recommended Textbooks and Websites

There is no primary textbook for this module. Thisset of lecture notes and the material covered in theproblems classes (including the worked solutions) rep-resent the primary reference. There are, however, awide range of recommended secondary textbooks (avail-able from the George Green library) and an extensivecollection of Fourier analysis-related websites (listed be-low). We strongly recommend that you make useof these secondary texts and websites to bolsterthe lecture- and problems class material.

1. A Student’s Guide to Fourier Transforms, JFJames (2nd Edition, Cambridge University Press,2002).

2. Chapters 10 and 11 of Mathematical Methodsfor Physics and Engineering, Riley, Hobson, andSpence (Cambridge University Press 1997)

3. Chapters 7 and 13 of Mathematical Methods in thePhysical Sciences, Mary L. Boas (John Wiley andSons, 1983)

4. Schaum’s outline of theory and problems of Fourieranalysis : with applications to boundary value prob-lems, Murray R. Spiegel. (New York : McGraw-Hill, 1974)

5. Fourier Analysis, Hwei P. Hsu (Simon and Schus-ter, Inc. 1970)

6. Fourier Series and Harmonic Analysis, KA Stroud(Stanley Thornes (Publishers) Ltd., 1984)

7. Lectures on Fourier Series, L. Solymar (Oxford Sci-ence Publications, 1988)

8. Introduction to Fourier Optics, JW Goodman(McGraw-Hill 1996)

9. The Fourier Transform and its Applications, RNBracewell (McGraw-Hill, 1986)

Links to some of the more useful Fourier-related web-sites are provided on the module web pages.

II. A GENTLE INTRODUCTION TO FOURIERANALYSIS

’Fourier’s theorem is not only one of the most beautifulresults of modern analysis, but it may be said to furnish

an indispensable instrument in the treatment of nearlyevery recondite question in modern physics.’

Lord Kelvin

Fourier analysis impacts on practically every area ofphysics. A knowledge of Fourier series and Fourier trans-forms thus forms an essential component of your trainingas a physicist. Lord Kelvin was particularly impressed byFourier’s achievements in analysing complex periodic andaperiodic waveforms (see quote above), and a particulargoal of this module is to impress upon you the remark-able ubiquity of Fourier analysis (as was recognized byKelvin). Although mathematics - and, in particular, thesolution of many integrals - will form a sizeable compo-nent of the module 1, a key objective is to teach youto look beyond the mathematics and develop an intu-ition for the relationships between waveforms/signals inthe time domain and their counterparts in the frequencydomain. (Or, alternatively, between real space and re-ciprocal space structures).

In the Applications of Fourier Analysis lecture coursewe will thus focus on the practical uses of Fourier meth-ods across a range of scientific disciplines. It is ex-tremely important to note that the problemsclasses are an integral component of the module.Your assessment on this module is based on your abilityto solve problems (rather than simply regurgitate facts/formulae). As such, the exam will have an ’open notes’format where you can bring these notes (annotated asyou wish) into the exam.

A. Overview

In the first lecture of the module, we will revise manyof the Fourier-related concepts that were encountered inlast semester’s Elements of Mathematical Physics module(Fourier series, Fourier transforms, Fourier coefficients,top-hat functions, Dirac-delta function). In addition torevisiting the mathematical basis of these areas, we willspend some time applying Fourier techniques to the anal-ysis of musical notes and sounds in general. A key topicwhich we will cover in some depth is the ability of Fourierseries and Fourier transforms to convert a signal from a

1 It is worthwhile quoting Kelvin again at this point: ’Once whenlecturing in class he [Kelvin] used the word ’mathematician’ andthen interrupting himself asked his class: ’Do you know what amathematician is?’ Stepping to his blackboard he wrote upon it:

∫ ∞

−∞exp(−x2)dx =

√π (1)

Then, putting his finger on what he had written, he turned tohis class and said, ’a mathematician is one to whom that is asobvious as that twice two makes four is to you.’” [Taken fromThe Life of Lord Kelvin, S. P. Thompson].

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FIG. 3 Frequency spectrum of a guitar note where the stringwas plucked near the bridge (top) or near the centre (bottom).

time domain to a frequency domain representation (or,as noted above, from real space to reciprocal space). Theconcept of the spectrum of a signal or a function underliesall of Fourier analysis.

B. The Basic Premise

Let’s start with a question (to be answered duringLecture 1):

What is the fundamental purpose of Fourier analysis?

A key application of Fourier analysis lies in the studyof sound and acoustics. We will explore this in somedepth, focusing on the use of Fourier techniques toanalyse the tonality or timbre of notes from variousmusical instruments. During the lecture you will heara sample of a G note played on a piano and a G noteplayed on a guitar. A question of some considerableimportance is:

What determines the timbre of a musical note? Why

does a G note on the piano sound different to a G noteon guitar?

The tone of a note on a given instrument also dependscritically on how the instrument is played. You’ll seein the lecture that there is a significant difference be-tween the tone of a note from a guitar when the stringis plucked at a position near the neck of the guitar com-pared to a note struck by picking closer to the bridge.The spectra shown in Fig. 3 and Fig. 4 provide some in-sight into why the note’s tone differs depending on wherethe string is struck. (In later lectures you will see howwe can use Fourier methods to solve partial differentialequations and thus treat the vibrations of a guitar string- or many other systems which may be treated as bound-ary value problems - from ’first principles’.) We returnto a discussion of the term ’spectrum’ below.

It is the harmonic content of the waveform that prin-cipally determines the tone/ timbre of the note. In Lec-ture 1 (and Problems Class 1) we will be concerned withthe representation of a waveform/ function in terms of aFourier series.

C. Fourier Series

The fundamental mathematics underlying the Fourierseries description of a function has been detailed in theMathematical Physics module last semester. As thismodule (F32SMS) is concerned with the applications ofFourier analysis, I will simply recap the key results de-rived in the Mathematical Physics module.

First, we posit that we can write a periodic functionas a sum of harmonic (sinusoidal) terms with differentamplitudes and phases - this is the fundamental premiseof Fourier analysis. (As described in Section 1.6 below,although not every mathematical function may be rep-resented as a Fourier series, the vast majority of func-tions of interest in the physical sciences are amenable toFourier analysis.) The trigonometric form of the Fourierseries is:

f(t) =A0

2+

∞∑n=1

(An cos(nω0t) + Bn sin(nω0t)) (2)

Having written f(t) in this form, we obviously needto know the values of the coefficients An and Bn beforewe can construct the function from its sine and cosinecomponents. Note the inclusion of the A0/2 term beforethe summation - this term is the ’DC level’ of the signalor function. That is, because the integral of a sine orcosine function over one period is zero, we need to takeaccount of any offset of the function f(t) (above or belowthe x-axis) that makes its mean value per period non-zero.

An important problem on the question sheet for Prob-lems Class 1 (PC1) deals with the calculation of a formula

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which yields the values of An and Bn. (The orthogonal-ity property of sine and cosine functions is required to dothis - again, see the question sheet for PC1). Once weknow these coefficients we can decompose the functionf(t) into its Fourier components (as given in Equations 3and 4 below).

An =2T

∫ t0+T

t0

f(t) cos(nω0t)dt (3)

Bn =2T

∫ t0+T

t0

f(t) sin(nω0t)dt (4)

In Fourier analysis: (i) we wish to synthesize a func-tion or signal f(t) from a series of harmonic components(Fourier synthesis), or (ii) given a signal or function f(t),we wish to extract the amplitudes, frequencies and/orphases of the harmonic components (this is known asFourier analysis or Fourier decomposition)

Example 1.1To illustrate how Fourier analysis works, let’s firstchoose the simplest possible example for our functionf(t): a pure sinusoidal tone. We’ll take f(t) = sin(ω0t).This is an odd function (i.e. f(t) = -f(-t)), so we canimmediately state that the An coefficients are zero.This is an extremely important observation to makeas it dramatically reduces the amount of work requiredto calculate a Fourier series. Get into the habit ofidentifying the symmetry of the function before youstart to determine the Fourier series coefficients.

For an odd function, the An coefficients are zero.For an even function, the Bn coefficients are zero.

We therefore need to calculate only the Bn coefficients.To do this we exploit the orthogonality of the set of func-tions sin(nωt)(see PC1) to show that the Bn coefficientsare non-zero only for n = 1. For n = 1, we have:

FIG. 4 A square wave. Its Fourier decomposition is given byEqn. 6

∫ T

0

sin2(ω0t)dt =12

∫ T

0

1− cos(2ω0t)dt =T

2(5)

From the expressions for the coefficients given inEquations 3 and 4 above, the only coefficient that is notzero is B1 and it has a value of 1. This is exactly aswe’d expect: as f(t) = sin(ω0t) we need only a singleharmonic to describe the function and the coefficientassociated with that harmonic has a value of 1 (becausethe amplitude of f(t) is 1).

Example 1.2A somewhat more useful example is the decompositionof a square wave (Fig. 4) into its Fourier components.You have encountered the Fourier decomposition of asquare wave in the Mathematical Physics module so I’mnot going to repeat the analysis here. (In addition, theFourier series expansion of a square wave is covered onp. 310-311 of Mathematical Methods in the Physical Sci-ences, Boas; in Fourier Series and Harmonic Analysis,Stroud; and in the 2nd year MATLAB course). Theresult of the analysis is that the square wave shown inFig. 4 has the following Fourier series expansion:

f(t) =12

+2π

(sinx +sin 3x

3+

sin 5x

5+ . . .) (6)

Note that the series has no cosine components (why? ).

Numerous other examples of Fourier series calcula-tions may be found in Boas and Stroud. In additionto a number of questions on the PC1 problem sheet(involving, for example, the application of Fourieranalysis to a rectified electrical signal), a good exampleto attempt is:

Calculate the Fourier series for the following (triangu-lar) function:

f(x) = −x − π < x < 0f(x) = x 0 < x < π (7)

D. Amplitude Spectra

An important method of visualizing the Fourier de-composition of a function is to plot a spectrum of the An

or Bn coefficients versus frequency (or, equivalently, har-monic number). In Lecture 1 we carried out this processfor the triangular wave (whose fundamental frequency is2 Hz) shown in Fig. 5. In Fig. 5(a), the triangular waveand the Fourier series representation resulting from thesummation of the first ten harmonic terms in the seriesare plotted. The spectrum shown in Fig.5(b) is a plot ofthe Bn coefficients versus frequency. (Why aren’t the An

coefficients also plotted?). Note that the coefficient foreach alternate harmonic component is of different sign -

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FIG. 5 A triangular wave (top) and its associated Bn coef-ficients (bottom). A Fourier series representation comprisingthe first 10 components is overlaid on the triangle function inthe upper.

what does this tell you about the phases of the sinusoidsthat comprise the triangular wave?

The important thing to grasp from Fig.5 is that thetime domain and frequency domain graphs are just dif-ferent representations of the same waveform or function.Nevertheless, transferring the information from the timedomain to the frequency domain in the form of a spec-trum makes interpretation of the waveform (and - as weshall see - the effects of filtering and processing on thewaveform) significantly easier. Moreover, visualizing thestructure of the function in terms of its Fourier compo-nents provides insights that are simply not available fromthe more conventional time domain picture (with whichyou are most familiar). In later lectures we shall see thatnot only amplitude-, but power- and phase spectra areextremely powerful tools in Fourier analysis.

E. The Gibbs Phenomenon

You will note from Fig. 5 that the Fourier seriesrepresentation of the function is particularly poor nearsharp edges (discontinuities): there is considerable over-shoot. One might suggest that this is because we havesimply not included enough terms in the Fourier seriesexpansion. It turns out that this is not the case - evenif we include an infinite number of terms in the Fourierexpansion (Equation 2), the overshoot persists. Indeed,this effect (which is called the Gibbs phenomenon)occurs for any function with discontinuities (see p. 315of Boas for an illustration of the effect in the Fouriersynthesis of a square wave). It should perhaps not beso surprising that the Gibbs phenomenon is observed -after all, we are attempting to represent a discontinuousfunction (such as a triangular or square wave) with aset of continuous functions. Nevertheless, as long aswe are careful when applying Fourier analysis to thebehaviour of a function near a discontinuity, Fourieranalysis remains a very powerful mathematical tool.

F. Dirichlet conditions

As noted above, not all functions may be decomposedinto a Fourier series. The issue of whether a Fourier seriesmay be used to represent a given function was addressedby Dirchlet who identified the following conditions:

1. f(x) is single-valued. That is, for every value of x,there is only one value of f(x). For example,y= x2

is single valued, but y2 = x is not.

2. f(x) is continuous or has a finite number of discon-tinuities within the periodic interval. For example,a triangular wave as shown in Fig. 6 is continuousand thus can be analyzed using Fourier methods.A square wave has a finite number of finite discon-tinuities (i.e. jumps) within a period so it too isamenable to Fourier analysis. The function f(x) =tan (x) does not satisfy condition 2 (non-finite dis-continuity at x =±π/2) and thus cannot be treatedusing conventional Fourier analysis.

3. The following integral must be finite:

∫ T

0

|f(x)|dx (8)

4. There must be a finite number of extrema (maximaand minima).

If these conditions are held then the Fourier seriesconverges to f(x) at all points where f(x) is continuous.At discontinuities (see discussion in Section II.E), theFourier series converges to the midpoint of the jump.

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FIG. 6 A triangular wave.

G. Complex Fourier series

A more compact and generally more easily-applied ver-sion of the Fourier series is the complex form expressedin Eqn. 9 below.

f(t) =+∞∑

n=−∞cn exp(inω0t) (9)

This form of the Fourier series expansion of a functionwas also covered in last semester’s Mathematical Physicsmodule. Note that, as for the trigonometric form, a func-tion (f(t)) has been decomposed into a series of harmonicterms with each term associated with an harmonic num-ber, n and a frequency nω0. The coefficients cn in thiscase are complex-valued and are derived using the follow-ing equation:

cn =1T

∫ t0+T

t0

exp(−inω0t)f(t)dt (10)

The coefficients cn are related to the coefficients for thetrigonometric series (Eqns. 3 and 4) via the followingrelations:

A0 = 2c0

An = cn + c−n

Bn = i(cn − c−n) (11)

For revision purposes, you should attempt the ques-tions on previous Mathematical Physics examination pa-pers related to the complex Fourier series. Furthermore,Q3 and Q5 on Examples Sheet 3 of the MathematicalPhysics course deal with the complex form of the Fourierseries. A number of questions on the problem sheets forPC1 and PC2 of this module also address the interpre-tation of the complex Fourier series expression given inequation 9 above. Q4, PC1 is particularly important asit deals with concepts such as negative frequencies, sym-metry, and phase- and amplitude spectra. Each of theseconcepts is briefly explained below.

FIG. 7 The complex coefficient spectrum for a cosine wave-form

Negative frequencies arise simply because any sine orcosine function can be expressed as the sum or differenceof exponential functions with complex exponents. Forexample,

cos(ω0t) =exp(+iω0t) + exp(−iω0t)

2(12)

Note that the exponential terms are associated with ei-ther a positive (+ω0) or negative (-ω0) angular frequency.Thus, as discussed in the solution to PC1 Q4, the com-plex coefficient spectrum comprises both positive andnegative frequencies (as shown in Fig. 7). It is importantto realize that the cn coefficients for a cosine functionare real whereas for a sine function the cn coefficientsare imaginary. This holds true for any odd or even func-tion (not just sines and cosines): a function with oddsymmetry will have only imaginary values in itsFourier series representation whereas a functionof even symmetry will have a Fourier series com-prising only real-valued coefficients.

If we are interested only in the amplitudes of theFourier components (and not their phases) we can plotthe magnitude (i.e. the modulus) of each of the (complex-valued) Fourier coefficients as a function of the harmonicnumber, n. We then obtain an amplitude spectrum whichis independent of the phase of the waveform. Hence, al-though Re(cn) and Im(cn) differ for a sine as comparedto a cosine function, the amplitude spectra for these func-tions are identical. This point is covered in depth in thesolution to PC1 Q4. (We will return to ideas very similarto these when we discuss Fourier transforms in Lecture2).

III. FOURIER TRANSFORMS AND CONJUGATEVARIABLES

The second lecture and problems class (and indeed thevast majority of the module) largely focus on Fourier

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FIG. 8 A simple periodic pulse waveform

transforms as opposed to Fourier series: we will startby discussing just why Fourier transforms are necessaryand what it is that distinguishes the transform from theFourier series expansion covered in Section I. As has beenhighlighted throughout Section I, a fundamental purposeof Fourier analysis is to enable a function to be convertedfrom a time domain to a frequency domain representa-tion. In Section II we will broaden this definition to covera number of other pairs of what are termed conjugatevariables (e.g. space and inverse space, or, in quantummechanics, the position and momentum of a particle). Ineach case, Fourier transformation can be used to convertbetween the two conjugate variable representations, i.e.time→ frequency, space→ inverse space, position→mo-mentum. Note that, in each case, the conjugate variablesare reciprocally related to each other. For example, ν=1/T, where ν and T are frequency (units: s−1) and pe-riod (units: s) respectively; k = 2π/λ , where k and λ arewavevector and wavelength respectively, and ∆x.∆px ∼h/2.You will recognise this latter equation as a statementof the Heisenberg Uncertainty Principle - as we shall see,the reciprocal relationship between conjugate variablesthat is entrenched in Fourier analysis manifests itself invery many areas of physics.

A. From Fourier Series to Fourier Transforms

Every function we have dealt with thus far has beenperiodic. There are, of course, very many functions ofcentral importance in fundamental and applied science(and, of course, mathematics) which are aperiodic - wewill discuss a number of these in later sections but for nowwe will focus on a consideration of the top-hat function.Importantly, an aperiodic function may be thought of asthe limiting case of a periodic function where the periodtends to infinity and thus the fundamental frequency (ω0)tends to zero. In this case, we can no longer representthe function as a Fourier series, but must describe it interms of a Fourier transform.

To show how the Fourier transform representation ofa function ’evolves’ naturally from the Fourier series ap-proach, consider the periodic function shown in Fig.8 (aregular series of pulses). We can determine the Fourierseries for this particular function either analytically (as

FIG. 9 The Fourier coefficients for Fig. 8

FIG. 10 A single pulse (otherwise known as a top-hat func-tion)

for the examples described in Section 1) or via a computerpackage such as LabVIEW or Matlab. (NB Although wecan’t analyse an aperiodic function using a Fourier se-ries approach, Fourier transforms may be applied to bothaperiodic and periodic functions). The magnitudes of theFourier series coefficients, cn, for the function shown inFig. 8 are shown in Fig. 9 (the fundamental frequencyof f(t) is 4 Hz in this case). As should be clear by now,we observe discrete frequency components.

The question we now ask is: what happens to theFourier series representation of the function as we in-crease the period of the function shown in Fig. 8? Inparticular, what is the result of increasing the period in-definitely so that instead of a regular, periodic train ofpulses we have a single, isolated pulse (i.e. a top-hatfunction), as shown in Fig. 10?

As the separation of the pulses increases (i.e. as T→ ∞, see Fig. 11), an increasing number of Fouriercomponents appear in the amplitude spectrum (Fig. 12).In the limit of an infinite pulse period (i.e. for a singleaperiodic pulse), the spacing between the Fourier (fre-quency) components becomes infinitesimally small andthe Fourier spectrum is no longer a discrete but a con-tinuous function of frequency (Fig. 13).

Equation 13 below is a statement, without proof, ofthe Fourier transform, F(ω) of a function f(t):

F (ω) =1√2π

∫ +∞

−∞f(t) exp(−iωt)dt (13)

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FIG. 11 The graphs in Figs. 12 and 13 (see below) show theeffects of increasing the pulse period on the Fourier compo-nents of the pulse train shown in this figure.

FIG. 12 Note that as the pulse period increases, the numberof Fourier components also increases. The graph in the lowerleft hand corner shows the Fourier components in the limitof an infinite pulse period: instead of discrete frequency com-ponents, the frequency spacing is infinitesimally small andthe Fourier series summation is replaced by a Fourier integral(i.e. a Fourier transform). Figs 11,12, and 13 are taken from’Fourier Analysis Part II: Continuous and Discrete FourierTransforms’, Maria Elena Angoletta. Link to website avail-able from the Module web pages.

We also define an inverse Fourier transform. WhileEquation 13 transforms from time to frequency space(i.e. we generate a Fourier spectrum, F(ω) for a func-tion f(t)), the inverse Fourier transform (Equation 14)takes us from the frequency domain to the time domain:

f(t) =1√2π

∫ +∞

−∞F (ω) exp(iωt)dω (14)

FIG. 13 The Fourier transform of an isolated pulse (i.e. apulse train with an infinite period).

If Equations 13 and 14 are compared with the expressionsfor the complex Fourier series given on page 7, it shouldbe clear that:

1. F(ω) corresponds to the cn coefficients (i.e. wehave replaced the discrete set of Fourier compo-nents with a Fourier integral)

2. The summation of discrete terms has been replacedby an integral whose limits are -∞ to +∞

3. nω0 has been replaced by ω

Problem 1 on the question sheet for PC2 deals with thecalculation of the Fourier transform for a top-hat functionsimilar to the isolated pulse shown in Fig. 9 . Hopefully,from the Mathematical Physics module you will recallthat the Fourier transform of a top-hat function is a sincfunction. A considerable amount of what we’ll cover inthe Applications of Fourier Analysis module will rely onthe concepts underlying the solution to Q1, PC2 so makesure that you fully understand the method and can dothe integration.

B. Conjugate Variables, Reciprocity, and the UncertaintyPrinciple

The result of Q1 for PC2 is that the following top-hatfunction:

f(t) = 1, |t| < τ

f(t) = 0, |t| > τ

has Fourier transform,

F (ω) =

√2π

sin(ωτ)ω

.

This sinc function has its first zero at the value of ω whereωτ = π. Thus, the wider the top-hat function/pulse, thenarrower the central lobe of the sinc function. I cannotstress too strongly the importance of this observation:a reciprocal relationship between conjugate vari-ables underlies all of Fourier analysis.

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1. Real Space and Reciprocal (Inverse) Space

As has been pointed out in Section 2A, with Fourieranalysis we are not restricted to time-frequency trans-formations. For example, we shall see in the followingsections that Fourier transforms are an extremely effec-tive tool for the study of diffraction phenomena. Indeed,whether we are interested in the diffraction of electronsfrom a solid surface, the scattering of X-rays from a crys-tal, or the study of the properties of an optical grating,the diffraction pattern in each case may be determinedvia an appropriate Fourier transform. We speak of thediffraction pattern as the ’inverse’ or ’reciprocal’ spaceequivalent of the ’real space’ lattice.

You will study Fraunhofer diffraction and diffractiongratings in the Optics module. Remarkably, the ampli-tude of a Fraunhofer diffraction pattern from a gratingis nothing more than the Fourier transform, F(k), of theaperture function of the grating 2. So, even if we have noknowledge of optics, given the aperture function (and,of course, an understanding of Fourier analysis) we cancalculate what the diffraction pattern should be!

The diffraction process Fourier transforms from theaperture function - which is described in terms of realspace co-ordinates, say (x, y) - to the reciprocal spacediffraction pattern where the peaks in intensity appearat certain values of kx and ky. The coordinates in realspace, (x, y) are inversely related to the reciprocal spacecoordinates: kx = 2π/x and ky = 2π/y. Thus, just aswe find for time-frequency transforms, if we ’stretch’ theaperture function in a particular direction in real space,it narrows in the corresponding direction in reciprocalspace. Lecture 4 and the associated problems class willdeal with this type of Fourier optics problem in detail.

Before we leave our discussion of real space-to-reciprocal space transformations for now, let me reiter-ate that it is not just the diffraction of optical photonsthat is described by Fourier transformation - for example,electrons, X-rays, or neutrons), the diffraction pattern isdirectly related to the Fourier transform of the real spacelattice. For example, Fig. 2.5(a) shows a scanning tun-neling microscope (STM) image of the atomic structureof a silicon surface. Each bright spot in the image repre-sents a single silicon atom. The atoms are separated byapproximately 0.8 nm. Low energy electrons (¡ 100 eV)have a de Broglie wavelength comparable to this spacingand scatter off the surface to form the diffraction patternseen in Fig. 14(a). (Note that low energy electron diffrac-tion (LEED) was an extremely important technique inthe elucidation of the wave-particle nature of the elec-tron in the early 20th century). The diffraction patternof Fig.14(b) is once again directly related to the Fourier

2 The aperture function describes which parts of the diffractingobject are transparent and which parts are opaque. Note alsothat the intensity of a diffraction pattern is given by the squareof the modulus of the Fourier transform (i.e. F (k)F (k)∗)

FIG. 14 Fig. 14(a) (on left) is an STM image of the atomicstructure of a silicon surface. Fig. 14(b) is a diffraction pat-tern from the surface. The diffraction pattern is directly re-lated to the Fourier transform of the real space structure ofthe surface (which is seen in the STM image.)

transform of the atomic lattice seen in the STM image(Fig. 14(a)).

2. Position-momentum transformations in quantum mechanicsand the Uncertainty Principle

The Heisenberg uncertainty principle is a natural con-sequence of the Fourier transform relationship betweenthe (conjugate) position and momentum representationsof a quantum mechanical wavefunction. In the FurtherExercises section of the question sheet for PC2, you areasked to consider a particle that is located in space witha Gaussian probability distribution centred at x0 and astandard deviation of a (see Fig. 15). We can write its(un-normalized) wavefunction as follows:

ψ = exp(−(x− x0)2)/2a2

)(15)

Just as time-frequency and real space-reciprocal spaceare conjugate variables which are inter-related via theirrespective Fourier transforms, so too are position andmomentum. We can therefore determine the momen-tum representation of the position wavefunction given in

FIG. 15 A Gaussian wavefunction centred at x0

11

Equation 15 by calculating the appropriate Fourier trans-form:

F (k) =1√2π

∫ +∞

−∞exp(−(x− x0)2)/2a2) exp(−ikx)dx

(16)The Further Exercises section of PC2 asks you to eval-

uate this integral. You will find that it is another Gaus-sian, but of standard deviation 1/a - yet again, we findthat there is a reciprocal relationship between the con-jugate variables. The momentum wavefunction has thesame Gaussian form as shown in Fig. 15 but it is nar-rower.

How does this relate to the uncertainty principle? Notethat if the uncertainty in the position of the particleincreases then the Gaussian describing ψ(x) becomeswider. The Gaussian associated with Ψ(k) correspond-ingly becomes narrower. This is the fundamental basisof the uncertainty principle - a small ∆x is associatedwith a large ∆k, and vice versa. You will explore in amore quantitative fashion the relationship of the uncer-tainty principle with Fourier transformation in the Fur-ther Exercises section of the question sheet for PC2. Notethat there is no reason to choose the spatial wavefunc-tion above as being Gaussian in character - we could havetaken ψ(x) as the top-hat function discussed earlier or, asin Q4 of PC2, we could consider spatially localized parti-cles. In all cases there is a reciprocal relationship betweenthe widths of the spatial and the momentum wavefunc-tions.

C. The Dirac delta function

The Dirac delta function (δ(x)) plays an exceptionallyimportant role in Fourier analysis. It can be (somewhatloosely) regarded as the limiting case of a top-hat func-tion - whose width approaches zero - of the type:

f(x) = 1/a, |x| < a

f(x) = 0, |x| > a (17)

As a approaches 0, the top-hat becomes increasinglynarrow and increasingly tall. However, its area alwaysremains equal to 1. The Dirac δ function therefore hasthe following characteristics:

δ(0) = ∞δ(x) = 0(unless x = 0) (18)

and, most importantly:∫ +∞

−∞δ(x) = 1 (19)

An extremely useful property of the Dirac delta func-tion is its sifting ability:

∫ +∞

−∞f(x)δ(x− a)dx = f(a) (20)

FIG. 16 The sifting property of the δ function.

It is straightforward to visualise the sifting propertyrepresented by Equation 20. Fig. 16 shows that becauseδ(x − a) is non-zero only at x = a, the product of f(x)and δ(x− a) is zero for all values of x except x = a. Theprocess represented by Equation 20 ’sifts’ the functionf(x) and returns the value of f(x) at the position of thedelta function.

Equation 20 leads naturally to the following definitionof the delta function:

∫ +∞

−∞f(x)δ(x) = f(0) (21)

Using Eqn. 21 we can determine the Fourier transformof the function:

F (k) =1√2π

∫ +∞

−∞δ(x) exp(−ikx)dx

⇒ F (k) =1√2π

exp(−ik0) =1√2π

(22)

That is: The Fourier transform of the Dirac deltafunction δ(x) is 1√

2π

We can rationalize this result by considering, as notedabove, the Dirac delta function as the limiting case of atop-hat function whose width approaches 0. We knowthat as the top-hat function’s width decreases, the widthof its Fourier transform (a sinc function) increases. Inthe limit of an infinitesimally narrow top-hat, the Fouriertransform will be infinitely wide. As the Fourier trans-form of the delta function is a constant,F (k) is indepen-dent of frequency and the transform is therefore a flat lineof value 1√

2π- i.e. an infinitely wide band of frequencies.

The Dirac-δ function also appears in the Fourier trans-form of a sine or a cosine wave. The inverse Fouriertransform of δ(ω − ω0)is:

f(t) =1√2π

∫ +∞

−∞δ(ω − ω0) exp(iωt)dω

12

⇒ f(t) =1√2π

exp(iω0t) (23)

From this result you should be able to write down andsketch the Fourier transform of the function cos(ω0t) (andthe Fourier transform of sin(ω0t)).

D. Magnitude, Phase, and Power Spectra

As the Fourier transform and inverse Fourier transformare generally complex-valued, we need to consider care-fully how we plot the Fourier representation of a function.We write:

F (ω) = <(F (ω)) + i=(F (ω)) (24)

where <(F (ω)) is the real part of the Fourier transformand =(F (ω)) is the imaginary part. (Note that it is pos-sible for the real or the imaginary parts to be zero. Com-pare the Fourier transforms - or, indeed, Fourier series(see Q4 of PC1)- of a sine and a cosine function). Wecould plot the real and imaginary parts of the transformseparately but this is generally an unwieldy method. Forexample, we might simply be concerned with the magni-tudes of the various Fourier components (as is the case inmuch of spectrum analysis). The quantity of interest inthat case is the modulus of the Fourier transform, |F (ω)|:

|F (ω)| =√

(<(F (ω)))2 + (=(F (ω)))2 (25)

It is important to realize that by plotting |F (ω)| we pro-vide no information on the phases of the Fourier compo-nents). Production of a phase spectrum requires that aplot is made of the following quantity 3:

φ(ω) = arctan(=(F (ω)<(F (ω)

)(26)

Any complex number can be written in the form z =r exp(iθ) where r is the magnitude (or modulus) of zand θ is the phase angle. From this expression (and alsofrom Eqn. 26), it is clear that if the Fourier transform ofa function is real, its phase spectrum will be uniformlyzero.

1. The Importance of Phase

The relative importance of the magnitude and phaseinformation depends very much on the application. Al-though the timbre of a musical note can be relatively in-sensitive to the phases of the Fourier components, phase

3 If you don’t know why the phase angle of a complex numberis given by Equation 26 then I suggest that you consult your1st year General Mathematics notes, download the informationfrom an appropriate website, or visit the library to consult anappropriate textbook.

FIG. 17 The effects of mixing Fourier phases and amplitudes(see text for details)

information plays an extremely important role in imageanalysis and interpretation. (The role of Fourier analysisin image processing will be discussed by Dr. Rourke inthe second half of the module and will be the subject offuture Problems Classes). Fig. 17 illustrates this point.

The photographs shown in the top half of Fig. 17 are ofKarle (left) and Hauptmann (right) who won the Nobelprize in 1985 for their work in the field of X-ray crystal-lography. (As discussed in Section 2.3, Fourier methodsare extremely important in X-ray diffraction/ crystallog-raphy). We take a Fourier transform of each image. Inthis case although it will be a 2D Fourier transform, themathematical principles are precisely the same as for the1D Fourier analysis we have covered thus far. The imagesshown in the bottom half of Fig. 17 are the result of mix-ing the phase spectrum from Hauptmann’s photographwith the magnitude spectrum from Karle’s photograph(A) and vice versa (B). It is clear that it is the phasespectrum rather than the magnitude spectrum that con-tains the majority of the visual information.

2. Fourier transforms, power spectra and symmetry

A final, very popular method of visualizing the Fouriertransform of a function or a signal is to plot a power spec-trum. A power spectrum is simply the square of the mod-ulus of the Fourier transform, |F (ω)|2. Note, however,that just as for the magnitude spectrum, a power spec-

13

trum contains no information on the phases of the Fouriercomponents. This is important not only in the contextof the discussion in the preceding paragraph but also interms of the symmetry of the function whose transformwe are calculating. A plot of the real or the imaginaryparts of a Fourier transform will be sensitive to the sym-metry (i.e. the degree of ’odd-ness’ or ’even-ness’) andphase of the original function f(t). The power spectrum,however, is not affected by whether the function is oddor even (or neither) nor by its phase. That this is thecase can be seen from the following shift theorem (whichwe state without proof):

FT (f(t− t0)) = exp(−iωt0)F (ω) (27)

In words, Equation 27 states that if we have a functionf(t) whose Fourier transform is F (ω), then if the func-tion is shifted by an arbitrary amount t0, the Fouriertransform of the shifted function (FT (f(t−t0)) is simplythe original transform multiplied by exp(−iωt0). Notethat the modulus of exp(−iωt0)F (ω) is simply |F (ω)| -the modulus of the Fourier transform of the original, un-shifted function. Thus, both the magnitude and powerspectra are insensitive to the symmetry/phase of thefunction or signal or interest. If we are interested onlyin the magnitudes of the Fourier components - as is thecase for very many applications - then this removal ofphase/symmetry sensitivity is very useful.

The reasons underlying just why |F (ω)|2 is called apower spectrum are outlined in the following section.

E. Parseval’s Theorem: Energy and Power

Parseval’s theorem simply states that the energy or, fora periodic signal, average power of a signal representedby a function f(t) is the same whether it is computed intime or frequency space. Let’s first consider a periodicfunction, f(t). To appreciate what is meant by the powercontent of this function, assume that f(t) is a voltagewaveform, V (t), applied across a 1Ω resistor. The currentis then given by:

I(t) =V (t)R

⇒ I(t) = V (t) (28)

The power is defined as:

P =1T

∫ T

0

V (t)I(t)dt =1T

∫ T

0

f(t)2dt (29)

Parseval’s theorem states:

P =1T

∫ T

0

f(t)2dt =+∞∑

n=−∞|cn|2 (30)

where cn are the the complex coefficients in the Fourierseries representation of the function.

For an aperiodic function, it is not possible to definepower as in Equation 3-. Instead, we write Parseval’stheorem in terms of the total energy of the function:

P =∫ +∞

−∞|f(t)|2dt =

∫ +∞

−∞|F (ω)|2dω (31)

Note that the magnitude (or modulus) of the Fouriertransform is associated with the energy or, for periodicsignals, power of the signal. This is the origin of theterm power spectrum.

F. Important Properties of Fourier Transforms and theDirac delta function

FT (f(t) + g(t)) = F (ω) + G(ω)FT (f(t− t0)) = exp(−iωt0)F (ω)

FT (df(t)dt

) = iω(F (ω))

FT (f(t) exp(iω0t)) = F (ω − ω0)∫ +∞

−∞f(x)δ(x− a)dx = f(a)

FT (δ(x)) =1√2π

2πδ(ω − ω0) =∫ +∞

−∞exp(−i(ω − ω0)t)dt

IV. MEASUREMENT, IMPULSE RESPONSE, ANDCONVOLUTION

The process of convolution underlies every measure-ment we make. Every instrument is associated both witha finite resolution and a response function which deter-mine how the measurement is influenced by the measur-ing device. In this section, we focus on the role thatFourier transforms play in measurement theory, intro-ducing, in particular, the concept of convolution. Weshall also see how the impulse response function of aninstrument is connected to the process of convolution.

A. Impulse response

Q1 on the question sheet for PC3 deals with the con-cept of an instrumental response function for a spec-trometer. Let’s consider a generic measuring instrumentwhich has an input and an output. We model the finiteresolution of the system with an impulse response func-tion. What exactly does this mean? Well, we take asour input a δ function (or impulse). As the system isnon-ideal, the δ function will not be transferred from theinput to the output without distortion. In particular, thesystem is associated with a finite frequency bandwidth sonot all frequency components comprising the input will

14

FIG. 18 Schematic illustration of convolution process

FIG. 19 Point spread function

be retained. The output of the system in response tothe δ function input is known as the impulse response4. As can be seen in Fig. 18, the output is generally a’smeared’ or ’degraded’ version of the input. In this case,the higher frequency components of the δ function havebeen removed to produce a broadened, ’blurred’ output.(It is worthwhile pointing out here that to produce sharpedges in a function, high frequency Fourier componentsmust be present. See Q3 of PC2.)

The system of Fig. 18 can be anything : an electroniccircuit (e.g. an audio amplifier), a spectrometer, a cam-era or other imaging system, or a mechanical system.The latter systems warrant particular mention at thispoint. An imaging system comprising lenses and/or mir-rors will never be perfect (indeed diffraction places a fun-damental limit on the imaging resolution, regardless ofthe quality of the optical components). Instead of theone-dimensional impulse depicted in Fig. 18, we considera point object (a delta-function ’spike’ on a 2D plane) andask what will be the image of this point (i.e. the ’out-put’ of the optical system)? As depicted schematically inFig. 19, instead of a point-like image we will have a small’smear’ of light which, for obvious reasons, is known asthe point spread function. For a mechanical system (say,the shock absorbers on a car), we also excite the systemwith an impulse - i.e. a sharp blow - and monitor theresponse as a function of time.

4 The impulse response of the system in many cases is known asa Green’s function after George Green (1793-1841), the famousNottingham-based mathematician.

FIG. 20 An arbitrary function represented as a stream ofimpulses

B. Convolution and Impulse Response

In general, however, the input function will not be animpulse. Rather, the input will be a significantly morecomplex periodic or aperiodic signal. Nevertheless, andas shown in Fig. 20, we can treat practically any functionas a succession of impulses. The question that now arisesis: having determined the impulse response function, howdo we use this to determine the output of the system for asuccession of impulses, each of different amplitude? Thatis, how do we calculate the response of the system for anarbitrary input function f(t)?

The response of a system (optical, audio, elec-trical, mechanical, etc..) to an arbitrary signalf(t) is the convolution of f(t) with the impulseresponse of the system.

What do we mean by convolution? The origins of theword lie in the Latin term convolvere which means ’toroll together’. As we’ll see, this is a rather good ’plainEnglish’ definition of convolution. The mathematical def-inition of the convolution of two functions f(x) and g(x)(denoted f ∗ g - do not confuse the symbol used for con-volution with that used for the complex conjugate of aquantity) is:

h(x′) =∫ +∞

−∞f(x)g(x′ − x)dx (32)

Equation 32 is best interpreted graphically (althoughgraphical convolution of two functions can of course be anunwieldy and time-consuming process). To carry out theprocess represented by Equation 32, the following stepsare required:

1. Sketch f(x)

2. Sketch the other function (g(x)) backwards on atransparency

3. Incrementally slide the transparency across thegraph paper in the +x direction.

4. At each point (x′), calculate the area under thecurve representing the product of the two functions.

15

I will not present examples of graphical convolution here- you’ll see these in the problems classes. In addition,you should visit the following site:

http://www.jhu.edu/~signals/convolve/

This website features a Java applet which enables thegraphical convolution of arbitrary functions to be carriedout.

Convolutions obey various arithmetic rules:

1. The commutative rule: f ∗ g = g ∗ f

2. The distributive rule: f ∗ [g + k] = f ∗ g + f ∗ k

3. The associative rule: f ∗ [g ∗ k] = [f ∗ g] ∗ k

C. The Convolution Theorem and Fourier Transforms

Both graphical convolution and the application ofEquation 32 can be tedious. Luckily, there is a remark-able theorem - called, appropriately, the convolution the-orem - which describes a convolution of two functions interms of their respective Fourier transforms. The theo-rem is expressed mathematically in Equation 33:

FT (f(t) ∗ g(t)) =√

2πF (ω)G(ω) (33)

Therefore, to calculate the Fourier transform of theconvolution of two functions we need only multiply theindividual Fourier transforms. This is an incredibly pow-erful theorem with applications in all areas of physics.In particular, a number of questions in PC3 deal withthe application of the convolution theorem to problemsrelated to electrical circuits, signal transmission, and op-tics.

1. Convolution of δ(x− x0) with a function f(x)

The convolution of any function, f(x), with a deltafunction simply yields the original function f(x) at theposition of the delta function. Consider equation 32:

h(x′) =∫ +∞

−∞f(x)g(x′ − x)dx

Take g(x) as an arbitrary function and place the deltafunction at x = x0. The convolution integral is thus:

h(x′) =∫ +∞

−∞g(x′ − x)δ(x− x0)dx (34)

Using the sifting property of delta-functions:

g(x) ∗ δ(x− x0) = g(x− x0) (35)

Note that if we now apply the convolution theoremto equation 35, we derive the shift theorem discussed inthe previous section (Eqn. 27).

2. The frequency convolution theorem

An important corollary to the convolution theo-rem (Eqn. 33) is the frequency convolution theorem.Whereas equation 33 shows that in order to get theFourier transform of a convolution we multiply the in-dividual transforms, the frequency convolution theoremstates that in order to get the Fourier transform of theproduct of two functions one must convolve the trans-forms. That is:

FT (f(t)g(t)) =1√2π

F (ω) ∗G(ω) (36)

A number of questions in PC3 and PC4 require theapplication of the frequency convolution theorem.

D. Impulse Response and Audio Signals

A rather striking example of the convolution processwill be presented in Lecture 3. First, a recording of theaudio impulse reponse of, for example, a concert hallor a cavern is acquired. This simply involves record-ing the sound of a hand-clap (or another appropriatelytime-limited signal) in the hall/ cavern. One now takesa recording of, say, a lecture or a piece of music and con-volves it with the impulse response function generatedby the hand-clap. When the convolution is carried out,the resulting waveform will sound as if it were recordedin the hall or the cavern (depending on which impulseresponse function was used). The samples used in thelecture to demonstrate this audio convolution are avail-able from the F32SMS website.

V. RECIPROCAL SPACE, DIFFRACTION, ANDFOURIER OPTICS

On a number of occasions in the module thus far,the importance of Fourier analysis in optics has beenstressed. In this section we focus solely on the area ofFourier optics and show how the various formulae andmethods outlined in previous sections may be appliedto the analysis of both diffraction patterns and images ingeneral. To treat two dimensional images, the 1D Fouriertransform covered in previous lectures is not sufficient -the 2D Fourier transform is required. Moreover, we willneed to consider reciprocal space in two dimensions. Al-though the concept of 2D reciprocal space may soundsomewhat daunting, my aim is to illustrate that not onlyis 2D inverse space not significantly more complicatedthan its one dimensional counterpart, but that by think-ing in terms of spatial frequencies significant insights maybe gained into the process of image formation.

16

A. Fraunhofer diffraction for single- and double slitexperiments

Throughout the module we focus on Fraunhoferdiffraction patterns where the following criteria are ful-filled: (i) the light appoaching the diffracting object isparallel and monochromatic, and (ii) compared to thesize of the diffracting object, the image plane is locatedat a large distance from the object.

Importantly, the amplitude of the Fraunhofer diffrac-tion pattern is the Fourier transform of the aperture func-tion. Hence, if we have a rectangular aperture such as aslit, the (amplitude of the) diffraction pattern will be asinc function (because the aperture transmission curve isequivalent to a top-hat function). To find the intensity ofthe diffraction pattern we need to take the expression forthe amplitude and multiply it by its complex conjugate.

In PC3 and PC4 you are asked to determine the diffrac-tion pattern observed for a pair of slits. By far the easi-est method of calculating the two slit diffraction patterninvolves the use of the convolution theorem. The twoslit aperture function is simply the single slit aperturefunction (whose Fourier transform you have calculatedad nauseum - it’s a sinc function) convolved with twodelta functions (one at each of the slit positions). It isessential that you understand the solution to Q1 of PC4as later problems involve ever more sophisticated appli-cations of the convolution theorem.

B. The diffraction grating

A diffraction grating may be thought of as a large arrayof slits. Let’s for now assume that we’re dealing with arather unrealistic infinite grating. To write down theaperture function for a grating we need to introduce theDirac comb function. As shown in Fig. 21, this comprisesan infinite number of equally spaced Dirac δ-functions.The aperture function for an infinite grating will simplybe the aperture function for a single slit convolved withthe Dirac comb function. As detailed in the solution toQ6 of PC4, the Fourier transform of a comb function isanother comb function. When we multiply this by thesinc function that is the Fourier transform of a single slit(and square to get the pattern intensity), we find thatthe diffraction pattern of an infinite grating is a sinc2

modulated series of equally spaced delta functions.PC4 also addresses the question of how the solution is

modified to cover the rather more realistic situation of afinite grating. In that case, the aperture function of theinfinite grating must be multiplied by a top-hat function(whose width determines the number of apertures) toyield a finite grating. From Section IV.C.2:

FT (f(t)g(t)) =1√2π

F (ω) ∗G(ω) (37)

That is, the Fourier transform of a product is the convo-lution of the transforms of the individual functions.

FIG. 21 The Dirac comb

FIG. 22 A simple 2D aperture

C. 2D Apertures and 2D Reciprocal Space

What happens to the diffraction pattern if instead ofhaving a 1D slit, we have a 2D aperture such as a squareor rectangular hole (see Fig. 22)? This necessitates aconsideration of 2D aperture functions and 2D Fouriertransforms. The two-dimensional Fourier transform is:

F (kx, ky) =∫ +∞

−∞

∫ +∞

−∞f(x, y) exp(−i(kxx + kyy))dxdy

(38)Note that the 2D transform involves integration with re-spect to both x and y and that, equivalently, two recip-rocal space coordinates (kx, ky) are required to describethe diffraction pattern. As the 2D function in this casemay be reduced to two 1D top-hat functions, it shouldn’tbe surprising to find that the diffraction pattern involvestwo sinc functions - one along the kx direction and theother along the ky direction. The form of the diffractionpattern is shown in Fig. 23. Note that the area nearthe centre of the diffraction pattern shown in Fig. 23 isassociated with low values of kx and ky - i.e. low spa-tial frequencies. The highest (absolute) values of kx andky are found at the corners of the photograph shown inFig. 23. This interpretation of the diffraction pattern(i.e. the square of the Fourier transform of the aperturefunction) in terms of its constituent spatial frequenciesis very important for the discussion of spatial filteringbelow.

17

FIG. 23 Fraunhofer diffraction pattern from rectangularaperture

FIG. 24 Fraunhofer diffraction pattern from rectangularaperture

As you should expect by now, note that there is a re-ciprocal relationship between the width of the aperturefunction along the x and y axes and the form of thediffraction pattern along the kx and ky directions in re-ciprocal space. If the aperture narrows along the x axis,then the diffraction pattern will broaden along kx andvice versa. [See Fig. 24]. (Note that in order to analyt-ically calculate the Fourier transform of a circular aper-ture, the use of Bessel functions is required. Although wewon’t cover Bessel functions in this module, you will useMatlab in PC5 to numerically determine the diffractionpattern for a circular aperture (this is the Airy functionyou have encountered in your Optics module).

D. Reciprocal Space, Spatial Frequencies, and ImageProcessing

Although at first it might perhaps be a rather difficultconcept to grasp, any image/ photograph is composedof a set of Fourier components with different spatial fre-quencies (see Fig. 25). We know that signals and func-tions in the time domain can be broken down into theircomposite frequency components via Fourier transforma-tion - precisely the same approach works for functionsof spatial coordinates (e.g. (x, y)) such as the image ofFourier shown in Fig. 25. For a function f(x, y), the cor-responding Fourier components are associated with spa-tial frequency. Just as a time-limited signal is associatedwith high frequency Fourier components (the ultimatetime-limited function being δ(t) whose Fourier transformis infinitely wide), sharp edges in an image are associatedwith high spatial frequencies.

Just as we have studied the effects of reduced band-width on the shape of electrical pulses (Q3, PC2), we canexplore - with any off-the-shelf image processing pack-age such as Paint Shop Pro or Photopaint - the changesin image quality that occur if certain spatial frequenciesare removed. Fig. 26 shows a photograph and its corre-sponding power spectrum (i.e. |F (kx, ky)|2). Note thatas the photograph does not have any periodic structure,there are no periodic features (dots or lines) in the powerspectrum. (The lines running vertically and horizontallythrough the centre of the image are artifacts associatedwith the computation of the Fourier transform). As forthe diffraction pattern shown in Fig. 23, the central re-gion of the power spectrum is associated with low spatialfrequencies, whereas as we move away from the centrethe values of kx, and ky get increasingly larger. If we

FIG. 25 Decomposition of portrait of Fourier into its compo-nent spatial frequencies

FIG. 26 Fourier spectrum of complicated image

18

FIG. 27 Low pass filtering

filter the image so that only frequencies below a cer-tain threshold are ’passed’ (as shown by the circle onthe power spectrum of Fig. 27), we have the imagingequivalent of an electrical low pass filter. A spatial lowpass filter removes higher Fourier components and thussharp edges and features that change on a short lengthscale are removed from the image. As seen in the post-filter image of Fig. 27, the photograph then appears veryblurry. We can also remove the lowest spatial frequencieswhile maintaining the higher frequency components. Inthis case only those Fourier components associated withhigh spatial frequencies are retained and we accentuateedges in the image. This is illustrated in Fig. 28.

It is also possible to filter an optical diffraction patternusing a suitably designed mask. The experimental appa-ratus required to carry out this spatial filtering process isshown in Fig. 29. A mask with an appropriate arrange-ment of holes is placed at the Fourier plane. By removingcertain diffraction spots (or, indeed, entire areas of thediffraction pattern) a filtering process analogous to thatdescribed for digital images may be carried out. A num-

ber of questions in PC 4 ask you to consider the resultsof applying spatial filters to the diffraction patterns ofa number of different objects. A very detailed descrip-tion of spatial filtering may be found in Optics, Hecht(2nd edition, Addison-Wesley Publishing (1987)) - thereare many copies of this textbook in the George Greenlibrary.

FIG. 28 High pass filtering

FIG. 29 Spatial filtering

To be continued...