Fourier and Wavelet Transforms

8/10/2019 Fourier and Wavelet Transforms

1/25

I

A Seminar Report on

Understanding Transforms

IN PARTIAL FULFILMENT OF REQUIREMENTS FOR THE DEGREE OF

BACHELOR OF ENGINEERING

IN

ELECTRONICS AND COMMUNICATIONS ENGINEERING

SUBMITTED BY:

AHTISHAM UL HAQ PAMPORI

ENROLLMENT NO: 32/11

ROLL NO: 3

DEPARTMENT OF ELECTRONICS AND COMMUNICATIONS ENGINEERING

NATIONAL INSTITUTE OF TECHNOLOGY SRINAGAR

HAZRATBAL, SRINAGAR, J&K 190006


2/25

II

INDEX

TITLE PAGE NO

ABSTRACT 1

1. INTRODUCTION 2

1.1. WHAT TRANSFORMS DO? 2

1.2. A BRIEF HISTORY 4

2. THE FOURIER TRANSFORM 6

2.1. INTRODUCTION 6

2.2. THE FOURIER SERIES 6

2.3. THE DISCRETE TIME FOURIER SERIES 7

2.4. THE FOURIER TRANSFORM 9

2.5. THE DISCRETE TIME FOURIER TRANSFORM 10

2.6. APPLICATIONS 11

3. THE FAST FOURIER TRANSFORM 13

3.1. INTRODUCTION 13

3.2. CALCULATING THE FFT 13

4. THE SHORT TIME FOURIER TRANSFORM 154.1. INTRODUCTION 15

4.2. EVALUATING THE STFT OF A SIGNAL 15

5. THE WAVELET TRANSFORMS 18

5.1. INTRODUCTION 18

5.2. SYNTHESIS AND ANALYSIS 18

6. FUTURE PROSPECTS 21

BIBLIOGRAPHY 22


3/25

1

Abstract

Electrical signals form the backbone of any electronic system capable of performingsomething productive. Be it a microprocessor processing digital signals, or a sound cardworking on analog music signals they are the carriers of information. It is signals that makethe hardware worth of doing what its designed to do. As important as they are, carryinginformation efficiently through signals is an area of unending research. Most of theinformation contained in analog signals is effectively in the frequencies they carry. Sincenature doesnt like digital, interpreting analog signals is the only way to extract informationabout all natural processes and understand nature better.

This paper discusses the two popular transforms which enable scientists and engineers toextract valuable information from signals their applications, drawbacks and the futurecourse of such transforms The Fourier Transforms (FT) and the rather recent WaveletTransforms (WT).

Fourier Transform (FT) was first introduced by the French mathematician J. Fourier who showed that any periodic function can be expressed as an infinite sum of complexexponential functions. His ideas were later generalized to first non-periodic functions, andthen periodic or non-periodic discrete time signals (now called the Discrete Time FourierTransform (DTFT)). It is after this generalization that it became a very suitable tool forcomputer calculations. In 1965, a new algorithm called Fast Fourier Transform (FFT) wasdeveloped and FT became even more popular.

What limits the use of Fourier Transform is its ability to detect the time distribution ofvarious frequency components. That is to say the Fourier Transform of a signal gives justits spectral components and not their time of occurrence. A solution would be to apply FT tothe signal taken small windows at a time. Such a solution is called a Short Time FourierTransform (STFT). As well see, the STFT has a disadvantage of having a fixed resolutionfor both high and low frequencies, and hence the time distribution is not too accurate.

Wavelet Transforms (WT) overcome the shortcomings of the STFT. This is done by

representing signals not as an infinite sum of periodic complex exponential functions but asan infinite sum of the daughter wavelets. These daughter wavelets are derived from a motherwavelet and form a complete orthonormal basis. That is to say, each and every function(provided it is square-integrable) can be represented by a linear combination of thesedaughter wavelets. The daughter wavelets unlike complex exponentials, have a finite widthwhich is variable. So, the high frequency components of a signal are well resolved by narrowwavelets while the low frequency components are well resolved by wider wavelets. This

provides for a good Multi Resolution Analysis (MRA) of signals as compared to STFT.


4/25

2

1. INTRODUCTION

1.1 WHAT TRANSFORMS DO?

Before delving into the woods of Fourier and Wavelet Transforms, lets consider the basic

question Why need transforms? . Take the Fourier transform as an example. What was

Fouriers discovery, and why is it useful? Imagine playing a note on a piano. When we press

the piano key, a hammer strikes a string that vibrates to and fro at a certain fixed rate (440

times a second for the A note). As the string vibrates, the air molecules around it bounce to

and fro, creating a wave of jiggling air molecules that we call sound. If we could watch the

air carry out this periodic dance, we d discover a smooth, undulating, en dlessly repeating

curve thats called a sinusoid, or a sine wave. (Clarification: In the example of the piano key,

there will really be more than one sine wave produced. The richness of a real piano note

comes from the many softer overtones that are produced in addition to the primary sine wave.

A piano note can be approximated as a sine wave, but a tuning fork is a more apt example of

a sound that is well -approximated by a single sinusoid.)


5/25

3

Now, instead of single key, say we play three keys together to make a chord. The resulting

sound wave isnt as pretty it looks like a complicated mess. But hidden in that messy sound

wave is a simple pattern. After all, the chord was just three keys struck together, and so the

messy sound wave that results is really just the sum of three notes (or sine waves).

Fouriers insight was that this isnt just a special property of musical chords, but applies more

generally to any kind of repeating wave, be it square, round, squiggly, triangular, whatever.

The Fourier transform is like a mathematical prism we feed in a wave and it spits out the

ingredients of that wave the notes (or sine waves) that when added together will reconstruct

the wave. [1]

Wavelet Transforms differ in the sense that they dont spit out sine wa ves but wavelets. That

is to say, just as a Fourier Transform decomposes signals into sinusoids of various

frequencies, Wavelet Transforms decompose them into daughter wavelets of various widths.


6/25

4

The reasons that compelled the scientific community to search for better transforms beyond

the Fourier transform will be discussed in Chapter 5.

1.2 A BRIEF HISTORY

Transforms are the basic tools used in spectral estimation of signals. The prehistory of

spectral estimation has its roots in ancient times with the development of the calendar and the

clock. The work of Pythagoras in 600 BC on the laws of musical harmony found

mathematical expression in the eighteenth century in terms of the wave equation. The

struggle to understand the solution of the wave equation was finally resolved by Jean Baptiste

Joseph de Fourier in 1807 with his introduction of the Fourier series. However, the

limitations of Fourier series regarding periodic functions was soon overcome by Sturm and

Liouville in 1836 when they extended the Fourier theory to the case of arbitrary orthogonal

functions. The Sturm-Liouville theory led to the greatest empirical success of spectral

analysis yet obtained, namely the formulation of quantum mechanics as given by Heisenberg

and Schrdinger in 1925 and 1926.

The modern history of spectral estimation begins with the breakthrough of J.W. Tukey in

1949, which is the statistical counterpart of the breakthrough of Fourier 142 years ago. This

result made possible an active development of empirical spectral analysis by research

workers in all scientific disciplines. However, spectral analysis was computationally

expensive. A major computational breakthrough occurred with the publication in 1965 of the

Fast Fourier Transform (FFT) algorithm by J.S. Cooley and J.W. Tukey. The Cooley-Tukey

method made it practical to do signal processing on waveforms in either the time or

frequency domain, something never practical with continuous systems. The Fourier

Transform became not just a theoretical description, but a tool. With the development of the

Fast Fourier Transform, the field of empirical spectral analysis grew from obscurity to

importance, and is now a major discipline. [2]

The newer generation of transforms included those which performed a time-frequency

analysis. These were required because most of the signals encountered were non-stationary

and the spectral composition of signals varied with time something Fourier transforms

could never point out. An immediate solution was to slice the given signal into small

windows and take the Fourier Transform of each window and then plot a time frequency

distribution of the signal. This was the Short Time Fourier Transform (STFT) [3] [4] .


7/25

5

Interesting as it was, STFT had an inherent problem It was difficult to determine the

appropriate window size for efficiently transforming non-deterministic signals whose

frequency range was unknown. This was later overcome by Wavelet transforms which

utilized a completely different basis function than Fourier Transforms - which had complex

exponentials as the orthonormal basis. From a historical point of view, the first reference to

the wavelet goes back to the early twentieth century when Alfred Haar wrote his dissertation

titled On the theory of the orthogonal function systems in 1909 to obtain his doctoral

degree at the University of G ttingen. His research on orthogonal systems of functions led to

the development of a set of rectangular basis functions. Later, an entire wavelet family, the

Haar wavelet, was named on the basis of this set of functions, and it is also the simplest

wavelet family developed till this date. [3][5]


8/25


9/25

7

[ ] where

[ ]

are the FS coefficients of the signal . We say that and [ ] are an FS pair anddenote the relationship as

[ ] Looking at the synthesis equation, we represent the signal as a sum of sinusoids each

with a frequency which is an integral multiple of some fundamental frequency . In effect, a

sinusoid with frequency is scaled by a factor [ ] and added up to form the time-domain signal .

The analysis equation on the other hand compares the signal with the sinusoid .

The degree of similarity of the two functions determines the value of the coefficient [ ]. Itmust however be kept in mind that the coefficient [ ] is a complex quantity with amagnitude and phase. This is because in the synthesis of a signal , it may be required

that a particular sinusoid (a particular frequency) be scaled as well as phase-shifted. Since

has a fixed phase and (unit) magnitude, the changes are reflected by the coefficient

[ ].2.3 THE DISCRETE TIME FOURIER SERIES (DTFS)

The DTFS representation of a periodic signal [ ] with fundamental period andfundamental frequency is given by

[ ] [ ] where

[ ] [ ] are the DTFS coefficients of the signal [ ]. We say that [ ] and [ ] are a DTFS pair anddenote this relationship as

[ ] [ ]


10/25

8

The DTFS of a signal is different from a FS in a sense that the sinusoids used are not

continuous but discrete sinusoids. The thing about discrete sinusoids though is that theyre

much more interesting than their continuous-time counterparts. Discrete sinusoids exhibit 2

counterintuitive properties that sets them apart from continuous-time sinusoids and makes

DTFS (and DTFT) much more interesting than FS (and FT).

2.3.1 PROPERTIES OF DISCRETE SINUSOIDS

1. ALIASINGConsider 2 discrete-time sinusoids with different frequencies:

[ ]

[ ] ( )

But note that

[ ] ( ) [ ] Two discrete sinusoids whose frequencies differ by a factor of radians/sec are

identical. Such sinusoids are said to be aliases of each other and the property is called

aliasing. Aliasing of discrete sinusoids leads to an important conclusion Discrete

sinusoids are unique only in a window of radians/sec .

2. PERIODICITYConsider a discrete-time sinusoid

[ ] With frequency where is the total no. of samples in the signal and is

any arbitrary integer i.e. must be an integral multiple of . Such sinusoids arecalled harmonic sinusoids (as their frequencies are the harmonics of the fundamental

frequency ). It turns out that only harmonic sinusoids are periodic with a period of

N or (provided is an integer).

[ ] [ ] Consider another sinusoid

[ ] With frequency . Such a sinusoid oscillates but is not periodic.


11/25

9

[ ] [ ] With these 2 properties of discrete sinusoids, we reach the following conclusions:

Sinusoids are unique only in the range of radians/sec. As such, it is conventional

to use sinusoids in the frequency range of [0 to ] or from [- to ] to describe theDTFS/DTFT. Frequency values near 0, indicate the lower frequencies. Valuesnear indicate higher frequencies.

The DTFS/DTFT expresses a signal as a linear combination of harmonic sinusoids .

The reason as to why only harmonic sinusoids are to be used follows in the next

section.

2.3.2 ORTHOGONAL BASES AND HARMONIC SINUSOIDS

A discrete-time signal can be well represented as a vector in a vector space. A basis { } for

a vector space is a collection of vectors from that are linearly independent and span

e.g. a vector containing the three unit vectors in the 3-dimentional real space form a basis

of . Any vector in can be represented as a linear combination of the three basis vectors.

In general, any vector in a vector space can be represented as a linear combination of its basis

vectors. A basis is said to be orthogonal if the basis vectors are mutually orthogonal and it is

said to be orthonormal if they are orthogonal and their -norm is unity.

It follows from linear algebra that orthonormal vectors (vectors whose -norm is unity

and whose inner product among themselves is 0 i.e. they are orthogonal) of length form an

orthonormal basis. An important property of harmonic sinusoids is that they are orthogonal.

normalized harmonic sinusoids of length thus form a n orthogonal basis. Since weve

used a vector representation for signals, it follows that any -dimensional signal can by

represented as a linear combination of length- harmonic sinusoids which behave as basis

vectors.

With the above discussions in place, we can now see (from Page 7) that the synthesis

equation expresses a time domain signal as a linear combination of harmonic sinusoids. The

analysis equation obtains a particular scaling value [ ] by taking the inner product of thetime domain signal [ ] with the harmonic sinusoid pertaining to that particular frequency.2.4 FOURIER TRANSFORM (FT)


12/25

10

In contrast to the case of the periodic signal, there are no restrictions on the period of the

sinusoids used to represent non-periodic signals. Hence, the Fourier transform representations

employ complex sinusoids having a continuum of frequencies ranging from to . The

signal is represented as a weighted integral of complex sinusoids where the variable of

integration is the sinusoids frequency. Discrete -time sinusoids are used to represent discrete-

time signals in the DTFT, while continuous-time sinusoids are used to represent continuous-

time signals in the FT [8] . Thus, the FT representation of a continuous-time signal involves an

integral over the entire frequency interval; that is

where

is the FT of the signal . Note that in the synthesis equation, we have expressed as a

weighted superposition of sinusoids having frequencies ranging from to . The

superposition is an integral, and the weight on each sinusoid is . We say that and are an FT pair and write

2.5 THE DISCRETE TIME FOURIER TRANSFORM (DTFT)

The DTFT is used to represent a discrete-time non-periodic signal as a superposition of

complex sinusoids. In the previous section, we reasoned that the DTFT would involve a

continuum of frequencies. These frequencies would however be in the range of .

When we talked about the DTFS, we noticed that the DTFS of a periodic signal is periodic

with a period and samples per period. Now, a non-periodic signal can be represented as

a periodic signal with . This implies that a Fourier representation for a non-periodic

signal (DTFT) would be a DTFS with i.e. each period would have infinite samples or

in other words, it would be a continuous function. However, the frequencies are given by

. Since both and tend to infinity, has a finite value and varies between

or , both legitimate ranges according to literature. As with the

previous representations, the DTFT of a time-domain signal involves an integral over

frequency, namely,


13/25

11

Taking the DTFS (DFT in the figure) and applying the limit N . Richard Baraniuk, Rice University.

[ ] where

( ) [ ] is the DTFT of the signal [ ]. We say that and [ ] are a DTFT pair and write

[ ]

2.6 APPLICATIONS

Fourier representations of signals find intense applications in almost every field of science

and engineering. The widespread among them are:

Spectroscopy (FTIR): In the Fourier Transform Infra Red spectroscopy, the IR afterabsorption/reflection is spectrum analyzed and its Fourier transform recorded. The spectrum

of a particular substance has a particular Fourier transform. This helps identifying unknown

substances by obtaining the FT of their spectra and comparing it with those of known

elements. [9]

MP3: The MP3 encoding uses a derivative of Fourier Transform called the Discrete Cosine

Transform (DCT). The raw audio signal is divided into frames and each frame is passed

through a filter bank performing DCT and FFT (Fast Fourier transform). The psychoacoustic


14/25

12

model is applied along the way to discard/attenuate all frequency components inaudible to the

human ear. The resulting transform is then digitally encoded and this is called noise

allocation. Finally headers are attached to noise allocated blocks for error checking and other

metadata. [10]

Speech Recognition: Speech recognition systems employ FFT along with other corrective

algorithms to recognize human speech. Words are recorded and their FFT is matched with a

repository which contains mappings from words to their FFTs. The best match is used at the

output or a choice of closely identical matches is provided to the user. [11]

Image compression: Formats such as JPEG utilize DCT to compress images. The image is

broken down into 8x8 sections and a 2D DCT of each section is computed. Instead of the 8

bits required to represent the color of each pixel, the DCT coefficients are now stored along

with some metadata which occupies much lesser space than the actual RAW format. Here a

tradeoff is made between storage space and computation. While the JPEG format is lesser in

size, reconstructing the image requires more computation to be performed than the lossless

case.


15/25

13

3. THE FAST FOURIER TRANSFORM

3.1 INTRODUCTION

The computation complexity of the DFT/DTFT is quadratic in nature i.e. if the length of the

signal doubles, the time required to compute the DFT/DTFT will be four times. The

algorithm is said to be Such a calculation is impractical to be performed on a

computer. It was in 1965 that J.S. Cooley and J.W. Tukey discovered the Fast Fourier

Transform. It is not a transform but an algorithm to compute the Fourier transform moer

efficiently. The algorithm was actually invented by Gauss in 1805 (Heideman, Johnson,

Burrus, 1984). The FFT algorithm takes the computation involved from to

a much efficient algorithm.

3.2 CALCULATING THE FFT

Over the years, scientists and engineers have molded the FFT in various ways to serve their

purpose in various applications. In order to understand the underlying principle in the

simplest way, we demonstrate a variant of FFT called the Radix-2, decimation-in-time

FFT . From the synthesis equation of DFT/DTFT, we have:

[ ] [ ]

We now define a conventional factor called the twiddle factor (used to make it a bit cleaner)

defined as

The synthesis equation can now be written as:


16/25

14

[ ] [ ]

Since the twiddle factors are periodic both in and :

The radix-2 type FFT, the length of the signal N is a power of 2.

The signal is then broke down into two sub-signals the even samples and the odd samples.

The resulting solution can be written as:

Now, [ ] and [ ] are also periodic with period . So, the process can be iterated againto get a DFT and so on.


17/25

15

4. THE SHORT TIME FOURIER TRANSFORM

4.1 INTRODUCTION

Although the Fourier transform was a huge success, it did have its demerits. The Fourier

transform could not distinguish between certain stationary and non-stationary signals which

were composed of the same frequency components but the former had them present all along

the signal while the latter had them at different instances of time (e.g. a chirp signal). The

immediate solution to the problem was to divide the signal into small fragments and calculate

the FT of each segment. This would provide a time-indexed frequency distribution of the

signal with a fixed resolution.

4.2 EVALUATING THE STFT OF A SIGNAL

The first step in evaluating the STFT of the signal is to choose a suitable window size to get

the correct balance in the frequency and time resolutions.


18/25

16

Figure: Different window sizes for STFT

The window is then slid across the signal and the FT computed for each fragment that falls

under the window. Different flavors of STFT are present in literature which differ from each

other in how the window is slid across the signal. Some versions of STFT move the window

such that each position is exclusive of the new position covered. Some move the window

such that the previous frame is overlapped to some degree by the new frame. The STFT of a

chirp signal with non-overlapping frames is shown in the following figure.


19/25

17

Figure: STFT of a chirp signal (narrow window)

The above STFT is obtained when the window chosen is of a small width. A small width

window offers a good time resolution but the tradeoff is that the frequency resolution is poor.

A wider width offers a good frequency resolution but a poor time resolution. The following

figure demonstrates the same.


20/25

18

Figure: STFT of a chirp signal (wide window)


21/25

19

5. THE WAVELET TRANSFORM

5.1 INTRODUCTION

The main drawback of STFT was the inability to perform a good Multi Resolution Analysis

(MRA). MRA, as implied by its name, analyzes the signal at different frequencies with

different resolutions. Every spectral component is not resolved equally as was the case in the

STFT. MRA is designed to give good time resolution and poor frequency resolution at high

frequencies and good frequency resolution and poor time resolution at low frequencies. The

continuous wavelet transform was developed as an alternative approach to the short time

Fourier transform to overcome the resolution problem. The wavelet analysis is done in a

similar way to the STFT analysis, in the sense that the signal is multiplied with a function,

(the wavelet), similar to the window function in the STFT, and the transform is computed

separately for different segments of the time-domain signal. However, there are two main

differences between the STFT and the CWT:

1. The Fourier transforms of the windowed signals are not taken, and therefore single

peak will be seen corresponding to a sinusoid i.e. negative frequencies are not

computed.

2. The width of the window is changed as the transform is computed for every single

spectral component, which is probably the most significant characteristic of the

wavelet transform.

5.2 Synthesis and Analysis

As with the FT, the transition from the time domain into the frequency-time domain is made

by a pair of equations. The synthesis equation works to recover the original time signal as a

combination of daughter wavelets combined both in time and scale. The analysis equation

on the other hand works to find the coefficient of a particular daughter wavelet by calculating

its inner product with the actual signal.


22/25

20

As seen in the analysis equation, the transformed signal is a function of two

variables, and , the translation and scale parameters, respectively. is the transforming

function, and it is called the mother wavelet . The term mother wavelet gets its name due to

two important properties of the wavelet analysis as explained below:

The term wavelet means a small wave. The smallness refers to the condition that this

(window) function is of finite length (compactly supported). The wave refers to the condition

that this function is oscillatory. The term mother implies that the functions with different

region of support that are used in the transformation process are derived from one main

function, or the mother wavelet. In other words, the mother wavelet is a prototype for

generating the other window functions called the daughter wavelets.

The term translation is used in the same sense as it is used in the STFT, it is related to the

location of the window, as the window is shifted through the signal. This term, obviously,

corresponds to time information in the transform domain. However, we do not have a

frequency parameter, as we had before for the STFT. Instead, we have scale parameter which


23/25

21

is defined as . The term frequency is reserved for the STFT.

Figure: Scales of different magnitudes

Much like the STFT, the output of a wavelet transform spans both time and frequency axes,

the information about time is given by the translation axis while the frequency information is

there in the scale axis. The wavelet transform of a chirp signal is shown below along with the


24/25

22

actual signal:

Figure: CWT of a chirp signal having 4 frequency components.

6. FUTURE PROSPECTS

The need for better transforms is a never ending hunt and continues to drive mathematicians

and engineers to search for better algorithms and improved mathematical tools. Advances

have been made in Wavelet transforms with the second generation wavelet transforms

already starting to show up in literature. These transforms work without actually going into

the frequency domain. Several others like the Discrete Tchebichef Transforms (DTT) use

Tchebichef polynomials and find interesting applications in speech recognition. A lot needs

to be done to reduce the computational complexity of existing algorithms to make them fasterand help process signals faster. An implementation of FFT using Sparce matrices at MIT has

led to significant improvement in the efficiency of the algorithm. These and other attempts to

improve the basic mathematical tools help build a strong toolset for engineers and scientists

to make use of abstractions and develop something beautiful.


25/25

BIBLIOGRAPHY[1] Bhatia Aatish , The Math Trick Behind MP3s, JPEGs, and Homer Simpsons Face ,

http://nautil.us/blog/the-math-trick-behind-mp3s-jpegs-and-homer-simpsons-face .

[2] Enders. A. Robinson, A Historical Perspective of Spectral Estimation , Proceedings of

the IEEE Vol.70 No.9 September 1982, Pg 885.

[3] Gao, R.X; Yan,R. , Wavelets Theory and Applications for manufacturing , Ch. 2,

http://www.springer.com/cda/content/document/cda_downloaddocument/9781441915443-c1.pdf?SGWID=0-0-45-1056240-p173924341

[4] Gabor D (1946), Theory of communication , J IE EE 93(3):429 457

[5] Haar A (1910), Zur theorie der orthogonalen funktionen systeme , Math Ann 69:331 371

[6] Herivel J (1975). Joseph Fourier. The man and the physicist. Clarendon Press, Oxford

[7] Fourier J (1822), The analytical theory of heat. (trans: Freeman A). Cambridge

University Press, London, p 1878

[8] Haykin Simon; Van Veen Barry, Signals and Systems , Wiley, Second Edition, Ch:3

[9] ThermoNicolet, Introduction to Fourier Transform Infrared Spectrometry ,

http://mmrc.caltech.edu/FTIR/FTIRintro.pdf

[10] Guckert John (Jake), The Use of FFT and MDCT in MP3 Audio Compression ,

http://www.math.utah.edu/~gustafso/s2012/2270/web-projects/Guckert-audio-compression-

svd-mdct-MP3.pdf

[11] Signal processing for speech recognition,

http://www.cs.rochester.edu/u/james/CSC248/Lec13.pdf
http://nautil.us/blog/the-math-trick-behind-mp3s-jpegs-and-homer-simpsons-facehttp://nautil.us/blog/the-math-trick-behind-mp3s-jpegs-and-homer-simpsons-facehttp://www.springer.com/cda/content/document/cda_downloaddocument/9781441915443-c1.pdf?SGWID=0-0-45-1056240-p173924341http://www.springer.com/cda/content/document/cda_downloaddocument/9781441915443-c1.pdf?SGWID=0-0-45-1056240-p173924341http://www.springer.com/cda/content/document/cda_downloaddocument/9781441915443-c1.pdf?SGWID=0-0-45-1056240-p173924341http://mmrc.caltech.edu/FTIR/FTIRintro.pdfhttp://mmrc.caltech.edu/FTIR/FTIRintro.pdfhttp://www.math.utah.edu/~gustafso/s2012/2270/web-projects/Guckert-audio-compression-svd-mdct-MP3.pdfhttp://www.math.utah.edu/~gustafso/s2012/2270/web-projects/Guckert-audio-compression-svd-mdct-MP3.pdfhttp://www.math.utah.edu/~gustafso/s2012/2270/web-projects/Guckert-audio-compression-svd-mdct-MP3.pdfhttp://www.cs.rochester.edu/u/james/CSC248/Lec13.pdfhttp://www.cs.rochester.edu/u/james/CSC248/Lec13.pdfhttp://www.cs.rochester.edu/u/james/CSC248/Lec13.pdfhttp://www.math.utah.edu/~gustafso/s2012/2270/web-projects/Guckert-audio-compression-svd-mdct-MP3.pdfhttp://www.math.utah.edu/~gustafso/s2012/2270/web-projects/Guckert-audio-compression-svd-mdct-MP3.pdfhttp://mmrc.caltech.edu/FTIR/FTIRintro.pdfhttp://www.springer.com/cda/content/document/cda_downloaddocument/9781441915443-c1.pdf?SGWID=0-0-45-1056240-p173924341http://www.springer.com/cda/content/document/cda_downloaddocument/9781441915443-c1.pdf?SGWID=0-0-45-1056240-p173924341http://nautil.us/blog/the-math-trick-behind-mp3s-jpegs-and-homer-simpsons-face

Documents

Fourier and Wavelet Transforms