Fourier analysis ; Introduction , History and Applications .

1. FOURIER ANALYSIS
• PRESENTED TO : DR MUHAMMAD WASEEM MUMTAZ
• PRESENTED BY : AYZA JABEEN
• DEPARTMENT OF CHEMISTRY
• UNIVERSITY OF GUJRAT

2. TABLE OF CONTENTS
SERIAL N0. TOPICS SLIDE NO.
1 Introduction to Fourier analysis 4
2 History of Fourier analysis 6
3 Dirichlet conditions 8
4 Variants of Fourier analysis 9
5 Applications of Fourier analysis 21
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3. Contents of figures
Serial no. figures Slide no
1 A Signal in time and frequency domain. 4
2 Sine and cosine wave. 5
3 Structure of inner eye. 6
4 Speech recognition. 7
5 Prism analogy. 8
6 Leonhard Euler , Jean Baptist Joseph and Dirichlet. 9,10,11
7 Heat transfer in a metal plate. 12
8 Even function. 15
9 Odd function. 16
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4. FOURIER ANALYSIS
 Method of defining periodic functions as well as aperiodic functions in terms of
trigonometric sums.
 Method gets its name from Jean Baptist Joseph Fourier.
 He showed that any signal could be made up by adding together a series of pure
tones of (sine wave) of appropriate amplitude and time.
FIG 01: this figure shows the signal in frequency and time domain.
Reference: https;//towardsdatascience-fourier-transformation-com
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5. PERIODIC SIGNAL
 DEFINITION
 A periodic function is just a function that repeats itself after a particular time period.
 Any function that satisfies;
 May be
 Contiuous time
 Discrete time
 FOR EXAMPLE
 Sine wave
 Cosine wave
FIG 2 : sine wave and cosine wave
Reference: www.pinterest.cl%.
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6. HOW DO WE HEAR?
 COCHLEA- spiral of tissues with liquid and
thousand of tiny hairs that gradually gets smaller.
 Each hair is connected to the nerve.
 The longest hair resonate with lower frequencies,
the shorter hair resonate with higher frequencies.
 Thus the time-domain air pressure signal is
transferred into frequency spectrum , which is
then processed by the brain.
 Our ear is a natural Fourier transform analyzer. FIG 3 : Structure of inner ear.
REFERENCE: www.delenalexy.blogspot.com.
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7. TIME DOMAIN
 For example: speech recognition
 Difficult to differentiate between
different sounds in time domain.
FIG 4 : Segment of sound.
Reference: www.researchgate.com
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8. PRISM ANALOGY
 A prism which splits white light into a spectrum
of colors.
 White light consists of all frequencies mixed
together.
 The prism breaks them apart so we can see
the separate frequencies.
FIG 5 : prism analogy.
REFERENCE: www.slideshare.net.brief-review-of-fourierppt
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9. HISTORY
 LEONHARD EULAR (1748)
 The configurations of a vibrating string can be
expressed as a linear combination of normal modes
is a string.
 JOSEPH – LOUIS LAGRANGE
 It is impossible to represent signal with corners as
trigonometric sums.
FIG 6 : Leonhard Euler.
REFERENCES: www.Wikipedia.com/history/fourier
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10. JEAN BAPTIST JOSEPH FOURIER
 French mathematician , Jean Baptist joseph Fourier in 1807
gave the Fourier analysis.
 He worked on propagation and diffusion of heat in
metal bodies.
 He gave the mathematical expressions for heat transfer.
 First announcement was made before French academy of
science.
 Sylvester François Lacroix, Gaspard monger, Pierre Simon
Laplace , joseph louis Lagrange were hired.
 The analytical theory of heat published in Day
 France in 1822.
FIG7 : Jean Baptist Joseph Fourier.
REFERENCES: www.Wikipedia.com.fourier/analysis.
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11. JOHANN PETER GUSTAY DIRICHLET
 He in 1829 provided precise conditions under which a
periodic signal can be expressed as Fourier series.
 It is famously known as Dirichlet condition.
 F(x) must be periodic, finite and single valued every where.
 It should have finite number of discontinuity to be a periodic
function.
 It should have finite number of maxima and minima.
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12. FOURIER SERIES
DEFINITION
 It is an expansion of a periodic function in
terms of an infinite sum of sines and cosines.
 Infinite sum of
sine wave + cosine wave = Fourier series
EXPRESSIONS FIG 8 : HEAT TRANSFER IN A METAL
PLATE
 Reference: www.slideshare.net-Ffourier-
series--by- gle.cmd-nazmul-islam&psig md-nazmul-
islam&psigv
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13. CONTI………
 Fourier series in terms of an even function f(θ) is expressed in terms of cosine
waves.
 Fourier series in terms of an odd function is expressed in terms(θ) of sine waves.
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14. EULER’S FORMULA
The Fourier series for the function f(x) in the interval a<x <a+2π is given by:
The a0 , an and bn are called the Fourier coefficients.
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15. EVEN FUNCTIONS
 The value of the function would be the same when
we walk equal distances along the x-axis in in opposite direction.
 Mathematically speaking
F FIG 9 : Even
function.
REFERENCE: www..mathsisfun.com%functions-odd-even
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16. ODD FUNCTIONS
 The value of function would change its sign with the
same magnitude when we walk walk equal distance
along the x-axis but in opposite direction .
 Mathematically speaking;
FIG 10 : Odd functions.
REFERENCE www..mathsisfun.com%functions-odd-even
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17. FOURIER TRANSFORM
 The representation of non- periodic functions as linear combination of
sinusoids is famously known as Fourier transform.
 It is used to find frequency contained by given time domain signal.
 Frequency is the no. of occurrences of a repeating event per unit time.
 It converts one function into another without the loss of energy.
 Mathematically: FT
 x(t) x(f)
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18. DISCRETE FOURIER TRANSFORM
 DFT is used for analyzing discrete-time finite duration signals in the frequency
domain.
 Discrete time data sets are converted into discrete frequency representation.
 Mathematically,
the DFT of discrete time sequence x(n) is denoted by x(k). It is given by,
Here k=0,1,2,……..N-1
Since this summation is taken for ‘N’ points ,it is called ‘N’ point DFT.
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19. CONTINUOUS FOURIER TRANSFORM
 It turns out a large class of aperiodic signals can also be written as linear
combination of trigonometric sums.
 It deals with continuous time aperiodic signals.
19

20. FIG 12 : variants of fourier analysis.
REFERENCES: www.slideshare/fourir/analysis.
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21. 21
Applications
Partial
differential
Equations
Telecomm
unication
Signal
processing
Acoustics
Image
processing
Geology

22. References
 https;//whatis.techtarget.com/definition/fourier( visited at 9:13pm , 27/11/2019)
 https;//en.m.Wikipedia.org/wiki/fourier analysis (visited at 10:43am , 27/11/2019)
 https;//www.math.purdue.edu/academic/files (visited at 8:12pm ,28/11/2019)
 https;// www.Britannica.com/science (visited at 9:31pm , 28/11/2019)
 https;//jegilles.sdsu.edu/doc/fouriercours(visited at 9:40 pm , 29/11/2019)
 https;//www.math.bgu.ac.il/leonid/code/schoenstadt fourier pdf(visited at 10: 13pm 30/11/2019)
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