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FourierTransform

Fourier Transform
By Dr.?Zhenyu He?and?Yao?Xu Autumn?2010

Outline
Introduction Fourier?Series Fourier Series Fourier?Transform
Continuous?Fourier?Transform Discrete?Fourier?Transform
Disadvantage Conclusion C l i

Introduction
The?conception?of?Fourier?transform?is?put y forward?by?the?French?mathematician?and physicist?Mr.?Jean?Baptiste?Joseph?Fourier. In?signal?analysis?and?signal?processing, In signal analysis and signal processing Fourier?transform?is?an?important?and?basis technology. It?can?transform?a?signal?in?time?domain?to It can transform a signal in time domain to another?in?frequency?domain.

Mr. Jean Baptiste Joseph Fourier 1768~1830 1768 1830

Fourier Series
Definition: f In?mathematics,?Fourier?series?decomposes , p any?periodic function?or?periodic?signal?into (p y ) p the?sum?of?a?(possibly?infinite)?set?of?simple oscillating?functions,?namely?sines?and cos es cosines. The?family?of?functions?{cosnωt,?sinnωt} constructs?an?orthonormal basis?of constructs an orthonormal basis of L2 [t ? T / 2, t + T / 2] .

Fourier Series
It?can?be?written?as:
f (t ) =

k
a k ek
Where
ak =
f (t ), ek (t )
Also,?more?detailedly: Also more detailedly:
f (t ) =
∑ (a
n=0

n ∞
c o s n ω 1t + b n s i n n ω 1t )
n
= a0 +
∑ (a
n =1
c o s n ω 1t + b n s i n n ω 1t )

Fourier Series
The?Fourier?coefficients?in?the?equation?are:
a0 an bn 1 = T 2 = T 2 = T
∫ ∫ ∫
T /2 ?T /2 T /2 ?T /2 T /2 ?T /2
f (t ) d t f (t ) c o s ( nω t ) d t f (t ) s in ( nω t ) d t

Fourier Series
The?periodic?function f (t ) should?satisfy?the Dirichilet?condition: must?have?a?finite?number?of?extrema in?any given?interval; given interval; must?have?a?finite?number?of?discontinuities in?any?given?interval; must?be?absolutely?integrable over?a?period; must be absolutely integrable over a period; must?be?bounded.

Fourier Series
Trigonometric?functions?are?orthogonal to each?other.
?0, n ≠ m ? ( ( sin( nω t ) sin( mω t ) dt = ? T ∫t ?T / 2 ?2 , n = m ? , ? 0, n ≠ m t +T / 2 ? cos( nω t ) cos( mω t ) dt = ? T ∫t ?T / 2 ?2 , n = m ?
t +T / 2

t +T / 2
t ?T / 2
sin( nω t ) cos( mω t ) dt = 0, m, n = 0,1,L , ∞.

Fourier Series
Another?expression?with?trigonometric ∞ functions: f (t ) = c0 + ∑ cn cos(nωt + φn ) (
n =1 ∞
f (t ) = d 0 + ∑ d n cos(nωt + θ n )
n =1
a0 = c0 = d 0
2 2 cn = d n = an + bn
an = cn cos φn = d n sin θ n bn = ?cn sin φn = d n cos θ n an bn tan θ n = , tan φn = ? bn an

Frequency Spectrum
The?spectral?lines?of?periodic?signal?only pp q g appear?at?the?frequencies?that?are?integral times?of?the?basis?frequency.
cn
? n (ω )
ω1
nω 1
ω1
nω 1

Complex Exponential Series
The?family?of?trigonometric?functions?can?be written?in?the?form?of?Euler's?function:
{e
jnω t
}
The?Fourier?series?can?be?written?as: Th F b
f (t ) =
n = 0 , ± 1 , ± 2 ,L


F ( nω )e
jnω t
where?
?∞
F (0) = a0
1 F (nω ) = (an ? jbn ) 2 Introduce?the?negative?frequency Introduce the negative frequency 1 F (?nω ) = (an + jbn ) 2

Complex Exponential Series
The?frequency?spectrogram?of?periodic p p g complex?exponential?signal?is?show?as follows:

Complex Exponential Series
The?coefficients?of?the?complex?Fourier series?will?be:
1 F ( nω ) = Fn = T

t0 + T / 2
t0 ? T / 2
f (t ) e ? jnω t dt

Fourier Series
Fourier?series?can?only?expand?the?periodic , g p function,?the?integral?domain?is?one?period of?the?function; For?non‐periodic function,?we?should For non periodic function we should expand?the?integral?domain?to?the?whole field,?i.e.?T→∞, then?we?get?the?Fourier transform.

Fourier Transform
For?any?signal?(periodic?or?not) the?Fourier?transform?is?defined?as:
F (t ) = ∫ j 2 = ?1
∞ ?∞
?f (t ) ∈ L2 ( R)
,?
f ( x)e? j 2π xt dx
The?inverse Fourier?transform?is: e ve se ou e t a s o s
1 f (t ) = 2π


?∞
F (u )e j 2π xt dx

Definition
Continuous?FT
1‐D?Case
F (u ) = ∫ j 2 = ?1
∞ ?∞
f (t )e ? j 2π ut dt
2‐D?Case
∞ ∞
F (u , v) = ∫
?∞ ?∞

f ( x, y )e ? j 2π ( ux + vy ) dxdy

Definition
Discrete?FT
1‐D?Case
F (u ) = ∑ f ( x)e ? j 2πux / N
x =0 M ?1
u = 0,1,..., N ? 1.
2‐D?Case
F (u , v) = ∑∑ f ( x, y )e ? j 2π ( ux / M + vy / N ) dxdy
x =0 y =0 M ?1 N ?1
u = 0,1,..., M ? 1; v = 0,1,..., N ? 1

(1-D) DEMO (1 D)
From?the?figure?we?see that?the?initial?signal?is?a pulse?signal.?FT?can?find its?frequency?easily. f l
A?signal?consisting?of?one A signal consisting of one frequqency f(t)=sin(2πt)

(1-D) DEMO (1 D)
From?the?figure?we?see that?FT?can?transform the?signal?in?time?domain to?signal?in?frequency l f domain.?FT?obviously finds?the?two?waves?of different?frequencies. diff tf i
A?signal?consisting?of?two?frequqencies A signal consisting of two frequqencies f(t)=sin(2*pi*t)+0.5*sin(2*pi*5*t)

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