Fourier Series & The Fourier Transform

What is the Fourier Transform?

Fourier Cosine Series for even functions and Sine Series for odd functions

The continuous limit: the Fourier transform (and its inverse)

The spectrum

Some examples and theorems

( ) ( ) exp( )F f t i t dtω ω

∞

−∞

= −∫1( ) ( ) exp( )2

f t F i t dω ω ωπ

∞

−∞

= ∫

What do we want from the Fourier Transform?

We desire a measure of the frequencies present in a wave. This will lead to a definition of the term, the “spectrum.”

Plane waves have only one frequency, ω.

This light wave has many frequencies. And the frequency increases in time (from red to blue).

It will be nice if our measure also tells us when each frequency occurs.

Ligh

t ele

ctric

fiel

d

Time

Lord Kelvin on Fourier’s theorem

Fourier’s theorem is not only one of the most beautiful results of modern analysis, but it may be said to furnish an indispensable instrument in the treatment of nearly every recondite question in modern physics.

Lord Kelvin

Joseph Fourier, our hero

Fourier was obsessed with the physics of heat and developed the Fourier series and transform to model heat-flow problems.

Anharmonic waves are sums of sinusoids.

Consider the sum of two sine waves (i.e., harmonic waves) of different frequencies:

The resulting wave is periodic, but not harmonic. Essentially all waves are anharmonic.

Fourier decomposing functions

Here, we write asquare wave as a sum of sine waves.

Any function can be written as thesum of an even and an odd function

( ) [ ( ) ( )] / 2

( ) [ ( ) ( )] / 2

( ) ( ) ( )

E x f x f x

O x f x f x

f x E x O x

≡ + −

≡ − −

⇓

= +

E(-x) = E(x)

O(-x) = -O(x)

E(x)

f(x)

O(x)

Let f(x) be any function.

Fourier Cosine Series

Because cos(mt) is an even function (for all m), we can write an even function, f(t), as:

where the set {Fm; m = 0, 1, … } is a set of coefficients that define the series.

And where we’ll only worry about the function f(t) over the interval (–π,π).

f( t) =1π

Fm cos(mt)m =0

∞

∑

The Kronecker delta function

,

1 if 0 if m n

m nm n

δ=⎧

≡ ⎨ ≠⎩

Finding the coefficients, Fm, in a Fourier Cosine Series

Fourier Cosine Series:

To find Fm, multiply each side by cos(m’t), where m’ is another integer, and integrate:

But:

So: only the m’ = m term contributes

Dropping the ‘ from the m: yields the coefficients for any f(t)!

0

1( ) cos( )m

m

f t F mtπ

∞

=

= ∑

f (t) cos(m' t) dt

− π

π

∫ =1π

m= 0

∞

∑ Fm cos(mt) cos(m' t) dt

− π

π

∫, '

'cos( ) cos( ' )

0 ' m m

if m mmt m t dt

if m m

π

π

ππ δ

−

=⎧= ≡⎨ ≠⎩∫

, '

0

1( ) cos( ' ) m m m

m

f t m t dt F

π

π

π δπ

∞

=−

= ∑∫( ) cos( )mF f t mt dt

π

π−

= ∫

Fourier Sine Series

Because sin(mt) is an odd function (for all m), we can write any odd function, f(t), as:

where the set {F’m; m = 0, 1, … } is a set of coefficients that define the series.

where we’ll only worry about the function f(t) over the interval (–π,π).

f (t) =1π

′ F m sin(mt)m= 0

∞

∑

Finding the coefficients, F’m, in a Fourier Sine Series

Fourier Sine Series:

To find Fm, multiply each side by sin(m’t), where m’ is another integer, and integrate:

But:

So: only the m’ = m term contributes

Dropping the ‘ from the m: yields the coefficients for any f(t)!

f (t) =1π

′ F m sin(mt)m= 0

∞

∑

0

1( ) sin( ' ) sin( ) sin( ' )mm

f t m t dt F mt m t dtπ π

π ππ

∞

=− −

′= ∑∫ ∫

, '

'sin( ) sin( ' )

0 ' m m

if m mmt m t dt

if m m

π

π

ππ δ

−

=⎧= ≡⎨ ≠⎩∫

, '0

1( ) sin( ' ) m m mm

f t m t dt Fπ

π

π δπ

∞

=−

′= ∑∫

( ) sin( )mF f t mt dtπ

π−

′ = ∫

Fourier Series

even component odd component

where

and

0 0

1 1( ) cos( ) sin( )m mm m

f t F mt F mtπ π

∞ ∞

= =

′= +∑ ∑

Fm = f (t) cos(mt) dt∫ ′ F m = f (t) sin(mt) dt∫

So if f(t) is a general function, neither even nor odd, it can be written:

We can plot the coefficients of a Fourier Series

We really need two such plots, one for the cosine series and another for the sine series.

Fm vs. m

m

5 252015

1030

1

.5

0

Discrete Fourier Series vs.Continuous Fourier Transform

Fm vs. m

m

Again, we really need two such plots, one for the cosine series and another for the sine series.

Let the integer m become a real number

and let the coefficients,

Fm, become a function F(m).

F(m)

The Fourier TransformConsider the Fourier coefficients. Let’s define a function F(m) that incorporates both cosine and sine series coefficients, with the sine series distinguished by making it the imaginary component:

Let’s now allow f(t) to range from –∞ to ∞, so we’ll have to integrate from –∞ to ∞, and let’s redefine m to be the “frequency,” which we’ll now call ω:

F(ω) is called the Fourier Transform of f(t). It contains equivalent information to that in f(t). We say that f(t) lives in the time domain, and F(ω) lives in the frequency domain. F(ω) is just another way of looking at a function or wave.

( ) cos( )f t mt dt∫ ( ) sin( )i f t mt dt− ∫F(m) ≡ Fm – i F’m =

( ) ( ) exp( )F f t i t dtω ω∞

−∞

= −∫ The FourierTransform

The Inverse Fourier TransformThe Fourier Transform takes us from f(t) to F(ω). How about going back?

Recall our formula for the Fourier Series of f(t) :

Now transform the sums to integrals from –∞ to ∞, and again replace Fm with F(ω). Remembering the fact that we introduced a factor of i(and including a factor of 2 that just crops up), we have:

'

0 0

1 1( ) cos( ) sin( )m mm m

f t F mt F mtπ π

∞ ∞

= =

= +∑ ∑

1( ) ( ) exp( )2

f t F i t dω ω ωπ

∞

−∞

= ∫Inverse Fourier

Transform

Fourier Transform Notation

There are several ways to denote the Fourier transform of a function.

If the function is labeled by a lower-case letter, such as f, we can write:

f(t) → F(ω)

If the function is labeled by an upper-case letter, such as E, we can write:

or: ( ) ( )E t E ω→ %( ) { ( )}E t E t→ Y

∩Sometimes, this symbol is used instead of the arrow:

The Spectrum

We define the spectrum, S(ω), of a wave E(t) to be:

2( ) { ( )}S E tω ≡ Y

This is the measure of the frequencies present in a light wave.

Example: the Fourier Transform of arectangle function: rect(t)

1/ 21/ 2

1/ 21/ 2

1( ) exp( ) [exp( )]

1 [exp( / 2) exp(

exp( / 2) exp(2

sin( sinc(

F i t dt i ti

i ii

i ii

ω ω ωω

ω ωω

ω ωω

ω ωω

−−

= − = −−

= − − /2)]−

1 − − /2)=

( /2)/2)

= ≡ /2)( /2)

∫

{rect( }sinc(

tω

)= /2)

F Imaginary Component = 0

F(ω)

ω

Example: the Fourier Transform of adecaying exponential: exp(-at) (t > 0)

0

0 0

0

( exp( )exp( )

exp( ) exp( [ )

1 1exp( [ ) [exp( ) exp(0)]

1 [0 1]

1

F at i t dt

at i t dt a i t dt

a i ta i a i

a i

a i

ω ω

ω ω

ωω ω

ω

ω

∞

∞ ∞

+∞

) = − −

= − − = − + ]

− −= − + ] = −∞ −

+ +−

= −+

=+

∫

∫ ∫

1(F iia

ωω

) = −−

A complex Lorentzian!

Example: the Fourier Transform of aGaussian, exp(-at2), is itself!

2 2

2

{exp( )} exp( )exp( )

exp( / 4 )

at at i t dt

a

ω

ω

∞

−∞

− = − −

∝ −

∫F

t0

2exp( )at−

ω0

2exp( / 4 )aω−

The details are a HW problem!

∩

Fourier Transform Symmetry PropertiesExpanding the Fourier transform of a function, f(t):

( ) Re{ ( )} cos( ) Im{ ( )}sin( )F f t t dt f t t dtω ω ω∞ ∞

−∞ −∞

= +∫ ∫ ←Re{F(ω)}

←Im{F(ω)}

↑ ↑

↓ ↓= 0 if Re{f(t)} is odd = 0 if Im{f(t)} is even

Even functions of ω Odd functions of ω

( ) [Re{ ( )} Im{ ( )}] [cos( ) sin( )]F f t i f t t i t dtω ω ω∞

−∞

= + −∫

Im{ ( )} cos( ) Re{ ( )}sin( )i f t t dt i f t t dtω ω∞ ∞

−∞ −∞

+ −∫ ∫= 0 if Im{f(t)} is odd = 0 if Re{f(t)} is even

↓ ↓

Expanding more, noting that: ( ) 0O t dt∞

−∞

=∫ if O(t) is an odd function

The Dirac delta function

Unlike the Kronecker delta-function, which is a function of two integers, the Dirac delta function is a function of a real variable, t.

if 0( )

0 if 0t

tt

δ∞ =⎧

≡ ⎨ ≠⎩t

δ(t)

The Dirac delta function

It’s best to think of the delta function as the limit of a series of peaked continuous functions.

if 0( )

0 if 0t

tt

δ∞ =⎧

≡ ⎨ ≠⎩

t

f1(t)

f2(t)

fm(t) = m exp[-(mt)2]/√π

f3(t)

δ(t)

Dirac δ−function Properties

( ) 1t dtδ∞

−∞

=∫ t

δ(t)

( ) ( ) ( ) ( ) ( )t a f t dt t a f a dt f aδ δ∞ ∞

−∞ −∞

− = − =∫ ∫

exp( ) 2 (

exp[ ( ) ] 2 (

i t dt

i t dt

ω π δ ω

ω ω π δ ω ω

∞

−∞

∞

−∞

± = )

′ ′± − = − )

∫

∫

ω

2πδ(ω)

The Fourier Transform of δ(t) is 1.

1exp( ) 2 (i t dtω π δ ω∞

−∞

− = )∫And the Fourier Transform of 1 is 2πδ(ω):

( ) exp( ) exp( [0]) 1t i t dt iδ ω ω∞

−∞

− = − =∫

t

δ(t)

ω

1

t

1

The Fourier transform of exp(iω0 t)

{ }0 0exp( ) exp( ) exp( )i t i t i t dtω ω ω∞

−∞

= −∫F

0exp( [ ] )i t dtω ω∞

−∞

= − −∫

The function exp(iω0t) is the essential component of Fourier analysis. It is a pure frequency.

Y {exp(iω0t)}

0 ω0

ω

02 ( )π δ ω ω= −

exp(iω0t)

0t

tRe

Im

0

The Fourier transform of cos(ω0 t)

{ }0 0cos( ) cos( ) exp( )t t i t dtω ω ω∞

−∞

= −∫F

[ ]0 01 exp( ) exp( ) exp( )2

i t i t i t dtω ω ω∞

−∞

= + − −∫0 0

1 1exp( [ ] ) exp( [ ] )2 2

i t dt i t dtω ω ω ω∞ ∞

−∞ −∞

= − − + − +∫ ∫0 0( ) ( )π δ ω ω π δ ω ω= − + +

+ω00−ω0

ω

0{cos( )}tωFcos(ω0t)t

0

The Modulation Theorem:The Fourier Transform of E(t) cos(ω0 t)

{ }0 0( ) cos( ) ( ) cos( ) exp( )E t t E t t i t dtω ω ω∞

−∞

= −∫F

0 01 ( ) exp( ) exp( ) exp( )2

E t i t i t i t dtω ω ω∞

−∞

= + − −⎡ ⎤⎣ ⎦∫0 0

1 1( ) exp( [ ] ) ( ) exp( [ ] )2 2

E t i t dt E t i t dtω ω ω ω∞ ∞

−∞ −∞

= − − + − +∫ ∫

{ }0 0 01 1( )cos( ) ( ) ( )2 2

E t t E Eω ω ω ω ω= − + +% %F

ωω00-ω0

{ }0( ) cos( )E t tωF

E(t) = exp(-t2)t

Example: 0( ) cos( )E t tω

Scale TheoremThe Fourier transform of a scaled function, f(at): { ( )} ( / ) /f at F a aω=F

{ ( )} ( ) exp( )f at f at i t dtω∞

−∞

= −∫F

{ ( )} ( ) exp( [ / ]) /f at f u i u a du aω∞

−∞

= −∫F

( ) exp( [ / ] ) /f u i a u du aω∞

−∞

= −∫( / ) /F a aω=

If a < 0, the limits flip when we change variables, introducing a minus sign, hence the absolute value.

Assuming a > 0, change variables: u = at

Proof:

The Scale Theorem in action

f(t) F(ω)

Shortpulse

Medium-lengthpulse

Longpulse

The shorter the pulse,

the broader the spectrum!

This is the essence of the Uncertainty Principle!

ω

ω

ω

t

t

t

The Fourier Transform of a sum of two functions

{ ( ) ( )}{ ( )} { ( )}a f t bg t

a f t b g t+ =+

FF F

Also, constants factor out.

f(t)

g(t)

t

t

t

ω

ω

ω

F(ω)

G(ω)

f(t)+g(t)F(ω) + G(ω)

Shift Theorem

( ){ }

( ) :

( ) exp( )

( ) exp( [ ])

exp( ) ( ) exp( )

f t a

f t a f t a i t dt

u t a

f u i u a du

i a f u i u du

ω

ω

ω ω

∞

−∞

∞

−∞

∞

−∞

−

− = − −

= −

− +

= − −

∫

∫

∫

F

The Fourier transform of a shifted function,

Proof :

Change variables :

exp( ) ( ) i a Fω ω= −

{ }( ) exp( ) ( )f t a i a Fω ω− = −F

Fourier Transform with respect to space

Y {f(x)} = F(k)

( ) ( ) exp( )F k f x ikx dx∞

−∞= −∫

If f(x) is a function of position,

We refer to k as the spatial frequency.

Everything we’ve said about Fourier transforms between the t and ωdomains also applies to the x and k domains.

k

x

The 2D Fourier Transform

Y (2){f(x,y)} = F(kx,ky)

= f(x,y) exp[-i(kxx+kyy)] dx dy

If f(x,y) = fx(x) fy(y),

then the 2D FT splits into two 1D FT's.

But this doesn’t always happen.

∫∫Y (2){f(x,y)}

xy

f(x,y)

The Pulse Width

There are many definitions of the "width" or “length” of a wave or pulse.

The effective width is the width of a rectangle whose height and area are the same as those of the pulse.

Effective width ≡ Area / height:

Advantage: It’s easy to understand.Disadvantages: The Abs value is inconvenient.

We must integrate to ± ∞.

1 ( )(0)efft f t dt

f

∞

−∞

Δ ≡ ∫t

f(0)

0

Δteff

t

Δt

(Abs value is unnecessary for intensity.)

The rms pulse width

The root-mean-squared width or rms width:

Advantages: Integrals are often easy to do analytically.Disadvantages: It weights wings even more heavily,so it’s difficult to use for experiments, which can't scan to ± )

1/ 2

2 ( )

( )rms

t f t dtt

f t dt

∞

−∞∞

−∞

⎡ ⎤⎢ ⎥⎢ ⎥Δ ≡⎢ ⎥⎢ ⎥⎣ ⎦

∫

∫

∞

t

Δt

The rms width is the “second-order moment.”

The Full-Width-Half-Maximum

Full-width-half-maximumis the distance between the half-maximum points.

Advantages: Experimentally easy.Disadvantages: It ignores satellite pulses with heights < 49.99% of the peak!

Also: we can define these widths in terms of f(t) or of its intensity, |f(t)|2.Define spectral widths (Δω) similarly in the frequency domain (t → ω).

t

ΔtFWHM

1

0.5

t

ΔtFWHM

The Uncertainty PrincipleThe Uncertainty Principle says that the product of a function's widthsin the time domain (Δt) and the frequency domain (Δω) has a minimum.

(Different definitions of the widths and the Fourier Transform yield different constants.)

1 1 (0)( ) ( ) exp( [0] )(0) (0) (0)

Ft f t dt f t i t dtf f f

∞ ∞

−∞ −∞

Δ ≥ = − =∫ ∫

(0) (0)2(0) (0)

f FtF f

ω πΔ Δ ≥ 2tω πΔ Δ ≥ 1tνΔ Δ ≥

Combining results:

or:

Define the widths assuming f(t) and F(ω) peak at 0:

1 1( ) ( )(0) (0)

t f t dt F df F

ω ω ω∞ ∞

−∞ −∞

Δ ≡ Δ ≡∫ ∫

1 1 2 (0)( ) ( ) exp((0) (0) (0)

fF d F i dF F F

πω ω ω ω ω ω∞ ∞

−∞ −∞

Δ ≥ = [0]) =∫ ∫

The Uncertainty Principle

For the rms width, Δω Δt ≥ ½

There’s an uncertainty relation for x and k: Δk Δx ≥ ½