Fourier Series & The Fourier Transform

What is the Fourier Transform?

Anharmonic Waves

Fourier Cosine Series for even functions

Fourier Sine Series for odd functions

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

( ) ( ) exp( )F f t i t dtω ω

∞

−∞

= −∫1( ) ( ) exp( )2

f t F i t dω ω ωπ

∞

−∞

= ∫

What do we hope to achieve with theFourier Transform?We desire a measure of the frequencies present in a wave. This willlead 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. Most 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)

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

F(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).

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.

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

( ) ( ) exp( )F f t i tω ω∞

−∞

= −∫ dt 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

The Fourier Transform and its Inverse

The Fourier Transform and its Inverse:

So we can transform to the frequency domain and back. Interestingly, these functions are very similar.

There are different definitions of these transforms. The 2π can occur in several places, but the idea is generally the same.

Inverse Fourier Transform

FourierTransform( ) ( ) exp( )F f t i t dtω ω∞

−∞

= −∫1( ) ( ) exp( )

2f t F i t dω ω ω

π

∞

−∞

= ∫

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 of a wave E(t) to be:

2{ ( )}E tY

This is our 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(

F i t dt i ti

i ii

i ii

ω ω ωω

ω ωω

ω ωω

ωω

−−

= − = −−

= − − /2)]−

1 − − /2)=

( /2)/2)

=( /2)

∫

( sinc(F ω ω) = /2) Imaginary Component = 0

F(ω)

ω

Sinc(x) and why it's important

Sinc(x/2) is the Fourier transform of a rectangle function.

Sinc2(x/2) is the Fourier transform of a triangle function.

Sinc2(ax) is the diffraction pattern from a slit.

It just crops up everywhere...

The Fourier Transform of the trianglefunction, Δ(t), is sinc2(ω/2)

ω0

2sinc ( / 2)ω1

t0

( )tΔ1

1/2-1/2

The triangle function is just what it sounds like.

∩

We’ll prove this when we learn about convolution.

Sometimes people use

Λ(t), too, for the triangle

function.

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!

∩

Some functions don’t have Fourier transforms.

The condition for the existence of a given F(ω) is:

Functions that do not asymptote to zero in both the +∞ and –∞directions generally do not have Fourier transforms.

So we’ll assume that all functions of interest go to zero at ±∞.

( )f t dt∞

−∞

< ∞∫

Expanding the Fourier transform of a function, f(t):

Expanding further:

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

−∞ −∞

= +∫ ∫ ←Re{F(ω)}

←Im{F(ω)}

↑ ↑

↓ ↓

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

Even functions of ω Odd functions of ω

Fourier Transform Symmetry Properties

( ) [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∞ ∞

−∞ −∞

+ −∫ ∫

Fourier Transform Symmetry Examples I

Fourier Transform Symmetry Examples II