Fourier Series & the Fourier Transform

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

∞ ∞ 1π ft()= F (ω )exp( iωω td ) F(ωω )=−ft ( ) exp( itdt ) 2 ∫ ∫ −∞ −∞ 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. Light electric field

Time

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. 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 a square wave as a sum of sine waves. Any function can be written as the sum of an even and an odd function

Let f(x) be any function. E(-x) = E(x) E(x) Ex()≡ [ f () x+− f ( x )]/2

O(x)

O(-x) = -O(x) Ox()≡−− [ f () x f ( x )]/2

⇓ f(x) fx()=+ Ex () Ox () Fourier Cosine Series

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

∞ 1 f(t) = F cos(mt) π ∑ m m =0

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). The Kronecker delta function

⎧1 if mn= δ mn, ≡ ⎨ ⎩0 if mn≠ Finding the coefficients, Fm, in a Fourier Cosine Series ∞ 1 f ()tFmt= cos( ) Fourier Cosine Series: π ∑ m m=0

To find Fm, multiply each side by cos(m’t), where m’ is another integer, and integrate: π ∞ π 1 f (t) cos(m' t) dt = F cos(mt) cos(m' t) dt ∫ π ∑ ∫ m − π m= 0 − π π π if m= m ' But: ⎧ cos(mt ) cos( m ' t ) dt =≡⎨ π δ mm,' ∫ ⎩0'if m≠ m − π π ∞ 1π ft()cos( mtdt ') = Fπ δ So: ∫ ∑ mmm,'Å only the m’ = m term contributes − m=0 π π Dropping the ‘ from the m: Å yields the Fm = ft()cos( mtdt ) ∫ coefficients for − any f(t)! π Fourier Sine Series

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

∞ 1 f (t) = F ′ sin(mt) π ∑ m m= 0

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

∞ 1 Fourier Sine Series: f (t) = F ′ sin(mt) π ∑ m m= 0

To find Fm, multiply each side by sin(m’t), where m’ is another integer, and integrate: ππ 1 ∞ f (tmtdt ) sin( ' )= Fmtmtdt′ sin( ) sin( ' ) ∫∫π ∑ m −−m=0 But: ππ π ⎧π if m= m' sin(mt ) sin( m ' t ) dt =≡⎨ π δmm,' ∫ 0'if m≠ m − ⎩ π π So: 1 ∞ ft()sin( mtdt ') = π F′ πδ ∫ ∑ mmm,'Å only the m’ = m term contributes − m=0 π

π Dropping the ‘ from the m: Fftmtdt′ = ()sin( ) Å yields the coefficients m ∫ − for any f(t)! π Fourier Series

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

11∞∞ ′ f ()tFmtFmt=+∑∑mm cos( ) sin( ) ππmm==00

even component odd component where

F = f (t) cos(mt) dt and F ′ = f (t) sin(mt) dt m ∫ m ∫ We can plot the coefficients of a Fourier Series

1

Fm vs. m

.5

0 52510 15 20 30 m

We really need two such plots, one for the cosine series and another for the sine series. Discrete Fourier Series vs. Continuous Fourier Transform

Fm vs. m Let the integer m become a F(m) real number and let the coefficients,

Fm, become a function F(m).

m

Again, we really need two such plots, one for the cosine series and another for the sine series. The Fourier Transform Consider 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:

F(m) ≡ F – iF’ = f ()cos(tmtdt ) − ift()sin( mtdt ) m m ∫ ∫ 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:

∞ The Fourier Fftitdt()ωω=− ()exp( ) ∫ Transform −∞

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. The Inverse Fourier Transform

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

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

11∞∞ f ()tFmtFmt=+ cos( )' sin( ) ∑∑mm ππmm==00

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:

∞ 1π Inverse f (tFitd )= (ω ) exp(ωω ) Fourier 2 ∫ 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:

E()tEt→ Y { ()} or: Et()→ E% (ω )

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 SEt()ω ≡ Y {()}

This is the measure of the frequencies present in a light wave. Example: the Fourier Transform of a rectangle function: rect(t)

1/2 1 Fitdtit(ωω )=−= exp( ) [exp( ω − )]1/2 ∫ −1/2 −1/2 −i 1 =−−/2)][exp(ii / 2) exp( −i ω 1−−/2)exp(ii /ωω 2) exp( ω = (/2) 2i ω ωω sin( /2) F(ω) =≡/2)sinc( (/2) ω ω ω Imaginary F {rect(t) } Component = 0

=/2)sinc(ω ω Example: the Fourier Transform of a decaying exponential: exp(-at) (t > 0)

∞ Fatitdt(ωω)=∫ exp( − )exp( − ) 0 ∞∞ =−−=−+]∫∫exp(at i t ) dt exp( [ a i t ) dt 00 ωω −−11+∞ =−+]=−∞−exp( [ait ) [exp( ) exp(0)] ai++0 ai −1 ωω =−[0 1] ai+ ω 1 = ω ai+ 1 Fi(ωω )=− − ia ω A complex Lorentzian! Example: the Fourier Transform of a Gaussian, exp(-at2), is itself!

∞ F {exp(−=at22 )}ω ∫ exp( − at )exp( − iω t ) dt −∞ 2 ∝−exp( / 4a ) The details are a HW problem!

exp(− at 2 ) exp(−ω 2 / 4a ) ∩

0 t 0 ω Fourier Transform Symmetry Properties

Expanding the Fourier transform of a function, f(t): ∞ F()ωωω=+∫ [Re{f ()ti} Im{f ()ttitdt}][cos( ) − sin( )] −∞ ∞ Expanding more, noting that: ∫ Ot() dt= 0 if O(t) is an odd function −∞

= 0 if Re{f(t)} is odd = 0 if Im{f(t)} is even ∞∞↓↓ F()ωω=+∫∫ Re{()}cos()ft tdt Im{()}sin() ft ω tdt ←Re{F(ω)} −∞ −∞ = 0 if Im{f(t)} is odd = 0 if Re{f(t)} is even ∞∞ ↓↓ +−i ∫∫Im{ f ()ttdti} cos(ωω ) Re{ f ()ttdt}sin( ) ←Im{F(ω)} −∞↑↑ −∞ Even functions of ω Odd functions of ω 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.

δ(t) ⎧∞=if t 0 δ ()t ≡ ⎨ ⎩0 if t ≠ 0 t ⎧∞ if t = 0 The Dirac delta function δ ()t ≡ ⎨ ⎩0 if t ≠ 0

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

2 fm(t) = m exp[-(mt) ]/√π

δ(t)

f3(t)

f2(t)

f1(t)

t Dirac δ−function Properties δ(t)

∞ ∫ δ ()tdt= 1 t −∞

∞∞ ∫∫δδ(t−=−= aftdt ) () ( t afadt ) () fa () −∞ −∞

∞ ∫ exp(±=itdtωπδω ) 2 ( ) −∞ ∞ ∫ exp[±−itdt (ω ωπδωω′ ) ] = 2 ( −)′ −∞ πδ ω =) )

ω ω ω ω ( itdt − is 1. 1 2πδ ) t ( δ 1exp( ) 2 ( ∫ ∞ −∞ ):

ω ( =− = δ 2π is 1 t t ) − t ( 1 δ titdti ( ) exp( ) exp( [0]) 1

δω ω ∫ ∞ −∞ The Fourier Transform of And the Fourier Transform of The Fourier transform of exp(iω0 t) ∞ F {}exp(itωωω )=− exp( it ) exp( itdt ) 00∫ ∞ −∞ =−−exp(itdt [ωω ] ) = 2(π δω− ω ) ∫ 0 0 −∞

exp(iω0t) Y {exp(iω0t)} Im t 0 ω Re t 0 ω0 0

The function exp(iω0t) is the essential component of Fourier analysis. It is a pure frequency. The Fourier transform of cos(ω0 t) ∞ F {}cos(ωωωttitdt )=− cos( ) exp( ) 00∫ −∞ ∞ 1 =+−−[]exp(itωωω ) exp( it ) exp( itdt ) 2 ∫ 00 −∞ ∞∞ 11 =−−+exp(i [ ωω ] t ) dt exp( −+ i [ ] t ) dt 22∫∫00 −∞ −∞

= π δω()−+ ω πδω00 ω () +

F {cos(t )} cos(ω0t) ω0 t −ω 0 ω 0 0 +ω0 The Modulation Theorem:

The Fourier Transform of E(t) cos(ω0 t) ∞ F E(tt )cos(ωωω )=− Ettitdt ( )cos( ) exp( ) {}00∫ ∞ −∞ 1 =+−−E(titititdt )⎡⎤ exp(ωωω ) exp( ) exp( ) 2 ∫ ⎣⎦00 ∞∞−∞ 11 =−−+−+E()exp(ti ωω [ωω ]) tdtEtitdt ()exp( [ ]) 22∫∫00 −∞ −∞ 11 F {}Et()cos(ω t )=−++ E%% (ωω ωω ) E ( ) 0022 0

Et()cos(ω t ) F {Et()cos(ω t )} Example: 0 0 E(t) = exp(-t2) t -ω0 0 ω0 ω Scale Theorem

The Fourier transform of a scaled function, f(at): F {(f at )}= F (/)/ω a a

∞ Proof: F {f ( at )}=−∫ f ( at ) exp( iω t ) dt −∞ Assuming a > 0, change variables: u = at ∞ F {(f at )}=−∫ f ()exp( u iω [/]) u a du / a −∞ ∞ =−∫ f ()exp([u iω /] audu ) / a −∞ = Faa(/)/ω

If a < 0, the limits flip when we change variables, introducing a minus sign, hence the absolute value. f(t) F(ω) The Scale

Theorem Short in action pulse t ω

The shorter Medium- the pulse, length the broader pulse the spectrum! t ω

This is the essence Long of the Uncertainty pulse Principle! t ω The Fourier Transform of a sum of two f(t) F(ω) functions t ω

g(t) G(ω) F {()af t+= bgt ()} t aftbgtFF{ ( )}+ { ( )} ω F(ω) + f(t)+g(t) G(ω)

Also, constants factor out. t ω Shift Theorem

The Fourier transform of a shifted function, f ():ta− F {ft(−= a )} exp( − iaFω ) (ω )

Proof : ∞ F {}ft()−= a∫ ft ( − a )exp( − itdtω ) −∞ Change variables : uta=−ω ∞ ∫ fu ( )exp(ωω−+ i [ u a ]) du −∞ ∞ =− exp(ia )∫ fu ( )exp( − iudu ) −∞ =− exp(iaFωω ) ( ) Fourier Transform with respect to space

If f(x) is a function of position,

∞ F(kfxikxdx )=− ( ) exp( ) ∫−∞ x

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

We refer to k as the spatial frequency. k

Everything we’ve said about Fourier transforms between the t and ω domains also applies to the x and k domains. The 2D Fourier Transform

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

= f(x,y) exp[-i(k x+k y)] dx dy y ∫∫ x y x

Y (2){f(x,y)} If f(x,y) = fx(x) fy(y),

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

But this doesn’t always happen. The Pulse Width Δt

There are many definitions of the t "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.

f(0) Effective width ≡ Area / height:

∞ Δteff 1 (Abs value is unnecessary Δ≡teff f ()tdt ∫ for intensity.) f (0) −∞ 0 t

Advantage: It’s easy to understand. Disadvantages: The Abs value is inconvenient. We must integrate to ± ∞. The rms pulse width Δt

The root-mean-squared width or t rms width:

∞ 1/2 ⎡ 2 ⎤ ⎢ ∫ tftdt() ⎥ Δ≡t ⎢ −∞ ⎥ rms ⎢ ∞ ⎥ ⎢ ∫ ftdt() ⎥ ⎣ −∞ ⎦

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

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 ±∞ ) The Full-Width- 1 Half-Maximum ΔtFWHM 0.5 Full-width-half-maximum is the distance between the half-maximum points. t

Advantages: Experimentally easy.

Disadvantages: It ignores satellite ΔtFWHM pulses with heights < 49.99% of the peak! t

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). The Uncertainty Principle

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

Define the widths 11∞∞ assumingωωωωωω f(t) and Δ≡tftdtFd( ) Δω ≡ (ωω ) fF(0)∫∫ (0) F(ω) peak at 0: −∞ −∞

11∞∞ F (0) Δ≥t∫∫ ftdt() = ft ()exp( − i [0]) tdt = f (0)−∞ff (0) −∞ (0)

11∞∞ 2(0)π f Δ≥∫∫Fd() = F ()exp( i [0])= d FF(0)−∞ (0) −∞ F (0)

(Different definitions of the widths and the Combining results: Fourier Transform yield different constants.) fF(0) (0) ΔΔ≥ωπt 2 or: Δω Δ≥t 2π Δν Δ≥t 1 Ff(0) (0) The Uncertainty Principle

For the rms width, Δω Δt ≥ ½ There’s an uncertainty relation for x and k: Δk Δx ≥ ½