1. The Fourier Transform What is the Fourier transform?
Fourier cosine and sine series for even and odd functions
The continuous limit: the Fourier transform (and its inverse) 
Some transform examples and theorems
The Uncertainty Principle
The Fourier transform with respect to position and the 2D Fourier transform w w0
F(w) t
f(t)
Prof. Rick Trebino, Georgia Tech,

2. What do we hope to achieve with the Fourier transform?
We desire a measure of the frequencies present in a wave.
This will lead to a definition of the spectrum.
Plane waves have only one frequency, w0.
Light electric field
Time
This light wave has many
frequencies. And the
frequency increases in
time (from red to blue).

3. Many waves are sums of sinusoids.
Sine wave #2
Sum
Consider the sum of two sine waves of different frequencies:
Time
Sine wave #1
Frequency #2
Frequency #1
Frequency
The spectrum of the sum of the two sine waves:
This is a common situation, especially when boundary conditions are present.

4. Joseph Fourier
Joseph Fourier brought the Rosetta stone back from Egypt when he accompanied Napoleon there, and he discovered the greenhouse effect of the earth’s atmosphere.
Fourier was obsessed with the physics of heat and developed the Fourier series and transform to model heat-flow problems.
Picture from
Joseph Fourier (1768 – 1830)

5. 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
Picture from

6. Fourier decomposing functions
Here, we write a
square wave as
a sum of sine waves of different frequencies.

7. Any function can be written as the sum of an even and an odd function.

8. 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 f(t) over the interval (–p,p).

9. Finding the coefficients Fm in a Fourier cosine series
To find Fm: Multiply each side by cos(m’t), where m’ is another integer, and integrate:
But:
only the m = m’ term contributes
So: 
 yields the coefficients for any even f(t)!
Dropping the ’ from the m:

10. 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 = 0, 1, … } is a set of coefficients that define the series. m
where we’ll only worry about the function f(t) over the interval (–p,p).

11. Finding the coefficients F ’ in a Fourier sine seriesm
To find F’ : Multiply each side by sin(m’t), where m’ is another integer, and integrate: m
But:
only the m = m’ term contributes
So: 
 yields the coefficients for any odd f(t)!
Dropping the ’ from the m:

12. Fourier series
If f(t) is a general function, neither even nor odd, it can be written:
even component odd component
where
and

13. We can plot the coefficients of a Fourier series.1
Fm vs. m
Let the integer m become a real number and let the coefficients Fm become a function F(m).
F(m) 20 30 10 m
We really need two such plots, one for the cosine series and another for the sine series.

14. The Fourier transform The Fourier Transform
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) º Fm – i F ’ = m
Now allow f(t) to range from –¥ to ¥, so we’ll have to integrate from –¥ to ¥, and redefine m to be the frequency, which we’ll now call w:
The Fourier
Transform
F(w) 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(w) lives in the frequency domain.
F(w) is just another way of looking at a function or wave.

15. The inverse Fourier transform
The Fourier Transform takes us from f(t) to F(w). 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(w).
And remembering the factor of i (and including a factor of ½ to account for double summing in integrating from –¥ to ¥, rather than 0 to ¥), we have:
Inverse
Fourier
Transform

16. The Fourier transform and its inverse
Inverse Fourier Transform
So we can transform to the frequency domain and back. Interestingly, these transformations are very similar.
These equations together are Fourier’s Theorem. Feel free to plug one into the other to verify that you get the function you started with!

17. 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(w)
If the function is already labeled by an upper-case letter, such as E, we can write:
or:
Sometimes, this symbol is used instead of the arrow: ∩

18. The spectrum We define the spectrum, S(w), of a wave E(t) to be:
This is the measure of the frequencies present in a wave.
What about the phase (j) of the Fourier transform—called the spectral phase—which we lose when we take its squared magnitude?
It yields the absolute phase of a given frequency’s sine wave (at t = 0), which yields information about when a particular frequency occurs (by computing dj/dw).

19. What exactly is the spectral phase?
The spectral phase is the phase (at t = 0) of each frequency in the pulse. t
All of these sine waves have zero phase at t = 0. So this pulse has:
j(w) = 0
Note that this wave-form sees constructive interference, and hence a peak, at t = 0.
And it has perfect cancellation everywhere else. w1 w2 w3 w4 w5 w6

20. Now try a linear spectral phase: j(w) = aw.
Linear spectral phase doesn’t change the pulse shape; it just delays in time. Let’s see how this works:
j(w1) = 0
j(w2) = 0.2 p
j(w3) = 0.4 p
j(w4) = 0.6 p
j(w5) = 0.8 p
j(w6) = p t

21. Example: the Fourier transform of a rectangle function, rect(t) 1
sinc(w/2)
zeros at w = 2np (n ≠ 0)
The sinc function is important in many fields. w

22. The Fourier transform of a Gaussian is a Gaussian!
Fourier transforms can be difficult to do. But here’s an important one you can try on your own if you like.
f(t) = exp(-t2/2) t w
f(t) and F(w) look similar, but remember that these functions live in different spaces or domains (that is, have different arguments)!

23. Scale Theorem The Fourier transform of a scaled function, f(at):
Proof:
Assuming a > 0, change variables: u = at, so t = u/a and dt = du/a
If a < 0, the limits flip when we change variables, introducing a minus sign, hence the absolute value.
Q.E.D.

24. The Scale Theorem in action
f(t)
F(w)
The Scale Theorem in action t w
Short pulse t w
The shorter the pulse, the broader the spectrum!
Medium- length pulse t w
This is the essence of the Uncertainty Principle!
Long pulse

25. The pulse width Dt Dteff
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: t
f(0)
Dteff
We can similarly define widths of F(w), the spectrum, or any other quantity.

26. The Uncertainty Principle
The Uncertainty Principle says that the product of a function's widths
in the time domain (Dt) and the frequency domain (Dw) is ≥ ~1.
Use effective widths assuming f(t) and F(w) peak at 0: 1 1
(Different definitions of the widths and the Fourier Transform yield different constants.)
Dn Dt = Time-Bandwidth Product
Combining results:
or:
Different width definitions yield different numbers (e.g., Dw Dt ≥ ½).

27. Shift Theorem The Fourier transform of a shifted function, f(t-a).
Proof:
Change variables: u = t - a, so t = u + a and dt = du:
Q.E.D.

28. The Fourier transform of a double pulset a -a w
f(t)
The Fourier transform of a double pulse
f(t) = rect[(t+a)/w] + rect[(t-a)/w]
F(w)
sinc(ww/2)
Use the scale and shift theorems simultaneously: w
Simplifying:

29. Spectra of single and double pulsesw
Single pulse
Double pulse
Rectangular pulses

30. The Fourier transform of a pulsed wave
Because the Fourier transform and inverse Fourier transform are so similar, any theorem for one also holds for the other, subject to a change in minus sign (and perhaps a factor of 2p).
Taking the Fourier transform of both sides:
Often called the Modulation Theorem
This means that shifting the frequency of a wave moves its spectrum:
f(t)
F(w) t w w0

31. Fourier Transform with respect to space
If f(x) is a function of position, x
f(x)
F {f(x)} = F(k) k
F(k)
We refer to k as the spatial frequency.
Everything we’ve said about Fourier transforms between the t and w domains also applies to the x and k domains.

32. The 2D Fourier Transform
Suppose we have a function of x and y. Then: x y
f(x,y)
F (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.
F (2){f(x,y)}

33. A 2D Fourier Transform: a square functionx y
f(x,y)
Consider a 2D rectangular function in the xy plane:
f(x,y) = rect(x) rect(y)
Its 2D Fourier transform splits into the product of two 1D Fourier transforms:
F {f(x,y)} = sinc(kx/2) sinc(ky/2)
This picture is an optical determination of the mag-squared Fourier transform of the 2D rectangular function!
|F (2){f(x,y)}|2 kx ky