Fourier Series & The Fourier Transform What is the Fourier transform?    Fourier cosine series for even functions   Fourier sine series for odd functions The continuous limit: the Fourier transform (and its inverse)  Some transform examples and the Dirac delta function Prof. Rick Trebino, Georgia Tech

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 term, 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). 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 Picture from http://www.theglasgowstory.com/image.php?inum=TGSE00939&add=99&t=&PHPSESSID=606c592262141b6658671cfabcebe1f2

Joseph Fourier Fourier was obsessed with the physics of heat and developed the Fourier series and transform to model heat-flow problems. Picture from http://www.sci.hkbu.edu.hk/scilab/math/fourier.html Joseph Fourier 1768 - 1830

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 sin(wt) sin(3wt) Here, we write a square wave as a sum of sine waves. sin(5wt)

Any function can be written as the sum of an even and an odd function. E(-x) = E(x) O(-x) = -O(x) E(x) O(x) f(x) 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).

The Kronecker delta function -5 -4 -3 -2 -1 0 1 2 3 4 … m n dm,n ~ 1 2

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: So:  only the m’ = m term contributes Dropping the ’ from the m:  yields the coefficients for any f(t)!

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).

Finding the coefficients, F’ , in a Fourier sine series m Fourier Sine Series:   To find F’ : 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)! m

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

We can plot the coefficients of a Fourier series 1 .5 Fm vs. m 5 10 25 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. the continuous Fourier transform 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 Again, we really need two such plots, one for the cosine series and another for the sine series.

The Fourier transform 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: 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 w: 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. F(m) º Fm – i F ’ = m The Fourier Transform

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). Remembering the fact that we introduced a factor of i (and including a factor of 2 to account for double summing in integrating from –¥ to ¥, rather than 0 to ¥), we have: Inverse Fourier Transform

The Fourier transform and its inverse   So we can transform to the frequency domain and back. Interestingly, these transformations are very similar. There are different definitions of these transforms. The 2π can occur in several places, but the idea is generally the same. Fourier Transform   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(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: ∩

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 light wave.

Example: the Fourier transform of a rectangle function, rect(t) -.5 0 .5 1 sinc(w/2) zeros at w = 2np (n ≠ 0) The sinc function is important in optics and many other fields. w

The Fourier transform of the triangle function, D(t), is sinc2(w/2) The triangle function is just what it sounds like. t 1 -1 1 ∩ w -2p 2p We’ll prove this when we learn about convolution.

Example: the Fourier transform of a decaying exponential, exp(-at) (t > 0) 0 for t < 0 exp(-at) for t > 0 t f(t) F(w) Imaginary part Real part w A complex Lorentzian!

Example: the Fourier transform of a Gaussian, exp(-at2), is itself! The details are a HW problem! t w ∩

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. t d(t) x d(x-y) y To see that the Dirac delta function is like the Kronecker delta function, we can also write:

The Dirac delta function It’s more rigorous to think of the delta function as the limit of a series of peaked continuous functions: fm(t) = m exp[-(mt)2]/√p d(t) f3(t) f2(t) f1(t) t Alternatively, we can use: fm(t) = m rect(mt)

Dirac d function properties d(t)

The Fourier transform of d(t) is 1. w 1 And the Fourier Transform of 1 is 2p d(w): t 1 2pd(w) w

The Fourier transform of exp(iw0 t) t Re Im F {exp(iw0t)} w w0 The function exp(iw0t) is the essential component of Fourier analysis. It is a pure frequency.

The Fourier transform of cos(w0 t) +w0 -w0 w

Fourier transform symmetry properties Expanding the Fourier transform of a function, f(t): Expanding more, noting that: if O(t) is an odd function = 0 if Re{f(t)} is odd = 0 if Im{f(t)} is even ¯ ¯ ¬Re{F(w)} = 0 if Im{f(t)} is odd = 0 if Re{f(t)} is even ¯ ¯ ¬Im{F(w)} ­ ­ Even functions of w Odd functions of w

Some functions don’t have Fourier transforms. The condition for the existence of a given F(w) 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 ±¥.