PDE Refresher II : Fourier Transform Indraneel Kasmalkar (Neel), ICME (ineel@stanford.edu)
IMPORTANT
IMPORTANT • Refresher classes are NOT comprehensive. • The best way to refresh you memory is to solve problems . • Problem sets available on the refresher course website. • Most of them are short, easy questions. • Need feedback to structure third PDE class: • http://freesuggestionbox.com/pub/oztvrxm • What you feel good about, what makes you uncomfortable, any topics you hate, or do not understand. The first few phrases that pop up in your mind when you think PDEs.
IMPORTANT • Refresher classes are NOT comprehensive. • The best way to refresh you memory is to solve problems . • Problem sets available on the refresher course website. • Most of them are short, easy questions. • Need feedback to structure third PDE class: • http://freesuggestionbox.com/pub/oztvrxm • What you feel good about, what makes you uncomfortable, any topics you hate, or do not understand. The first few phrases that pop up in your mind when you think PDEs.
IMPORTANT • Refresher classes are NOT comprehensive. • The best way to refresh you memory is to solve problems . • Problem sets available on the refresher course website. • Most of them are short, easy questions. • Need feedback to structure third PDE class: • http://freesuggestionbox.com/pub/oztvrxm • What you feel good about, what makes you uncomfortable, any topics you hate, or do not understand. The first few phrases that pop up in your mind when you think PDEs.
IMPORTANT • Refresher classes are NOT comprehensive. • The best way to refresh you memory is to solve problems . • Problem sets available on the refresher course website. • Most of them are short, easy questions. • Need feedback to structure third PDE class: • http://freesuggestionbox.com/pub/oztvrxm • What you feel good about, what makes you uncomfortable, any topics you hate, or do not understand. The first few phrases that pop up in your mind when you think PDEs.
IMPORTANT • Refresher classes are NOT comprehensive. • The best way to refresh you memory is to solve problems . • Problem sets available on the refresher course website. • Most of them are short, easy questions. • Need feedback to structure third PDE class: • http://freesuggestionbox.com/pub/oztvrxm • What you feel good about, what makes you uncomfortable, any topics you hate, or do not understand. The first few phrases that pop up in your mind when you think PDEs.
What is a Fourier Transform?
What is a Fourier Transform? It is a different perspective of looking at functions
The Traditional Input-Output Perspective
The Traditional Input-Output Perspective x f(x) 0 0 0.1 0.01 2 4 r r 2
Fourier Perspective: As a sum of special functions
Fourier Perspective: As a sum of special functions g(x) = Ae ix + Be -2ix + Ce 2.7ix
Fourier Perspective: As a sum of special functions g(x) = Ae ix + Be -2ix + Ce 2.7ix The coefficients are more important than the output value.
Fourier Perspective: As a sum of special functions g(x) = A e ix + B e -2ix + C e 2.7ix The coefficients are more important than the output value.
What are these special functions?
What are these special functions? ζ ranging over all real numbers.
Why are they special? ζ ranging over all real numbers.
Why are they special? ζ ranging over all real numbers.
Why are they special? ζ ranging over all real numbers. DIFFERENTIATION IS SIMPLER!
Why are they special? ζ ranging over all real numbers. g(x) = Ae ix + Be -2ix + Ce 2.7ix DIFFERENTIATION IS SIMPLER!
Why are they special? ζ ranging over all real numbers. g(x) = Ae ix + Be -2ix + Ce 2.7ix g'(x) = iAe ix + -2iBe -2ix + 2.7iCe 2.7ix DIFFERENTIATION IS SIMPLER!
Why are they special? ζ ranging over all real numbers. g(x) = Ae ix + Be -2ix + Ce 2.7ix g'(x) = iAe ix + -2iBe -2ix + 2.7iCe 2.7ix So easy to get new coefficients!! DIFFERENTIATION IS SIMPLER!
Infinite sum of special functions
Infinite sum of special functions
Infinite sum of special functions More formally, as an integral:
Infinite sum of special functions More formally, as an integral: The coefficients A( ζ ) can be thought of as a function. This function is the Fourier transform of ƒ:
Infinite sum of special functions More formally, as an integral: The coefficients A( ζ ) can be thought of as a function. This function is the Fourier transform of ƒ: or
Formal definition of Fourier Transform
Formal definition of Fourier Transform
Formal definition of Fourier Transform Multiplying by inverse special function to (sort of) isolate the coefficient The coefficients
With the above definition, hopefully you get back ƒ as sum of Fourier coefficients times special functions.
With the above definition, hopefully you get back ƒ as sum of Fourier coefficients times special functions.
Fourier Transform
Fourier Transform Inverse Transform
Fourier Transform Inverse Transform
Other Forms
Other Forms If ƒ is a discrete sequence:
Other Forms Discrete Fourier Transform If ƒ is a discrete sequence:
Other Forms Discrete Fourier Transform If ƒ is a discrete sequence:
Other Forms
Other Forms If ƒ is 2 π -periodic function:
Other Forms Fourier Series If ƒ is 2 π -periodic function:
Other Forms Fourier Series If ƒ is 2 π -periodic function:
Other Forms Fourier Series If ƒ is 2 π -periodic function: !! Only true for smooth functions.
Differentiation
Differentiation
Differentiation
Proof:
Proof:
Proof:
Proof: ...Integration by parts
Proof: ...Integration by parts ... ƒ decays
Proof: ...Integration by parts ... ƒ decays
But this should not be surprising
But this should not be surprising g(x) = Ae ix + Be -2ix + Ce 2.7ix g'(x) = iAe ix + -2iBe -2ix + 2.7iCe 2.7ix
But this should not be surprising g(x) = Ae ix + Be -2ix + Ce 2.7ix g'(x) = iAe ix + -2iBe -2ix + 2.7iCe 2.7ix
But this should not be surprising g(x) = Ae ix + Be -2ix + Ce 2.7ix g'(x) = iAe ix + -2iBe -2ix + 2.7iCe 2.7ix Caution: Just an analogy. Fourier Transforms are integrals, not discrete sums.
Shifting
Shifting g(x) = ƒ(x+a)
Shifting g(x) = ƒ(x+a) g
Proof Intuition: use the analogy
Proof Intuition: use the analogy
Proof Intuition: use the analogy
Proof Intuition: use the analogy New factor in coefficient
Shifting g(x) = ƒ(x+a) g
Example Fourier Transforms
Example Fourier Transforms 0 function
Example Fourier Transforms 0 function 0 function
Example Fourier Transforms
Example Fourier Transforms
Example Fourier Transforms
Example Fourier Transforms How would I have even guessed this??
Example Fourier Transforms
Example Fourier Transforms
Example Fourier Transforms This is one of those special functions!
Example Fourier Transforms ? This is one of those special functions!
Fourier Transform is coefficients of special functions.
Fourier Transform is coefficients of special functions. How would you draw the Fourier Transform?
CONFUSION ZONE
FOURIER TRANSFORM CANDIDATE A
FOURIER TRANSFORM CANDIDATE A
FOURIER TRANSFORM CANDIDATE A is equal to 1 for ζ = 3.2, and 0 for everything else.
FOURIER TRANSFORM CANDIDATE A is equal to 1 for ζ = 3.2, and 0 for everything else. 1 ζ = 3.2
Does this work with our summation intuition?
Does this work with our summation intuition? is equal to 1 for ζ = 3.2, and 0 for everything else.
Does this work with our summation intuition? is equal to 1 for ζ = 3.2, and 0 for everything else. f is the 'infinite sum' of coefficients times special functions
Does this work with our summation intuition? is equal to 1 for ζ = 3.2, and 0 for everything else. f is the 'infinite sum' of coefficients times special functions Integrand equals zero almost everywhere!
Does this work with our summation intuition? is equal to 1 for ζ = 3.2, and 0 for everything else. f is the 'infinite sum' of coefficients times special functions Integrand equals zero almost everywhere! After integral, ƒ(x) = 0?
FOURIER TRANSFORM CANDIDATE A is equal to 1 for ζ = 3.2, and 0 for everything else. 1 ζ = 3.2
FOURIER TRANSFORM CANDIDATE A is equal to 1 for ζ = 3.2, and 0 for everything else. 1 ζ = 3.2
FOURIER TRANSFORM CANDIDATE B
FOURIER TRANSFORM CANDIDATE B
FOURIER TRANSFORM CANDIDATE B Delta "function".
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