Parsevals Theorem and The Convolution Theorem

Question 1

How is power related to the amplitude?

Answer 1

P(x) ∝ E(x)^2, where E is the amplitude, or I ∝ u^2

Question 2

What is the equation for the total power/intensity? What is the u part equal to?

Answer 2

I(total) = int from -inf to inf of dx |u(x)|^2

|u(x)|^2 = uu*

Question 3

How can we incorporate fourier transforms into this power integral?

Answer 3

-Use inverse transform to sub in for u(x), and use dummy variables for u*(x), and sub them both into the integral equation.

Question 4

How can we incorporate the kronecker delta into the power integral?

Answer 4

Once rearranged, we have 1/2π * integral from -inf to inf of exp(ix(k-k’)) dx, which is equal to 𝛿(k-k’), so we can sub this in. We can then set k=k’ to remove this delta.

Question 5

What result to we get after subbing in the kronecker delta and removing it?

Answer 5

1/2π * integral from -inf to inf of dk * u(k)u(k)

Question 6

What is Parcevals’s Theorem?

Answer 6

integral from -inf to inf of dx |u(x)|^2 = 1/2π * integral from -inf to inf of dk * |u(k)|^2

Question 7

What is a convolution?

Answer 7

Where something convolutes a signal, for example a microscope convolutes the image signal to make it larger.

Question 8

What is the equation for a convolution and what do the letters mean?

Answer 8

f(x) = integral from -inf to inf of dx’ s(x’)*G(x-x’), where x’ is the image plane, x is what we see in microscope, s is the source, f is the image and G is the optical spread function

Question 9

What is the optical spread function often found to be approximated by?

Answer 9

A gaussian: N*exp(-(x-x’)^2/2σ^2)

Question 10

What is the used notation for a convolution?

Answer 10

f(x) = s* * G(x)

Question 11

How do we compute the fourier transform of a convolution?

Answer 11

• Sub it into the f(k) equation and then sub in the inverse transform for the G part
• Separate the integrals into different variables and sub in kronecker delta
• Set k=k’ and then see if any parts equal the fourier transform equation
• Have the answer

Question 12

What is the fourier transform of a convolution equal to?

Answer 12

f(k) = G(k)*S(k)

(the product of the two transformed functions) This is the convolution theorem

Question 13

What is the first step in solving a differential equation (eg. diffusion equation) using fourier transforms?

Answer 13

-Transform both sides of the equation and take d/dt out of the left hand side, leaving just transforms on each side

Question 14

What is the second step in solving a differential equation (eg. diffusion equation) using fourier transforms?

Answer 14

Trial solution: u(k,t) = A(k)*exp(-Dk^2t)

• Transform this back into real space to find u(x,t)
• Set A(k) = 1 and find this as the transform of a gaussian
• Is a solution to diffusion equation

Question 15

How can we use fourier transforms in more than one dimension?

Answer 15

Can transform twice (or more)- first in x, second in y etc.

Question 16

What is the general forward fourier transform for any number of dimensions?

Answer 16

f(k) = integral from -inf to inf of exp(-ikr)*f(r) d^d r (r and k suitable generalised to d dimensions)

Question 17

What is the general inverse fourier transform for any number of dimensions?

Answer 17

f(r) = 1/(2π) * d * integral from -inf to inf of exp(ikr) * f(k) * d^2 k