![]() ![]() Then equation becomes single partial derivative equation (easy to solve), so you can solve for U in terms of k and t (make sure to find the constant with respect to t (A(k)). ![]() Conjugation - F(f*(t))=g*(-omega) What is the reality condition (useful)? If f is real, then you for g(omega) you don't need to specify negative frequencies (since they are related to the positive frequencies by complex conjugates) (means you can store information in half the space of f is real) How do you explain the reality condition mathematically (what property of Fourier transforms does this condition use)? F(f*(t))=F(t), =g(omega)=g*(-(omega)) (so the complex conjugate of the positive frequencies are the negative frequencies - don't need to store the negative frequencies as information (half the degrees of freedom)) (the conjugation property) What is Parseval's Identity for Fourier Transforms (what is the right hand side of this equation equal to?)? ∫(-infinity, infinity)f(x)g*(x)dx = ∫(-infinity, infinity)F(omega)G*(omega)d(omega) Give an application of Fourier Transform when helping with partial differential equations If you have a d^2()/dx^2 and d()/dt and the function has an x and t dependence, then Fourier transform U to remove x dependence (so now has k (reciprocal of spatial dimension) and t dependence) means for d^2()/dx^2 term you only need to differentiate the complex exponential term of the transform. Translation - F(f(t-t0))=e^i(omega)t0g(omega) (phase shift induced). What does the Fourier transform of a Gaussian function result in (explains) (how do you get to)? Another Gaussian, however the new Gaussian has reciprocal width (explains why you trade off one error for another) (Gaussian is f(t), which is (1/(2(pi)(sigma)))e^-(1/2)(t^2/(sigma)^2)(differentiate each side of the Fourier transform in terms of (omega), then end up with first order differential equation to solve) What is the Fourier transform of the delta function (meaning) (real life representation)? Constant (has the same value at every frequency) (spectrum has a value in all frequencies with the same power) (real life represented by very thin Gaussian (gives reciprocal width Gaussian)) What are the Fourier transform properties? 1. ![]()
0 Comments
Leave a Reply. |