Fourier Transforms Part 1

Question 1

What is the equation for Fourier Series?

Answer 1

f(x) = a0/2 + sum from n=1 to infinity of a(n)cos(2πnx/L)+b(n)sin(2πnx/L)

Question 2

What is the first step to derive the Fourier transform from the fourier series?

Answer 2

Turn the fourier series into a form involving complex exponentials. Use the equations cos(θ) = (exp(iθ)+exp(-iθ))/2, sin(θ) = (exp(iθ)-exp(-iθ))/2i

Question 3

What do you do after inputting the complex exponentials to the fourier series?

Answer 3

Multiply it all out and rearrange to have the exponentials outside the brackets. Set the parts inside the brackets equal to c(n), where c0 = a0/2, c(n>1) = (a(n)-ib(n))/2, c(n<1) = (a(n)+ib(n))/2

Question 4

What is the final equation for f(x) after the first two steps?

Answer 4

f(x) = sum from n=-infinity to infinity of c(n)*exp(i2πnx/L)

Question 5

What are we concerned with in this equation for f(x) and what does this mean?

Answer 5

Need real part of f(x), so f(x) = f(x), so c(-n) = c(n), and c(-n)* = c(n)

Question 6

What is the orthogonality relationship for the fourier series?

Answer 6

1/L * integral from -L/2 to L/2 of exp(i*2π(n-m)x/L) dx = 𝛿(nm)

Question 7

How do we exploit the orthogonality condition in this case?

Answer 7

Multiply both sides by exp(i2πmx/L) and integrate, giving integral from -L/2 to L/2 of f(x)exp(i2πmx/L) dx = sum from n=-infinity to infinity of the integral of exp(i2π(n-m)x/L)c(n)

Last part is equal to 𝛿(nm)

Question 8

What do we do once we have the orthogonality condition implemented?

Answer 8

Summation only non-zero for n=m due to the kronecker delta. Sub this in to find equation for c(m).

Question 9

What is the equation for c(m)? What does this equation do?

Answer 9

c(m) = 1/L * integral from -L/2 to L/2 of f(x) * exp(-i*2πmx/L) dx

This equation maps f(x) to its complex Fourier coefficients

Question 10

What is the reverse transform for c(m)?

Answer 10

f(x) = sum from n = -inf to inf of c(n)exp(i2πnx/L)

Question 11

What do we do next to define the functions on the entire real axis -inf<0

Answer 11

• Take limit L->inf so periodic nature disappears
• As L increases, spacing between k values decreases so we can set k = dk = 2π/L
• 1 = dk*L/2π
• Insert this into the sum

Question 12

What do we get in the sum after inserting our new finding?

Answer 12

sum over k of dkL/2πc(k)*exp(ikx)

Question 13

What is the last step in deriving the Fourier transform?

Answer 13

-Use equation found before for c(m):
Lc(m) = integral from -L/2 to L/2 of f(x)exp(-i2πmx/L) dx
-Set k = 2πm/L, sub this in and take the limit for L->infinity
-Define Lim as L->inf of Lc(k) as the fourier transform of f(x)

Question 14

What is the equation for the Forward fourier transform?

Answer 14

transform f(k) = integral from -inf to inf of f(x)*exp(-ikx) dx

Question 15

How do we find the reverse fourier transform?

Answer 15

• use the equation we found before for the sum after defining the function which is not periodic
• take limit to infinity of right hand side
• find that f(x) = 1/2π * integral from -inf to inf of dk transform f(k) * exp(ikx)