Fourier Transforms

Question 1

What is a FT?

Answer 1

When a Fourier series is replaced with a continuous
distribution of frequencies with the discrete
summations replaced with an integral.

Question 2

How do you measure the width of an FT?

Answer 2

First two zeros
When the argument of sine OR cosine = 0
The distance between them is the width

Question 3

What does FT tell us?

Answer 3

The Fourier transform of any time varying signal tells us the frequency components present in that signal and their amplitudes

Question 4

What is the width relation?

Answer 4

For any function and its Fourier transform
Δ𝑥Δ𝑘=𝑐𝑜𝑛𝑠𝑡
Δ𝑡Δ𝜔=𝑐𝑜𝑛𝑠𝑡
The broader is f(t) the narrower is F(𝜔) and vice versa
The width of the original function to the transform is inversely proportional

The more values of k we add the greater is Δk but the smaller Δx becomes

Question 5

FT of an even function?

Answer 5

If f(x) is even then the equation contains cos(kx)and F(k) will be real
2/√2𝜋 ∫(0to∞)𝑓(𝑥)cos⁡( 𝑘𝑥)𝑑𝑥

Question 6

FT of an odd function?

Answer 6

If f(x) is odd then the equation contains sin(kx) and F(k) will be complex

Question 7

What are the applications of FT?

Answer 7

Diffraction of light
Nuclear Physics: scattering of electrons
Optical fibre data transmission

Question 8

What is the delta function?

Answer 8

The delta function 𝛿(x) has the value zero everywhere except at x = 0 where its value is infinitely large in such a way that its total integral is 1.

∫(−∞ to ∞)𝛿(𝑥)𝑑𝑥=1

Question 9

What is the product of the delta function?

Answer 9

The product of the delta function) with any function f(x) is zero for all x except where x = x0.

∫_(−∞)(∞)𝑓(𝑥)𝛿(𝑥−𝑥_0)𝑑𝑥=𝑓(𝑥_0)

Question 10

Draw a sinc function

Answer 10

sinc(x)= sinx/x
Periodic functions contain all possible frequency
Most intensity is between the first two zero points

Question 11

What happens to the FT if you have a short time?

Answer 11

Increases the width of the FT so there is a greater spread of frequency