Fourier Transform

Question 1

the fourier transform and inverse fourier transform allow us to

Answer 1

move between conjugate spaces, such as time and frequency

Question 2

the two most important integral transforms for physics

Answer 2

fourier transform and integral transform

Question 3

fourier and laplace transforms both have the property that

Answer 3

the dimension of the original variable is the reciprocal of the target variable

Question 4

considerations to get from fourier series to fourier transform

Answer 4

consider a function of time with period T
angular frequency of the rth component is wr=r2pi/T
delta w=2pi/T
make notation replacements to complex fourier series and complex fourier coefficients and combine

Question 5

why is the 1/2pi split into two factors of 1/sqrt(2pi)

Answer 5

to make definitions of the fourier transform and inverse fourier transform as symmetrical as possible

Question 6

The only requirement f(t) has to satisfy in order for its Fourier transform to exist is

Answer 6

integral from -infinity to infinity of |f(t)|dt is finite

Question 7

The Fourier transform of a periodic function with frequency ω0 consists of

Answer 7

a series of narrow peaks
at the discrete frequencies nω0 (n = 1, 2, 3, . . .), which are exactly the frequencies represented
by the Fourier series expansion of f(t).

Question 8

the fourier transform of spoke sound determines

Answer 8

what vowel we perceive it as

Question 9

how is the fourier transform used to compress an image

Answer 9

the bits of the image that matter (the boundaries between different regions) have different frequency
components to the more homogenous parts, which can be removed without ruining our ability to identify shapes.

Question 10

fourier transforming a function twice

Answer 10

gives almost the original function back

Question 11

properties of the fourier transform

Answer 11

linearity
scaling
translation
differentiation
integration

Question 12

linearity of fourier transforms

Answer 12

FT[af(t)+bg(t)] = aFT[f(t)] + bFT[g(t)]

Question 13

scaling of fourier transforms

Answer 13

FT[f(at)]= 1/|a| FT(w/z)

Question 14

translation of fourier transforms

Answer 14

FT[f(t+a)] = exp(iwa)FT(w)

Question 15

fourier transform differentiation

Answer 15

FT[f’(t)]=iwFT(w)

FT[f^(n)(t)]=(iw)^n FT(w)

Question 16

integration of FT

Answer 16

=1/iw FT(w) +2pi c delta(w)

where 2pi c delta(w) is the FT of the constant of integration

Question 17

solving differential equation using FTs steps

Answer 17

1. get differential equation in required form
2. apply the FT
3. exploit linearity
4. use derivatives of the FT
5. n can be brought outside sum
6. apply inverse FT