fourier 1

Question 1

fourier

original motivation:
basis functions: oscillations (sine and cosine)
describe a signal by its ……… spectrum

Answer 1

original motivation: representation by easier-to-handle functions

basis functions: oscillations (sine and cosine)

describe a signal by its frequency spectrum

Question 2

spatial frequency

Answer 2

𝑠𝑝𝑎𝑡𝑖𝑎𝑙 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 = 𝑐𝑦𝑐𝑙𝑒𝑠/𝑢𝑛𝑖𝑡𝑠 𝑜𝑓 𝑠𝑝𝑎𝑐𝑒

inverse of the spacial periodic length

Question 3

A ………., ……… function
f(t) in time or f(x) in space,
can be expanded into sine and cosine

Answer 3

A periodic, continuous function
f(t) in time or f(x) in space,
can be expanded into sine and cosine

Question 4

fourier series

Answer 4

𝑓(𝑡) =sumk=0 to inf(Akcos(wkt)+Bksin(wkt))
with k = 2pi k /T

Question 5

How can we get the coefficients Ak, Bk in fourier series ?

Answer 5

The functions coswkt, sinwkt form an orthogonal basis, using these features of an orthogonal basis we can extract the Fourier coefficients Ak, Bk hence the amplitude of the respective cosine or sinus function

Question 6

polar representation

Answer 6

𝑧 = 𝑎 + 𝑏 𝑖
= 𝐴 exp(i 𝜑)

𝐴 = sqrt(a^2+b^2)
𝜑 = arctan(b/a)

Question 7

eulars formula

Answer 7

exp(iwt) = cos(wt) + i sin(wt)

Question 8

fourier series in polar form

Answer 8

f(t) = sum k form -inf to inf (Ck exp(iwt))

Ck = (integral from -T/2 to T/2 (f(t) exp(-iwt) dt))/T

Question 9

fourier series to fourier transform

Answer 9

from continuous and periodic function values to continuous and non-periodic functions

T goes to infinity
wk becomes continous w

Question 10

fourier transform defn

Answer 10

F(w) = integral form -inf to inf(f(t) exp(-iwt)dt)

f(t) = integral form -inf to inf(F(w) exp(iwt)dw) / 2pi

Question 11

cos fourier

Answer 11

cos (2𝜋𝑢0𝑥) ⇔(1/2) * (𝛿( 𝑢 + 𝑢0) + 𝛿 (𝑢 − 𝑢0))

Question 12

sin fourier

Answer 12

sin (2𝜋𝑢0𝑥) ⇔(i/2) * (𝛿( 𝑢 + 𝑢0) - 𝛿 (𝑢 − 𝑢0))

Question 13

Basic properties of the FT

Answer 13

Linearity (because of integration)

Scaling (because of integration, substitution)

shifting/modulation (shift in one space is rotation in the other)

rotation

Question 14

convolution defn

Answer 14

𝑓 (𝑥) ⊗𝑔(𝑥) =integral from -inf to inf (f(s) g(x-s) ds) )

Question 15

Correlation of f and g:

Answer 15

𝑓 (𝑥) * 𝑔(𝑥) =integral from -inf to inf (f*(s) g(x+s) ds) )

note the +s
and 𝑓*( 𝑠) being the
conj. complex of 𝑓 (𝑠)

Question 16

power spectrum

Parseval’s theorem (energy conservation)

Answer 16

P(u) = |F(u)|^2

integral from -inf to inf (|f(x)^2|dx) =
integral from -inf to inf (|F(u)^2|du)

Question 17

what happens if a function has a discontinuity ?

Answer 17

• ringing, overshoots, ‘Gibbs-Phenomenon’

the Gibbs-Phenomenon happens for finite Fourier series and for
truncated Fourier transform

Question 18

The discrete Fourier transform in 1D

Answer 18

Fk = (1/N) sum(n from -N/2+1 to N/2 (fn exp(-i 2pi k n/N)))

fn = sum(k from -N/2+1 to N/2 (Fk exp(i 2pi k n/N)))

Question 19

Ft Fs DFT properties in spatial domain*** frequency domain

Answer 19

FS: continuous , periodic** discrete, infinite
FT: continuous** continuous
DFT: discrete, periodic **discrete, periodic

Question 20

Fast Fourier Transform (FFT)

Answer 20

DFT is computationally very slow (O(N^2) operations

FFT O(N log2 N)

Split DFT into 2, odd and even
l Apply this recursively

Question 21

FFT

Answer 21

Fk = (1/N) sum(n from 0 to N-1 (fn exp(-i 2pi k n/N)))

Question 22

Amplitude vs. phase reconstruction

Answer 22

phase reconstruction works better

Question 23

… and ….are special cases of the ….,
obtained by applying restrictions of ……,

Answer 23

FS and DFT are special cases of the FT,
obtained by applying restrictions of periodicity,
finiteness and discreteness

Question 24

the DFT is periodic in

Answer 24

the DFT is periodic in both space and
frequency