Lecture 5/6 Fourier transform, properties and application

Question 1

When is a function periodic?

Answer 1

A function f(x) is periodic if:
- it is defined for all real x, and
- if there is some positive number T (the ‘period’) such that f(x + T) = f(x)

Question 2

What are the time-varying quantities that a continuous signal contains?

Answer 2

A continuous signal that contains time-varying quantities
– always smooth and infinite temporal resolution
– carries information and energy for video and audio

Question 3

What are some issues with the transfer of analogue signals?

Answer 3

Analogue signals in communication carry repeated information
– easily affected by noise, and hard to analyse

Question 4

What is the Fourier series equation?

Answer 4

If is a periodic function with period,
the function can be represented
using the Fourier series:

Question 5

Integration of product of sines

Answer 5

Question 6

Integration of product of cosines

Answer 6

Question 7

Integration of product of sine and cosine

Answer 7

Question 8

Compute the a_0

Answer 8

Question 9

Compute the a_n

Answer 9

Question 10

Compute the b_n

Answer 10

Question 11

What are the motivations for the fourier transform?

Answer 11

Transformations are useful for analysing signals
– Natural to analyse audio signals by decomposing into frequencies
– Can also analyse images using frequencies in x- and y-directions

Question 12

What are some applications of the Fourier transform?

Answer 12

• Low and high-pass filtering
  – Fast linear filtering using the convolution theorem
  – Removing structured noise
  – Image compression (JPEG)

Question 13

How can a histogram inform image filtering?

Answer 13

A histogram is used to get
insights about the intensity
domain and design a point
filter

Question 14

For constants, a and b and functions f and g, the linearity property of the Fourier transform implies …

Answer 14

Question 15

for a constant a, if g(x) = f(x - a), then the shifting property of the fourier transform implies …

Answer 15

Question 16

for a constant a, if g(x) = e^iax f(x) then the modulation property of the fourier transform implies …

Answer 16

Question 16

for a constant a, if g(x) = f(ax) then the scaling property of the fourier transform implies …

Answer 16

Question 17

What are the four properties of the Fourier transform?

Answer 17

Linearity
Shifting
Modulation
Scaling

Question 18

The convolution f * g of two functions and is given by:

Answer 18

Question 19

Why is multiplication preferable over convolution?

Answer 19

Convolution becomes
multiplication in the
Fourier domain

Multiplication is much
more efficient:
– can save a lot of time
– particularly for large kernels

Question 20

What is the relationship between the Fourier and signal domain?

Answer 20

Question 21

What is the Discrete Fourier transform (DFT) used for?

Answer 21

Discrete analogue to the continuous Fourier transform
– deals with finite sampled signals, such as audio, images
– N values decomposed into N frequency components

Question 22

What is the DFT of f * g ?

Answer 22

F[ k ] · G[ k ]

Question 23

How does structured noise removal work?

Answer 23

Discard regions in spectrum associated with patterned noise

Question 24

How does a low pass filter work?

Answer 24

A low pass filter passes signals with frequencies lower than a threshold – Removes fine details and hence blurs an image Example using hard cut-off in frequency space: – Ringing artefacts in image space Better low-pass filter: Gaussian blur – Removes ringing artefacts

Question 25

How does a high pass filter work?

Answer 25

A high pass filter passes signals with frequencies higher than a threshold – Removes low frequencies (~overall shape) – get edge image Example using hard cut-off in frequency space: – Ringing artefacts in image space Better high-pass filter: 1–Gaussian blur – Removes ringing artefacts

Question 26

Integration of Sin and Cos

Answer 26

Question 27

The Fourier transform of a function f: R ---> R is given by:

Answer 27

Question 28

The inverse Fourier transform restores f from :

Answer 28

Question 29

Describes complex numbers on the unit circle:

Answer 29

Question 30

Euler's formula

Answer 30

Question 31

What is the convolution theorem?

Answer 31

Question 32

What is the Discrete Fourier Transform (DFT)?

Answer 32

Question 33

What is the Inverse Discrete Fourier Transform (DFT)?

Answer 33

Question 34

What are the properties of the discrete Fourier transform

Answer 34

Question 35

What is the discrete Fourier transform in 2D

Answer 35

Question 36

How does Discrete Cosine Tranform work, and what is it used for?

Answer 36

The Fourier transform of an even, real-valued signal is even and real-valued (i.e. no imaginary part) – Audio signals and images are real-valued – We can make them even by reflection around or even – This cancels out the purely imaginary, sine-related terms Used for audio compression in MP3, image compression in JPEG

Question 37

How many arithmetic operations are needed for the computation of DFT?

Answer 37

Naïve computation of DFT is expensive and complex:

Question 38

What are the advantages of Fast Fourier Transform?

Answer 38

Question 39

What are the dis-advantages of Fast Fourier Transform?

Answer 39

Question 40

What is the overall computational complexity of Fast Fourier Transform (FFT)?

Answer 40

Question 41

Write out the four properties of the Fourier transform

Answer 41