Fourier Transform + 1D signal

Question 1

What is Fourier Transform?

Answer 1

Breaks a waveform (a function or signal) into alternate representation, characterised by sine and cosines

Question 2

What does Fourier Transform show?

Answer 2

Any waveform can be written as sum of sinusoidal functions

Question 3

What are all waveforms?

Answer 3

Sum of simple sinusoids of different frequencies (continuous range of frequencies)

Question 4

What does Fourier Series break down?

Answer 4

A periodic function into sum of sinusoidal functions

Question 5

When a function is a periodic?

Answer 5

With fundamental period T, if the following is true for all t:
f(t+T)=f(t)

Question 6

What will a function of time with period T have?

Answer 6

same value in T seconds

Question 7

What is a fundamental period?

Answer 7

Value of T (>0) that is the smallest possible T for which (t+T) is always true

Question 8

What is Fourier Series (With Period T) and what does it include?

Answer 8

Infinite sum of sinusoidal functions (cosine+sine)
each with an integer multiple of 1/T

Includes constant

Question 9

What do coefficients of Fourier series determine?

Answer 9

Relative weight for each of sinusoids

Question 10

When can you approximate f(t)?

Answer 10

exactly when f(t) is continuous and ‘smooth’

Question 11

What is am+bn?

Answer 11

How much of each frequency is included to create g(t)

Question 12

The first term in fourier series?

Answer 12

Average value of the function

Question 13

For modelling odd functions

Answer 13

Use the sine terms

Question 14

For modelling even functions

Answer 14

Use cosine terms

Question 15

What happens when the waveform fluctates with respect to time?

Answer 15

The wave can be characterised by frequency

number of cycles passing a given point each second

Question 16

How are spatial frequency expressed?

Answer 16

Line pairs per min

Question 17

How can square wave be constructed?

Answer 17

Adding a large number of sine waves of different frequencies and amplitude

Question 18

How can fourier series also be represented as ?

Answer 18

Frequency spectrum

Question 19

What is plotted against spatial frequency?

Answer 19

The amplitude of frequency component for square wave

Question 20

What can fourier spectrum be used to identify?

Answer 20

Frequencies and amplitide of sine wave which contribute to make up a given waveform

Question 21

Spatial domain representation

Answer 21

Plots of amplitude versus distance

Question 22

Frequency domain representation

Answer 22

Plots of amplitude vs spatial frequency

Question 23

Why is the Fourier Transform widely used throughout medical imaging?

Answer 23

1. Determining the spatial resolution of imaging system
2. Spatial localisation in magnetic resonance Imaging
3. Analysis of Doppler ultrasound signal
4. Image filtering in emission + transmission computed tomography

Question 24

What is inverse Fourier transform?

Answer 24

Mathematical technique for converting data in opposite direction
frequency domain –> spatial domain

Question 25

What is Fourier pairs?

Answer 25

The transformation can go from time domain to frequency domain or opposite direction…it’s reversible

Question 26

What are known as Fourier pairs?

Answer 26

Two functions g(t) and G(t)

One is the fourier transform of other

Question 27

What is fourier pairs?

Answer 27

Two functions

The frequency domain form and corresponding time domain form

Question 28

What is a straight line?

Answer 28

The fourier transform of a delta function (mini-boxcar)

Question 29

The more coefficients included in approximation

Answer 29

The closer the approximation to original signal