Fourier Transform + 1D signal Flashcards
What is Fourier Transform?
Breaks a waveform (a function or signal) into alternate representation, characterised by sine and cosines
What does Fourier Transform show?
Any waveform can be written as sum of sinusoidal functions
What are all waveforms?
Sum of simple sinusoids of different frequencies (continuous range of frequencies)
What does Fourier Series break down?
A periodic function into sum of sinusoidal functions
When a function is a periodic?
With fundamental period T, if the following is true for all t:
f(t+T)=f(t)
What will a function of time with period T have?
same value in T seconds
What is a fundamental period?
Value of T (>0) that is the smallest possible T for which (t+T) is always true
What is Fourier Series (With Period T) and what does it include?
Infinite sum of sinusoidal functions (cosine+sine)
each with an integer multiple of 1/T
Includes constant
What do coefficients of Fourier series determine?
Relative weight for each of sinusoids
When can you approximate f(t)?
exactly when f(t) is continuous and ‘smooth’
What is am+bn?
How much of each frequency is included to create g(t)
The first term in fourier series?
Average value of the function
For modelling odd functions
Use the sine terms
For modelling even functions
Use cosine terms
What happens when the waveform fluctates with respect to time?
The wave can be characterised by frequency
number of cycles passing a given point each second
How are spatial frequency expressed?
Line pairs per min
How can square wave be constructed?
Adding a large number of sine waves of different frequencies and amplitude
How can fourier series also be represented as ?
Frequency spectrum
What is plotted against spatial frequency?
The amplitude of frequency component for square wave
What can fourier spectrum be used to identify?
Frequencies and amplitide of sine wave which contribute to make up a given waveform
Spatial domain representation
Plots of amplitude versus distance
Frequency domain representation
Plots of amplitude vs spatial frequency
Why is the Fourier Transform widely used throughout medical imaging?
- Determining the spatial resolution of imaging system
- Spatial localisation in magnetic resonance Imaging
- Analysis of Doppler ultrasound signal
- Image filtering in emission + transmission computed tomography
What is inverse Fourier transform?
Mathematical technique for converting data in opposite direction
frequency domain –> spatial domain
What is Fourier pairs?
The transformation can go from time domain to frequency domain or opposite direction…it’s reversible
What are known as Fourier pairs?
Two functions g(t) and G(t)
One is the fourier transform of other
What is fourier pairs?
Two functions
The frequency domain form and corresponding time domain form
What is a straight line?
The fourier transform of a delta function (mini-boxcar)
The more coefficients included in approximation
The closer the approximation to original signal
