Lecture 5/6 Fourier transform, properties and application Flashcards
When is a function periodic?
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)
What are the time-varying quantities that a continuous signal contains?
A continuous signal that contains time-varying quantities
– always smooth and infinite temporal resolution
– carries information and energy for video and audio
What are some issues with the transfer of analogue signals?
Analogue signals in communication carry repeated information
– easily affected by noise, and hard to analyse
What is the Fourier series equation?
If is a periodic function with period,
the function can be represented
using the Fourier series:
Integration of product of sines
Integration of product of cosines
Integration of product of sine and cosine
Compute the a_0
Compute the a_n
Compute the b_n
What are the motivations for the fourier transform?
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
What are some applications of the Fourier transform?
- Low and high-pass filtering
– Fast linear filtering using the convolution theorem
– Removing structured noise
– Image compression (JPEG)
How can a histogram inform image filtering?
A histogram is used to get
insights about the intensity
domain and design a point
filter
For constants, a and b and functions f and g, the linearity property of the Fourier transform implies …
for a constant a, if g(x) = f(x - a), then the shifting property of the fourier transform implies …
for a constant a, if g(x) = e^iax f(x) then the modulation property of the fourier transform implies …
for a constant a, if g(x) = f(ax) then the scaling property of the fourier transform implies …
What are the four properties of the Fourier transform?
Linearity
Shifting
Modulation
Scaling
The convolution f * g of two functions and is given by:
Why is multiplication preferable over convolution?
Convolution becomes
multiplication in the
Fourier domain
Multiplication is much
more efficient:
– can save a lot of time
– particularly for large kernels
What is the relationship between the Fourier and signal domain?
What is the Discrete Fourier transform (DFT) used for?
Discrete analogue to the continuous Fourier transform
– deals with finite sampled signals, such as audio, images
– N values decomposed into N frequency components
What is the DFT of f * g ?
F[ k ] · G[ k ]
How does structured noise removal work?
Discard regions in spectrum associated with patterned noise
How does a low pass filter work?
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
How does a high pass filter work?
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
Integration of Sin and Cos
The Fourier transform of a function f: R —> R is given by:
The inverse Fourier transform restores f from :
Describes complex numbers on the unit circle:
Euler’s formula
What is the convolution theorem?
What is the Discrete Fourier Transform (DFT)?
What is the Inverse Discrete Fourier Transform (DFT)?
What are the properties of the discrete Fourier transform
What is the discrete Fourier transform in 2D
How does Discrete Cosine Tranform work, and what is it used for?
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
How many arithmetic operations are needed for the computation of DFT?
Naïve computation of DFT is expensive and complex:
What are the advantages of Fast Fourier Transform?
What are the dis-advantages of Fast Fourier Transform?
What is the overall computational complexity of Fast Fourier Transform (FFT)?
Write out the four properties of the Fourier transform
