Reference Text: The evolution of Applied harmonics analysis by Elena Prestini

Transcription

1 Notes for July 14. Filtering in Frequency domain. Reference Text: The evolution of Applied harmonics analysis by Elena Prestini It all started with: Jean Baptist Joseph Fourier ( ) Mathematician, Physicist, Historian, and Politician Best known for the Fourier transform Auxerre, France In 1807, his introductory manuscript Theory of the propagation of heat in solid bodies In 1822, published "Theacuteorie analytique de la chaleur" Established the mathematical theory of heat diffusion Introduced the representation of a function as a series of sines and cosines known as Fourier series - Joseph Fourier was responsible for developing a method in which any periodic function can be expressed as the sum of sines and/or cosines of different frequencies, each multiplied by a different coefficient. - Some theorize that he became interested in heat distribution after living in the Alps and being really cold. He apparently had a box made that would go around his body to stay warm.

2 Quote by Fourier Heat : like gravity, penetrates every substance of the universe, its rays occupy all parts of space. The object of our work is to set forth the mathematical laws which this element obeys. The theory of heat will hereafter form one of the most important branches of general physics. Analytical Theory of Heat Fourier transforms have an extraordinary amount of uses: from MRI machines to I- Pods. Group Project 5: Where the FT is used? Find a real application of the Fourier transform that may look surprising and attractive for high school students. Write a 3-page hand-out for the students explaining the background of the application and how the Fourier transform is used in the application. Please include the references used in the hand-out and for further reading - assignment should be 2 pages long (not including the pictures) - it is optional and worth (2%) - upload the assignment to the website - include the references used in the handout and for further reading Pure Tones and Their Frequencies Blue: 262 Hz Red: 524 Hz

3 - Pure tones are pure sine or cosine functions - Periodicity of the blue tone is /0.6 = /T = 262HZ - Red is double the frequency of the blue - Low pitch with low frequency - High pitch with high frequency - Pure tones do not sound good to human ears -Here we are representing the signal as some quantity vs. time (time domain) Combining the blue and red tones: tone: Blue (262 Hz) + Red (524 Hz) -This tone is more complex and is sounds more pleasant -The more complicated a tones sound nicer

4 -This is a guitar tone, it is more complicated, and is easier to listen to - Since this tone is more complicated, it is harder to figure out its frequency The Fourier function rewrites the complicated function as a sum of sine or cosine waves A Tour in the 1D Fourier Transform: Fourier Transform of a Signal Inner Product of Two Vectors: A projection of vector B onto vector A measures similarity between vector B and vector A -In a similar way Fourier Transform can find similarities between a complex function and sine waves with different frequencies - The important property of Fourier transform is that a function can be reconstructed back completely via an inverse process with no loss of information The Fourier Transform (1807) f + Fourier transform ( x) F ( u) = f ( x) exp( j2π ux)dx

5 f + = ( x) F( u) exp( j2π ux) du F( u) Inverser Fourier Transform -where e j2πux = cos2πux jsin 2πux - the integral acts like the inner product - f(x) is the signal function - ux is the frequency variable Review of Complex Numbers -F(u) is defined in the Complex domain

6 Examples of functions in time domain and frequency domain Ex.1 Pure tone - the area under the bars in the frequency domain should approach to 1 as the hight goes to infinity due to the property of the impulse function? Ex.2 Notice that this function is more complex, the Fourier transform breaks it up into functions with two distinct frequencies

7 -The height of the bars represents the contribution of each function to the amplitude of the function in the time domain - there is no bar at 0, which means the average intensity of the signal is 0 Ex. 3 : Guitar note - there is the bar at 0, which is the average of the signal intensity - this complex guitar tone is decomposed into functions with higher and lower frequencies - the height of each bar represents the contribution of each to the amplitude of the tone in time domain - the contribution high tones is a lot smaller than the contribution from the low tones - the low frequencies give the general shape while the high frequencies provide the small details - telephone systems cut out the high frequencies from our voices to save on the bandwidth - (Low Pass Filter) We can get rid of the high frequencies to smooth out the small details - (High Pass Filter) Get rid of low frequencies to capture details, edge information.

8 Lower and higher frequencies Below is what the curve looks like if we get rid of the higher frequencies - The details in the curve are gone - Low Pass Filter

9 Below is what the curve looks like when the lower frequencies are eliminated, lots of small details are left. High Pass Filter. Note: the curve has been shifted up for comparison purposes, the 0 frequency(middle line) in the frequency domain has been deleted, the curve should be averaged over 0. What the Fourier Transform of other functions look like? Rectangular Window( Time domain) Sinc (frequency domain)

10 Above: The FT of a rectangular window is sin c( πu) - - sinc is defined as sin(x)/x sinc(m) = sin(pi*m)/(pi*m) The Fourier Transform of a Gaussian is a Gaussian exp 2 2 ( π x ) exp( πu )

11 -The Fourier Transformation of a unit impulse located at the origin of the spatial domain is a constant in the frequency domain. - this function is apparently used for sampling of continuous functions. Continuous functions have to be converted into a sequence of discrete values before they can be processed in a computer. 2. A Tour in the 1D Fourier Transform: Important basic properties Uniqueness: t = h t H () () ( u) H ( u) h1 2 1 = 2 - this is why there is no loss of information, every function has a unique Fourier Transform, and the inverse will get the original function back Linearity: ah1 t + bh2 t ah1 u + bh 2 () () ( ) ( u) Linear combination hold for functions and their Fourier transform - by the property of integrals

12 Convolution Theorem: Convolution Multiplication h1 ( x) h2 ( x) H 1( u) H 2( u) x h x u H u 1 ( ) ( ) H1( ) 2( ) h 2 - The Fourier transform of a Convolution of two functions in the spatial domain is equal to the product in the frequency domain of the Fourier transforms of the two functions. This is very useful because convolution is a process that goes through every point and is time consuming, whereas it is much quicker to perform multiplication. This is why MATLAB can perform certain operations very fast - In this example we have corresponding cases of one dimensional and two dimensional Fourier Transforms. In one dimension, the Fourier transform breaks up a complicated curve into a composition of basic sinusoidal graphs. In two dimensions, the Fourier Transform uses spatial sinusoids The higher frequency ones are used for finer details of a picture The lower frequency ones are used for the overall shape of the picture Since they are in two dimensions, they will be at different angles.

13 In this slide we see the directions and frequencies of the Fourier Spectra. The further away from the centre, we have higher frequencies. The closer to the centre we have lower frequencies. The spatial sinusoids are vertical or horizontal on the axis, but slanted when not on the axis. In this slide two dots were added to the frequency domain, producing the image with incorrect spatial sinusoid.

14 In this slide, all the high frequency information was deleted, with only low frequency information remaining. We can see that the picture has the same general shape but no fine details. Because only low frequency spatial sinusoids are used. This is the opposite of the previous slide. The low frequency information was deleted. The fine details are left in the picture, because only the high frequency spatial sinusoids are used.