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Page 1 Objectives_template file:///C/Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture34/34_1.htm[6/14/2012 3:53:53 PM] Module 7:Data Representation Lecture 34: DFT and DCT The Lecture Contains: Fourier analysis Discrete Fourier Transform (DFT) Properties Coefficients Dimensionality reduction Discrete Cosine Transform (DCT) Page 2 Objectives_template file:///C/Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture34/34_1.htm[6/14/2012 3:53:53 PM] Module 7:Data Representation Lecture 34: DFT and DCT The Lecture Contains: Fourier analysis Discrete Fourier Transform (DFT) Properties Coefficients Dimensionality reduction Discrete Cosine Transform (DCT) Objectives_template file:///C/Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture34/34_2.htm[6/14/2012 3:53:53 PM] Module 7:Data Representation Lecture 34: DFT and DCT Fourier analysis Fourier analysis represents a periodic wave as a sum of (infinite) sine and cosine waves Way to analyze the frequency components in a signal Fourier transform is a transformation from time domain to frequency domain can be obtained from g(u) by the inverse transformation and form a Fourier transform pair Page 3 Objectives_template file:///C/Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture34/34_1.htm[6/14/2012 3:53:53 PM] Module 7:Data Representation Lecture 34: DFT and DCT The Lecture Contains: Fourier analysis Discrete Fourier Transform (DFT) Properties Coefficients Dimensionality reduction Discrete Cosine Transform (DCT) Objectives_template file:///C/Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture34/34_2.htm[6/14/2012 3:53:53 PM] Module 7:Data Representation Lecture 34: DFT and DCT Fourier analysis Fourier analysis represents a periodic wave as a sum of (infinite) sine and cosine waves Way to analyze the frequency components in a signal Fourier transform is a transformation from time domain to frequency domain can be obtained from g(u) by the inverse transformation and form a Fourier transform pair Objectives_template file:///C/Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture34/34_3.htm[6/14/2012 3:53:54 PM] Module 7:Data Representation Lecture 34: DFT and DCT Discrete Fourier Transform (DFT) For discrete case, assume vector x has N components DFT of x, denoted by X, has N components given by Inverse transformation (IDFT) is given by To avoid using separate scaling factors, both can be taken as Page 4 Objectives_template file:///C/Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture34/34_1.htm[6/14/2012 3:53:53 PM] Module 7:Data Representation Lecture 34: DFT and DCT The Lecture Contains: Fourier analysis Discrete Fourier Transform (DFT) Properties Coefficients Dimensionality reduction Discrete Cosine Transform (DCT) Objectives_template file:///C/Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture34/34_2.htm[6/14/2012 3:53:53 PM] Module 7:Data Representation Lecture 34: DFT and DCT Fourier analysis Fourier analysis represents a periodic wave as a sum of (infinite) sine and cosine waves Way to analyze the frequency components in a signal Fourier transform is a transformation from time domain to frequency domain can be obtained from g(u) by the inverse transformation and form a Fourier transform pair Objectives_template file:///C/Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture34/34_3.htm[6/14/2012 3:53:54 PM] Module 7:Data Representation Lecture 34: DFT and DCT Discrete Fourier Transform (DFT) For discrete case, assume vector x has N components DFT of x, denoted by X, has N components given by Inverse transformation (IDFT) is given by To avoid using separate scaling factors, both can be taken as Objectives_template file:///C/Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture34/34_4.htm[6/14/2012 3:53:54 PM] Module 7:Data Representation Lecture 34: DFT and DCT Properties Parseval's theorem: , i.e., length of vectors is preserved When both scaling factors are Contractive mapping Invertible, linear transformation Essentially, a rotation in Ndimensional space Page 5 Objectives_template file:///C/Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture34/34_1.htm[6/14/2012 3:53:53 PM] Module 7:Data Representation Lecture 34: DFT and DCT The Lecture Contains: Fourier analysis Discrete Fourier Transform (DFT) Properties Coefficients Dimensionality reduction Discrete Cosine Transform (DCT) Objectives_template file:///C/Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture34/34_2.htm[6/14/2012 3:53:53 PM] Module 7:Data Representation Lecture 34: DFT and DCT Fourier analysis Fourier analysis represents a periodic wave as a sum of (infinite) sine and cosine waves Way to analyze the frequency components in a signal Fourier transform is a transformation from time domain to frequency domain can be obtained from g(u) by the inverse transformation and form a Fourier transform pair Objectives_template file:///C/Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture34/34_3.htm[6/14/2012 3:53:54 PM] Module 7:Data Representation Lecture 34: DFT and DCT Discrete Fourier Transform (DFT) For discrete case, assume vector x has N components DFT of x, denoted by X, has N components given by Inverse transformation (IDFT) is given by To avoid using separate scaling factors, both can be taken as Objectives_template file:///C/Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture34/34_4.htm[6/14/2012 3:53:54 PM] Module 7:Data Representation Lecture 34: DFT and DCT Properties Parseval's theorem: , i.e., length of vectors is preserved When both scaling factors are Contractive mapping Invertible, linear transformation Essentially, a rotation in Ndimensional space Objectives_template file:///C/Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture34/34_5.htm[6/14/2012 3:53:54 PM] Module 7:Data Representation Lecture 34: DFT and DCT Coefficients Expanding , First coefficient is (scaled) sum or average Other coefficients define the frequency components (cosine and sine) at frequencies forRead More
1. What is the difference between DFT and DCT? 
2. How does the DFT work? 
3. What are the applications of DCT? 
4. How does the DCT differ from the Fourier Transform? 
5. What are some limitations of the DFT and DCT? 

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