dft - DISCRETE FOURIER TRANSFORM JOHN RANDALL Contents 1....

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DISCRETE FOURIER TRANSFORM JOHN RANDALL Contents 1. Representations of polynomials 1 2. Comparison of the representations 2 3. Roots of unity 2 4. Discrete Fourier Transform 2 5. Fast Fourier Transform 2 1. Representations of polynomials A polynomial f ( x ) of degree less than n can be represented in coefficient form as f ( x ) = a 0 + a 1 x + a 2 x 2 + ··· + a n - 1 x n - 1 , or, more succinctly, as the sequence of n numbers a = ( a 0 ,...,a n - 1 ). This is not the only representation. If we choose n distinct points x = x 0 ,...,x n , we can calculate the values y 0 = f ( x 0 ) ,...,y n - 1 = f ( x n - 1 ) . The sequence of n numbers y = ( y 0 ,...,y n - 1 ) is called the point-value form of f with respect to the points x . The coefficient form and the point-value form are equivalent. To recover a from y , we find the the unique polynomial of degree less than n that interpolates the points ( x 0 ,y 0 ) ,..., ( x n - 1 ,y n - 1 ). Note that a and y are related by x 0 0 x 1 0 ... x n - 1 0 . . . . . . x
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This note was uploaded on 10/10/2010 for the course MATH 01:640:111 taught by Professor Carey during the Spring '10 term at Saint Louis.

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dft - DISCRETE FOURIER TRANSFORM JOHN RANDALL Contents 1....

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