n f n f f C c a b c c a 2 b 2 Cookbook for data scientists Fourier analysis

N f n f f c c a b c c a 2 b 2 cookbook for data

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n . f n , [ f ( f [ C ( [ [ * ] [ ] [ ] * . [ . [ [ [ * ] [ ] [ ] * . [ c a [ b [ c [ c a 2 [ b 2 [ ] [ ] Cookbook for data scientists Fourier analysis Fourier Transform (FT) Let x : R C such that + -∞ | x ( t ) | d t < Fourier transform X : R C is defined as X ( u ) = F [ x ]( u ) = + -∞ x ( t ) e - i 2 πut t x ( t ) = F - 1 [ X ]( t ) = + -∞ X ( u ) e i 2 πut where u is referred to as the frequency. Properties of continuous FT F [ ax + by ] = a F [ x ] + b F [ y ] (Linearity) F [ x ( t - a )] = e - i 2 πau F [ x ] F [ x ( at )]( u ) = 1 | a | F [ x ]( u/a ) (Modulation) F [ x * ]( u ) = F [ x ]( - u ) * (Conjugation) F [ x ](0) = + -∞ x ( t ) d t (Integration) + -∞ | x ( t ) | 2 d t = + -∞ | X ( u ) | 2 d u (Parseval) F [ x ( n ) ]( u ) = (2 πiu ) n F [ x ]( u ) (Derivation) F [ e - π at ]( u ) = 1 πa e - u /a (Gaussian) x is real X ( ε ) = X ( - ε ) * (Real Hermitian) Properties with convolutions ( x y )( t ) = -∞ x ( s ) y ( t - s ) d s (Convolution) F [ x y ] = F [ x ] F [ y ] (Convolution theorem) Multidimensional Fourier Transform Fourier transform is separable over the different d dimensions, hence can be defined recursively as F [ x ] = ( F 1 ◦ F 2 . . . ◦ F d )[ x ] where F k [ x ]( t 1 . . . , ε k , . . . , t d ) = F [ t k x ( t 1 , . . . , t k , . . . , t d ε k ) and inherits from above properties (same for DFT). u [ ] u n 1 t t e i 2 t 1 [ ] t 1 n 1 u k e i 2 1 e i 2 * 1 1 / 2 [ a [ b [ y ] [ t a e i 2 [ ] [ * ] u [ ] * n u n [ ] 0 n 1 t t || || 2 2 1 || || 2 2 || || 1 || || 1 || || 1 || || || || 1 || || 1 || || 1 u * n u n ( * y ) t n s s y ( t s n * y y y ) y ( t s n [ * y ] u [ ] u [ y ] u ( ) ( 2 ) 1 Cookbook for data scientists Linear algebra II Eigenvalues / eigenvectors If λ C and e C n (= 0) satisfy Ae = λe λ is called the eigenvalue associated to the eigenvector e of A . There are at most n distinct eigenvalues λ i and at least n linearly independent eigenvectors e i (with norm 1 ). The set λ i necessarily distinct) eigenvalues is called the spectrum of A (for a proper definition see characteristic polynomial, multiplicity, eigenspace). This set has exactly r = rank A non zero values. Eigendecomposition ( ) If it exists E C n × n , and a diagonal matrix Λ C n × n st A = E Λ E - 1 A is said diagonalizable and the columns of the n eigenvectors e i of A with corresponding eigenvalues Λ i,i = λ i . Properties of eigendecomposition ( ) If, for all i , Λ i,i = 0 , then A is invertible and A - 1 = E Λ - 1 E - 1 with Λ - 1 i,i = (Λ ) 1 If A is Hermitian ( A = A * ), such decomposition always exists, the eigenvectors of E can be chosen orthonormal such that E is unitary ( E - 1 * λ i are real. If A is Hermitian ( A = A * ) and λ i > 0 , positive definite, and for all x = 0 , xAx * 0 . n n n * r k k k v * k r k . . * * 2 . * [ { i \ i ) } ) { v i n \ i ( r ) } ) * ) 1 if ii 0 , 0 1 . || || p x ; || x || || || p , || || 2 k k , || || * k k , || || 2 F | a | 2 * k 2 k Cookbook for data scientists Charles Deledalle Linear algebra I Notations x , y , z , . . . : vectors of C n a , b , c , . . . : scalars of C A , B , C : matrices of C m × n Id : identity matrix i = 1 , . . . , m and j = 1 , . . . , n Matrix vector product ( Ax ) i = n k =1 A i,k x k ( AB ) i,j = n k =1 A i,k B k,j Basic properties A ( ax + by ) = aAx + bAy A Id = Id A = A Inverse ( m = n ) A is said invertible, if it exists B st AB = BA = Id . B is unique and called inverse of A . We write B = A - 1 . Adjoint and transpose ( A t ) j,i = A i,j , A t C m × n ( A * ) j,i = ( A i,j ) * , A * C m × n Ax, y = x, A * y Trace and determinant ( m = n ) tr A = n i =1 A i,i = n i =1 λ i det A = n i =1 λ i tr A = tr A * tr AB = tr BA det A * = det A det A - 1 = (det A ) - 1 det AB = det A det B A is invertible det A = 0 λ i = 0 , i
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