5.6 Convolution and FFT

2 Fast Fourier Transform: Applications Applications. n Optics, acoustics, quantum physics, telecommunications, control systems, signal processing, speech recognition, data compression, image processing. n DVD, JPEG, MP3, MRI, CAT scan. n Numerical solutions to Poisson's equation. The FFT is one of the truly great computational developments of this [20th] century. It has changed the face of science and engineering so much that it is not an exaggeration to say that life as we know it would be very different without the FFT.

3 Fast Fourier Transform: Brief History Gauss (1805, 1866). Analyzed periodic motion of asteroid Ceres. Runge-König (1924). Laid theoretical groundwork. Danielson-Lanczos (1942). Efficient algorithm. Cooley-Tukey (1965). Monitoring nuclear tests in Soviet Union and tracking submarines. Rediscovered and popularized FFT. Importance not fully realized until advent of digital computers.

4 Polynomials: Coefficient Representation Polynomial. [coefficient representation] Add: O(n) arithmetic operations. Evaluate: O(n) using Horner's method. Multiply (convolve): O(n 2 ) using brute force.

5 Polynomials: Point-Value Representation Fundamental theorem of algebra. A degree n polynomial with complex coefficients has n complex roots. Corollary. A degree n-1 polynomial A(x) is uniquely specified by its evaluation at n distinct values of x. x y xjxj y j = A(x j )

6 Polynomials: Point-Value Representation Polynomial. [point-value representation] Add: O(n) arithmetic operations. Multiply: O(n), but need 2n-1 points. Evaluate: O(n 2 ) using Lagrange's formula.

7 Converting Between Two Polynomial Representations Tradeoff. Fast evaluation or fast multiplication. We want both! Goal. Make all ops fast by efficiently converting between two representations. Coefficient Representation O(n 2 ) Multiply O(n) Evaluate Point-valueO(n)O(n 2 ) coefficient representation point-value representation O(n log n)

8 Converting Between Two Polynomial Representations: Brute Force Coefficient to point-value. Given a polynomial a 0 + a 1 x a n-1 x n-1, evaluate it at n distinct points x 0,..., x n-1. Point-value to coefficient. Given n distinct points x 0,..., x n-1 and values y 0,..., y n-1, find unique polynomial a 0 + a 1 x a n-1 x n-1 that has given values at given points. Vandermonde matrix is invertible iff x i distinct O(n 3 ) for Gaussian elimination O(n 2 ) for matrix-vector multiply

9 Coefficient to Point-Value Representation: Intuition Coefficient to point-value. Given a polynomial a 0 + a 1 x a n-1 x n-1, evaluate it at n distinct points x 0,..., x n-1. Divide. Break polynomial up into even and odd powers. n A(x) = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + a 4 x 4 + a 5 x 5 + a 6 x 6 + a 7 x 7. n A even (x) = a 0 + a 2 x + a 4 x 2 + a 6 x 3. n A odd (x) = a 1 + a 3 x + a 5 x 2 + a 7 x 3. n A(-x) = A even (x 2 ) + x A odd (x 2 ). n A(-x) = A even (x 2 ) - x A odd (x 2 ). Intuition. Choose two points to be  1. n A(-1) = A even (1) + 1 A odd (1). n A(-1) = A even (1) - 1 A odd (1). Can evaluate polynomial of degree  n at 2 points by evaluating two polynomials of degree  ½n at 1 point.

10 Coefficient to Point-Value Representation: Intuition Coefficient to point-value. Given a polynomial a 0 + a 1 x a n-1 x n-1, evaluate it at n distinct points x 0,..., x n-1. Divide. Break polynomial up into even and odd powers. n A(x) = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + a 4 x 4 + a 5 x 5 + a 6 x 6 + a 7 x 7. n A even (x) = a 0 + a 2 x + a 4 x 2 + a 6 x 3. n A odd (x) = a 1 + a 3 x + a 5 x 2 + a 7 x 3. n A(-x) = A even (x 2 ) + x A odd (x 2 ). n A(-x) = A even (x 2 ) - x A odd (x 2 ). Intuition. Choose four points to be  1,  i. n A(-1) = A even (-1) + 1 A odd ( 1). n A(-1) = A even (-1) - 1 A odd (-1). n A(-i) = A even (-1) + i A odd (-1). n A(-i) = A even (-1) - i A odd (-1). Can evaluate polynomial of degree  n at 4 points by evaluating two polynomials of degree  ½n at 2 points.

11 Discrete Fourier Transform Coefficient to point-value. Given a polynomial a 0 + a 1 x a n-1 x n-1, evaluate it at n distinct points x 0,..., x n-1. Key idea: choose x k =  k where  is principal n th root of unity. Discrete Fourier transform Fourier matrix F n

12 Roots of Unity Def. An n th root of unity is a complex number x such that x n = 1. Fact. The n th roots of unity are:  0,  1, …,  n-1 where  = e 2  i / n. Pf. (  k ) n = (e 2  i k / n ) n = (e  i ) 2k = (-1) 2k = 1. Fact. The ½n th roots of unity are: 0, 1, …, n/2-1 where = e 4  i / n. Fact.  2 = and (  2 ) k = k.  0 = 0 = 1 11  2 = 1 = i 33  4 = 2 = -1 55  6 = 3 = -i 77 n = 8

13 Fast Fourier Transform Goal. Evaluate a degree n-1 polynomial A(x) = a a n-1 x n-1 at its n th roots of unity:  0,  1, …,  n-1. Divide. Break polynomial up into even and odd powers. n A even (x) = a 0 + a 2 x + a 4 x 2 + … + a n/2-2 x (n-1)/2. n A odd (x) = a 1 + a 3 x + a 5 x 2 + … + a n/2-1 x (n-1)/2. n A(x) = A even (x 2 ) + x A odd (x 2 ). Conquer. Evaluate degree A even (x) and A odd (x) at the ½n th roots of unity: 0, 1, …, n/2-1. Combine. n A(  k+n ) = A even ( k ) +  k A odd ( k ), 0  k < n/2 n A(  k+n ) = A even ( k ) -  k A odd ( k ), 0  k < n/2  k+n = -  k k = (  k ) 2 = (  k+n ) 2

14 fft(n, a 0,a 1,…,a n-1 ) { if (n == 1) return a 0 (e 0,e 1,…,e n/2-1 )  FFT(n/2, a 0,a 2,a 4,…,a n-2 ) (d 0,d 1,…,d n/2-1 )  FFT(n/2, a 1,a 3,a 5,…,a n-1 ) for k = 0 to n/2 - 1 {  k  e 2  ik/n y k+n/2  e k +  k d k y k+n/2  e k -  k d k } return (y 0,y 1,…,y n-1 ) } FFT Algorithm

15 FFT Summary Theorem. FFT algorithm evaluates a degree n-1 polynomial at each of the n th roots of unity in O(n log n) steps. Running time. T(2n) = 2T(n) + O(n)  T(n) = O(n log n). assumes n is a power of 2 O(n log n) coefficient representation point-value representation

16 Recursion Tree a 0, a 1, a 2, a 3, a 4, a 5, a 6, a 7 a 1, a 3, a 5, a 7 a 0, a 2, a 4, a 6 a 3, a 7 a 1, a 5 a 0, a 4 a 2, a 6 a0a0 a4a4 a2a2 a6a6 a1a1 a5a5 a3a3 a7a7 "bit-reversed" order perfect shuffle

17 Point-Value to Coefficient Representation: Inverse DFT Goal. Given the values y 0,..., y n-1 of a degree n-1 polynomial at the n points  0,  1, …,  n-1, find the unique polynomial a 0 + a 1 x a n-1 x n-1 that has given values at given points. Inverse DFT Fourier matrix inverse (F n ) -1

18 Claim. Inverse of Fourier matrix is given by following formula. Consequence. To compute inverse FFT, apply same algorithm but use  -1 = e -2  i / n as principal n th root of unity (and divide by n). Inverse FFT

19 Inverse FFT: Proof of Correctness Claim. F n and G n are inverses. Pf. Summation lemma. Let  be a principal n th root of unity. Then Pf. n If k is a multiple of n then  k = 1  sums to n. n Each n th root of unity  k is a root of x n - 1 = (x - 1) (1 + x + x x n-1 ). n if  k  1 we have: 1 +  k +  k(2)  k(n-1) = 0  sums to 0. ▪ summation lemma

20 Inverse FFT: Algorithm ifft(n, a 0,a 1,…,a n-1 ) { if (n == 1) return a 0 (e 0,e 1,…,e n/2-1 )  FFT(n/2, a 0,a 2,a 4,…,a n-2 ) (d 0,d 1,…,d n/2-1 )  FFT(n/2, a 1,a 3,a 5,…,a n-1 ) for k = 0 to n/2 - 1 {  k  e -2  ik/n y k+n/2  (e k +  k d k ) / n y k+n/2  (e k -  k d k ) / n } return (y 0,y 1,…,y n-1 ) }

21 Inverse FFT Summary Theorem. Inverse FFT algorithm interpolates a degree n-1 polynomial given values at each of the n th roots of unity in O(n log n) steps. assumes n is a power of 2 O(n log n) coefficient representation O(n log n) point-value representation

22 Polynomial Multiplication Theorem. Can multiply two degree n-1 polynomials in O(n log n) steps. O(n) point-value multiplication O(n log n) FFT inverse FFT O(n log n) coefficient representation

Course summary Major computational problems:  DFS  BFS  Dijkstra’s algorithm  Bellman-Ford algorithm  Scheduling algorithms (e.g. interval partitioning)  Knapsack – fractional, 0,1-version, integer version  String similarity and edit distance – Smith-Waterman algorithm  RNA folding  Network flow  k-connectivity  Maximum matching in a bipartite graph  Sorting  Integer multiplication (Karatsuba’s algorithm)  Fast Fourier Transform  Polynomial multplication 23

Course summary Major computational problems (color coded by technique):  DFS  BFS  Dijkstra’s algorithm  Bellman-Ford algorithm  Scheduling algorithms (e.g. interval partitioning)  Knapsack – fractional, 0,1-version, integer version  String similarity and edit distance – Smith-Waterman algorithm  RNA folding  Network flow  k-connectivity  Maximum matching in a bipartite graph  Sorting  Integer multiplication (Karatsuba’s algorithm)  Fast Fourier Transform  Polynomial multiplication 24 Generic Greedy Dynamic programming Iterative improvement Divide and conquer