Lecture 10: The Fast Fourier Transform (FFT)

Background Material

• Complex numbers (see complex.ppt or complex.pdf).
• Elements of Algebra and Algebraic Computing, John D. Lipson, Benjamin Cummings Publishing Co., 1981.

Reading

• Chapter 9 (pp. 293-308) of Lipson (handout, see above for reference).

Topics

1. Review of evaluation, pointwise product, interpolation algorithm for polynomial multiplication.
2. Review of matrix interpretation of evaluation and interpolation.
3. Horner's method for polynomial evaluation.
4. Evaluating a polynomial at a point and its negative.
5. Review of complex numbers
   • Arithmetic and norm
   • Polar coordinates and Euler's identity
6. Roots of unity.
7. Discrete Fourier Transform as a Vandermonde matrix.
8. Evaluating the Vandermonde matrix with even/odd symmetry
9. Fast Fourier Transform

Lecture Slides

1. fft.ppt - power point slides containing lecture notes on the FFT.

Maple worksheets and programs

• interp.mws - Worksheet containing an implementation of the evaluate, pointwise product, interpolate algorithm for polynomial multiplication.
  
• fft.mws - Worksheet containing an implementation of a recursive FFT.

Assignment

• Assignment 3