# 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

- Review of evaluation, pointwise product, interpolation algorithm
for polynomial multiplication.
- Review of matrix interpretation of evaluation and interpolation.
- Horner's method for polynomial evaluation.
- Evaluating a polynomial at a point and its negative.
- Review of complex numbers
- Arithmetic and norm
- Polar coordinates and Euler's identity

- Roots of unity.
- Discrete Fourier Transform as a Vandermonde matrix.
- Evaluating the Vandermonde matrix with even/odd symmetry
- Fast Fourier Transform

### Lecture Slides

- 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

Created: May 18, 2004 by jjohnson@mcs.drexel.edu