The Scientist and Engineer's Guide to
Digital Signal Processing
By Steven W. Smith, Ph.D.

Chapter 8: The Discrete Fourier Transform

Fourier analysis is a family of mathematical techniques, all based on decomposing signals into sinusoids. The discrete Fourier transform (DFT) is the family member used with digitized signals. This is the first of four chapters on the real DFT, a version of the discrete Fourier transform that uses real numbers to represent the input and output signals. The complex DFT, a more advanced technique that uses complex numbers, will be discussed in Chapter 29. In this chapter we look at the mathematics and algorithms of the Fourier decomposition, the heart of the DFT.

• The Family of Fourier Transform
• Notation and Format of the Real DFT
• The Frequency Domain's Independent Variable
• DFT Basis Functions
• Synthesis, Calculating the Inverse DFT
• Analysis, Calculating the DFT
• Duality
• Polar Notation
• Polar Nuisances