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

Chapter 31: The Complex Fourier Transform

Although complex numbers are fundamentally disconnected from our reality, they can be used to solve science and engineering problems in two ways. First, the parameters from a real world problem can be substituted into a complex form, as presented in the last chapter. The second method is much more elegant and powerful, a way of making the complex numbers mathematically equivalent to the physical problem. This approach leads to the complex Fourier transform, a more sophisticated version of the real Fourier transform discussed in Chapter 8. The complex Fourier transform is important in itself, but also as a stepping stone to more powerful complex techniques, such as the Laplace and z-transforms. These complex transforms are the foundation of theoretical DSP.

• The Real DFT
• Mathematical Equivalence
• The Complex DFT
• The Family of Fourier Transforms
• Why the Complex Fourier Transform is Used