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

Chapter 30: Complex Numbers

Complex numbers are an extension of the ordinary numbers used in everyday math. They have the unique property of representing and manipulating two variables as a single quantity. This fits very naturally with Fourier analysis, where the frequency domain is composed of two signals, the real and the imaginary parts. Complex numbers shorten the equations used in DSP, and enable techniques that are difficult or impossible with real numbers alone. For instance, the Fast Fourier Transform is based on complex numbers. Unfortunately, complex techniques are very mathematical, and it requires a great deal of study and practice to use them effectively. Many scientists and engineers regard complex techniques as the dividing line between DSP as a tool, and DSP as a career. In this chapter, we look at the mathematics of complex numbers, and elementary ways of using them in science and engineering. The following three chapters discuss important techniques based on complex numbers: the complex Fourier transform, the Laplace transform, and the z-transform. These complex transforms are the heart of theoretical DSP. Get ready, here comes the math!

• The Complex Number System
• Polar Notation
• Using Complex Numbers by Substitution
• Complex Representation of Sinusoids
• Complex Representation of Systems
• Electrical Circuit Analysis