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The Scientist and Engineer's
Guide to Digital Signal Processing
by Steven W. Smith
California Technical Publishing
ISBN 0-9660176-3-3 (1997)
Chapter 30. Complex Numbers
- The Complex Number System
- Polar Notation
- Using Complex Numbers by Substitution
- Complex Representation of Sinusoids
- Complex Representation of Systems
- Electrical Circuit Analysis
- Summary of the key concepts
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!
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