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!
CHAPTER
Complex Numbers
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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 b