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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 31. The Complex Fourier Transform
- The Real DFT
- Mathematical Equivalence
- The Complex DFT
- The Family of Fourier Transforms
- Why the Complex Fourier Transform is Used
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.
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