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
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CHAPTER
The Complex Fourier Transform
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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 tr