``Proof of Euler's Identity'' derives Euler's identity
in detail. This is an important tool for working with complex
numbers, and one of the critical elements of the DFT definition we
need to understand.
``Geometric Signal Theory'' provides an introduction to
vector spaces, inner products, orthogonality, projection of one signal
onto another, norms, and elementary vector space operations. In this
setting, the DFT can be regarded as a change of coordinates from one
basis set (shifted impulses) to another (sinusoids at different
``The DFT Derived'' derives the DFT as a projection of a
length signal onto the set of sampled complex
sinusoids generated by the th roots of unity.
``Example Applications of the DFT'' illustrates
practical FFT analysis in Matlab
and Octave (an open-source
matlab) through a series of examples. The various Fourier theorems of
the preceding chapter provide a ``thinking vocabulary'' for
understanding these applications.