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Understanding the 'Phasing Method' of Single Sideband Demodulation
Complex Digital Signal Processing in Telecommunications
Introduction to Sound Processing
Introduction of C Programming for DSP Applications
Search Mathematics of the DFT
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In §2.10, De Moivre's theorem was introduced as a consequence of Euler's identity:
Proof:
To establish the ``basis'' of our mathematical induction proof, we may
simply observe that De Moivre's theorem is trivially true for
. Now assume that De Moivre's theorem is true for some positive
integer . Then we must show that this implies it is also true for
, i.e.,
From trigonometry, we have the following sum-of-angle identities:
These identities can be proved using only arguments from classical geometry.^{3.8}Applying these to the right-hand side of Eq.(3.3), with and , gives Eq.(3.2), and so the induction step is proved.
De Moivre's theorem establishes that integer powers of lie on a circle of radius 1 (since , for all ). It therefore can be used to determine all of the th roots of unity (see §3.12 above). However, no definition of emerges readily from De Moivre's theorem, nor does it establish a definition for imaginary exponents (which we defined using Taylor series expansion in §3.7 above).