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Mathematics of the DFT
Sinusoids and Exponentials
Complex Sinusoids
Phasor and Carrier Components of Sinusoids
Why Phasors are Important
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Why Phasors are Important
Linear, time-invariant (LTI) systems can be said to perform only four operations on a signal: copying, scaling, delaying, and adding. As a result, each output is always a linear combination of delayed copies of the input signal(s). (A linear combination is simply a weighted sum, as discussed in §5.6.) In any linear combination of delayed copies of a complex sinusoid
is a weighting factor, 
is the 
th delay, and
can be ``factored out'' of the linear combination:
![\begin{eqnarray*}
y(n) &=& \sum_{i=1}^N g_i e^{j[\omega (n-d_i)T]}
= \sum_{i=1}...
...e^{-j \omega d_i T}
= x(n) \sum_{i=1}^N g_i e^{-j \omega d_i T}
\end{eqnarray*}](http://www.dsprelated.com/josimages/mdft/img634.png)
The operation of the LTI system on a complex sinusoid is thus reduced to a calculation involving only phasors, which are simply complex numbers.
Since every signal can be expressed as a linear combination of complex sinusoids, this analysis can be applied to any signal by expanding the signal into its weighted sum of complex sinusoids (i.e., by expressing it as an inverse Fourier transform).
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Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.
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