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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

$\displaystyle y(n) = \sum_{i=1}^N g_i x(n-d_i)

where $ g_i$ is a weighting factor, $ d_i$ is the $ i$ th delay, and

$\displaystyle x(n)\isdef e^{j\omega nT}

is a complex sinusoid, the ``carrier term'' $ e^{j\omega nT}$ can be ``factored out'' of the linear combination:

y(n) &=& \sum_{i=1}^N g_i e^{j[\omega (n-d_i)T]}
= \sum_{i=1}^N g_i e^{j\omega nT}e^{-j\omega d_i T}\\
&=& e^{j\omega n T} \sum_{i=1}^N g_i e^{-j \omega d_i T}
= x(n) \sum_{i=1}^N g_i e^{-j \omega d_i T}

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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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8
Copyright © 2024-04-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University