Linear Time Invariant Systems

Notice, the ``carrier term'' $x(n) = e^{j\omega nT}$ can be factored out to obtain

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

$\displaystyle =$ $\displaystyle \sum^N_{i=1}g_ie^{j[\omega(n-d_i)T]}$

$\displaystyle =$ $\displaystyle \sum^N_{i=1}g_ie^{j\omega nT}e^{-j\omega d_iT}$

$\displaystyle =$ $\displaystyle x(n)\sum^N_{i=1}g_ie^{-j\omega d_iT},$

showing an LTI system can be reduced to a calculation involving only the sum phasors.
Since every digital signal can be expressed as a linear combination of complex sinusoids, this analysis can be applied to any digital signal.

``Music 270a: Complex Exponentials and Spectrum Representation'' by Tamara Smyth, Department of Music, University of California, San Diego (UCSD).

$\displaystyle y(n)$	$\displaystyle =$	$\displaystyle \sum^N_{i=1}g_i x(n-d_i)$
	$\displaystyle =$	$\displaystyle \sum^N_{i=1}g_ie^{j[\omega(n-d_i)T]}$
	$\displaystyle =$	$\displaystyle \sum^N_{i=1}g_ie^{j\omega nT}e^{-j\omega d_iT}$
	$\displaystyle =$	$\displaystyle x(n)\sum^N_{i=1}g_ie^{-j\omega d_iT},$