I'm reading and pondereing about the convolution sumation, properties and how this is related to discrete LTI systems.
I'm using the book Signals and Systems by Alan V. Oppenheim, and on the chapter 3.2 "Response of LTI Systems to complex exponentials" we start using as inputs for the Systems complex exponentials of the form :
$$ z^n \rightarrow H(z)z^n $$
And it is defined the same as in a convolution of a real valued input signal $x[n]$ to a sistem with impulse response $h[n]$
$$ y[n]=\sum_{k=-\infty}^{\infty}{x[k]h[n-k]} $$
Due to the conmutativity of the convolution :
$$ y[n]=\sum_{k=-\infty}^{\infty}{x[n]h[n-k]} = \sum_{k=-\infty}^{\infty}{h[k]x[n-k]} $$ $$ y[n]=\sum_{k=-\infty}^{\infty}{h[k]z^{n-k}} = z^{n}\sum_{k=-\infty}^{\infty}{h[k]z^{-k}} $$
I get the idea that the System is behaving as a linear system for complex input signals, how ever I dont understand why using complex valued signals as inputs.
It seems that the simplification in the calculation of the convolution is the main goal of using complex exponentials as a basis for a linear combination of the system inputs.