How can real and imaginary part of $e^{ix}$ be used in complex tasks? I am aware that sine is expressed as imaginary part and cosine as real part, but I am confused when there is some sort of operation that is applied on these, how should I proceed? I suppose this comes from not knowing properties of this, should I say "operator" ($\text{Re},\text{Im}$).
Here is an example of my confusion: I have sequence $x_n = \sin^3(n\alpha)$, where $n$ is integer number, and $\alpha$ is some given angle. Using $\text{Im}$ function, that is same as $x_n = \text{Im}\{e^{in\alpha}\}^3$
With known system function $H(z)$, I should be able to find system response by plugging known $z$ and multiplying with $z^n$ but trouble is that I don't know what is my $z$ now. Is it $e^{i\alpha}$, where should I put that number 3?
I was thinking what if there is no number 3, let's say I just had to find response to sine, how should I use this $\text{Im}$ function (is this even right term?) when calculation $H(e^{i\alpha})$, if $H(z)$ looks something like $$H(z)=\frac{z(z+1)}{(z+2)(z+3)(z+4)}$$
Should I apply $\text{Im}$ function in front of fraction, or apply to every $z$ occurrence, or something third?
Best Answer
Can you continue?