Sinusoidal Response for SD Systems

For the following example…
$r[k]=\cos ({\omega }_{0}kT)=\cos (k\theta )$, in steady state: $y[k]=\frac{1}{T}{\sum }_{n=0}^{\infty }|T_{ry}[e^{j\omega T}]||P(j{\omega }_n)||H(j{\omega }_n)|\cos ({\omega }_{n}t+<Tru[e^{j{\omega }_{0}T}]+<P(j{\omega }_n)+<H(j{\omega }_n))$

Defining ${{\omega }_n}_{n=0}^{\infty }$ we have the pattern $=n{\omega }_s-{\omega }_0$ and $=n{\omega }_s+{\omega }_0$

The $n=0$ term corresponds to the output of an LTI system.
The frequency content of $y(t)$ is at ${\omega }_0$, $n{\omega }_s\pm {\omega }_0\forall n\ge 1$.

Note

If ${\omega }_N\ge {\omega }_{bw}$, ${\omega }_N>{\omega }_0$ and ${\omega }_N\ge \max ({\omega }_{bw},{\omega }_0)$ and ${\omega }_s=2\max ({\omega }_{bw},{\omega }_0)$.

Also in practice $|P(j\omega )|\ne 0\forall \omega >{\omega }_{bw}$ which ties into Sampling Time.