# Polar form of eigenvalue

I greatly appreciate the opportunity afforded by this forum to submit a query.

It has been suggested that for a sinusoidal input, represented as 2 phasors, into a Linear Shift-invariant system, the output...

...can also be written as follows:

where $$H(\Omega) = \int_{-\infty}^{\infty} h(\lambda) e^{-j\Omega\lambda} d\lambda$$

Accompanying brief comment suggests the above to be on account of "polar form representation of H".

Although I am cognizant of polar form representation of complex numbers i.e.

$$z = |z|(\cos(\theta)+i \sin(\theta)) = |z|e^{i \theta}$$

...I am completely unable to fathom the alternative above.

All advice would be greatly appreciated.

Best regards,

wirefree


I think all it's saying is that we can write $H(\Omega)$ as:
$$H(\Omega) = \left| H(\Omega) \right | e^{j \angle H(\Omega) }$$
It also seems to be assuming that $h(t)$ is real-valued so that $H(-\Omega) = H^*(\Omega)$.
The function $H(\Omega)$ is just a map from $\mathbf{R} \mapsto \mathbf{C}$ (from the reals to the complex numbers). For any specific value of $\Omega$ this generates a single complex number $H(\Omega)$. And, as you say, any complex number resolves to its exponential notation.