# Poles of the transfer function

Why are the poles of the transfer function of the sinusoidal input signal always on the left side? I know that it makes the system bounded, but is there a way to understand it conceptually without going through the maths?

The input signal is irrelevant. All the poles of a TF must have a negative real part because each contributes $e^{\alpha t}$ to the response, where $\alpha$ is the real value of a pole, and $\alpha$ must be negative to give a decaying exponential. A positive value means the response goes to infinity.
• An imaginary pair of poles, i.e. zero real part, will give a steady state sinusoidal response, $e^{\pm j \omega t}=cos(\omega t)\pm j\:sin(\omega t)$ – Chu Apr 15 '16 at 7:19