Exponential Class Signals

Right sided CT signal is of exponential class if $|x(t)| ≤ Me^{\alpha t}$

LTI systems with impulse response $h$

If the input is the $e^{st}$ then the output is also proportional to $e^{st}$, but scaled by the “transfer function” $H(s)$

Transfer Function Methods

CTFT is not always well-defined, it turns out that $H(s)$will exist in many situations where $H(j\omega)$ does not

• Theory here can be developed but two-sided signals and for non-causal LTI systems, we will study only:
  • CT signals that are right-sided
  • CT systems that are LTI and causal

Right-sided Laplace transform

Laplace transform of $x$ is complex-valued function:

$$ X: R_x -> C, X(s) = \int _{0^-} ^{\infty} x(t)e^{-st} dt $$

$X(s)$ is only defined by $s \in \R _x$, the region of convergence.

LT and CTFT connection

When $$2

Important Laplace transforms

Properties of the Laplace transform

Response of a causal LTI system and the LT

The LT can be used to compute the response of a causal LTI CT system to a general right-sided input

• $T$ causal LTI, impulse response $h$ with LT $H(s)$ and RoC $R_h$
• $x$ right-sided with LT $X(s)$ and RoC $R_x$