Exponential Signals

Continuous Time

The exponential signal is given as:

$$e^{st}$$

where $s=\sigma +j\omega$. This expands to:

$$\begin{array}{rl}{e}^{st}& =e^{(\sigma +j\omega )t}=e^{\sigma t}{e}^{j\omega t}\\ & =e^{\sigma t}(\cos \omega t+j\sin \omega t)\end{array}$$

This function is very general and can be used for a large variety of functions:

• A constant $k=k{e}^{0t}$, where $s=0$
• A monotonic exponential $e^{\sigma t}$, where $\omega =0,s=\sigma$
• A sinusoid $\cos \omega t$, where $\sigma =0,s=\pm j\omega$
• An exponentially varying sinusoid $e^{\sigma t}\cos \omega t$, where $s=\sigma \pm j\omega$

Discrete Time

In discrete time, we write:

$$e^{st}={\gamma }^t$$

such that $e^s=\gamma$. Conversely, we get:

$$s=\ln \gamma$$

Expanding $\gamma$, we have:

$$\begin{array}{rl}\gamma & =e^s=e^{\sigma +j\omega }=e^{\sigma }{e}^{j\omega }\\ |\gamma |& =|e^{\sigma }||e^{j\omega }|\\ |\gamma |& =|e^{\sigma }|\end{array}$$

Where did the |e^jw| term go?

$$\begin{array}{r}{e}^{j\omega }=\cos \omega +j\sin \omega \\ |e^{j\omega }|=\sqrt{{\cos }^2\omega +{\sin }^2\omega }\\ |e^{j\omega }|=1\end{array}$$

Cases:

• If $\sigma =0$, $|\gamma |=1$ (Sinusoid)
• If $\sigma <0$, $|\gamma |=|e^{\sigma }|<1$ (Exponential delay)
• If $\sigma >0$, $|\gamma |=|e^{\sigma }|>1$ (Exponential growth)

In the diagram, $\lambda$ is used instead of $s$.