Complex sinusoid

A complex sinusoid generalizes the real sinusoid and the real exponential at once:

$g(t)=A{e}^{({\sigma }_0+j{\omega }_0)t}=A{e}^{{\sigma }_{0}t}[\cos ({\omega }_{0}t)+j\sin ({\omega }_{0}t)],$

where the second equality uses Euler’s formula. The pair $({\sigma }_0,{\omega }_0)$ fully describes a complex sinusoid: ${\omega }_0$ is the radian frequency of oscillation, and ${\sigma }_0$ controls whether the envelope decays, stays constant, or grows.

• ${\sigma }_0<0$: oscillation inside a decaying envelope — the bumper-after-the-speed-bump signal, or the ring of a struck tuning fork. This is the shape of an underdamped second-order system’s transient (in the second-order sense, “underdamped” means damping ratio $\zeta <1$, which puts the poles off the real axis and produces exactly this kind of damped oscillation).
• ${\sigma }_0=0$: pure undamped oscillation, $A{e}^{j{\omega }_{0}t}$, tracing a circle in the complex plane.
• ${\sigma }_0>0$: oscillation inside a growing envelope — physically usually means instability.

The point in the s-plane

The pair $({\sigma }_0,{\omega }_0)$ is exactly a point in the complex s-plane: real part ${\sigma }_0$, imaginary part ${\omega }_0$. The complex sinusoid is what the Laplace transform returns when you query “what time-domain signal has a pole at $s={\sigma }_0+j{\omega }_0$?” Pole locations and complex sinusoids are two views of the same object.

A pole in the left half-plane → decaying sinusoid. On the imaginary axis → pure oscillation. In the right half-plane → growing sinusoid. This pole-location-to-signal-shape mapping is one of the most important pictures in the course.

Why we use them even for real signals

Complex exponentials multiply, differentiate, and integrate cleanly:

$\frac{d}{dt}{e}^{st}=s{e}^{st},\int e^{st}dt=\frac{1}{s}{e}^{st}.$

Real sinusoids are messier — products of cosines give sums of cosines via half-angle identities. A typical derivation strategy is to replace a real cosine by a sum of two complex exponentials (via Euler’s formula), do the algebra, then collapse back to a real cosine.

Complex sinusoids are also the eigenfunctions of LTI systems — feed one in, get the same complex sinusoid out, scaled by a complex constant $H(s_0)$. This is the entire foundation of frequency-domain analysis.

A note on $j$ versus $i$

Electrical engineering writes the imaginary unit as $j=\sqrt{-1}$, while mathematicians and physicists use $i$. The EE convention exists because $i$ already means current. The two are the same object — just different notation.