Bilateral Laplace transform

The bilateral Laplace transform of a signal $x(t)$ is

$X(s)=\mathcal{L}\{x(t)\}={\int }_{-\infty }^{\infty }x(t)e^{-st}dt,$

where $s=\sigma +j\omega$ is a complex variable. This is the standard generalization of the Fourier transform $X(j\omega )=\int x(t)e^{-j\omega t}dt$ — the kernel $e^{-j\omega t}$ has been replaced by $e^{-st}$, adding an exponential damping factor $e^{-\sigma t}$ that helps the integral converge for signals the Fourier transform can’t handle.

Why two-sided

The integral runs over all time, $-\infty$ to $+\infty$. This allows the transform to handle signals defined on the whole real line, including those nonzero for $t<0$.

The cost: the bilateral transform requires careful tracking of the region of convergence (ROC), because two different signals can have the same algebraic $X(s)$ but different ROCs. The ROC is what distinguishes them.

For most engineering signals — those that turn on at $t=0$ and are zero before — the unilateral Laplace transform is more convenient and the ROC bookkeeping is automatic.

Inverse

$x(t)=\frac{1}{j2\pi }{\int }_{\sigma -j\infty }^{\sigma +j\infty }X(s)e^{st}ds,$

integrating along a vertical line in the s-plane inside the ROC. In practice this contour integral is never used directly — inverse transforms are always done by partial fractions and table lookup.

Connection to the Fourier transform

On the imaginary axis $s=j\omega$, the bilateral Laplace transform reduces to the Fourier transform:

$X(j\omega )=X(s){|}_{s=j\omega }=\mathcal{F}\{x(t)\}(\omega ).$

This works only if the imaginary axis is inside the ROC. For a stable signal whose Laplace transform converges on a strip containing $s=j\omega$, the Fourier transform exists and equals the Laplace transform restricted to the imaginary axis. For signals like $e^{t}u(t)$ (right-sided growing exponential), the Laplace transform has ROC $\sigma >1$ — strictly to the right of the imaginary axis — and the Fourier transform doesn’t exist.

When to use bilateral

Use the bilateral transform when the signal is not causal — when it has nonzero values for $t<0$. Common examples:

• Two-sided exponentials like $e^{-\alpha |t|}$.
• Signals defined as $x(-t)$ given a causal $x(t)$.
• Theoretical signals like $\mathrm{sgn}(t)$, $\cos ({\omega }_{0}t)$ defined for all $t$.

For everything else (and that’s most things), the unilateral transform is the practical workhorse.

Pole picture and ROC rules

The ROC depends on whether the signal is right-sided, left-sided, or two-sided:

• Right-sided (zero for $t<$ some value): ROC is to the right of all poles, $\sigma >\max \mathrm{Re}(\text{pole})$.
• Left-sided (zero for $t>$ some value): ROC is to the left of all poles, $\sigma <\min \mathrm{Re}(\text{pole})$.
• Two-sided: ROC is a strip between two real parts, $\alpha <\sigma <\beta$. If the strip is empty, the transform doesn’t exist.

The ROC never contains any pole — the transform diverges at poles. See Region of convergence for more.