SC · Laplace

The Laplace Transform

The Laplace transform adds a convergence factor e^(−σt) to the Fourier transform, widening the frequency jω into a complex s = σ + jω. See firsthand how that σ tames a growing signal, so even signals Fourier cannot handle get transformed.

Does the weighted signal die down

The top shows a growing signal e^(at) (faint) and the weighted signal (bold) after multiplying by the convergence factor e^(−σt). Below is the running integral of that weighted signal. Raise σ until the growing signal dies to 0 and the integral settles at a ceiling. That is when the Laplace transform exists.

growing signal e^(at)weighted e^((a−σ)t)

running integral

convergence factor σ0.50

Transform and ROC

X(s) = 1/(s − 1)   ROC: Re(s) > 1  · ∞

σ is below the growth rate a, so the weighted signal e^((a−σ)t) still grows. The integral diverges and no transform exists. This is the region where Fourier (σ=0) fails.

Diverges

Add a convergence factor to Fourier

The Fourier transform multiplied a signal by e^(−jωt) and integrated. The Laplace transform multiplies by one more real exponential, e^(−σt). Combining the two factors gives e^(−st), where s = σ + jω is the complex frequency. The jω part still picks out oscillation, but the σ part, through the ever-shrinking e^(−σt), forces the signal to die down. So a signal that grew without bound and made the Fourier integral diverge can, with σ raised enough, make the integral converge and have a transform.

ObserveX(s) = ∫ x(t) e^−st dt

The transform multiplies by e^−st and integrates.

Chooses = ?

The complex frequency is s = σ + jω.

Fill ine^atu(t): ROC Re(s) ?

e^atu(t) converges for Re(s) > a.

On your ownσ = 0: s = ? (Fourier)

At σ=0 Laplace is Fourier.

The region of convergence

The Laplace integral does not converge for every σ. For a growing signal e^(at)u(t) the weighted signal is e^((a−σ)t), which decays to 0 only when σ > a, making the integral converge. The range of σ where it converges is the region of convergence, the ROC. This signal’s transform is X(s) = 1/(s−a) with ROC Re(s) > a. Since the same formula with a different ROC means a different signal, a Laplace transform is complete only as the pair X(s) and its ROC.

Fourier is one slice of Laplace

s = σ + jω is a point in a plane. Set σ to 0 and s = jω, the vertical imaginary axis, and there the Laplace transform becomes exactly the Fourier transform — but only when that axis lies in the ROC. So Fourier is the slice along the imaginary axis of the whole s-plane that Laplace sees. Being free to choose σ is what gives Laplace its power over transients and stability. The next unit marks poles and zeros on this s-plane to read a system.

Back to the first screen

When σ was small the weighted signal still grew and the integral diverged — exactly where the Fourier transform fails. Raising σ past the growth rate a made e^((a−σ)t) decay to 0, the integral settled at the ceiling 1/(σ−a), and the Laplace transform began to exist. What Laplace adds to Fourier is precisely this choosable σ, a real exponential that forces convergence. The range of σ it creates is the ROC, and a transform is the pair X(s) and its ROC.

The Laplace transform X(s) = ∫₀^∞ x(t) e^−st dt uses the complex frequency s = σ + jω. In the factor e^−st = e^−σt e^−jωt, the e^−σt is a convergence factor that tames growing signals so even those Fourier cannot handle get transformed. The range of σ where the integral converges is the region of convergence (ROC). For e^atu(t), X(s) = 1/(s−a), ROC: Re(s) > a. At σ=0 (s=jω), if the imaginary axis lies in the ROC, Laplace reduces to Fourier.

On to the next unit

If s is a point in a plane, then the points where X(s) blows up to infinity and where it goes to 0 tell almost the whole character of a system. The next unit, the s-plane and poles and zeros, marks the pole s = a that makes the denominator of X(s) = 1/(s−a) zero. A pole to the left of the imaginary axis (Re < 0) means a decaying response; to the right, a diverging one. The a < 0 stability condition from B4 reappears as exactly this pole position. Reading a system from a few points on the plane opens there.