Unit 5: The laplace transform

The Laplace transform generalises the continuous-time Fourier transform by inserting a real exponential weighting, letting signals that grow or fail to converge in the Fourier sense still be analysed and giving a compact algebraic tool for solving differential equations and characterising LTI systems.

• Core definition: Maps a time signal x(t) to a complex-frequency function X(s) where s = σ + jω; σ (real part) is the damping/growth rate, ω the radian frequency.
• Relation to Fourier: The Fourier transform is the Laplace transform evaluated on the imaginary axis, s = jω, whenever that axis lies in the region of convergence.
• Two forms: Bilateral (limits −∞ to ∞) suits general signals; unilateral (limits 0⁻ to ∞) suits causal systems and initial-condition problems.
• Why it matters: Convolution in time becomes multiplication in s; differential equations become polynomial equations, solved by algebra then inverted.

II. The Laplace Transform

Definition and mechanics of the transform integral

The bilateral Laplace transform decomposes a signal into complex exponentials e^{st}.

A. The laplace transform

The defining integral converts x(t) into a function of the complex variable s.

• Bilateral definition: the transform integral is

X(s) = ∫_{-∞}^{∞} x(t) e^{-st} dt ,   s = σ + jω

• Symbols: X(s) is the transform; e^{-st} the analysing kernel; σ sets exponential weighting so the integral converges.
• Notation: written x(t) ⟷ X(s), the transform pair.
• Fourier connection: setting σ = 0 gives X(jω) = ∫ x(t)e^{-jωt}dt, the Fourier transform — valid only if jω is inside the ROC.
• Worked pair: for x(t) = e^{-at}u(t),

X(s) = ∫_0^{∞} e^{-at} e^{-st} dt = 1/(s+a),  Re{s} > -a

The pole sits at s = -a; convergence requires σ > -a.

III. The Region of Convergence for Laplace Transforms

The set of s for which the integral exists

X(s) is meaningful only where the defining integral converges; that set of s-values is the ROC and must always accompany the algebraic expression.

A. The region of convergence for laplace transforms

The ROC is a strip or half-plane in the s-plane bounded by poles, never containing any pole.

• Strips parallel to jω: the ROC depends only on Re{s} = σ, so it appears as vertical strips or half-planes.
• Right-sided signals: ROC is a right half-plane, Re{s} > σ_max, to the right of the rightmost pole — e.g. e^{-at}u(t) gives Re{s} > -a.
• Left-sided signals: ROC is a left half-plane, to the left of the leftmost pole — e.g. -e^{-at}u(-t) gives the same 1/(s+a) but Re{s} < -a.
• Two-sided signals: ROC is a bounded strip between two poles, or empty if poles conflict.
• No poles in ROC: because the integral diverges at a pole; the ROC boundaries are pole locations.
• Fourier existence test: the Fourier transform exists iff the ROC includes the jω-axis (σ = 0).
• Stability link: for a causal system, including the jω-axis in the ROC ⇔ all poles in the left half-plane ⇔ stability.

IV. The Inverse Laplace Transform

Recovering x(t) from X(s)

Inversion returns the time signal, done in practice by partial fractions and a table rather than the defining contour integral.

A. The inverse laplace transform

Two routes exist: the formal complex integral and the practical algebraic method.

1. Formal synthesis integral: the exact inversion is a contour integral along a vertical line inside the ROC,

x(t) = (1/2πj) ∫_{σ-j∞}^{σ+j∞} X(s) e^{st} ds

where σ is any value in the ROC.

2. Partial-fraction method (practical): expand rational X(s) = N(s)/D(s) into simple terms, then invert each using known pairs.

• Procedure: factor D(s), write X(s) = Σ A_k/(s − p_k), find residues A_k = (s − p_k)X(s)|_{s=p_k}.
• ROC decides direction: each term A_k/(s − p_k) inverts to A_k e^{p_k t}u(t) (right-sided) or −A_k e^{p_k t}u(−t) (left-sided) according to the ROC.
• Worked example: X(s) = 1/[(s+1)(s+2)], ROC Re{s} > −1.

X(s) = 1/(s+1) − 1/(s+2)
x(t) = (e^{-t} − e^{-2t}) u(t)

V. Geometric Evaluation of the Fourier Transform from the Pole-Zero Plot

Reading magnitude and phase off the s-plane

The frequency response can be estimated visually by treating each pole and zero as a vector to the point s = jω.

A. Geometric evaluation of the fourier transform from the pole zero plot

Magnitude and phase at frequency ω come from vector lengths and angles measured from poles and zeros to the point jω on the imaginary axis.

• Vector construction: for H(s) = K·Π(s − z_i)/Π(s − p_k), draw a vector from each zero z_i and each pole p_k to the test point s = jω.
• Magnitude rule:

|H(jω)| = |K| · (Π length of zero vectors)/(Π length of pole vectors)

• Phase rule:

∠H(jω) = ∠K + Σ(zero vector angles) − Σ(pole vector angles)

• Behaviour near a pole: as jω passes close to a pole, its vector length shrinks, so |H| peaks — this explains resonant humps.
• Behaviour near a zero: as jω approaches a zero, that vector length shrinks, driving |H| toward a notch.
• Design insight: poles near the jω-axis sharpen resonance; zeros on the axis create exact nulls.

VI. Properties of the Laplace Transform

Operational rules that replace calculus with algebra

Each property maps a time-domain operation to a simple s-domain manipulation, with the ROC tracked at every step.

A. Properties of the laplace transform

The properties parallel those of the Fourier transform but carry ROC bookkeeping.

• Linearity: ax₁(t)+bx₂(t) ⟷ aX₁(s)+bX₂(s); ROC at least the intersection.
• Time shift: x(t−t₀) ⟷ e^{-st₀}X(s); ROC unchanged.
• s-shift (modulation): e^{s₀t}x(t) ⟷ X(s−s₀); ROC shifted by Re{s₀}.
• Time scaling: x(at) ⟷ (1/|a|)X(s/a); ROC scaled by a.
• Differentiation in time: dx/dt ⟷ sX(s) — the key to turning ODEs into algebra.
• Differentiation in s: −t·x(t) ⟷ dX(s)/ds.
• Integration in time: ∫_{-∞}^{t}x(τ)dτ ⟷ X(s)/s; adds a pole at s = 0.
• Convolution: x(t)*h(t) ⟷ X(s)H(s) — the property behind transfer-function analysis.
• Initial-value theorem: x(0⁺) = lim_{s→∞} sX(s) (for causal x).
• Final-value theorem: lim_{t→∞} x(t) = lim_{s→0} sX(s), valid only if all poles of sX(s) are in the left half-plane.

VII. Analysis and Characterisation of LTI Systems Using the Laplace Transform

The transfer function as system fingerprint

An LTI system is fully described by the transform of its impulse response, and its pole-zero pattern encodes causality, stability, and dynamics.

A. Analysis and characterisation of LTI systems using the laplace transforms

The transfer function H(s) links input and output algebraically and its geometry reveals system behaviour.

• Transfer function: H(s) = Y(s)/X(s) = L{h(t)}; output follows from Y(s) = H(s)X(s).
• From differential equation: applying dⁿ/dtⁿ ⟷ sⁿ to

Σ a_k y^{(k)}(t) = Σ b_m x^{(m)}(t)

gives the rational form

H(s) = (Σ b_m s^m)/(Σ a_k s^k)

• Causality: for a causal system the ROC is a right half-plane to the right of the rightmost pole.
• Stability: BIBO stable ⇔ ROC includes the jω-axis; for causal systems ⇔ every pole has Re{p_k} < 0.
• Frequency response: obtained as H(jω) = H(s)|_{s=jω} when the axis is in the ROC.
• First-order example: H(s) = 1/(s+a) has one pole at −a; causal and stable for a > 0, time constant 1/a.

VIII. Software Simulation of System Representation and Pole-Zero Analysis

Numerical tools for modelling and visualising H(s)

Software packages represent H(s) by its coefficient vectors and compute poles, zeros, responses, and stability automatically.

A. Software simulation of system representation and pole zero analysis

Systems are entered as numerator/denominator polynomials, then analysed with built-in routines.

• Representation: store H(s) as coefficient arrays num = [b_m … b_0], den = [a_n … a_0].
• MATLAB/Octave build: create the model and inspect poles/zeros:

num = [1];            % numerator coefficients
den = [1 3 2];        % s^2 + 3s + 2
H   = tf(num, den);   % transfer function
pzmap(H)              % plots poles (x) and zeros (o)
[p, z] = pole(H), zero(H)

• Python (SciPy) equivalent:

from scipy import signal
H = signal.TransferFunction([1], [1, 3, 2])
poles, zeros = H.poles, H.zeros

• Frequency response: bode(H) or freqs(num,den) returns magnitude and phase versus ω.
• Stability check: confirm every returned pole has negative real part; here poles at s = −1, −2 ⇒ stable.
• Time response: step(H) and impulse(H) simulate the system's reaction, matching the analytic inverse transform.
• Design use: dragging poles/zeros in an interactive pzmap shows resonance and damping change in real time, closing the loop with the geometric evaluation of Section V.