Engineering Hub/Signal Processing & Communication/Bilateral Laplace & Z-Transform Pole-Zero Stability Workbench

Signal Processing & CommunicationDiscipline: Signals and Systems (EEE2006)

Bilateral Laplace & Z-Transform Pole-Zero Stability Workbench

Interactive multi-domain laboratory exploring continuous s-plane and discrete z-plane pole-zero geometry, Region of Convergence (ROC) topology, BIBO stability theorems, and the conformal mapping z = e^(s Ts).

Instrument State:Live Analytical DSP Engine

EEE2006 Signals & Systems LabUnit 4: Complex Planes & Stability

Bilateral Laplace & Z-Transform Pole-Zero Stability Workbench

Interactive multi-domain laboratory investigating continuous s-plane & discrete z-plane pole-zero geometry, Region of Convergence (ROC) topology, BIBO stability theorems, and the conformal mapping z = e^(s Ts).

System Architecture Preset

Damping σ: -1.0

σ < 0: Stable LHP | σ > 0: Unstable RHP

Oscillation ωd: 2.0 rad/s

Natural ringing frequency

Convergence Region (ROC)

BIBO Stability

Stable (jω-axis included in ROC)

System Causality

Causal (Right-Sided Strip: h(t)=0 for t<0)

Dominant Pole

s = -1.00 ± j2.00

Complex s-Plane (s = σ + jω)ROC: Re(s) > -1.0

σ (Real)jω (Imag)

Poles s = -1.0 ± j2.0ROC Strip

Continuous Frequency Response |H(jω)| (along jω-axis)0 to 6 rad/s

Time-Domain Impulse Response h(t) = L⁻¹{H(s)}t: 0 to 5s

LTI Stability Theorem: A continuous LTI system with transfer function H(s) is BIBO stable if and only if its Region of Convergence contains the imaginary axis s = jω. For causal systems, this requires all poles to strictly lie in the Open Left-Half Plane (Re(p) < 0).

Technical Instrument Specifications

s-Plane AnalysisComplex pole-zero placement with ROC half-plane/strip shading & jω stability check

z-Plane AnalysisDiscrete pole placement with Unit Circle |z|=1 & annular ROC stability evaluation

Conformal MappingReal-time z = e^(s Ts) transformation mapping LHP to Unit Disk with aliasing indicator

Transient & FrequencyReal-time impulse responses h(t) & h[n] with continuous & discrete Bode magnitude plots

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