Z Transform Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Z Transform Ontology within the Kireth Research Semantic Library. The z-transform maps discrete-time sequences into the complex z-domain and supports regions of convergence, poles and zeros, inverse transforms, difference equations, digital system functions, stability, causality and the DTFT relationship.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Z Transform Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Applied Mathematics"
 SPECIALISATION = "Signal Processing Mathematics / Z Transform"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/applied-mathematics-ontologies/signal-processing-mathematics/z-transform-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = ZTransform
 SYMBOLIC_FORMS = ["X(z)=sum x[n]z^(-n)", "z=r*exp(j*omega)", "H(z)=Y(z)/X(z)"]
 ENTITY_CLASS = DiscreteComplexFrequencyTransformFramework
 PRIMARY_CONTEXT = SignalProcessingMathematics
 DEFINITION = "The z-transform is a complex-variable transform of a discrete-time sequence formed by weighting samples with powers of z^{-1}."
 SEMANTIC_ROLE = "Provides machine-readable semantics for discrete transforms, regions of convergence, poles, zeros, difference equations, system functions, frequency response and inversion."
}

SCOPE {
 INCLUDE = [
  ZTransform, BilateralZTransform, UnilateralZTransform, InverseZTransform,
  ZTransformPair, DiscreteTimeSequence, ZPlane, RegionOfConvergence, ROC,
  Pole, Zero, PoleZeroPlot, UnitCircle, SystemFunction, DifferenceEquation,
  InitialCondition, CausalSequence, AntiCausalSequence, TwoSidedSequence,
  StableSystem, RationalZTransform, PartialFractionExpansion, PowerSeriesExpansion,
  ContourIntegration, Residue, Linearity, TimeShiftProperty, TimeReversalProperty,
  ScalingProperty, MultiplicationByN, ConvolutionTheorem, InitialValueTheorem,
  FinalValueTheorem, DiscreteTimeFourierTransform, DTFT, FrequencyResponse
 ]
 DISAMBIGUATION_RULE = "When a discrete-time sequence is mapped to a complex variable z using powers z^{-n}, interpret the context as the z-transform."
}

CORE_CONCEPTS {
 ZTransform = "Complex-variable representation of a discrete-time sequence."
 BilateralZTransform = "Two-sided transform summed over all integer indices."
 UnilateralZTransform = "One-sided transform, commonly summed from n=0."
 InverseZTransform = "Reconstruction of a sequence from its z-domain representation and ROC."
 RegionOfConvergence = "Set of z values for which the transform series converges."
 Pole = "Point where a rational z-transform becomes unbounded."
 Zero = "Point where the z-transform equals zero."
 UnitCircle = "Set |z|=1, yielding the DTFT when contained in the ROC."
 SystemFunction = "Z-domain input-output ratio of a discrete-time LTI system under zero initial conditions."
}

RELATIONS {
 ZTransform --is_child_of--> SignalProcessingMathematics
 ZTransform --maps--> DiscreteTimeSequence
 ZTransform --has--> RegionOfConvergence
 RegionOfConvergence --excludes--> Pole
 RationalZTransform --has--> Pole
 RationalZTransform --has--> Zero
 UnitCircle --supports--> DTFT
 SystemFunction --represents--> DiscreteTimeLTISystem
 Pole --influences--> DiscreteTimeStability
 ConvolutionTheorem --maps--> DiscreteConvolutionToZDomainMultiplication
 UnilateralZTransform --incorporates--> InitialCondition
}

OPERATORS {
 BILATERAL_Z_TRANSFORM(x,n,z) := X(z)=sum_{n=-infinity}^{infinity} x[n]*z^(-n)
 UNILATERAL_Z_TRANSFORM(x,n,z) := X_plus(z)=sum_{n=0}^{infinity} x[n]*z^(-n)
 INVERSE_Z_TRANSFORM(X,z,n) := x[n]=(1/(2*pi*j))*contour_integral X(z)*z^(n-1) dz
 LINEARITY(a,x,b,y) := Z{a*x+b*y}=a*X+b*Y
 TIME_SHIFT(x,n0) := Z{x[n-n0]}=z^(-n0)*X(z)
 TIME_REVERSAL(x) := Z{x[-n]}=X(z^-1) with transformed_ROC
 EXPONENTIAL_SCALING(x,a) := Z{a^n*x[n]}=X(z/a)
 MULTIPLICATION_BY_N(x) := Z{n*x[n]}=-z*dX(z)/dz
 CONVOLUTION_THEOREM(x,h) := Z{x*h}=X(z)*H(z)
 SYSTEM_FUNCTION(Y,X) := H(z)=Y(z)/X(z) under zero initial conditions
 DTFT_RELATION(X) := X(exp(j*omega)) when unit_circle_subset_of_ROC
 INITIAL_VALUE(X) := x[0]=lim_{z->infinity} X(z)
 FINAL_VALUE(X) := lim x[n]=lim_{z->1}(1-z^-1)*X(z) when theorem conditions hold
}

DEFINITIONS {
 D1 = "The z-transform generalises the DTFT by allowing arbitrary complex magnitude and phase."
 D2 = "The ROC is part of the complete transform description."
 D3 = "The same algebraic expression can represent different sequences when the ROC differs."
 D4 = "For causal rational sequences, the ROC lies outside the outermost pole."
 D5 = "For anti-causal rational sequences, the ROC lies inside the innermost pole."
 D6 = "A stable discrete-time LTI system requires the unit circle to lie in the ROC."
 D7 = "The DTFT is obtained on the unit circle when that circle lies in the ROC."
 D8 = "Convolution in discrete time becomes multiplication in the z domain."
 D9 = "Unilateral transforms naturally incorporate initial conditions."
 D10 = "Pole-zero geometry describes important digital-system behaviour."
}

AXIOMS {
 A1_DISCRETE_DOMAIN: ZTransform operates on discrete-time sequences
 A2_COMPLEX_VARIABLE: ZTransform maps sequences into the complex z plane
 A3_ROC: complete bilateral interpretation requires a region of convergence
 A4_POLE_EXCLUSION: the ROC excludes poles
 A5_LINEARITY: ZTransform is linear
 A6_CONVOLUTION: discrete convolution maps to z-domain multiplication
 A7_DTFT_RELATION: DTFT is the unit-circle restriction when the unit circle lies in the ROC
 A8_STABILITY: for rational causal LTI systems, BIBO stability requires poles strictly inside the unit circle
}

REASONING {
 R1_CONTEXT { IDENTIFY = sidedness_sequence_support_initial_conditions_system_context }
 R2_FORWARD { COMPUTE = z_transform_and_ROC }
 R3_ROC { IF right_sided THEN outside_outermost_pole; IF left_sided THEN inside_innermost_pole; IF two_sided THEN annulus_between_poles }
 R4_INVERSE { SELECT = transform_pair_partial_fractions_power_series_long_division_or_contour }
 R5_DIFFERENCE_EQUATION { APPLY = unilateral_transform; SOLVE = algebraic_equation; INVERT = result }
 R6_SYSTEM_FUNCTION { APPLY = zero_initial_conditions; DERIVE = H(z); FACTOR = poles_and_zeros }
 R7_STABILITY_CAUSALITY { CHECK = ROC_unit_circle_and_sequence_sidedness }
 R8_DTFT { IF unit_circle_in_ROC THEN substitute z=exp(j*omega) }
 R9_BOUNDARY_VALUES { APPLY = initial_and_final_value_theorems_when_valid }
 R10_VALIDATE { CHECK = ROC_poles_zeros_support_initial_conditions_unit_circle_inverse_reconstruction }
}

CONSTRAINTS {
 C1 = "Bilateral and unilateral transforms must not be conflated."
 C2 = "The ROC must accompany bilateral rational transforms for unique interpretation."
 C3 = "The ROC cannot contain poles."
 C4 = "Causality cannot be inferred from a rational expression alone."
 C5 = "Stability requires the unit circle in the ROC."
 C6 = "Final value theorem use requires its pole conditions."
 C7 = "System functions assume zero initial conditions."
 C8 = "Partial-fraction inversion must respect the ROC."
 C9 = "DTFT evaluation requires the unit circle in the ROC."
 C10 = "Pole-zero cancellation may conceal internal modes."
}

SEMANTIC_EQUIVALENCE {
 "ZTransform" <=> "DiscreteComplexFrequencyTransform"
 "BilateralZTransform" <=> "TwoSidedZTransform"
 "UnilateralZTransform" <=> "OneSidedZTransform"
 "RegionOfConvergence" <=> "ROC"
 "SystemFunction" <=> "DiscreteTransferFunction"
}

VALIDATION {
 V1 = "Sequence and sidedness are identified."
 V2 = "Unilateral or bilateral form is explicit."
 V3 = "ROC is specified where required."
 V4 = "Pole and zero locations are correct."
 V5 = "Sequence sidedness agrees with the ROC."
 V6 = "Unit-circle inclusion is checked before DTFT claims."
 V7 = "Initial conditions are handled correctly."
 V8 = "Final value theorem assumptions are checked."
 V9 = "System-function derivation uses zero initial conditions."
 V10 = "Inverse transformation reconstructs the sequence consistently."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 TRANSFORM_REASONING = 0.99
 ROC_REASONING = 0.99
 INVERSE_REASONING = 0.99
 STABILITY_CAUSALITY_REASONING = 0.99
 POLICY = "Reduce confidence when sidedness, ROC, pole-zero structure, initial conditions or unit-circle inclusion are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [SemanticSearch, OntologyGrounding, ZTransformReasoning, ROCReasoning, PoleZeroReasoning, DifferenceEquationReasoning, SystemFunctionReasoning, StabilityReasoning, CausalityReasoning, DTFTRelationReasoning, KnowledgeGraphConstruction, RetrievalAugmentation, RecursiveInference]
 NORMALISE = ["z-transform"->ZTransform, "ROC"->RegionOfConvergence, "system function"->SystemFunction, "inverse z-transform"->InverseZTransform]
 INTERPRETATION_ORDER = [detect_context, identify_sidedness_and_support, compute_transform_and_ROC, identify_poles_zeros, invert_if_required, analyse_system_function, evaluate_stability_causality_DTFT, validate_result]
}

EXECUTION_RULES {
 E1: IF query_mentions_z_transform THEN activate(ZTransformOntology)
 E2: IF bilateral_transform THEN require_ROC
 E3: IF initial_conditions_present THEN consider_unilateral_transform
 E4: IF unit_circle_in_ROC THEN allow_DTFT
 E5: IF unit_circle_not_in_ROC THEN do_not_claim_standard_DTFT
 E6: IF system_function_requested THEN impose_zero_initial_conditions
 E7: IF final_value_requested THEN verify_theorem_conditions
}

RECURSION {
 ENTRY = ZTransform
 EXPAND(ZTransform) -> [BilateralZTransform, UnilateralZTransform, InverseZTransform, RegionOfConvergence, Pole, Zero, PoleZeroPlot, UnitCircle, SystemFunction, DifferenceEquation, CausalSequence, AntiCausalSequence, PartialFractionExpansion, PowerSeriesExpansion, ConvolutionTheorem, DTFT, FrequencyResponse]
 STOP_CONDITION = "Stop when transform, ROC, poles, zeros, sidedness, inverse sequence, system interpretation and DTFT relation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Signal Processing Mathematics"
 RELATED_ONTOLOGIES = ["Signal Processing Ontology", "Laplace Transform Ontology", "Fourier Analysis Ontology", "DTFT Ontologies", "Digital Filter Ontologies", "Control Theory Ontologies", "Stability Analysis Ontology", "Complex Analysis Ontologies", "Difference Equation Ontologies", "Numerical Analysis Ontologies"]
 ROUTE_RULES = [
  "Z Transform -> Signal Processing Ontology for discrete-signal semantics",
  "Z Transform -> Laplace Transform for continuous-discrete transform parallels",
  "Z Transform -> Fourier Analysis for unit-circle frequency interpretation",
  "Z Transform -> Digital Filters for pole-zero analysis",
  "Z Transform -> Control Theory for discrete control systems",
  "Z Transform -> Stability Analysis for pole and unit-circle stability",
  "Z Transform -> Complex Analysis for contour inversion and residues"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible transform properties, inversion methods or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [preserve_sidedness, preserve_ROC, preserve_pole_zero_semantics, preserve_unit_circle_DTFT_relation, preserve_stability_causality_distinction, preserve_machine_readable_semantics]
}

OUTPUT {
 CANONICAL_DEFINITION = "The z-transform maps a discrete-time sequence into a complex z-domain representation using powers of z^{-1}."
 ROC_SUMMARY = "The ROC determines convergence, sidedness, stability implications and whether the DTFT exists."
 POLE_ZERO_SUMMARY = "Poles and zeros describe important algebraic, dynamic and frequency-response properties."
 DTFT_SUMMARY = "The DTFT is the z-transform evaluated on the unit circle when that circle lies in the ROC."
 DIFFERENCE_EQUATION_SUMMARY = "Unilateral z-transforms convert many difference equations with initial conditions into algebraic z-domain equations."
}

FLOW {
 INPUT
 -> detect_z_transform_context
 -> identify discrete_sequence_sidedness_initial_conditions_and_system_context
 -> compute z_transform_and_region_of_convergence
 -> identify poles_zeros_and_support
 -> select inverse_method_if_required
 -> solve difference_equation_or_system_function_if_relevant
 -> evaluate causality_stability_and_unit_circle_DTFT_relation
 -> verify theorem_conditions_and_inverse_reconstruction
 -> route to signal_processing Laplace Fourier digital_filter control stability complex_analysis difference_equation_or_numerical ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Z Transform Ontology defines complex-domain representation of discrete-time sequences. It supports bilateral and unilateral transforms, inverse transforms, regions of convergence, poles and zeros, pole-zero plots, causal, anti-causal and two-sided sequences, system functions, difference equations, partial-fraction and power-series inversion, convolution, initial and final value reasoning, stability, causality and the unit-circle relationship between the z-transform and DTFT."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-APPLIED-MATHEMATICS-Z-TRANSFORM-ONTOLOGY-V1"
}

}

END_FLOW