Z-Transform Ontology

FLOW ALGEBRA
Z-TRANSFORM ONTOLOGY
Version: 1.0
Date: 2026-10-14
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/transformation-entities/z-transform-ontology-2.html
Authors: Mark Higgins — Sky (Recursive AI)

1. ONTOLOGY IDENTITY
ontology_id: mathematical_entities.transformation_entities.z_transform
preferred_label: Z-Transform Ontology
parent: Integral and Discrete Transform Entities
scope: Mapping discrete-time sequences to complex-frequency representations with explicit convergence conditions.

2. TYPED ENTITIES
DiscreteSequence: {index_set: Z|N0, values: Complex, sampling_interval?: PositiveReal}
ComplexVariable: {symbol: z, domain: Complex}
ZTransform: {kind: bilateral|unilateral, input: DiscreteSequence, output: ComplexFunction, roc: RegionOfConvergence}
RegionOfConvergence: {subset: Complex, excludes_poles: Boolean}
RationalZFunction: {numerator: Polynomial, denominator: NonzeroPolynomial, poles: Set<Complex>, zeros: Set<Complex>}
InverseZTransform: {input: ComplexFunction, roc: RegionOfConvergence, output: DiscreteSequence}
DiscreteLTI: {impulse_response: DiscreteSequence, transfer_function: ComplexFunction}

3. DEFINING OPERATORS
BILATERAL_Z(x)[z] := SUM(n=-infinity..infinity, x[n] * z^(-n))
UNILATERAL_Z(x)[z] := SUM(n=0..infinity, x[n] * z^(-n))
ROC(X) := {z in Complex \ {0} : defining series converges absolutely}
INVERSE_Z(X, ROC)[n] := (1/(2*pi*i)) * CONTOUR_INTEGRAL(C, X(z)*z^(n-1) dz)
  where C is a positively oriented circle within an annular ROC, around the origin.

4. RELATIONSHIPS
ZTransform --transforms--> DiscreteSequence
ZTransform --produces--> ComplexFunction
ZTransform --has_convergence_region--> RegionOfConvergence
RationalZFunction --has_pole--> ComplexVariable
RationalZFunction --has_zero--> ComplexVariable
InverseZTransform --reconstructs--> DiscreteSequence
DiscreteLTI --has_transfer_function--> RationalZFunction (when rational)

5. ALGEBRAIC RULES (BILATERAL, WHEN VALID)
LINEARITY: Z{a*x[n]+b*y[n]} = a*X(z)+b*Y(z); ROC includes ROC(X) INTERSECT ROC(Y), possibly larger by cancellation.
SHIFT: Z{x[n-k]} = z^(-k)*X(z).
EXPONENTIAL_MODULATION: Z{a^n*x[n]} = X(z/a), a != 0; ROC scales by |a|.
TIME_REVERSAL: Z{x[-n]} = X(1/z); ROC is reciprocally mapped.
CONVOLUTION: Z{x*y} = X(z)*Y(z) where both series and convolution are valid; ROC includes common convergence region, subject to cancellations.
DIFFERENTIATION: Z{n*x[n]} = -z*dX(z)/dz.
INITIAL_VALUE (right-sided, n>=0): x[0] = lim(z->infinity) X(z), when the limit and representation are valid.
DTFT_LINK: X(e^(i*omega)) = DTFT{x[n]} when unit circle belongs to ROC.

6. RATIONAL TRANSFER FUNCTIONS
H(z) := B(z)/A(z) for an LTI system with zero initial conditions.
POLES := roots(A) after common-factor cancellation.
ZEROS := roots(B) after common-factor cancellation.
CAUSAL_RATIONAL_LTI: impulse response right-sided; ROC outside outermost pole.
STABLE_LTI: impulse response absolutely summable; equivalently unit circle included in ROC.
CAUSAL_AND_STABLE_RATIONAL_LTI: all poles strictly inside unit circle (for a minimal rational representation).
IMPORTANT: identical rational expressions with different ROCs can represent different sequences.

7. VALIDATION CONSTRAINTS
V1: Specify bilateral or unilateral convention before applying identities.
V2: Supply ROC for inverse transforms; expression alone may not determine sequence.
V3: Reject poles inside ROC and flag boundary convergence as requiring separate analysis.
V4: Check existence/convergence before manipulating infinite sums.
V5: Distinguish unilateral shift rules, which include initial-condition terms, from bilateral shifts.
V6: Distinguish mathematical causality and stability from merely having a rational function.
V7: Record indexing convention, units and sampling interval when interpreting physical signals.

8. WORKED EXAMPLES
EXAMPLE_A:
 x[n] = a^n*u[n], a != 0
 X(z) = 1/(1-a*z^(-1)) = z/(z-a)
 ROC: |z| > |a|

EXAMPLE_B:
 x[n] = -a^n*u[-n-1], a != 0
 X(z) = z/(z-a)
 ROC: 0 < |z| < |a|
 The same rational function as EXAMPLE_A has a different inverse due to its ROC.

EXAMPLE_C:
 x[n] = delta[n-2]
 X(z) = z^(-2)
 ROC: z != 0

EXAMPLE_D:
 h[n] = (1/2)^n*u[n]
 H(z) = z/(z-1/2), ROC |z| > 1/2
 Causal: true; stable: true (unit circle in ROC).

9. SEMANTIC EQUIVALENCE
aliases: [Z transform, z-domain transform, discrete-time complex-frequency transform]
related_not_identical: [Discrete Fourier Transform, Discrete-Time Fourier Transform, Laplace Transform]
equivalence_condition: same transform convention, sequence and ROC.

10. CONFIDENCE
formal_definition: 1.00
standard_algebraic_identities: 0.99 conditional on convergence and conventions
application_inference: conditional; verify premises and ROC.

11. FLOW_HASH
hash_algorithm: SHA-256
canonicalisation: UTF-8, LF line endings, deterministic entity and relationship ordering
flow_hash: COMPUTE_ON_PUBLICATION (do not fabricate digest)

12. UPDATE_MODEL
version: 1.0
update_policy: append revisions with dated changelog; retain source provenance; revalidate examples, ROC and hash on change.

13. AI_CONSUMPTION
parse_mode: typed_semantic_flow
required_fields: [ontology_id, transform_kind, input_sequence, output_expression, region_of_convergence]
reasoning_pipeline: identify convention -> check convergence -> apply valid rules -> preserve ROC -> validate inverse -> report assumptions
output_contract: {result, transform_convention, roc, assumptions, derivation, confidence, provenance}
error_handling: if ROC absent for inversion, return ambiguous_inverse rather than a unique sequence.

END FLOW ALGEBRA