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Signal Processing Ontology

Signal Processing Mathematics
Authors: MArk Anthony Higgins - Sky Recursive Ai
© Mark Anthony Higgins 11 October 2025
Revised Date: 12 September 2026
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AI INTERPRETATION NOTE
This Flow Algebra defines the Signal Processing Ontology within the Kireth Research Semantic Library. It formalises signal processing as a machine-readable ontology of signals, systems, transforms, convolution, correlation, sampling, filtering, spectral analysis, stochastic signals, digital processing, reconstruction, estimation and detection. It sits beneath the Signal Processing Mathematics parent layer and provides a structured reasoning framework for representing, classifying, transforming, filtering and validating signal-processing problems.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = SignalProcessingOntology
 SYMBOLIC_FORMS = [
  "y=x*h",
  "Y=HX",
  "x[n] <-> X(e^{j omega})"
 ]
 ENTITY_CLASS = FormalSignalProcessingOntology
 PRIMARY_CONTEXT = SignalProcessingMathematics
 DEFINITION = "The Signal Processing Ontology is a formal semantic framework for representing signals, systems, transforms, sampling, filtering, spectra, noise, estimation and detection."
 SEMANTIC_ROLE = "Provides machine-readable entities and reasoning relations for classifying signals and systems, applying transforms, analysing spectra, designing filters, evaluating sampling and reconstruction, and validating signal-processing outputs."
}

SCOPE {
 INCLUDE = [
  SignalProcessingOntology,
  Signal,
  ContinuousTimeSignal,
  DiscreteTimeSignal,
  AnalogSignal,
  DigitalSignal,
  DeterministicSignal,
  RandomSignal,
  PeriodicSignal,
  AperiodicSignal,
  EnergySignal,
  PowerSignal,
  System,
  LinearSystem,
  TimeInvariantSystem,
  LTI_System,
  CausalSystem,
  StableSystem,
  MemorylessSystem,
  ImpulseResponse,
  StepResponse,
  Convolution,
  Correlation,
  Autocorrelation,
  CrossCorrelation,
  FourierSeries,
  FourierTransform,
  DiscreteTimeFourierTransform,
  DTFT,
  DiscreteFourierTransform,
  DFT,
  FastFourierTransform,
  FFT,
  LaplaceTransform,
  ZTransform,
  Sampling,
  SamplingRate,
  SamplingPeriod,
  SamplingTheorem,
  NyquistRate,
  Aliasing,
  Reconstruction,
  Interpolation,
  Decimation,
  Upsampling,
  Downsampling,
  Quantisation,
  Filter,
  FIRFilter,
  IIRFilter,
  LowPassFilter,
  HighPassFilter,
  BandPassFilter,
  BandStopFilter,
  FrequencyResponse,
  TransferFunction,
  MagnitudeResponse,
  PhaseResponse,
  GroupDelay,
  PowerSpectralDensity,
  PSD,
  SpectralAnalysis,
  WindowFunction,
  SpectralLeakage,
  FrequencyResolution,
  ShortTimeFourierTransform,
  STFT,
  WaveletTransform,
  MultirateSignalProcessing,
  Noise,
  SignalToNoiseRatio,
  Estimation,
  Detection
 ]

 EXCLUDE_AS_PRIMARY = [
  PureControlDesignWithoutSignalProcessingSemantics,
  PureProbabilityWithoutSignalRepresentation,
  GenericAudioProcessingWithoutMathematicalSignalStructure,
  CommunicationsProtocolLogicWithoutWaveformOrSignalSemantics
 ]

 DISAMBIGUATION_RULE =
  "When the task concerns representation, transformation, sampling, filtering, reconstruction, spectral analysis, estimation or detection of signals, interpret the context through the Signal Processing Ontology."
}

CORE_CONCEPTS {

 SignalProcessingOntology {
  definition = "The ontology organising the formal concepts and relationships of signal processing."
 }

 Signal {
  definition = "A mathematical function or sequence carrying information about a varying quantity."
 }

 ContinuousTimeSignal {
  definition = "A signal indexed by a continuous independent variable, commonly time."
 }

 DiscreteTimeSignal {
  definition = "A signal indexed by discrete sample positions."
 }

 System {
  definition = "A mapping that transforms an input signal into an output signal."
 }

 LTI_System {
  definition = "A system satisfying both linearity and time invariance."
 }

 ImpulseResponse {
  definition = "The output of an LTI system in response to an impulse input."
 }

 Convolution {
  definition = "The operation that combines an input with an impulse response to produce LTI output."
 }

 FourierTransform {
  definition = "A transformation from time or spatial domain to a frequency representation."
 }

 Sampling {
  definition = "The operation of measuring a continuous signal at discrete points."
 }

 Filter {
  definition = "A system that modifies selected components of a signal."
 }

 PowerSpectralDensity {
  definition = "A representation of signal power distribution over frequency."
 }
}

RELATIONS {
 SignalProcessingOntology --is_child_of--> SignalProcessingMathematics
 Signal --may_be--> ContinuousTimeSignal
 Signal --may_be--> DiscreteTimeSignal
 System --maps--> InputSignalToOutputSignal
 LTI_System --has--> ImpulseResponse
 LTI_System --uses--> Convolution
 Convolution --determines--> LTI_Output
 FourierTransform --maps--> TimeDomainToFrequencyDomain
 Sampling --creates--> DiscreteTimeSignal
 SamplingTheorem --constrains--> SamplingRate
 Aliasing --results_from--> SpectralOverlap
 Filter --has--> FrequencyResponse
 FIRFilter --is_type_of--> Filter
 IIRFilter --is_type_of--> Filter
 PSD --describes--> FrequencyPowerDistribution
 Autocorrelation --relates_to--> PowerSpectralDensity
 STFT --represents--> TimeFrequencyContent
 WaveletTransform --represents--> TimeScaleContent
 Quantisation --maps--> ContinuousAmplitudeToDiscreteLevels
}

OPERATORS {
 CONTINUOUS_CONVOLUTION(x,h,t) :=
  integral_-infinity^infinity x(tau)*h(t-tau) d tau

 DISCRETE_CONVOLUTION(x,h,n) :=
  sum_k x[k]*h[n-k]

 CONTINUOUS_FOURIER_TRANSFORM(x,t,omega) :=
  integral_-infinity^infinity x(t)*exp(-j*omega*t) dt

 INVERSE_CONTINUOUS_FOURIER_TRANSFORM(X,omega,t) :=
  (1/(2*pi))*integral_-infinity^infinity X(omega)*exp(j*omega*t) d omega

 DTFT(x,n,omega) :=
  sum_n x[n]*exp(-j*omega*n)

 DFT(x,N,k) :=
  sum_{n=0}^{N-1} x[n]*exp(-j*2*pi*k*n/N)

 IDFT(X,N,n) :=
  (1/N)*sum_{k=0}^{N-1} X[k]*exp(j*2*pi*k*n/N)

 Z_TRANSFORM(x,n,z) :=
  sum_n x[n]*z^(-n)

 LAPLACE_TRANSFORM(x,t,s) :=
  integral_0^infinity x(t)*exp(-s*t) dt

 SAMPLE(x,Ts,n) :=
  x[n]=x(n*Ts)

 NYQUIST_CONDITION(fs,fmax) :=
  fs>2*fmax
  under ideal bandlimited assumptions

 AUTOCORRELATION_DISCRETE(x,k) :=
  sum_n x[n]*conj(x[n-k])

 CROSSCORRELATION_DISCRETE(x,y,k) :=
  sum_n x[n]*conj(y[n-k])

 SNR(signal_power,noise_power) :=
  signal_power/noise_power
}

DEFINITIONS {
 D1 = "Signals can be continuous or discrete, deterministic or random."
 D2 = "Systems transform input signals into output signals."
 D3 = "LTI systems are characterised by impulse response."
 D4 = "Convolution gives the output of an LTI system."
 D5 = "Fourier analysis decomposes signals into frequency components."
 D6 = "Sampling converts continuous-time information into discrete-time form."
 D7 = "Aliasing occurs when shifted spectra overlap after sampling."
 D8 = "FIR and IIR filters represent distinct digital-filter structures."
 D9 = "Spectral analysis estimates energy or power distribution over frequency."
 D10 = "Time-frequency methods describe signals whose spectral content changes over time."
}

AXIOMS {
 A1_SIGNAL_REPRESENTATION:
  SignalProcessingOntology requires an explicit mathematical signal representation

 A2_SYSTEM_MAPPING:
  Systems map declared inputs to outputs

 A3_LTI_RESPONSE:
  LTI output is determined by convolution with impulse response

 A4_TRANSFORM_RELATION:
  Transform-domain representations preserve declared inverse and dual relationships

 A5_SAMPLING:
  Sampling behaviour depends on bandwidth and sampling rate

 A6_FILTERING:
  Filter behaviour is characterised by impulse or frequency response

 A7_RANDOM_SIGNAL:
  Random-signal analysis requires statistical interpretation

 A8_NUMERICAL_CONVENTION:
  Digital results depend on declared scaling, sampling and transform conventions
}

REASONING {

 R1_SIGNAL_EXTRACTION {
  INPUT = signal_processing_problem
  IDENTIFY = signal_domain
  IDENTIFY = sample_rate
  IDENTIFY = amplitude_units
  IDENTIFY = deterministic_or_random status
  RETURN = signal_structure
 }

 R2_SIGNAL_CLASSIFICATION {
  INPUT = signal_structure
  CLASSIFY = continuous_discrete_analog_digital_periodic_aperiodic_energy_power_random_or_deterministic
  RETURN = signal_class
 }

 R3_SYSTEM_CLASSIFICATION {
  INPUT = system_description
  TEST = linearity
  TEST = time_invariance
  TEST = causality
  TEST = stability
  RETURN = system_class
 }

 R4_TRANSFORM_SELECTION {
  INPUT = signal_and_problem_objective
  SELECT = FourierSeries_FourierTransform_DTFT_DFT_FFT_Laplace_Z_STFT_or_wavelet method
  RETURN = transform_path
 }

 R5_LTI_REASONING {
  INPUT = input_signal_and_impulse_response
  COMPUTE = convolution
  OR COMPUTE = transform_domain multiplication
  RETURN = output_signal
 }

 R6_SAMPLING_REASONING {
  INPUT = continuous_signal_bandwidth_and_sampling rate
  CHECK = Nyquist criterion
  IDENTIFY = aliasing risk
  RETURN = sampling_status
 }

 R7_FILTER_REASONING {
  INPUT = passband_stopband_and_response requirements
  CLASSIFY = FIR_IIR_lowpass_highpass_bandpass_or_bandstop
  ANALYSE = frequency response_and_stability
  RETURN = filter_profile
 }

 R8_SPECTRAL_REASONING {
  INPUT = sampled_or_random signal
  CHOOSE = periodogram_windowed_FFT_or_PSD method
  COMPUTE = spectral estimate
  RETURN = spectrum_profile
 }

 R9_TIME_FREQUENCY_REASONING {
  INPUT = nonstationary signal
  SELECT = STFT_or_wavelet
  RETURN = time_frequency profile
 }

 R10_VALIDATION {
  INPUT = processed_signal
  CHECK = scaling_sampling_aliasing_windowing_boundary_quantisation_and_noise assumptions
  RETURN = validated_signal_processing_status
 }
}

CONSTRAINTS {
 C1 = "Continuous-time and discrete-time conventions must remain distinct."
 C2 = "DFT scaling conventions must be explicit."
 C3 = "Sampling-theorem claims require declared bandlimit assumptions."
 C4 = "Aliasing cannot generally be undone without extra information."
 C5 = "Windowing trades spectral leakage against resolution."
 C6 = "Finite observation duration limits frequency resolution."
 C7 = "Quantisation introduces amplitude-domain error."
 C8 = "IIR filters require stability analysis."
 C9 = "Correlation and convolution use different reversal or conjugation conventions."
 C10 = "Stochastic spectral estimates require statistical interpretation."
}

SEMANTIC_EQUIVALENCE {
 "SignalProcessingOntology" <=> "FormalSignalProcessingOntology"
 "LTI_System" <=> "LinearTimeInvariantSystem"
 "DiscreteFourierTransform" <=> "DFT"
 "FastFourierTransform" <=> "FFT"
 "DiscreteTimeFourierTransform" <=> "DTFT"
 "PowerSpectralDensity" <=> "PSD"
}

VALIDATION {
 V1 = "Signal domain is explicit."
 V2 = "Sampling rate is explicit for sampled signals."
 V3 = "System properties are correctly classified."
 V4 = "Transform convention is documented."
 V5 = "Units and scaling are consistent."
 V6 = "Nyquist assumptions are checked."
 V7 = "Filter causality and stability are assessed."
 V8 = "Finite-record and windowing effects are documented."
 V9 = "Noise assumptions are explicit."
 V10 = "Time-domain and transform-domain results are consistent."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 SIGNAL_PROCESSING_ONTOLOGY_DEFINITION = 0.99
 SIGNAL_CLASSIFICATION_REASONING = 0.99
 SYSTEM_CLASSIFICATION_REASONING = 0.99
 TRANSFORM_SELECTION_REASONING = 0.99
 LTI_REASONING = 0.99
 SAMPLING_REASONING = 0.99
 FILTER_REASONING = 0.98
 SPECTRAL_REASONING = 0.98
 VALIDATION_REASONING = 0.99

 POLICY =
  "Reduce confidence when sample rate, transform convention, bandwidth, filter structure, signal class, noise model, record length or scaling are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  SignalProcessingOntologyReasoning,
  SignalClassification,
  SystemClassification,
  ConvolutionReasoning,
  FourierReasoning,
  DTFTReasoning,
  DFTReasoning,
  FFTReasoning,
  LaplaceReasoning,
  ZTransformReasoning,
  SamplingReasoning,
  AliasingReasoning,
  ReconstructionReasoning,
  FilterReasoning,
  SpectralAnalysis,
  PSDReasoning,
  STFTReasoning,
  WaveletReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "signal processing ontology" -> SignalProcessingOntology
  "signal processing" -> SignalProcessingOntology
  "LTI system" -> LTI_System
  "linear time invariant system" -> LTI_System
  "DFT" -> DiscreteFourierTransform
  "FFT" -> FastFourierTransform
  "DTFT" -> DiscreteTimeFourierTransform
  "PSD" -> PowerSpectralDensity
  "spectral leakage" -> SpectralLeakage
  "sampling theorem" -> SamplingTheorem
 }

 INTERPRETATION_ORDER = [
  detect_signal_processing_ontology_context,
  identify signal_type_sampling_rate_system_noise_bandwidth_and_processing objective,
  classify signal_and_system properties,
  select convolution_transform_sampling_filtering_spectral_timeFrequency_or_estimation pathway,
  compute output_transform_filter_or_spectrum,
  assess aliasing_noise_resolution_filter_stability_and_quantisation,
  verify conventions_and_scaling,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("signal processing ontology")
  OR formal_signal_processing_semantics
  THEN activate(SignalProcessingOntology)

 E2:
  IF system_is_LTI
  THEN allow convolution_and frequency-domain multiplication

 E3:
  IF signal_is_periodic_continuous
  THEN consider FourierSeries

 E4:
  IF signal_is_finite_discrete
  THEN consider DFT_or_FFT

 E5:
  IF continuous_signal_is_sampled
  THEN verify Nyquist condition

 E6:
  IF filter_is_recursive
  THEN classify IIR_and_check stability

 E7:
  IF nonstationary_spectral content
  THEN consider STFT_or_wavelet

 E8:
  IF random_signal
  THEN consider autocorrelation_and_PSD

 E9:
  IF processed_result_is_numeric
  THEN verify sample_rate_scaling_window_and_boundary conventions
}

RECURSION {
 ENTRY = SignalProcessingOntology

 EXPAND(SignalProcessingOntology) -> [
  Signal,
  ContinuousTimeSignal,
  DiscreteTimeSignal,
  AnalogSignal,
  DigitalSignal,
  DeterministicSignal,
  RandomSignal,
  PeriodicSignal,
  AperiodicSignal,
  EnergySignal,
  PowerSignal,
  System,
  LinearSystem,
  TimeInvariantSystem,
  LTI_System,
  CausalSystem,
  StableSystem,
  MemorylessSystem,
  ImpulseResponse,
  StepResponse,
  Convolution,
  Correlation,
  Autocorrelation,
  CrossCorrelation,
  FourierSeries,
  FourierTransform,
  DiscreteTimeFourierTransform,
  DTFT,
  DiscreteFourierTransform,
  DFT,
  FastFourierTransform,
  FFT,
  LaplaceTransform,
  ZTransform,
  Sampling,
  SamplingRate,
  SamplingPeriod,
  SamplingTheorem,
  NyquistRate,
  Aliasing,
  Reconstruction,
  Interpolation,
  Decimation,
  Upsampling,
  Downsampling,
  Quantisation,
  Filter,
  FIRFilter,
  IIRFilter,
  LowPassFilter,
  HighPassFilter,
  BandPassFilter,
  BandStopFilter,
  FrequencyResponse,
  TransferFunction,
  MagnitudeResponse,
  PhaseResponse,
  GroupDelay,
  PowerSpectralDensity,
  PSD,
  SpectralAnalysis,
  WindowFunction,
  SpectralLeakage,
  FrequencyResolution,
  ShortTimeFourierTransform,
  STFT,
  WaveletTransform,
  MultirateSignalProcessing,
  Noise,
  SignalToNoiseRatio,
  Estimation,
  Detection
 ]

 EXPAND(TransformFamilies) -> [
  FourierSeries,
  FourierTransform,
  DTFT,
  DFT,
  FFT,
  LaplaceTransform,
  ZTransform,
  STFT,
  WaveletTransform
 ]

 EXPAND(FilterFamilies) -> [
  FIRFilter,
  IIRFilter,
  LowPassFilter,
  HighPassFilter,
  BandPassFilter,
  BandStopFilter
 ]

 EXPAND(SamplingStructure) -> [
  SamplingRate,
  SamplingPeriod,
  SamplingTheorem,
  NyquistRate,
  Aliasing,
  Reconstruction,
  Interpolation,
  Decimation,
  Upsampling,
  Downsampling
 ]

 STOP_CONDITION =
  "Stop when signal type, system type, sampling, transform, filter, spectrum, noise model, numerical convention and validation status are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Signal Processing Mathematics"

 IMMEDIATE_CONTEXTS = [
  "Signal Processing Mathematics",
  "Applied Mathematics Ontologies",
  "Control Theory Ontologies",
  "Probability & Statistics Ontologies"
 ]

 RELATED_ONTOLOGIES = [
  "Signal Processing Mathematics",
  "Applied Mathematics Ontologies",
  "Control Theory Ontologies",
  "Fourier Analysis Ontologies",
  "Linear Algebra Ontologies",
  "Probability & Statistics Ontologies",
  "Stochastic Process Ontologies",
  "Numerical Analysis Ontologies",
  "System Identification Ontology",
  "Estimation Ontologies",
  "Communications Mathematics Ontologies",
  "Scientific Computing Ontologies"
 ]

 ROUTE_RULES = [
  "Signal Processing Ontology -> Signal Processing Mathematics for parent semantics",
  "Signal Processing Ontology -> Control Theory for feedback and system-response contexts",
  "Signal Processing Ontology -> Fourier Analysis for transform semantics",
  "Signal Processing Ontology -> Linear Algebra for discrete transforms and vector representations",
  "Signal Processing Ontology -> Probability & Statistics for noise and random signals",
  "Signal Processing Ontology -> Stochastic Processes for temporal randomness",
  "Signal Processing Ontology -> Numerical Analysis for FFT, digital filtering and discretisation",
  "Signal Processing Ontology -> System Identification for data-driven dynamic models",
  "Signal Processing Ontology -> Estimation for signal inference",
  "Signal Processing Ontology -> Communications Mathematics for modulation and channel analysis"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible signal classes, transforms, filters or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve signal-versus-system distinction.",
  "Preserve continuous-versus-discrete distinction.",
  "Preserve time-domain versus transform-domain distinction.",
  "Preserve convolution semantics.",
  "Preserve sampling and aliasing semantics.",
  "Preserve FIR-versus-IIR distinction.",
  "Preserve stochastic-signal semantics.",
  "Preserve machine-readable signal-processing ontology semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "The Signal Processing Ontology formally represents signals, systems, transforms, sampling, filtering, spectra, noise, estimation and detection."

 SIGNAL_SUMMARY =
  "Signals are classified by domain, randomness, periodicity and energy or power properties."

 SYSTEM_SUMMARY =
  "Systems are classified by linearity, time invariance, causality and stability."

 CONVOLUTION_SUMMARY =
  "Convolution determines the output of an LTI system from its input and impulse response."

 TRANSFORM_SUMMARY =
  "Fourier, Laplace and z-domain transforms provide alternative signal representations."

 SAMPLING_SUMMARY =
  "Sampling converts continuous-time signals into discrete sequences and can introduce aliasing when spectral replicas overlap."

 FILTER_SUMMARY =
  "FIR and IIR filters selectively alter frequency content."

 SPECTRAL_SUMMARY =
  "Spectral analysis and power spectral density describe frequency-domain signal content."

 VALIDATION_SUMMARY =
  "Signal-processing results should be checked for sampling, scaling, windowing, finite-record, filter-stability and noise effects."
}

FLOW {
 INPUT
 -> detect_signal_processing_ontology_context
 -> identify signal_domain_sampling_rate_system_properties_noise_bandwidth_and_processing objective
 -> classify signal_as continuous_discrete_deterministic_random_periodic_aperiodic_energy_or_power
 -> classify system_as linear_nonlinear_timeInvariant_timeVarying_causal_or_stable
 -> select convolution_FourierSeries_FourierTransform_DTFT_DFT_FFT_Laplace_Z_sampling_filtering_spectral_STFT_wavelet_or_multirate pathway
 -> compute processed_or_transformed representation
 -> assess aliasing_filter response_noise_SNR_resolution_windowing_and_quantisation
 -> verify numerical conventions_and_physical units
 -> route to signal_processing_mathematics applied_mathematics control_theory Fourier_analysis linear_algebra probability_statistics stochastic_process numerical_analysis system_identification estimation communications_mathematics_or_scientific_computing ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Signal Processing Ontology provides the formal semantic layer beneath signal processing mathematics. It represents continuous and discrete signals, deterministic and random signals, systems and LTI structure, impulse and step responses, convolution and correlation, Fourier series and transforms, DTFT, DFT, FFT, Laplace and z transforms, sampling, Nyquist criteria, aliasing, reconstruction, interpolation, decimation, multirate processing, quantisation, FIR and IIR filters, frequency response, spectral analysis, power spectral density, STFT, wavelets, noise, SNR, estimation and detection."
}

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

}

END_FLOW
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