Signal Processing Mathematics

AI INTERPRETATION NOTE
This Flow Algebra defines the Signal Processing Mathematics layer within the Kireth Research Semantic Library. Signal processing mathematics studies the representation, transformation, analysis, filtering, reconstruction and interpretation of signals in continuous and discrete domains. This parent ontology connects time-domain analysis, frequency-domain analysis, Fourier methods, Laplace and z transforms, sampling theory, convolution, filtering, spectral analysis, stochastic signals, digital signal processing, estimation, wavelets, multirate systems and relationships with control theory, probability, linear algebra, numerical analysis and communications mathematics.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = SignalProcessingMathematics
 SYMBOLIC_FORMS = [
  "y(t)=x(t)*h(t)",
  "Y(omega)=X(omega)H(omega)",
  "x[n] <-> X(e^{j omega})"
 ]
 ENTITY_CLASS = SignalRepresentationAndTransformationFramework
 PRIMARY_CONTEXT = AppliedMathematics
 DEFINITION = "Signal processing mathematics is the quantitative study of signals and systems through transformations, filtering, sampling, spectral analysis, estimation and reconstruction."
 SEMANTIC_ROLE = "Provides the parent semantic framework for continuous and discrete signals, convolution, Fourier analysis, transforms, filters, sampling, spectra, stochastic signals and digital processing."
}

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

 EXCLUDE_AS_PRIMARY = [
  PureControlDesignWithoutSignalAnalysis,
  PureProbabilityWithoutSignalContext,
  AudioEngineeringWithoutMathematicalSignalSemantics,
  CommunicationsProtocolDesignWithoutSignalRepresentation
 ]

 DISAMBIGUATION_RULE =
  "When mathematical operations are used to represent, transform, filter, sample, reconstruct or estimate signals, interpret the context as signal processing mathematics."
}

CORE_CONCEPTS {

 SignalProcessingMathematics {
  definition = "The mathematical framework for analysing, transforming and processing signals."
 }

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

 ContinuousTimeSignal {
  definition = "A signal defined for a continuous time variable."
 }

 DiscreteTimeSignal {
  definition = "A signal defined on discrete sample indices."
 }

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

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

 Convolution {
  definition = "The operation combining an input signal with an impulse response to produce LTI system output."
 }

 FourierTransform {
  definition = "A transform mapping a signal between time and frequency representations."
 }

 Sampling {
  definition = "The conversion of a continuous-time signal into a discrete sequence."
 }

 Filter {
  definition = "A system designed to alter selected signal components while preserving or suppressing others."
 }

 PowerSpectralDensity {
  definition = "A description of signal or random-process power distribution over frequency."
 }

 SignalToNoiseRatio {
  definition = "The ratio of signal power to noise power."
 }
}

RELATIONS {
 SignalProcessingMathematics --is_branch_of--> AppliedMathematics
 Signal --may_be--> ContinuousTimeSignal
 Signal --may_be--> DiscreteTimeSignal
 LTI_System --has--> ImpulseResponse
 LTI_System --uses--> Convolution
 Convolution --maps--> InputSignalToOutputSignal
 FourierTransform --maps--> TimeDomainToFrequencyDomain
 Sampling --maps--> ContinuousTimeSignalToDiscreteTimeSignal
 SamplingTheorem --constrains--> SamplingRate
 Aliasing --results_from--> InsufficientSampling
 Filter --has--> FrequencyResponse
 FIRFilter --is_type_of--> Filter
 IIRFilter --is_type_of--> Filter
 PowerSpectralDensity --describes--> FrequencyPowerDistribution
 CrossCorrelation --measures--> SignalSimilarity
 WaveletTransform --represents--> TimeScaleContent
 STFT --represents--> TimeFrequencyContent
 Quantisation --maps--> AmplitudeToDiscreteLevels
}

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

 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)

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

 AUTOCORRELATION(x,tau) :=
  integral x(t)*conj(x(t-tau)) dt

 CROSS_CORRELATION(x,y,tau) :=
  integral x(t)*conj(y(t-tau)) dt

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

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

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

DEFINITIONS {
 D1 = "Signals may be continuous, discrete, deterministic or stochastic."
 D2 = "LTI systems are completely characterised by their impulse responses."
 D3 = "Convolution describes LTI system input-output behaviour."
 D4 = "Fourier methods decompose signals into frequency components."
 D5 = "Sampling converts continuous-time signals into discrete sequences."
 D6 = "Aliasing occurs when spectral components overlap after insufficient sampling."
 D7 = "Filters shape signal spectra."
 D8 = "FIR and IIR filters differ in impulse-response duration and recursion structure."
 D9 = "Spectral analysis describes signal energy or power across frequency."
 D10 = "Time-frequency methods represent nonstationary spectral content."
}

AXIOMS {
 A1_SIGNAL_DOMAIN:
  SignalProcessingMathematics operates on explicitly represented signals

 A2_SYSTEM_MAPPING:
  Systems map input signals to output signals

 A3_LTI_CONVOLUTION:
  LTI system output equals convolution of input with impulse response

 A4_TRANSFORM_DUALITY:
  Signal transformations preserve mathematically defined relationships between domains

 A5_SAMPLING:
  Reconstruction guarantees depend on signal bandwidth and sampling rate

 A6_FILTERING:
  Filtering modifies signal content according to a system frequency response

 A7_NOISE:
  Signal interpretation may depend on explicit noise models

 A8_VALIDATION:
  Numerical signal-processing results require consistent sampling, scaling and transform conventions
}

REASONING {

 R1_SIGNAL_CLASSIFICATION {
  INPUT = signal_description
  CLASSIFY = continuous_or_discrete
  CLASSIFY = deterministic_or_random
  CLASSIFY = periodic_or_aperiodic
  CLASSIFY = energy_or_power
  RETURN = signal_class
 }

 R2_SYSTEM_CLASSIFICATION {
  INPUT = system_description
  TEST = linearity
  TEST = time_invariance
  TEST = causality
  TEST = stability
  RETURN = system_class
 }

 R3_LTI_ANALYSIS {
  INPUT = input_and_impulse_response
  COMPUTE = convolution
  OR transform_to_frequency_domain_and_multiply
  RETURN = output_signal
 }

 R4_FOURIER_ANALYSIS {
  INPUT = signal
  SELECT = FourierSeries_FourierTransform_DTFT_DFT_or_FFT
  COMPUTE = spectral_representation
  RETURN = frequency_profile
 }

 R5_SAMPLING_REASONING {
  INPUT = continuous_signal_bandwidth_and_sampling_rate
  CHECK = Nyquist condition
  IF insufficient
   IDENTIFY = aliasing risk
  RETURN = sampling_profile
 }

 R6_FILTER_REASONING {
  INPUT = desired_passband_stopband_and_signal characteristics
  SELECT = FIR_IIR_lowpass_highpass_bandpass_or_bandstop structure
  RETURN = filter_model
 }

 R7_SPECTRAL_ANALYSIS {
  INPUT = finite_or_random signal
  SELECT = periodogram_windowed_spectrum_or_PSD estimator
  RETURN = spectral_density profile
 }

 R8_TIME_FREQUENCY_REASONING {
  INPUT = nonstationary signal
  SELECT = STFT_or_wavelet transform
  RETURN = time_frequency representation
 }

 R9_NOISE_AND_ESTIMATION {
  INPUT = signal_plus_noise
  ESTIMATE = signal_parameters_or_state
  COMPUTE = SNR_and_uncertainty
  RETURN = estimation_profile
 }

 R10_NUMERICAL_VALIDATION {
  INPUT = processed_signal
  CHECK = sampling_rate_transform_scaling_windowing_quantisation_and_boundary effects
  RETURN = validated_signal_result
 }
}

CONSTRAINTS {
 C1 = "Continuous and discrete transform conventions must not be mixed."
 C2 = "DFT frequency bins depend on sample rate and transform length."
 C3 = "Sampling theorems require declared bandlimit assumptions."
 C4 = "Aliased information cannot generally be uniquely reconstructed."
 C5 = "Windowing changes spectral leakage and resolution."
 C6 = "Finite record length limits frequency resolution."
 C7 = "Quantisation introduces amplitude error."
 C8 = "IIR stability depends on pole locations."
 C9 = "Correlation and convolution differ in reversal and conjugation conventions."
 C10 = "Spectral estimates require careful normalisation and statistical interpretation."
}

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

VALIDATION {
 V1 = "Signal domain is identified."
 V2 = "Sampling rate is explicit for discrete sampled signals."
 V3 = "Transform convention is explicit."
 V4 = "Units and scaling are consistent."
 V5 = "LTI assumptions are verified before convolution-based simplification."
 V6 = "Nyquist assumptions are checked."
 V7 = "Filter stability and causality are evaluated."
 V8 = "Windowing and finite-record effects are documented."
 V9 = "Noise model is explicit where relevant."
 V10 = "Time-domain and frequency-domain results are mutually consistent."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 SIGNAL_PROCESSING_DEFINITION = 0.99
 SIGNAL_CLASSIFICATION_REASONING = 0.99
 LTI_REASONING = 0.99
 FOURIER_REASONING = 0.99
 SAMPLING_REASONING = 0.99
 FILTER_REASONING = 0.98
 SPECTRAL_REASONING = 0.98
 TIME_FREQUENCY_REASONING = 0.98
 NUMERICAL_VALIDATION_REASONING = 0.99

 POLICY =
  "Reduce confidence when sampling rate, transform convention, bandwidth, noise model, filter structure, finite-record length or scaling are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  SignalProcessingReasoning,
  SignalClassification,
  LTIReasoning,
  ConvolutionReasoning,
  FourierReasoning,
  DFTReasoning,
  FFTReasoning,
  LaplaceTransformReasoning,
  ZTransformReasoning,
  SamplingReasoning,
  AliasingReasoning,
  FilterReasoning,
  SpectralAnalysis,
  PSDReasoning,
  TimeFrequencyReasoning,
  NoiseReasoning,
  EstimationReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "signal processing mathematics" -> SignalProcessingMathematics
  "signal processing" -> SignalProcessingMathematics
  "linear time-invariant system" -> LTI_System
  "LTI system" -> LTI_System
  "discrete Fourier transform" -> DiscreteFourierTransform
  "DFT" -> DiscreteFourierTransform
  "fast Fourier transform" -> FastFourierTransform
  "FFT" -> FastFourierTransform
  "power spectral density" -> PowerSpectralDensity
  "PSD" -> PowerSpectralDensity
 }

 INTERPRETATION_ORDER = [
  detect_signal_processing_context,
  identify signal_domain_sampling_system_noise_and_objective,
  classify signal_and_system properties,
  select convolution_Fourier_DFT_FFT_Laplace_Z_sampling_filtering_spectral_or_timeFrequency pathway,
  compute transformed_filtered_or_estimated representation,
  assess aliasing_noise_resolution_and_stability,
  validate conventions_and_scaling,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("signal processing")
  OR asks_to_transform_filter_sample_or_analyse signals
  THEN activate(SignalProcessingMathematicsOntology)

 E2:
  IF LTI_system_and_impulse_response_known
  THEN use convolution_or_transform multiplication

 E3:
  IF periodic_continuous_signal
  THEN consider FourierSeries

 E4:
  IF finite_discrete_signal
  THEN consider DFT_or_FFT

 E5:
  IF sampled_continuous_signal
  THEN verify Nyquist condition

 E6:
  IF filter_design_or_analysis
  THEN classify FIR_or_IIR_and_frequency response

 E7:
  IF nonstationary_frequency_content
  THEN consider STFT_or_wavelet transform

 E8:
  IF stochastic_signal
  THEN consider autocorrelation_and_PSD

 E9:
  IF numerical_result produced
  THEN verify scaling_sampling_and_boundary conventions
}

RECURSION {
 ENTRY = SignalProcessingMathematics

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

 EXPAND(TransformMethods) -> [
  FourierSeries,
  FourierTransform,
  DiscreteTimeFourierTransform,
  DiscreteFourierTransform,
  FastFourierTransform,
  LaplaceTransform,
  ZTransform,
  WaveletTransform,
  ShortTimeFourierTransform
 ]

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

 EXPAND(SamplingMethods) -> [
  Sampling,
  Reconstruction,
  Interpolation,
  Decimation,
  Upsampling,
  Downsampling,
  MultirateSignalProcessing
 ]

 STOP_CONDITION =
  "Stop when the signal class, system class, sampling, transform, filter, spectral representation, noise model and numerical conventions are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Applied Mathematics Ontologies"

 IMMEDIATE_CONTEXTS = [
  "Applied Mathematics Ontologies",
  "Control Theory",
  "Probability & Statistics Ontologies",
  "Numerical Analysis Ontologies"
 ]

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

 ROUTE_RULES = [
  "Signal Processing Mathematics -> Applied Mathematics Ontologies for parent semantics",
  "Signal Processing Mathematics -> Control Theory for feedback and system-response contexts",
  "Signal Processing Mathematics -> Fourier Analysis for transform theory",
  "Signal Processing Mathematics -> Linear Algebra for discrete transforms and state representations",
  "Signal Processing Mathematics -> Probability & Statistics for random signals and noise",
  "Signal Processing Mathematics -> Stochastic Processes for random temporal models",
  "Signal Processing Mathematics -> Numerical Analysis for FFT, filtering and discretisation",
  "Signal Processing Mathematics -> Differential Equations for continuous-system models",
  "Signal Processing Mathematics -> System Identification for data-driven model inference",
  "Signal Processing Mathematics -> Communications Mathematics for modulation and channel contexts"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

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

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

OUTPUT {
 CANONICAL_DEFINITION =
  "Signal processing mathematics provides the mathematical framework for representing, transforming, filtering, sampling, reconstructing and analysing signals."

 SIGNAL_SUMMARY =
  "Signals may be continuous or discrete, deterministic or random, periodic or aperiodic."

 SYSTEM_SUMMARY =
  "Linear time-invariant systems are characterised by impulse response, convolution and frequency response."

 FOURIER_SUMMARY =
  "Fourier methods represent signals as combinations of frequency components."

 SAMPLING_SUMMARY =
  "Sampling converts continuous-time signals to discrete sequences and requires sufficient rate to avoid aliasing under bandlimited assumptions."

 FILTER_SUMMARY =
  "Filters selectively modify signal frequency content."

 SPECTRAL_SUMMARY =
  "Spectral analysis describes energy or power distribution over frequency."

 TIME_FREQUENCY_SUMMARY =
  "STFT and wavelet methods represent signals whose spectral content changes with time."

 NUMERICAL_SUMMARY =
  "Digital signal-processing results depend on sample rate, finite record length, scaling, windowing and quantisation."
}

FLOW {
 INPUT
 -> detect_signal_processing_mathematics_context
 -> identify signal_type_time_domain_sampling_rate_system_model_noise_bandwidth_and_processing objective
 -> classify continuous_discrete_deterministic_random_periodic_aperiodic_energy_or_power signal
 -> classify system_linearity_time invariance_causality_and_stability
 -> select convolution_FourierSeries_FourierTransform_DTFT_DFT_FFT_Laplace_Z_sampling_filtering_spectral_STFT_wavelet_or_multirate pathway
 -> compute transformed_filtered_sampled_reconstructed_or_estimated signal
 -> assess aliasing_frequency response_noise_SNR_resolution_windowing_and_quantisation
 -> validate mathematical conventions_and_numerical scaling
 -> route to applied_mathematics control_theory Fourier_analysis linear_algebra probability_statistics stochastic_process numerical_analysis differential_equations 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 Mathematics layer defines mathematical analysis and transformation of signals. It supports continuous and discrete signals, deterministic and random signals, LTI systems, 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, FIR and IIR filters, frequency response, power spectral density, spectral analysis, STFT, wavelets, quantisation, noise, signal-to-noise ratio, estimation and detection."
}

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

}

END_FLOW