Digital Signal Processing Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Digital Signal Processing Ontology within the Kireth Research Semantic Library. Digital signal processing studies the representation, transformation, filtering, analysis and reconstruction of discrete-time signals using numerical algorithms. This ontology supports sampling, quantisation, discrete-time systems, FIR and IIR filters, convolution, correlation, DFT and FFT methods, z-transform reasoning, multirate processing, spectral analysis, windowing, digital filter design, finite word length, fixed-point and floating-point implementation, real-time processing, stability, causality and relationships with signal processing mathematics, Fourier analysis, z-transforms, numerical analysis and control systems.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = DigitalSignalProcessing
 SYMBOLIC_FORMS = [
  "y[n]=sum_k h[k]x[n-k]",
  "X[k]=sum_{n=0}^{N-1}x[n]e^{-j2pi kn/N}",
  "H(z)=Y(z)/X(z)"
 ]
 ENTITY_CLASS = DiscreteSignalAlgorithmicProcessingFramework
 PRIMARY_CONTEXT = SignalProcessingMathematics
 DEFINITION = "Digital signal processing is the mathematical and algorithmic processing of discrete-time and quantised signals using numerical methods."
 SEMANTIC_ROLE = "Provides the machine-readable framework for sampling, quantisation, discrete-time systems, digital filtering, transforms, spectral analysis, multirate processing, numerical implementation and real-time DSP."
}

SCOPE {
 INCLUDE = [
  DigitalSignalProcessing,
  DSP,
  DiscreteTimeSignal,
  DigitalSignal,
  Sample,
  SamplingRate,
  SamplingPeriod,
  Quantisation,
  QuantisationLevel,
  QuantisationNoise,
  ADC,
  DAC,
  DiscreteTimeSystem,
  LinearTimeInvariantSystem,
  CausalSystem,
  StableSystem,
  ImpulseResponse,
  DifferenceEquation,
  Convolution,
  Correlation,
  Autocorrelation,
  CrossCorrelation,
  FIRFilter,
  IIRFilter,
  LowPassFilter,
  HighPassFilter,
  BandPassFilter,
  BandStopFilter,
  DigitalFilterDesign,
  WindowMethod,
  BilinearTransformMethod,
  ImpulseInvarianceMethod,
  PoleZeroDesign,
  FrequencyResponse,
  DiscreteFourierTransform,
  DFT,
  FastFourierTransform,
  FFT,
  InverseFFT,
  ZTransform,
  SpectralAnalysis,
  PowerSpectralDensity,
  Periodogram,
  WindowFunction,
  SpectralLeakage,
  FrequencyResolution,
  MultirateSignalProcessing,
  Decimation,
  Interpolation,
  Upsampling,
  Downsampling,
  PolyphaseStructure,
  FilterBank,
  FixedPointArithmetic,
  FloatingPointArithmetic,
  FiniteWordLength,
  Overflow,
  RoundoffNoise,
  LimitCycle,
  RealTimeDSP,
  BlockProcessing,
  StreamingProcessing,
  Latency,
  ComputationalComplexity
 ]

 EXCLUDE_AS_PRIMARY = [
  AnalogSignalProcessingWithoutDiscreteRepresentation,
  ContinuousFourierTransformWithoutDigitalImplementation,
  GenericNumericalComputationWithoutSignalSemantics,
  EmbeddedProgrammingWithoutDSPMathematics
 ]

 DISAMBIGUATION_RULE =
  "When sampled or quantised signals are processed through discrete-time numerical algorithms, interpret the context as digital signal processing."
}

CORE_CONCEPTS {

 DigitalSignalProcessing {
  definition = "The mathematical and numerical processing of discrete-time digital signals."
 }

 DigitalSignal {
  definition = "A discrete-time signal represented using a finite or countable amplitude representation."
 }

 Quantisation {
  definition = "The mapping of signal amplitudes to a finite set of representable levels."
 }

 DiscreteTimeSystem {
  definition = "A system mapping discrete-time input sequences to discrete-time output sequences."
 }

 FIRFilter {
  definition = "A digital filter with a finite-duration impulse response and no recursive feedback in its standard direct form."
 }

 IIRFilter {
  definition = "A digital filter with an impulse response of theoretically infinite duration, usually implemented recursively."
 }

 DifferenceEquation {
  definition = "An equation relating current and past output samples to current and past input samples."
 }

 FastFourierTransform {
  definition = "An efficient algorithm for computing the DFT."
 }

 MultirateSignalProcessing {
  definition = "Signal processing involving more than one sampling rate."
 }

 FiniteWordLength {
  definition = "The numerical limitation caused by representing values with a finite number of bits."
 }
}

RELATIONS {
 DigitalSignalProcessing --is_child_of--> SignalProcessingMathematics
 DigitalSignalProcessing --uses--> DiscreteTimeSignal
 DigitalSignalProcessing --uses--> Quantisation
 DiscreteTimeSystem --may_be--> LinearTimeInvariantSystem
 LinearTimeInvariantSystem --has--> ImpulseResponse
 ImpulseResponse --determines--> ConvolutionResponse
 DifferenceEquation --represents--> DiscreteTimeSystem
 FIRFilter --is_type_of--> DigitalFilter
 IIRFilter --is_type_of--> DigitalFilter
 DigitalFilterDesign --produces--> FIRFilter
 DigitalFilterDesign --produces--> IIRFilter
 FFT --computes--> DFT
 ZTransform --represents--> DiscreteTimeSystem
 Decimation --changes--> SamplingRate
 Interpolation --changes--> SamplingRate
 PolyphaseStructure --supports--> EfficientMultirateProcessing
 FiniteWordLength --causes--> RoundoffNoise
 FixedPointArithmetic --may_cause--> Overflow
 IIRFilter --may_exhibit--> LimitCycle
}

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

 FIR_DIFFERENCE_EQUATION(b,x,n) :=
  y[n]=sum_{k=0}^{M} b_k*x[n-k]

 IIR_DIFFERENCE_EQUATION(a,b,x,y,n) :=
  y[n]=sum_{k=0}^{M} b_k*x[n-k]-sum_{k=1}^{N} a_k*y[n-k]

 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)

 FREQUENCY_BIN(k,fs,N) :=
  f_k=k*fs/N

 QUANTISATION_STEP(xmax,xmin,L) :=
  Delta=(xmax-xmin)/L

 UNIFORM_QUANTISATION_ERROR_BOUND(Delta) :=
  abs(error)<=Delta/2
  under ideal rounding

 SNR_QUANTISATION(bits) :=
  approximately 6.02*bits+1.76 dB
  for ideal full-scale sinusoid

 DECIMATE(x,M) :=
  y[n]=x[M*n]
  after appropriate anti_alias filtering

 INTERPOLATE(x,L) :=
  upsample_by_L_then_lowpass_filter

 Z_SYSTEM_FUNCTION(B,A,z) :=
  H(z)=B(z)/A(z)

 STABILITY_TEST_IIR(poles) :=
  all abs(p_i)<1

 LINEAR_CONVOLUTION_FFT(x,h) :=
  IFFT(FFT(zero_pad(x))*FFT(zero_pad(h)))
}

DEFINITIONS {
 D1 = "DSP operates on sampled signals represented numerically."
 D2 = "Quantisation converts continuous amplitudes into finite-resolution digital values."
 D3 = "Discrete-time LTI systems are characterised by impulse response and convolution."
 D4 = "FIR filters can be implemented without recursive feedback."
 D5 = "IIR filters use recursion and require pole-based stability analysis."
 D6 = "DFT and FFT methods analyse finite-length sequences in discrete frequency bins."
 D7 = "Multirate processing uses decimation and interpolation to change sample rates."
 D8 = "Finite-word-length effects include roundoff, overflow and coefficient quantisation."
 D9 = "Digital filter performance depends on both mathematical design and numerical implementation."
 D10 = "Real-time DSP imposes latency and computational constraints."
}

AXIOMS {
 A1_DISCRETE_REPRESENTATION:
  DSP operates on discrete-time numerical sequences

 A2_SAMPLING:
  Digital signals derive from explicit or implicit sampling

 A3_QUANTISATION:
  Digital amplitude representation has finite numerical precision

 A4_LTI_CONVOLUTION:
  Discrete-time LTI output equals convolution of input with impulse response

 A5_TRANSFORM:
  DFT and FFT provide finite discrete-frequency representations

 A6_STABILITY:
  Rational causal IIR stability requires poles inside the unit circle

 A7_MULTIRATE:
  Downsampling requires alias control and upsampling requires imaging control

 A8_IMPLEMENTATION:
  Numerical architecture can alter realised DSP behaviour through finite precision
}

REASONING {

 R1_DSP_CONTEXT_EXTRACTION {
  INPUT = signal_processing_problem
  IDENTIFY = sample_rate
  IDENTIFY = signal_length
  IDENTIFY = numeric_precision
  IDENTIFY = real_time_or_offline context
  RETURN = DSP_context
 }

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

 R3_FILTER_CLASSIFICATION {
  INPUT = difference_equation_or_impulse_response
  IF no_recursive_terms
   CLASSIFY = FIR
  ELSE
   CLASSIFY = IIR
  RETURN = filter_class
 }

 R4_FILTER_DESIGN_REASONING {
  INPUT = passband_stopband_ripple_and_order requirements
  SELECT = FIR_window_or_equiripple OR IIR_prototype_transform method
  RETURN = digital_filter
 }

 R5_SPECTRAL_REASONING {
  INPUT = finite_sequence_and_sample_rate
  SELECT = DFT_or_FFT
  APPLY = window_if_required
  MAP = bins_to_frequency
  RETURN = spectral_profile
 }

 R6_MULTIRATE_REASONING {
  INPUT = rate_change factor
  IF downsampling
   APPLY = anti_alias_filter_then_decimate
  IF upsampling
   APPLY = zero_insertion_then_anti_imaging_filter
  RETURN = multirate_signal
 }

 R7_FIXED_POINT_REASONING {
  INPUT = coefficient_and_signal word lengths
  ANALYSE = range_quantisation_roundoff_overflow
  RETURN = finite_precision_profile
 }

 R8_IIR_STABILITY_REASONING {
  INPUT = denominator_coefficients
  COMPUTE = poles
  CHECK = unit_circle
  RETURN = stability_profile
 }

 R9_REAL_TIME_REASONING {
  INPUT = operations_per_sample_sample_rate_hardware limits
  COMPUTE = computational load_and_latency
  RETURN = feasibility_profile
 }

 R10_DSP_VALIDATION {
  INPUT = processed_signal_or_filter
  CHECK = sampling_aliasing_frequency response_stability_precision_latency_and_reconstruction
  RETURN = validated_DSP_result
 }
}

CONSTRAINTS {
 C1 = "Sampling rate must be explicit when physical frequency interpretation is required."
 C2 = "Quantisation model must match numerical representation."
 C3 = "FIR and IIR stability semantics differ."
 C4 = "IIR coefficient quantisation can move poles."
 C5 = "Downsampling without anti-alias filtering can destroy information."
 C6 = "Upsampling creates spectral images that generally require filtering."
 C7 = "FFT output frequency mapping depends on sample rate and transform length."
 C8 = "Fixed-point overflow and scaling must be checked."
 C9 = "Real-time feasibility depends on worst-case processing time, not average time alone."
 C10 = "Zero padding increases frequency-grid density but not intrinsic resolution."
}

SEMANTIC_EQUIVALENCE {
 "DigitalSignalProcessing" <=> "DSP"
 "DiscreteTimeSystem" <=> "DigitalSignalSystem"
 "FIRFilter" <=> "FiniteImpulseResponseFilter"
 "IIRFilter" <=> "InfiniteImpulseResponseFilter"
 "FiniteWordLength" <=> "FinitePrecisionRepresentation"
 "RealTimeDSP" <=> "RealTimeDigitalSignalProcessing"
}

VALIDATION {
 V1 = "Sampling rate and sample units are explicit."
 V2 = "Quantisation or numerical precision is documented."
 V3 = "System classification is correct."
 V4 = "Filter class is correctly identified."
 V5 = "IIR stability is checked."
 V6 = "DFT scaling and frequency bins are correct."
 V7 = "Multirate anti-alias and anti-imaging filtering are represented."
 V8 = "Finite-word-length effects are considered."
 V9 = "Real-time computational load is feasible."
 V10 = "Output is validated against expected signal or system behaviour."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 DSP_DEFINITION = 0.99
 SYSTEM_CLASSIFICATION_REASONING = 0.99
 FILTER_REASONING = 0.99
 SPECTRAL_REASONING = 0.99
 MULTIRATE_REASONING = 0.99
 FIXED_POINT_REASONING = 0.98
 STABILITY_REASONING = 0.99
 REAL_TIME_REASONING = 0.98
 VALIDATION_REASONING = 0.99

 POLICY =
  "Reduce confidence when sample rate, filter structure, numerical precision, pole locations, rate-change filters, latency requirements or transform conventions are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  DigitalSignalProcessingReasoning,
  DiscreteTimeSystemReasoning,
  FIRReasoning,
  IIRReasoning,
  DigitalFilterDesignReasoning,
  DFTReasoning,
  FFTReasoning,
  ZTransformReasoning,
  MultirateReasoning,
  QuantisationReasoning,
  FixedPointReasoning,
  SpectralAnalysis,
  RealTimeDSPReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "digital signal processing" -> DigitalSignalProcessing
  "DSP" -> DigitalSignalProcessing
  "FIR filter" -> FIRFilter
  "IIR filter" -> IIRFilter
  "fixed point" -> FixedPointArithmetic
  "floating point" -> FloatingPointArithmetic
  "decimation" -> Decimation
  "interpolation" -> Interpolation
  "real-time DSP" -> RealTimeDSP
 }

 INTERPRETATION_ORDER = [
  detect_DSP_context,
  identify sampling_rate_signal_length_numeric_precision_and_processing objective,
  classify discrete_system_and_filter structure,
  select convolution_differenceEquation_DFT_FFT_zTransform_filterDesign_multirate_or_spectral pathway,
  compute processed_signal_or_filter response,
  assess stability_aliasing_quantisation_roundoff_and_computational load,
  validate numerical implementation_and_signal behaviour,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("digital signal processing")
  OR involves sampled_signal_numerical processing
  THEN activate(DigitalSignalProcessingOntology)

 E2:
  IF recursive_output_terms_present
  THEN classify IIRFilter

 E3:
  IF no_recursive_output_terms
  THEN classify FIRFilter

 E4:
  IF spectral_analysis_requested
  THEN use DFT_or_FFT_with_correct bin mapping

 E5:
  IF downsampling
  THEN apply anti_alias filtering

 E6:
  IF upsampling
  THEN apply anti_imaging filtering

 E7:
  IF fixed_point_implementation
  THEN assess range_roundoff_overflow_and_limit cycles

 E8:
  IF IIR_filter
  THEN verify poles_inside_unit_circle_for stability

 E9:
  IF real_time_requirement
  THEN verify computational_deadline_and_latency
}

RECURSION {
 ENTRY = DigitalSignalProcessing

 EXPAND(DigitalSignalProcessing) -> [
  DiscreteTimeSignal,
  DigitalSignal,
  SamplingRate,
  Quantisation,
  QuantisationNoise,
  DiscreteTimeSystem,
  LinearTimeInvariantSystem,
  ImpulseResponse,
  DifferenceEquation,
  Convolution,
  Correlation,
  FIRFilter,
  IIRFilter,
  DigitalFilterDesign,
  FrequencyResponse,
  DFT,
  FFT,
  ZTransform,
  SpectralAnalysis,
  WindowFunction,
  SpectralLeakage,
  MultirateSignalProcessing,
  Decimation,
  Interpolation,
  PolyphaseStructure,
  FilterBank,
  FixedPointArithmetic,
  FloatingPointArithmetic,
  FiniteWordLength,
  Overflow,
  RoundoffNoise,
  LimitCycle,
  RealTimeDSP,
  Latency,
  ComputationalComplexity
 ]

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

 EXPAND(TransformMethods) -> [
  DFT,
  FFT,
  ZTransform,
  FrequencyResponse,
  SpectralAnalysis
 ]

 EXPAND(ImplementationIssues) -> [
  Quantisation,
  FiniteWordLength,
  FixedPointArithmetic,
  FloatingPointArithmetic,
  Overflow,
  RoundoffNoise,
  LimitCycle,
  Latency,
  ComputationalComplexity
 ]

 STOP_CONDITION =
  "Stop when sampling, system class, filter or transform method, multirate structure, numerical implementation, stability and validation status are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Signal Processing Mathematics"

 IMMEDIATE_CONTEXTS = [
  "Signal Processing Ontology",
  "Fourier Analysis Ontology",
  "Fast Fourier Transform Ontology",
  "Z Transform Ontology"
 ]

 RELATED_ONTOLOGIES = [
  "Signal Processing Ontology",
  "Signal Processing Mathematics",
  "Fourier Analysis Ontology",
  "Fast Fourier Transform Ontology",
  "Z Transform Ontology",
  "Wavelet Transform Ontology",
  "Digital Filter Ontologies",
  "Multirate Signal Processing Ontologies",
  "Numerical Analysis Ontologies",
  "Control Theory Ontologies",
  "System Identification Ontology",
  "Communications Mathematics Ontologies"
 ]

 ROUTE_RULES = [
  "Digital Signal Processing -> Signal Processing Ontology for core signal semantics",
  "Digital Signal Processing -> Fourier Analysis for spectral representations",
  "Digital Signal Processing -> FFT for efficient frequency-domain computation",
  "Digital Signal Processing -> Z Transform for discrete-system pole-zero analysis",
  "Digital Signal Processing -> Wavelet Transform for multiscale processing",
  "Digital Signal Processing -> Digital Filters for filter design",
  "Digital Signal Processing -> Multirate Processing for rate conversion",
  "Digital Signal Processing -> Numerical Analysis for finite-precision computation",
  "Digital Signal Processing -> Control Theory for digital controllers",
  "Digital Signal Processing -> System Identification for data-driven discrete models"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible DSP methods, architectures or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve discrete-time semantics.",
  "Preserve sampling and quantisation semantics.",
  "Preserve FIR-versus-IIR distinction.",
  "Preserve transform and frequency-bin semantics.",
  "Preserve multirate anti-aliasing semantics.",
  "Preserve finite-word-length semantics.",
  "Preserve real-time implementation semantics.",
  "Preserve machine-readable DSP semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Digital signal processing applies numerical algorithms to sampled and quantised signals for filtering, transformation, analysis and reconstruction."

 SIGNAL_SUMMARY =
  "Digital signals are discrete in time and represented with finite numerical precision."

 FILTER_SUMMARY =
  "FIR and IIR filters implement discrete-time frequency-selective systems."

 TRANSFORM_SUMMARY =
  "DFT, FFT and z-transform methods provide complementary representations of discrete-time signals and systems."

 MULTIRATE_SUMMARY =
  "Decimation and interpolation change sampling rate using anti-alias and anti-imaging filtering."

 NUMERICAL_SUMMARY =
  "Finite word length introduces quantisation, roundoff, overflow and coefficient sensitivity."

 REALTIME_SUMMARY =
  "Real-time DSP requires processing to complete within the available sample or block deadline."

 VALIDATION_SUMMARY =
  "DSP outputs should be validated for sampling, stability, spectral scaling, numerical precision and implementation constraints."
}

FLOW {
 INPUT
 -> detect_digital_signal_processing_context
 -> identify signal_sampling_rate_quantisation_signal_length_system_structure_and_processing objective
 -> classify discrete_system_as linear_nonlinear_timeInvariant_timeVarying_causal_or_stable
 -> classify filter_as FIR_or_IIR where relevant
 -> select convolution_differenceEquation_DFT_FFT_Z_transform_filterDesign_multirate_wavelet_or_spectral pathway
 -> compute processed_signal_filter_response_or_transform
 -> assess aliasing_imaging_stability_quantisation_roundoff_overflow_and_frequency resolution
 -> if_real_time evaluate computational_complexity_and_latency
 -> route to signal_processing Fourier FFT Z_transform wavelet digital_filter multirate numerical_analysis control system_identification_or_communications ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Digital Signal Processing Ontology defines mathematical and algorithmic processing of discrete-time digital signals. It supports sampling, quantisation, discrete-time LTI systems, impulse responses, difference equations, convolution and correlation, FIR and IIR filters, digital filter design, DFT and FFT methods, z-transform reasoning, spectral analysis, windowing, multirate processing, decimation, interpolation, polyphase structures, filter banks, fixed-point and floating-point arithmetic, finite-word-length effects, overflow, roundoff noise, limit cycles, real-time processing, latency and computational complexity."
}

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

}

END_FLOW