Ontological Layer: The Mathematical Model
Signal Foundations
Representation, Classification, and Transformation
The Language of Variation
Before a system can process information, that information must be modeled mathematically as a Signal. This foundation realm establishes the absolute definitions of continuity and discreteness. Here, we observe how physical phenomena translate into functions of time x(t) or integer sequences x[n]. We manipulate them through shifts, scales, and reflections, and categorize them by their energy, periodicity, and determinism—preparing the mathematical stage for the LTI systems of the subsequent realms.
The 14 Foundation Nodes
Ontological Mapping: Characteristics and Manipulations
Nature of a Signal
- Information representation
- Independent variable (Time, Space)
- Dependent variable (Amplitude)
- Mathematical function model
Continuous-Time
- x(t) notation
- Defined at every instant 't'
- Continuous independent variable
- Analog physical origins
Discrete-Time
- x[n] notation
- Defined only at integer indices 'n'
- Sequence representation
- Often derived via sampling
Analog vs Digital
- Continuous time/amplitude (Analog)
- Discrete time/quantized amp (Digital)
- The sampling & quantization bridge
- Conceptual distinction
Deterministic
- Completely specified behavior
- Predictable future values
- Explicit mathematical description
- Sinusoids, exponentials, steps
Random / Stochastic
- Unpredictable exact future realization
- Described via probabilities/statistics
- Noise, natural fluctuations
- Ensembles and random processes
Periodic & Aperiodic
- x(t) = x(t + T) for all t
- Fundamental period T
- Discrete periodicity constraints
- Transient vs repeating waveforms
Even & Odd Symmetry
- Even: x(t) = x(-t) (y-axis symmetry)
- Odd: x(t) = -x(-t) (Origin symmetry)
- Decomposition: x(t) = x_e(t) + x_o(t)
- Fourier analysis implications
Energy Signals
- Finite total energy (0 < E < ∞)
- Average power evaluates to zero
- Finite-duration pulses
- Decaying transients
Power Signals
- Finite non-zero average power
- Total energy is infinite
- Periodic signals (Sinusoids)
- Constant DC signals
Elementary Signals
- Unit Step u(t)
- Unit Impulse δ(t) (Dirac/Kronecker)
- Ramp r(t)
- Building blocks for complex waveforms
Time Shifting
- Delay: x(t - t₀) shifts right
- Advance: x(t + t₀) shifts left
- Discrete equivalent x[n - k]
- Rigid translation along time axis
Time Scaling & Reversal
- Compression: x(at) where a > 1
- Expansion: x(at) where a < 1
- Time Reversal: x(-t) reflects on y-axis
- Order of operations: Shift then Scale
01 — The Signal Laboratory
Signals carry the universe's information. By mastering how to mathematically define, categorize, and warp these waveforms, you lay the absolute foundation. A signal by itself is just data—its true nature is revealed only when it is passed through a System. This prepares you directly for Realm 02: Systems & Convolution.