1 Causality

1.1 Definition of causal behavior

A system is causal when its output at any time depends only on the input values at the same time and earlier (for continuous time, “earlier” means times strictly less than the current time; for discrete time, it means current and past samples). Equivalently, the system’s response cannot use information from the input at future times. In modeling terms, this prevents the output from “anticipating” future input.

1.2 Non-causality and intuitive examples

Non-causal behavior occurs when an output uses future input information. A common intuition is a predictor: if the output at time \(t\) is computed using the input at times \(>t\), then causality is violated. In signal-processing contexts, an operation like a zero-phase filter (applied by processing forward and backward through data) can be viewed as non-causal when interpreted as a real-time mapping, because it effectively uses future samples. Similarly, formula-based convolution kernels whose support extends to negative time correspond to non-causal influence.

1.3 Causal systems in time-domain models

In time-domain descriptions, causality is reflected in how the system’s impulse response or kernel is supported. For a continuous-time linear time-invariant (LTI) system, a standard characterization is that the impulse response must be zero for negative time. For discrete-time LTI systems, the impulse response must be zero for negative indices. In state-space form, causality is typically ensured when the system equations evolve forward in time without requiring future input values.

For general (possibly nonlinear or time-varying) models, causality is expressed by an input-output rule in which the output at time \(t\) is a function of input history up to \(t\). Any dependence on an input segment beyond \(t\) implies non-causal structure.

1.4 Causality checks in common representations

1.4.1 Transfer function causality criteria

For LTI systems expressed by a transfer function, causality is tied to the analytic structure of the corresponding function in the complex frequency domain. Roughly, the mapping from Laplace variable to time response must correspond to a realizable time-domain response; in continuous-time, this is often connected to whether the transfer function is consistent with a causal impulse response. A common practical approach is to verify that a rational transfer function can be realized by a proper state-space model with no need to “precompute” future input.

While exact conditions vary with convention (e.g., whether one uses Laplace transforms with zero initial conditions), the main idea remains: the transfer function must correspond to an impulse response that does not extend to negative time.

1.4.2 Impulse response support tests

A direct and widely used check is to examine whether the impulse response has support only for allowable times. For continuous-time systems, one tests whether \(h(t)=0\) for \(t<0\). For discrete-time systems, one checks whether \(h[k]=0\) for \(k<0\). If the impulse response exhibits nonzero values at forbidden times, the model cannot be implemented as a causal real-time filter under the stated time indexing.

This method is often straightforward when the impulse response can be derived analytically, or when it is computed numerically and inspected for nonzero pre-response.

2 Properness

2.1 Definition of properness

Properness is a constraint related to the “instantaneous” dependence between input and output implied by a system representation, especially for rational models. In transfer-function language, properness compares the degree of the numerator polynomial to that of the denominator polynomial. Intuitively, proper models do not require unbounded or instantaneous differentiation of the input to produce the output.

In more physical terms, properness ensures that the computation implied by the transfer function does not demand an algebraic inversion of time at the same instant. This matters when implementing continuous-time models, designing controllers, or approximating behaviors with discrete-time schemes.

2.2 Strict properness vs proper (relative degree)

A transfer function is strictly proper when the numerator degree is strictly less than the denominator degree. It is proper (but not strictly proper) when the numerator degree is equal to the denominator degree. The difference between these degrees is sometimes called the relative degree and determines whether there is a direct feedthrough term.

Strict properness corresponds to a system whose output cannot respond instantaneously to the current input; the effect arrives through dynamics. A merely proper transfer function may include a nonzero direct gain, meaning the output includes an instantaneous term proportional to the input at the same time.

2.3 Implications of improper or non-proper models

Improper (numerator degree greater than denominator degree) models typically imply that the output depends on derivatives of the input of higher order than the model’s dynamics would physically support. In time-domain terms, this can manifest as impulse responses containing distributions such as derivatives of delta functions or other singular behavior at the origin. For practical implementation, such models can be problematic or undefined without additional regularization, sampling assumptions, or reformulation.

In control design and simulation, improper models may produce numerical instability, require unrealistic high-frequency behavior, or yield artifacts when discretized. Even when a theoretical derivation produces an improper transfer function, engineers often reformulate it to obtain a realizable and well-posed model.

2.4 Properness checks in mathematical forms

2.4.1 Rational transfer functions and degree tests

For rational transfer functions \(G(s)=\frac{N(s)}{D(s)}\), properness can be checked by polynomial degrees:

  • Strictly proper: \(\deg N < \deg D\)
  • Proper: \(\deg N = \deg D\)
  • Improper: \(\deg N > \deg D\)

When the transfer function is expressed in factored form, the same degree comparison can be made by counting zeros and poles at infinity. In practice, it is also common to simplify cancellations carefully: pole-zero cancellations can change apparent degrees and therefore the properness classification if the simplification corresponds to a genuine model reduction rather than an artifact.

2.4.2 Realization consistency (state-space vs transfer function)

Properness in a transfer-function description should match what is implied by a state-space realization. In a standard state-space model, the direct feedthrough term \(D\) determines whether an instantaneous input-output path exists. If a realization produces a nonzero direct term, the associated transfer function is typically proper rather than strictly proper. If \(D=0\), the transfer function should be strictly proper, assuming the realization is minimal and the algebraic simplifications are consistent.

Checking consistency between representations helps catch modeling mistakes such as incorrect normalization, forgotten direct terms, or incorrect algebraic reduction.

3 Relationship between causality and properness

3.1 How properness supports implementability

Causality restricts the time direction of influence, while properness restricts the “rate” or differentiation order needed to compute outputs from inputs. Together they support implementability: a causal system describes a feasible information flow, and a proper transfer function avoids requiring instantaneous infinite bandwidth or derivative computations at a single time instant.

For many common engineering models (particularly rational LTI systems), causality and properness are intertwined: causal realizations often correspond to transfer functions that are proper, because realizable dynamics do not generate singular derivative requirements without corresponding state dynamics.

3.2 Causality requirements for realization

When constructing a realization (for example, from a transfer function), causality is ensured by selecting a structure that evolves state variables forward using current and past inputs only. This may require choosing a state-space form that does not introduce algebraic dependence on future values. In linear time-invariant settings, canonical realizations and minimal realizations generally support causal behavior when the associated impulse response is causal.

If a transfer function suggests non-causal impulse behavior, any realization derived from it will typically inherit that non-causality, unless one modifies the model to represent a different time indexing or filtering convention.

3.3 Edge cases: algebraic loops and direct feedthrough

3.3.1 Direct term (D) and instantaneous effects

A direct feedthrough term \(D\) creates an immediate mapping from the current input to the current output. This is compatible with causality and usually aligns with a transfer function that is proper but not strictly proper. However, direct feedthrough can create algebraic loops when coupled with other systems or when the input itself is generated through feedback paths that depend instantaneously on outputs.

Algebraic loops are not inherently “non-causal,” but they can make the system equations ill-posed unless the loop gain structure is handled carefully (e.g., by requiring that the algebraic equations at each time step have a unique solution). In discrete-time implementations, these loops often appear as implicit update rules that require solving a small system at each step.

4 Implementation and design considerations

4.1 Realizability in discretized systems

Discretization transforms time dynamics into sampled-time dynamics. Causality is typically preserved by using convolution sums and state updates that reference only current and past samples. Properness affects how the discretized model represents high-frequency components: an improper continuous-time transfer function may yield discrete-time behavior that effectively demands nonlocal dependence or unstable difference operators.

In digital filter design, one often starts from a realizable analog prototype (proper and causal) and then applies discretization methods (such as bilinear transformation or impulse-invariant mappings) that preserve the intended time-direction behavior.

4.2 Properness in digital filter design

In discrete-time signal processing, properness is closely related to whether the filter has a direct term and whether the difference equation requires future samples. A realizable causal IIR filter uses past outputs and inputs only, with coefficients that avoid singularities at the implementation level. If a model corresponds to an improper continuous-time system, naive discretization can produce filters that are numerically sensitive or that behave as if they require non-causal operations.

Engineers often ensure that the discrete-time transfer function is causal by enforcing that its realization can be written in a standard causal form, such as a recursion with \(y[n]\) depending on \(x[n]\) and earlier, optionally plus a direct gain \(x[n]\) term when appropriate.

4.3 Avoiding non-causal artifacts in simulation

Simulation environments sometimes compute entire trajectories and therefore can mask non-causal behavior. When using such outputs to emulate a real-time controller or estimator, the model may inadvertently rely on future samples due to offline processing or improper indexing. One mitigation is to implement filters and observers with explicit step-by-step updates that only use available information.

Properness also matters for simulation correctness: an improper model can create derivative-like numerical requirements, leading to spikes at time instants where the simulation introduces impulses or starts with inconsistent initial conditions. Regularizing the model or using a strictly proper design can reduce these artifacts.

4.4 Minimal realizations and numerical stability

4.4.1 Choosing realizations that respect constraints

Minimal realizations remove redundant state dynamics while preserving input-output behavior. However, minimality alone does not guarantee good numerical behavior: poorly conditioned realizations can amplify rounding errors or produce unstable internal dynamics even when the input-output transfer function is stable.

A practical design strategy is to choose realizations that:

  • preserve causality and the intended direct feedthrough structure,
  • maintain correct properness properties,
  • use numerically stable forms (e.g., balanced truncation or well-conditioned canonical structures when appropriate).

This helps ensure that the implemented model remains faithful to its mathematical specification across finite-precision computation.

5 Mathematical tools and criteria

5.1 Laplace/z-domain conventions

Causality and properness checks depend on transform conventions. For continuous-time systems, Laplace transforms assume a specific relationship between time-domain support and complex-frequency behavior. In discrete-time, the z-transform uses its own mapping between time indices and the complex plane, and causality corresponds to analyticity properties with respect to the region of convergence.

A reliable workflow keeps track of assumed initial conditions, time indexing, and transform definitions to avoid incorrectly concluding that a system is causal or proper when the representation was shifted or transformed under different assumptions.

5.2 Impulse response and Markov parameter interpretation

For causal systems, the impulse response (or discrete-time impulse sequence) provides direct evidence of time support. For properness and direct feedthrough, Markov parameters—the first coefficients in the series expansion of the transfer function—are informative. The presence or absence of an instantaneous term is typically reflected in the first Markov parameter at time zero:

  • if the instantaneous term exists, the model is not strictly proper,
  • if it is absent, the model’s earliest response begins through dynamics.

This interpretation ties algebraic representation to physically meaningful timing: how quickly influence from input reaches output.

5.3 Frequency-domain viewpoints

Frequency-domain analysis provides additional intuition. Improper systems often exhibit growth at high frequencies, implying that the model would require unbounded resources or differentiation to match behavior. Proper (and especially strictly proper) transfer functions tend to have more controlled high-frequency behavior, which aligns with implementable dynamics and finite bandwidth expectations.

These viewpoints are not a substitute for time-domain causality checks, but they help diagnose whether a model is likely to cause implementation difficulties even before deriving impulse responses.

5.4 Structural properties for automatic verification

Automation can verify causality and properness by analyzing a model’s structure:

  • checking numerator and denominator degrees for rational transfer functions,
  • testing support properties of computed impulse responses,
  • verifying consistency between transfer-function coefficients and state-space parameters (such as whether a direct term appears when expected).

When systems are stored in symbolic form or in standardized block representations, automated checks can also detect algebraic loop risks by examining direct feedthrough connections and feedback interconnections.

6 Practical examples

6.1 Designing a causal low-pass filter

A low-pass filter is commonly designed as a stable, causal IIR or FIR system that attenuates high frequencies. For an analog prototype, a typical goal is a strictly proper transfer function so that the output does not depend instantaneously on the input. After confirming properness, the designer selects a realization (e.g., direct form IIR structure or state-space form) that yields causal recursion.

In digital implementation, the filter is then discretized or designed directly in discrete-time, using a causal difference equation and ensuring that coefficients correspond to a valid recursion that only uses current and past samples.

6.2 Handling strict properness for noise robustness

Strictly proper models delay the output response through internal dynamics, which can reduce sensitivity to measurement noise and limit amplification of high-frequency components. In practical controller design, using a strictly proper plant model or prefilter can prevent instantaneous gain from magnifying noise at the sensor-to-actuator interface.

When strict properness is required, designers may modify an existing model by adding dynamics (e.g., a roll-off prefilter) or by enforcing a realization structure where the direct term is removed or set to zero.

6.3 Correcting improper models in controller synthesis

In some synthesis workflows, the algebraic manipulation of transfer functions can lead to an improper intermediate controller or plant model description. Such a controller may appear reasonable in frequency-domain algebra but becomes unrealizable when implemented. A common correction is to rewrite the interconnection so that the resulting controller transfer function becomes proper, often by moving factors between numerator and denominator, adding appropriate filters, or implementing a realization with proper dynamic order.

Another approach is to work directly in state-space and constrain the design to yield a realizable realization with the correct direct-feedthrough structure, ensuring that the computed controller does not imply instantaneous differentiation of signals.

6.4 Verifying a state-space realization’s direct path

To verify whether a state-space realization implies direct feedthrough, one checks the direct term matrix \(D\). If \(D\neq 0\), the output contains an instantaneous component proportional to the current input. This corresponds to a transfer function that is proper but not strictly proper. If \(D=0\), the earliest influence should arrive through state dynamics, corresponding to strict properness.

Beyond inspecting \(D\), a consistency check compares the transfer function computed from the realization against the target model. Discrepancies can reveal errors such as incorrect output matrix selection, missing feedthrough terms, or unintended cancellations that changed the effective order and therefore the properness classification.