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Linear time-invariant system

A linear time-invariant (LTI) system is a mathematical model of a dynamical system that satisfies both linearity—meaning its response to a sum of inputs is the sum of the individual responses, and scaling an input by a constant scales the output by the same constant—and time-invariance, where a time shift in the input produces an identical time shift in the output without altering the system's behavior.^[1]^[2] These properties make LTI systems a foundational concept in fields such as signal processing, control theory, and electrical engineering, as many physical processes, including mechanical vibrations and electrical circuits, can be approximated by LTI models under small-signal conditions.^[3]^[4] The linearity property, rooted in the principle of superposition, allows LTI systems to be analyzed using techniques like convolution, where the output y(t) is given by the integral y(t) = \int_{-\infty}^{\infty} u(\tau) h(t - \tau) \, d\tau, with h(t) as the system's impulse response.^[2] Time-invariance ensures that the impulse response h(t) remains fixed, enabling efficient frequency-domain representations via the Fourier or Laplace transform, where complex exponentials act as eigenfunctions and the system's frequency response H(\omega) scales the input amplitude and phase without changing the frequency.^[1]^[5] In state-space form, continuous-time LTI systems are described by differential equations \dot{x}(t) = A x(t) + B u(t) and y(t) = C x(t) + D u(t), with constant matrices A, B, C, D, facilitating analysis of stability, controllability, and observability.^[4] LTI systems are widely applied in engineering to model and design filters, amplifiers, and feedback controllers, such as in audio processing where low-pass filters remove high-frequency noise while preserving signal integrity, or in robotics for predicting electromechanical responses.^[5]^[2] For instance, a spring-mass-damper system follows LTI dynamics m \ddot{x} + c \dot{x} + k x = f(t), allowing simulation of vibrations in structures or vehicles.^[4] Their computational tractability supports discrete-time implementations in digital signal processing, underpinning technologies like image enhancement and communication systems.^[5]

Overview

Definition

A linear time-invariant (LTI) system is a dynamical system that satisfies both the linearity and time-invariance properties, enabling a complete characterization of its input-output behavior through simple mathematical operations.^[6] Linearity is defined by the superposition principle, which encompasses additivity and homogeneity: additivity requires that the system's response to the sum of two inputs equals the sum of the individual responses, while homogeneity stipulates that scaling an input by a constant factor scales the output by the same factor.^[7] These properties assume the system maps input functions to output functions without additional constraints like initial conditions affecting the mapping in a non-superposable way.^[8] Time-invariance complements linearity by ensuring the system's behavior does not depend on absolute time: if a time-shifted input produces an output, then the same shift applied to that output yields the response to the shifted input.^[9] Formally, for an input x(t) producing output y(t), a shifted input x(t - \tau) must produce y(t - \tau) for any delay \tau.^[10] This axiom holds for both continuous- and discrete-time systems, presupposing familiarity with signal representations as functions of time. The general input-output relationship for any LTI system is given by convolution of the input with the system's impulse response h(t), which fully specifies the system's dynamics under these properties.^[11] Unlike nonlinear systems, where superposition fails and impulse responses lack predictive utility, or time-varying systems, whose parameters change with time and preclude fixed convolution forms, LTI systems allow predictable, time-independent transformations.^[12]

Historical context and significance

The mathematical foundations of linear time-invariant (LTI) system theory were established in the late 18th and early 19th centuries by Pierre-Simon Laplace, who developed the Laplace transform around 1809 for solving linear differential equations, and by Joseph Fourier, who introduced the Fourier series in 1822, enabling the decomposition of signals into frequency components crucial for LTI analysis.^[13]^[14] In the late 19th century, British engineer Oliver Heaviside developed operational calculus as a practical method for solving linear differential equations with constant coefficients, particularly in electromagnetic theory and electrical transmission problems.^[15] This approach treated differentiation as an algebraic operator, enabling efficient analysis of systems that exhibit linear responses independent of time shifts, laying early groundwork for modeling dynamic behaviors in physical systems.^[16] In the 1940s, American mathematician Norbert Wiener advanced the field through his work on cybernetics, introducing concepts of feedback and statistical prediction that highlighted the role of linear systems in control and communication processes.^[17] Wiener's developments during World War II, including optimal filtering techniques to minimize prediction errors in noisy environments, demonstrated the power of linear models for handling stochastic inputs, influencing signal processing and control theory.^[18] His 1948 book Cybernetics: Or Control and Communication in the Animal and the Machine formalized these ideas, bridging biological and engineered systems via linear approximations.^[18] The formalization of LTI theory in signal processing occurred post-1950s, as digital computation enabled broader applications, with seminal texts like Alan V. Oppenheim and Ronald W. Schafer's Digital Signal Processing (1975) synthesizing convolution-based representations and frequency-domain methods for analysis and design.^[19] This era solidified LTI systems as essential building blocks for modeling physical phenomena, such as electrical circuits through transfer functions, acoustic wave propagation in rooms, and mechanical vibrations in structures like helicopter rotors.^[20] These models facilitate frequency-domain analysis, including Fourier and Laplace transforms, crucial for filter design and system stability assessment.^[21] LTI approximations prove effective for many real-world systems because nonlinear behaviors can be linearized around an operating point using small-signal analysis, where perturbations are minor enough to neglect higher-order terms, yielding time-invariant responses.^[22] Additionally, when system parameters vary slowly compared to signal dynamics, time-invariance holds as a valid assumption, simplifying complex phenomena into tractable forms.^[23] In modern contexts, LTI principles underpin digital signal processing for applications like audio equalization and noise reduction, while inspiring machine learning architectures such as convolutional neural networks, where convolution operations mimic LTI filtering to extract spatial features from data.^[24]

Fundamental properties

Linearity

In linear time-invariant (LTI) systems, the linearity property enables the application of the superposition principle, whereby the system's response to a sum of inputs equals the sum of the responses to each input individually, and scaling an input by a constant factor scales the corresponding output by the same factor. This principle underpins the analysis of complex signals by breaking them down into simpler components whose responses can be computed separately and then combined.^[25] Formally, linearity is characterized by two axioms: additivity and homogeneity, expressed using the system operator H, where the output y(t) = H[x(t)] for an input signal x(t). Additivity states that if y_1(t) = H[x_1(t)] and y_2(t) = H[x_2(t)], then H[x_1(t) + x_2(t)] = y_1(t) + y_2(t). Homogeneity requires that for any scalar constant a, H[a x(t)] = a H[x(t)]. These axioms together imply the full superposition principle: for scalars a and b, and inputs x_1(t) and x_2(t),

\begin{align} H[a x_1(t) + b x_2(t)] &= H[a x_1(t)] + H[b x_2(t)] \quad \text{(by additivity)} \\ &= a H[x_1(t)] + b H[x_2(t)] \quad \text{(by homogeneity)}. \end{align}

This derivation shows how additivity and homogeneity combine to yield superposition.^[25] A key implication of linearity is that a zero input produces a zero output: setting a = 0 in the homogeneity axiom gives H[0 \cdot x(t)] = 0 \cdot H[x(t)], or

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. Furthermore, linearity facilitates the decomposition of arbitrary input signals into sums of basis functions or components, allowing the overall response to be obtained as the corresponding sum of individual responses, which simplifies computational and analytical approaches in signal processing.^[25] From a mathematical perspective, the space of input signals (and outputs) can be viewed as a vector space over the real numbers, with addition and scalar multiplication defined pointwise; under this interpretation, the system operator H acts as a linear transformation that preserves vector addition and scalar multiplication. This vector space framework aligns LTI systems with linear algebra, enabling techniques such as basis expansions for system representation. Linearity is complemented by the time-invariance property, which ensures consistent behavior under time shifts.^[25]

Time-invariance

A time-invariant system is characterized by the property that any time shift applied to the input signal produces an identical time shift in the output signal. Formally, if the response of the system \mathcal{H} to an input x(t) is the output y(t) = \mathcal{H}\{x(t)\}, then the response to a shifted input x(t - \tau) is y(t - \tau) for any time shift \tau. This property ensures that the system's behavior does not change over time, independent of when the input is applied.^[26]^[5] The time-invariance property commutes with linearity, meaning that the system's response to time-shifted linear combinations of inputs equals the linear combination of the time-shifted responses. Specifically, if the system satisfies linearity—where \mathcal{H}\{\alpha x_1(t) + \beta x_2(t)\} = \alpha \mathcal{H}\{x_1(t)\} + \beta \mathcal{H}\{x_2(t)\} for scalars \alpha, \beta—then shifting the inputs preserves this superposition: \mathcal{H}\{\alpha x_1(t - \tau) + \beta x_2(t - \tau)\} = \alpha y_1(t - \tau) + \beta y_2(t - \tau). This commutativity underpins the formation of linear time-invariant (LTI) systems, allowing the properties to be analyzed independently yet jointly.^[27]^[28] In LTI systems, outputs are determined solely by the relative time differences between inputs and outputs, exhibiting no dependence on absolute time or the specific origin of the time axis. This translation invariance implies that the system's operation is consistent regardless of the temporal reference frame, facilitating predictable behavior across different time scales.^[28] Unlike LTI systems, time-varying systems incorporate explicit dependence on absolute time, altering their response based on when the input occurs. For instance, the system defined by y(t) = t x(t) is linear, as it satisfies superposition and homogeneity, but it is not time-invariant: applying a shift \tau to the input yields y(t - \tau) = (t - \tau) x(t - \tau), which differs from the shifted output t x(t - \tau). Such systems arise in applications like time-dependent amplifiers or seasonally varying filters, where parameters evolve with time.^[9]^[29]

Continuous-time LTI systems

Impulse response and convolution

The unit impulse in continuous time is the Dirac delta function \delta(t), defined such that \int_{-\infty}^{\infty} \delta(t) \, dt = 1 and \delta(t) = 0 for t \neq 0. This function serves as a fundamental input for characterizing linear time-invariant (LTI) systems. The impulse response h(t) of a continuous-time LTI system is the output produced when the input is \delta(t). It completely determines the system's behavior for any input, leveraging the linearity and time-invariance properties.^[30] Any arbitrary continuous-time input signal x(t) can be expressed as an integral of shifted and scaled unit impulses:

x(t) = \int_{-\infty}^{\infty} x(\tau) \delta(t - \tau) \, d\tau.

Due to the linearity property, which allows superposition of responses, the output y(t) to this input is the integral of the responses to each individual term x(\tau) \delta(t - \tau). Time-invariance ensures that the response to the shifted impulse \delta(t - \tau) is simply the shifted impulse response h(t - \tau). Therefore, the overall output is

y(t) = \int_{-\infty}^{\infty} x(\tau) h(t - \tau) \, d\tau,

or equivalently,

y(t) = \int_{-\infty}^{\infty} h(\tau) x(t - \tau) \, d\tau.

This expression, known as the convolution integral, provides the time-domain representation of the system's response and fully characterizes continuous-time LTI systems.^[31] The support of the impulse response h(t) refers to the set of t where h(t) \neq 0. Systems with impulse responses of finite duration are idealizations, while most physical systems have responses extending indefinitely, though decaying. A continuous-time LTI system is causal if its impulse response satisfies h(t) = 0 for all t < 0, meaning the output at time t depends only on current and past inputs.^[32] From a computational perspective, the convolution integral can be evaluated numerically using methods like direct integration or fast Fourier transform (FFT) approximations, though analytical solutions are preferred for design. This continuous convolution integral is the foundation for analyzing systems like filters and control loops in the time domain.^[33]

Eigenfunctions and response to exponentials

In linear time-invariant (LTI) systems, an eigenfunction is a signal that, when input to the system, produces an output that is a scalar multiple of the input signal itself. For continuous-time LTI systems, complex exponential signals of the form e^{st}, where s is a complex number, serve as such eigenfunctions. Specifically, if the input is x(t) = e^{st}, the output is y(t) = H(s) e^{st}, where H(s) is a complex-valued scalar known as the eigenvalue, which depends on s and characterizes the system's response at that frequency.^[34] To demonstrate this property, consider the output of a continuous-time LTI system given by the convolution integral y(t) = \int_{-\infty}^{\infty} h(\tau) x(t - \tau) \, d\tau, where h(t) is the impulse response. Substituting the exponential input x(t) = e^{st} yields

y(t) = \int_{-\infty}^{\infty} h(\tau) e^{s(t - \tau)} \, d\tau = e^{st} \int_{-\infty}^{\infty} h(\tau) e^{-s\tau} \, d\tau,

assuming the integral converges. The integral defines H(s) = \int_{-\infty}^{\infty} h(\tau) e^{-s\tau} \, d\tau, so y(t) = H(s) e^{st}, confirming that e^{st} is an eigenfunction with eigenvalue H(s). This holds for s = \sigma + j\omega, where \sigma and \omega are real, and the real part \sigma > 0 ensures convergence for growing exponentials, linking directly to the s-plane in Laplace analysis.^[34] This eigenfunction property is particularly useful because arbitrary input signals can be decomposed as integrals (or sums) of complex exponentials, allowing the system's output to be expressed as the corresponding integral of scaled exponentials. This decomposition underpins frequency-domain techniques, such as the Fourier and Laplace transforms, for analyzing LTI system responses.^[34]

Frequency and Laplace domain analysis

The frequency response of a continuous-time linear time-invariant (LTI) system characterizes its steady-state behavior to sinusoidal inputs and is obtained via the Fourier transform. For an input signal x(t) with Fourier transform X(j\omega), the output y(t) has Fourier transform Y(j\omega) = H(j\omega) X(j\omega), where H(j\omega) is the frequency response, defined as the Fourier transform of the impulse response h(t):

H(j\omega) = \int_{-\infty}^{\infty} h(t) e^{-j\omega t} \, dt.

This multiplicative property in the frequency domain simplifies analysis of system effects on signal spectra, such as amplification or phase shift at different frequencies \omega.^[35]^[36] The Laplace transform extends this analysis to a broader class of signals, including those with exponential growth or decay, by introducing the complex variable s = \sigma + j\omega. The transfer function H(s) is the Laplace transform of h(t):

H(s) = \int_{-\infty}^{\infty} h(t) e^{-st} \, dt,

and the output transform is Y(s) = H(s) X(s), assuming appropriate regions of convergence (ROC). The bilateral Laplace transform applies to signals over all t, while the unilateral form,

\mathcal{L}\{x(t)\} = \int_{0}^{\infty} x(t) e^{-st} \, dt,

is used for causal systems with zero initial conditions, facilitating solutions to differential equations describing LTI dynamics. The ROC, a vertical strip in the s-plane where the integral converges, determines the transform's validity and relates to system stability; for stable causal systems, it includes the imaginary axis \sigma = 0.^[37]^[38]^[39] The inverse Laplace transform recovers the time-domain impulse response via the Bromwich integral:

h(t) = \frac{1}{2\pi j} \int_{\sigma - j\infty}^{\sigma + j\infty} H(s) e^{st} \, ds,

where the contour lies in the ROC. For practical computation, especially with rational transfer functions H(s) = \frac{P(s)}{Q(s)} (polynomials P and Q with \deg Q \geq \deg P), partial fraction expansion decomposes H(s) into simpler terms:

H(s) = \sum_{k} \frac{A_k}{s - p_k} + \sum_{m} \frac{B_m s + C_m}{s^2 + \alpha_m s + \beta_m},

for distinct poles p_k and quadratic factors, allowing inversion term-by-term using standard tables. This method is essential for finding closed-form expressions for h(t) in physical systems like RLC circuits.^[38]^[37]^[40] Pole-zero diagrams visualize rational H(s) by plotting zeros (roots of P(s)) as open circles and poles (roots of Q(s)) as crosses in the s-plane. The diagram reveals key behaviors: proximity of poles to the imaginary axis indicates resonance or instability risks, while zero locations shape high-frequency roll-off. For example, in a second-order low-pass filter H(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}, poles at -\zeta\omega_n \pm j\omega_n\sqrt{1-\zeta^2} determine damping and natural frequency. These plots, combined with partial fractions, enable efficient system design and response prediction without full time-domain simulation.^[41]^[42]^[43]

Examples

A prominent example of a continuous-time LTI system is the RC low-pass filter, consisting of a resistor R and capacitor C in series, with output across the capacitor. The differential equation is RC \dot{y}(t) + y(t) = x(t), and the transfer function is H(s) = \frac{1}{RC s + 1}, with impulse response h(t) = \frac{1}{RC} e^{-t/RC} u(t), where u(t) is the unit step function. This system attenuates high frequencies with cutoff \omega_c = 1/RC. Another example is the ideal integrator, where y(t) = \int_{-\infty}^t x(\tau) \, d\tau, governed by \dot{y}(t) = x(t), with transfer function H(s) = 1/s and impulse response h(t) = u(t). Integrators accumulate signals and are used in control systems but require compensation for stability.^[44] The spring-mass-damper system models mechanical vibrations with equation m \ddot{y}(t) + c \dot{y}(t) + k y(t) = x(t), where m is mass, c damping, k stiffness, and x(t) force. The transfer function is H(s) = \frac{1}{m s^2 + c s + k}, and for underdamped cases (c^2 < 4mk), the impulse response involves damped sinusoids, illustrating oscillatory behavior in structures or vehicles.^[4] The differentiator, \dot{y}(t) = x(t), has H(s) = s and h(t) = \delta(t), but ideal differentiators amplify noise and are approximated in practice. These examples demonstrate how LTI models apply to electrical and mechanical systems.^[45]

Causality and stability

In continuous-time linear time-invariant (LTI) systems, causality refers to the property that the output at any time t depends only on the input values at the current time and past times, not on future inputs. For such systems, this condition is equivalent to the impulse response h(t) being zero for all negative time, i.e., h(t) = 0 for t < 0.^[46] This right-sided nature of the impulse response ensures that the system's response to an input signal x(t) up to time t is fully determined by the convolution integral y(t) = \int_{0}^{\infty} h(\tau) x(t - \tau) \, d\tau, without requiring knowledge of future input values.^[47] Stability in continuous-time LTI systems is typically analyzed through bounded-input bounded-output (BIBO) stability and asymptotic stability. A system is BIBO stable if every bounded input signal produces a bounded output signal, which holds if and only if the impulse response is absolutely integrable: \int_{-\infty}^{\infty} |h(t)| \, dt < \infty.^[48] This condition guarantees that the convolution integral does not amplify bounded inputs indefinitely, as the total contribution from the impulse response remains finite. For rational transfer functions H(s), asymptotic stability—where the system's zero-input response decays to zero as t \to \infty—requires all poles to lie strictly in the left half of the s-plane, meaning the real parts of all poles satisfy \operatorname{Re}(s) < 0.^[49] BIBO stability implies asymptotic stability for causal LTI systems, but the converse does not always hold. To assess asymptotic stability without explicitly finding the roots of the characteristic polynomial, the Routh-Hurwitz criterion provides an algebraic procedure. For a polynomial D(s) = \sum_{k=0}^{N} a_k s^k with a_N > 0, the criterion constructs a Routh array and checks that all elements in the first column are positive, ensuring no roots with positive real parts. This method is essential for higher-order systems in control design.

Discrete-time LTI systems

Derivation from continuous-time systems

Discrete-time linear time-invariant (LTI) systems are typically derived from continuous-time LTI systems through the process of sampling, which converts analog signals into discrete sequences while aiming to preserve the underlying linear and time-invariant properties. The sampling operation defines the discrete-time input signal x = x_c(nT), where x_c(t) is the continuous-time signal and T is the fixed sampling period. Similarly, the discrete-time unit impulse \delta arises as the sampled version of the continuous-time Dirac delta function \delta_c(t), scaled appropriately to approximate its sifting property in the discrete domain. This derivation assumes ideal uniform sampling, but practical considerations require adherence to the Nyquist-Shannon sampling theorem to ensure faithful representation.^[50] The Nyquist-Shannon sampling theorem states that a continuous-time bandlimited signal with maximum frequency f_{\max} can be perfectly reconstructed from its samples if the sampling frequency f_s = 1/T satisfies f_s \geq 2 f_{\max}, known as the Nyquist rate. Failure to meet this criterion leads to aliasing, where higher-frequency components fold into the lower-frequency band, potentially distorting the system's frequency response and compromising the preservation of LTI characteristics in the discrete domain. The Nyquist criterion, foundational to this theorem, emphasizes the minimum sampling rate needed to avoid such spectral overlap.^[51] Several discretization methods transform the continuous-time impulse response h_c(t) or transfer function H(s) into their discrete counterparts while maintaining LTI properties. The impulse invariance method sets the discrete impulse response as h = T h_c(nT) for n \geq 0, ensuring that the discrete system's response to a sampled impulse matches the continuous system's at sampling instants. This approach is particularly useful for IIR filter design but can introduce aliasing if h_c(t) is not bandlimited.^[52] The bilinear transform provides an alternative by conformally mapping the s-plane to the z-plane via the substitution s = \frac{2}{T} \frac{1 - z^{-1}}{1 + z^{-1}}, yielding H(z) directly from H(s). This method preserves stability for stable continuous systems and avoids aliasing through nonlinear frequency warping, though it compresses the frequency axis, often requiring prewarping for critical frequencies. Introduced for analyzing linear systems in terms of time series, it is a standard technique in digital control and signal processing. In control systems, the zero-order hold (ZOH) approximation models the hold operation in digital-to-analog conversion, where the control signal is held constant between samples. The ZOH equivalent discrete-time model exactly reproduces the continuous-time output for piecewise-constant inputs, with the discrete transfer function derived as H(z) = (1 - z^{-1}) \mathcal{Z} \left\{ \frac{H(s)}{s} \right\}, where \mathcal{Z} denotes the z-transform. This method is essential for sampled-data control, ensuring accurate simulation of physical plants interfaced with digital controllers.

Impulse response and convolution

The unit impulse sequence in discrete time, denoted \delta, is defined as \delta = 1 for n = 0 and \delta = 0 otherwise. This sequence serves as a fundamental input for characterizing linear time-invariant (LTI) systems. The impulse response h of a discrete-time LTI system is the output sequence produced when the input is \delta. It completely determines the system's behavior for any input, leveraging the linearity and time-invariance properties. Any arbitrary discrete-time input signal x can be expressed as a linear combination of shifted and scaled unit impulses:

x = \sum_{k=-\infty}^{\infty} x \delta[n - k].

Due to the linearity property, which allows superposition of responses, the output y to this input is the sum of the responses to each individual term x \delta[n - k]. Time-invariance ensures that the response to the shifted impulse \delta[n - k] is simply the shifted impulse response h[n - k]. Therefore, the overall output is

y = \sum_{k=-\infty}^{\infty} x h[n - k],

or equivalently,

y = \sum_{k=-\infty}^{\infty} h x[n - k].

This expression, known as the convolution sum, provides the time-domain representation of the system's response and fully characterizes discrete-time LTI systems. The support of the impulse response h refers to the set of n where h \neq 0. Systems with finite support are classified as finite impulse response (FIR) systems, where h is nonzero only over a finite range of n. In contrast, infinite impulse response (IIR) systems have impulse responses with infinite support, extending indefinitely in at least one direction. A discrete-time LTI system is causal if its impulse response satisfies h = 0 for all n < 0, meaning the output at time n depends only on current and past inputs. From a computational perspective, FIR systems are implemented via direct evaluation of the finite convolution sum, requiring a fixed number of multiplications and additions per output sample. IIR systems, however, typically use recursive difference equations derived from h, enabling efficient computation despite the infinite support, though they may introduce feedback that affects numerical stability. This discrete convolution sum is analogous to the continuous-time convolution integral used for LTI systems in that domain.

Eigenfunctions and z-domain analysis

In discrete-time linear time-invariant (LTI) systems, sequences of the form x = z^n, where z is a complex constant, serve as eigenfunctions. The output corresponding to such an input is y = H(z) z^n, with H(z) denoting the system's transfer function evaluated at z. This property arises because the convolution operation defining the system's response multiplies the input by a scalar factor H(z) when the input is an eigenfunction.^[53]^[54] The z-transform provides a frequency-domain representation for discrete-time signals and systems, enabling analysis of LTI systems through algebraic manipulation. For an input signal x and output y, their z-transforms satisfy Y(z) = H(z) X(z), where H(z) is the transfer function. The unilateral z-transform of a sequence y is defined as

Y(z) = \sum_{n=0}^{\infty} y z^{-n},

with the bilateral form extending the sum from n = -\infty to \infty; convergence occurs within a region of convergence (ROC) in the complex z-plane, which depends on the signal's properties such as causality or duration.^[55]^[56] The discrete-time Fourier transform (DTFT) emerges as a special case of the z-transform evaluated on the unit circle in the z-plane, where |z| = 1 or z = e^{j\omega}. Thus, the frequency response of the system is H(e^{j\omega}), which characterizes the system's steady-state response to sinusoidal inputs and relates directly to the magnitude and phase alterations at frequency \omega. This connection allows the z-transform to generalize the DTFT for broader analytic continuation beyond the unit circle.^[57]^[58] The transfer function H(z) is typically a rational function expressed as

H(z) = \frac{\sum_{k=0}^{M} b_k z^{-k}}{1 + \sum_{k=1}^{N} a_k z^{-k}},

with zeros at the roots of the numerator polynomial and poles at the roots of the denominator. These poles and zeros are plotted in the complex z-plane, providing geometric insight into system behavior; for instance, zeros attenuate specific exponential components, while poles amplify them. The ROC must exclude poles to ensure convergence.^[41]^[59] To recover the time-domain sequence from its z-transform, the inverse z-transform employs a contour integral over a closed path C in the ROC:

y = \frac{1}{2\pi j} \oint_C Y(z) z^{n-1} \, dz,

where the integral is taken counterclockwise. This formulation leverages complex analysis, such as the residue theorem, to compute the inverse for rational functions by summing residues at poles inside C.^[60]^[61]

Examples

A prominent example of a discrete-time LTI system is the finite impulse response (FIR) filter, which computes each output sample as a finite weighted sum of current and past input samples.^[62] The difference equation for an FIR filter of length M+1 is given by

y = \sum_{k=0}^{M} b_k x[n-k],

where b_k are the filter coefficients and the impulse response h is finite in duration, specifically h = b_n for $0 \leq n \leq M and zero otherwise.^[63] This structure ensures the system is inherently stable and can implement linear-phase filtering when coefficients are symmetric.^[64] In contrast, the infinite impulse response (IIR) filter incorporates feedback, relying on both past inputs and past outputs to produce the current output, resulting in a potentially infinite-duration impulse response.^[62] The general form of the difference equation for an IIR filter is

y = \sum_{k=0}^{M} b_k x[n-k] - \sum_{k=1}^{N} a_k y[n-k],

where a_k and b_k are coefficients, and the system function in the z-domain is a rational function H(z) = \frac{\sum_{k=0}^{M} b_k z^{-k}}{1 + \sum_{k=1}^{N} a_k z^{-k}}.^[63] IIR filters achieve sharper frequency responses with fewer coefficients compared to FIR filters but require careful design to ensure stability.^[62] A specific instance of an FIR filter is the moving average filter, which smooths a signal by averaging a fixed number of consecutive input samples.^[64] For an N-point moving average, the impulse response is h = \frac{1}{N} for $0 \leq n < N and zero elsewhere, yielding the output y = \frac{1}{N} \sum_{k=0}^{N-1} x[n-k].^[5] This filter attenuates high-frequency components, acting as a low-pass system, and its output can be computed via the convolution sum with the input.^[64] The unit delay system represents a basic building block for more complex discrete-time LTI systems, simply shifting the input sequence by one sample.^[65] Its difference equation is y = x[n-1], with the z-domain transfer function H(z) = z^{-1}.^[66] This system introduces a phase shift proportional to frequency without altering amplitudes, and its impulse response is h = \delta[n-1].^[65]

Causality and stability

In discrete-time linear time-invariant (LTI) systems, causality refers to the property that the output at any time index n depends only on the input values at the current time and past times, not on future inputs. For such systems, this condition is equivalent to the impulse response h being zero for all negative time indices, i.e., h = 0 for n < 0.^[67] This right-sided nature of the impulse response ensures that the system's response to an input sequence x up to time n is fully determined by the convolution sum y = \sum_{k=0}^{\infty} h x[n-k], without requiring knowledge of future input values.^[68] Stability in discrete-time LTI systems is typically analyzed through two key concepts: bounded-input bounded-output (BIBO) stability and asymptotic stability. A system is BIBO stable if every bounded input sequence produces a bounded output sequence, which holds if and only if the impulse response is absolutely summable: \sum_{n=-\infty}^{\infty} |h| < \infty.^[68] This condition guarantees that the convolution integral does not amplify bounded inputs indefinitely, as the total contribution from the impulse response remains finite. For rational transfer functions H(z), asymptotic stability—where the system's zero-input response decays to zero as n \to \infty—requires all poles to lie strictly inside the unit circle in the z-plane, meaning the magnitudes of all poles satisfy |z| < 1.^[69] BIBO stability implies asymptotic stability for causal LTI systems, but the converse does not always hold. To assess asymptotic stability without explicitly finding the roots of the characteristic polynomial, the Jury stability test offers an algebraic procedure analogous to the Routh-Hurwitz criterion for continuous-time systems. Developed by E.I. Jury, the test constructs a tabular array from the coefficients of the polynomial D(z) = \sum_{k=0}^{N} a_k z^k and checks a set of inequalities to verify that all roots have magnitude less than one.^[70] For a polynomial of degree N, the table is built iteratively by computing determinants of submatrices, and stability is confirmed if the leading coefficients satisfy specific sign and magnitude conditions, such as |a_0| < a_N and positivity of certain table entries. This method is particularly useful for higher-order systems where root-finding algorithms are computationally intensive.^[71]

Applications and extensions

In signal processing and control theory

In signal processing, linear time-invariant (LTI) systems form the foundation for designing filters that selectively attenuate or amplify specific frequency components of a signal. Low-pass filters, which allow low-frequency signals to pass while suppressing higher frequencies, are commonly used for noise removal in applications such as audio smoothing or image denoising, as they model the system's response through convolution with an impulse response that preserves desirable signal content.^[72] High-pass filters, conversely, eliminate low-frequency noise or offsets, enabling edge detection in signals or DC component removal, and both types leverage the time-invariance property to ensure consistent performance across signal durations.^[5] These filters are analyzed in the frequency domain using Fourier transforms, where the system's transfer function directly multiplies the input spectrum to yield the output, facilitating efficient design and implementation.^[73] To accelerate the convolution operation central to LTI systems, the fast Fourier transform (FFT) is employed, converting time-domain multiplication into frequency-domain multiplication, which reduces computational complexity from O(N²) to O(N log N) for large signals. This technique is pivotal in real-time signal processing tasks, such as implementing long impulse response filters without excessive latency.^[74] In control theory, LTI systems are integral to feedback loops, where transfer functions describe the relationship between input commands and output responses, enabling stability analysis via tools like Bode plots. These loops use LTI models to predict system behavior under closed-loop operation, ensuring controlled dynamics in processes like motor speed regulation. Proportional-integral-derivative (PID) controllers approximate LTI systems by combining proportional, integral, and derivative actions into a linear transfer function, providing robust performance for setpoint tracking in industrial automation despite minor nonlinearities in real plants.^[42]^[75] In audio processing, equalizers function as LTI systems by applying frequency-selective gains to balance spectral content, such as boosting bass or cutting harsh highs in music signals, often realized through cascaded biquad filters that maintain phase coherence.^[76] Digital implementations of LTI systems rely on specialized digital signal processing (DSP) chips, which execute convolution and filtering algorithms at high speeds using fixed-point or floating-point arithmetic optimized for real-time operations in embedded devices like smartphones or hearing aids. These chips, such as those from Texas Instruments, handle the matrix operations inherent to LTI transforms with low power consumption, supporting applications from echo cancellation to spectral analysis.^[77] Despite their utility, LTI systems face limitations in nonlinear regimes, where real-world phenomena like saturation or hysteresis violate linearity assumptions, leading to distorted outputs and inaccurate predictions that require hybrid or adaptive extensions for robustness. In such cases, LTI approximations hold only near operating points, necessitating nonlinear control strategies to maintain performance under varying conditions.^[78]

State-space representations

State-space representations provide a time-domain framework for modeling linear time-invariant (LTI) systems using first-order differential or difference equations that describe the evolution of internal state variables.^[79] For continuous-time LTI systems, the standard form consists of the state equation \dot{x}(t) = A x(t) + B u(t) and the output equation y(t) = C x(t) + D u(t), where x(t) \in \mathbb{R}^n is the state vector, u(t) \in \mathbb{R}^r is the input vector, y(t) \in \mathbb{R}^m is the output vector, and A, B, C, D are constant matrices of appropriate dimensions.^[79] In the discrete-time case, the representation is given by the state update x[n+1] = A x + B u and output y = C x + D u, where n denotes the discrete time index.^[80] This state-space form is equivalent to the transfer function representation obtained via the Laplace transform for continuous-time systems or the z-transform for discrete-time systems.^[79] Specifically, the transfer function matrix is H(s) = C (sI - A)^{-1} B + D, which connects the input-output behavior to the state dynamics.^[79] For discrete-time systems, the analogous form is H(z) = C (zI - A)^{-1} B + D.^[81] One key advantage of state-space representations is their natural handling of multi-input multi-output (MIMO) systems, where multiple inputs and outputs are described through matrix coefficients without requiring separate transfer functions for each pair.^[79] Additionally, these models facilitate analysis of internal system properties such as controllability and observability.^[81] Controllability refers to the ability to drive the state from any initial value to any final value in finite time using the input, while observability means that the initial state can be determined from the input and output over a finite interval.^[82] The Kalman decomposition theorem partitions the state space into four subspaces—controllable and observable, controllable but unobservable, observable but uncontrollable, and neither—via a similarity transformation, enabling structured analysis of system behavior.^[81] In realization theory, state-space models are constructed from transfer functions or impulse responses, with a minimal realization defined as the lowest-order model that is both controllable and observable.^[81] All minimal realizations of a given transfer function are related by similarity transformations and share the same order, equal to the degree of the transfer function for SISO systems or the rank of the Hankel matrix for MIMO systems.^[83] For non-diagonalizable state matrices A, the Jordan canonical form is used to represent the system, where A is block-diagonal with Jordan blocks corresponding to eigenvalues, allowing computation of solutions via matrix exponentials or powers even when algebraic and geometric multiplicities differ.^[83]

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