The bilinear transform is a special case of a conformal mapping ( namely, the Möbius transformation ), often used to convert a transfer function of a linear, time-invariant ( LTI ) filter in the continuous-time domain ( often called an analog filter ) to a transfer function of a linear, shift-invariant filter in the discrete-time domain ( often called a digital filter although there are analog filters constructed with switched capacitors that are discrete-time filters ).

The transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a linear time-invariant solution of the nonlinear differential equations describing the system.

If we assume the controller C, the plant P, and the sensor F are linear and time-invariant ( i. e., elements of their transfer function C ( s ), P ( s ), and F ( s ) do not depend on time ), the systems above can be analysed using the Laplace transform on the variables.

The impulse response h of a linear time-invariant causal filter specifies the output that the filter would produce if it were to receive an input consisting of a single impulse at time 0.

The impulse response h completely characterizes any linear time-invariant ( or shift-invariant in the discrete-time case ) filter.

LTI system theory describes linear time-invariant ( LTI ) filters of all types.

Therefore, the behavior of a linear time-invariant system can be analyzed at each frequency independently.

In physics and engineering it is used for analysis of linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems.

In applied mathematics, semigroups are fundamental models for linear time-invariant systems.

A transfer function ( also known as the system function or network function ) is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a linear time-invariant system with zero initial conditions and zero-point equilibrium.

The term is often used exclusively to refer to linear, time-invariant systems ( LTI ), as covered in this article.

Let be the input to a general linear time-invariant system, and be the output, and the bilateral Laplace transform of and be

is input to a linear time-invariant system, then the corresponding component in the output is:

Note that, in a linear time-invariant system, the input frequency has not changed, only the amplitude and the phase angle of the sinusoid has been changed by the system.

Although arbitrary wave shapes will propagate unchanged in lossless linear time-invariant systems, in the presence of dispersion the sine wave is the unique shape that will propagate unchanged but for phase and amplitude, making it easy to analyze.

A convolutional encoder is a discrete linear time-invariant system.

The transfer function for a linear, time-invariant, digital filter can be expressed as a transfer function in the Z-domain ; if it is causal, then it has the form:

In the case of linear time-invariant FIR filters, the impulse response is exactly equal to the sequence of filter coefficients:

They are discrete linear time-invariant systems.

To understand a system with an input and an output, such as an audio amplifier, we start with an ideal system where the transfer function is linear and time-invariant.

In any linear time-invariant system, all of the currents and voltages can be expressed with the same s parameter as the input to the system, allowing the time-varying complex exponential term to be canceled out and the system described algebraically in terms of the complex scalars in the current and voltage waveforms.

A high-pass filter is usually modeled as a linear time-invariant system.

There are many methods of analysis developed specifically for linear time-invariant ( LTI ) deterministic systems.

From a mathematical viewpoint, continuous-time IIR LTI filters may be described in terms of linear differential equations, and their impulse responses considered as Green's functions of the equation.

Most real systems have non-linear input / output characteristics, but many systems, when operated within nominal parameters ( not " over-driven ") have behavior that is close enough to linear that LTI system theory is an acceptable representation of the input / output behavior.

A third way to specify an LTI system is by its characteristic linear differential equation ( for analog systems ) or linear difference equation ( for digital systems ).

Because memristors are time-variant by definition, they are not included in linear time-invariant ( LTI ) circuit models.

( For mathematical details about these systems and their behavior see harmonic oscillator and linear time invariant ( LTI ) system.

Any system in a large class known as linear, time-invariant ( LTI ) is completely characterized by its impulse response.

Suppose that the system is a discrete-time, linear, time-invariant ( LTI ) system described by the impulse response.

For a continuous time linear time invariant ( LTI ) system, the condition for BIBO stability is that the impulse response be absolutely integrable, i. e., its L < sup > 1 </ sup > norm exist.

* LTI system theory ( linear time-invariant system theory ), an engineering theory that investigates the response of a linear, time-invariant system to an arbitrary input signal

Any system that can be modeled as a linear homogeneous differential equation with constant coefficients is an LTI system.

Most LTI system concepts are similar between the continuous-time and discrete-time ( linear shift-invariant ) cases.

If a linear time invariant ( LTI ) system's impulse response is to be measured using a MLS, the response can be extracted from the measured system output y by taking its circular cross-correlation with the MLS.

The complex-valued eigenfunctions of any linear time-invariant ( LTI ) system are of the following forms:

The Routh – Hurwitz stability criterion is a necessary and sufficient method to establish the stability of a single-input, single-output ( SISO ), linear time invariant ( LTI ) control system.

Indeed, a linear, time-invariant system ( see LTI system theory ) is said to be BIBO stable if and only if bounded inputs produce bounded outputs ; this is equivalent to requiring that the denominator of its transfer function ( which can be proven to be rational ) is stable.

By examining a simple integrator circuit it can be demonstrated that when a function is put into a linear time-invariant ( LTI ) system, an output can be characterized by a superposition or sum of the Zero Input Response and the zero state response.

* A " linear " filter is a linear transformation of input samples ; other filters are " non-linear ".

The output of a linear digital filter to any given input may be calculated by convolving the input signal with the impulse response.

Block diagram of a feedback linear oscillator ; an amplifier A with its output v < sub > o </ sub > fed back into its input v < sub > f </ sub > through a electronic filter | filter, β ( jω ).

The most common form of linear oscillator is an electronic amplifier such as a transistor or op amp connected in a feedback loop with its output fed back into its input through a frequency selective electronic filter to provide positive feedback.

Consider a physical system that acts as a linear filter, such as a system of springs and masses, or an analog electronic circuit that includes capacitors and / or inductors ( along with other linear components such as resistors and amplifiers ).

In digital signal processing, linear prediction is often called linear predictive coding ( LPC ) and can thus be viewed as a subset of filter theory.

Since the late 1970s, most non-musical vocoders have been implemented using linear prediction, whereby the target signal's spectral envelope ( formant ) is estimated by an all-pole IIR filter.

In linear prediction coding, the all-pole filter replaces the bandpass filter bank of its predecessor and is used at the encoder to whiten the signal ( i. e., flatten the spectrum ) and again at the decoder to re-apply the spectral shape of the target speech signal.