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Encyclopedia > LTI system theory

In electrical engineering, specifically in signal processing and control theory, LTI system theory investigates the response of a linear, time-invariant system to an arbitrary input signal. Electrical engineers design power systems. ... Signal processing is the processing, amplification and interpretation of signals and deals with the analysis and manipulation of signals. ... In engineering and mathematics, control theory deals with the behavior of dynamical systems over time. ... A linear system is a model of a system based on some kind of linear operator. ... A time-invariant system is one whose output does not depend explicitly on time. ...


The defining properties of any LTI system are, of course, linearity and time invariance:

  • Linearity means that the relationship between the input and the output of the system satisfies the scaling and superposition properties. Formally, a linear system is a system which exhibits the following property: if the input of the system is
x(t) = Ax_1(t) + Bx_2(t) ,
then the output of the system will be
y(t) = Ay_1(t) + By_2(t) ,
for any constants A and B.
  • Time invariance means that whether we apply an input to the system now or t seconds from now, the output will be identical, except a time delay of the t seconds. More specifically, an input affected by a time delay should effect a corresponding time delay in the output, hence time-invariant.


The fundamental result in LTI system theory is that all LTI systems can be characterized entirely by a single function called the system's impulse response. The output of the system is simply the convolution of the input to the system and the system's impulse response. This method of analysis is often called the time domain point-of-view. In the language of mathematics, the impulse response of a linear transformation is the image of Diracs delta function under the transformation. ... For the computer science usage see convolution (computer science) . In mathematics and in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version... Time-domain is a term used to describe the analysis of mathematical functions, or real-life signals, with respect to time. ...

Relationship between the time domain and the frequency domain
Relationship between the time domain and the frequency domain

Equivalently, any LTI systems can be characterized in the frequency domain by the system's transfer function, which is simply the Laplace transform of the system's impulse response. As a result of the properties of the Laplace transform, the output of the system in the frequency domain is the product of the transfer function and the Laplace transform of the input. In other words, convolution in the time domain is equivalent to multiplication in the frequency domain. Image File history File links Download high resolution version (1728x1296, 97 KB) Summary I created this image myself in Mocrosoft PowerPoint and saved it as a PNG file. ... Image File history File links Download high resolution version (1728x1296, 97 KB) Summary I created this image myself in Mocrosoft PowerPoint and saved it as a PNG file. ... Frequency domain is a term used to describe the analysis of mathematical functions with respect to frequency. ... A transfer function is a mathematical representation of the relation between the input and output of a linear time-invariant system. ... In mathematics, the Laplace transform is a powerful technique for analyzing linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems, to name just a few. ...


For all LTI systems, the eigenfunctions are complex exponentials, which are superpositions of sinusoids. As a result, if the input to the system is a sinusoid, then the output of the system will also be a sinusoid, perhaps with a different amplitude and a different phase, but always with the same frequency. They are called eigenfunctions because, if we view the system as a linear operator on some Hilbert space (such as a matrix for example), then when we apply the operator on a complex exponential (input), the output is also a complex exponential with a scalar in front of it. This is mathematically identical to the notion of eigenvectors and eigenvalues. In this transformation of the Mona Lisa, the blue vector has been rotated, but the red one has not. ... Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form a + bi, where a and b are real numbers, and i stands for the square root of negative one (−1), which cannot be represented by any real number. ... The exponential function is one of the most important functions in mathematics. ... A sine wave or sinusoid is a waveform whose graph is identical to the generalized sine function y = Asin[ω(x − α)] + C where A is the amplitude, ω is the angular frequency (2π/P where P is the wavelength), α is the phase shift, and C is the vertical offset. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, of elements of a ring-like algebraic structure. ... In linear algebra, the eigenvectors (from the German eigen meaning inherent, characteristic) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ...


Due to the simplicity offered by the impulse response in describing the input/output relationship of the system, linear time invariant systems are often convenient models for a wide range of applications including: communication channels that model the propagation media in communications, and all circuits involving resistors, capacitors and inductors, among many others.

Contents


Time invariance and linear transformation

Let us start with a time-varying system whose impulse response is a two dimensional function and see how the condition of time-invariance helps us reduce it to one dimension. For example, suppose the input signal is x(t) where its index set is the real line, i.e., t in mathbb{R}. The linear operator mathbb{H} represents the system operating on the input signal. The appropriate operator for this index set is a two-dimensional function A parameter is a measurement or value on which something else depends. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...

h(t_1, t_2) mbox{ where } t_1, t_2 in mathbb{R}

Since mathbb{H} is a linear operator, the action of the system on the input signal x(t) is a linear transformation represented by the following superposition integral In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... The word linear comes from the Latin word linearis, which means created by lines. ...

y(t_1) = int_{-infty}^{infty} h(t_1, t_2) , x(t_2) , d t_2

If the linear operator mathbb{H} is also time-invariant, then In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... A time-invariant system is one whose output does not depend explicitly on time. ...

h(t_1, t_2) = h(t_1 + tau, t_2 + tau) qquad forall , tau in mathbb{R}

If we let

tau = -t_2 ,

then it follows that

h(t_1, t_2) = h(t_1 - t_2, 0) ,

We usually drop the zero second argument to h(t1,t2) for brevity of notation so that the superposition integral now becomes the familiar convolution integral used in filtering For the computer science usage see convolution (computer science) . In mathematics and in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version...

y(t_1) = int_{-infty}^{infty} h(t_1 - t_2) , x(t_2) , d t_2 = (h * x) (t_1)

Thus, the convolution integral represents the effect of a linear, time-invariant system on any input function. For a finite-dimensional analog, see the article on a circulant matrix. For the computer science usage see convolution (computer science) . In mathematics and in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version... The word linear comes from the Latin word linearis, which means created by lines. ... A time-invariant system is one whose output does not depend explicitly on time. ... In linear algebra, a circulant matrix is a special kind of Toeplitz matrix where each row vector is shifted one element to the right relative to the preceding row vector. ...


Impulse response

If we input a Dirac delta function to this system, the result of the LTI transformation is known as the impulse response since the delta function is an ideal impulse. We illustrate this idea as follows: The Dirac delta function, introduced by Paul Dirac, can be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere, and a total integral of one. ... In the language of mathematics, the impulse response of a linear transformation is the image of Diracs delta function under the transformation. ...

(h * delta) (t) = int_{-infty}^{infty} h(t - tau) , delta (tau) , d tau = h(t) (by definition of the delta function)

Note that The Dirac delta function, introduced by Paul Dirac, can be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere, and a total integral of one. ...

h(t) = h(t_1 - t_2, 0) ,!mbox{ where } t = t_1 - t_2

so that h(t) is the impulse response of the system. We can also think of the delta function as the identity operator for LTI operators. In linear algebra, the identity matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere. ...


Complex exponentials as eigenfunctions

Recall that j =i =the square root of -1. The complex exponential functions e^{j omega t} mbox{ where } omega in mathbb{R} are eigenfunctions of a linear, time-invariant operator. A simple proof illustrates this concept. The exponential function is one of the most important functions in mathematics. ... In this transformation of the Mona Lisa, the blue vector has been rotated, but the red one has not. ... The word linear comes from the Latin word linearis, which means created by lines. ... A time-invariant system is one whose output does not depend explicitly on time. ...


Suppose the input is x(t) = ,!e^{j omega t}. The transformation of this function is then

int_{-infty}^{infty} h(t - tau) , e^{j omega tau} , d tau

which is equivalent to the following by the commutative property of convolution For the computer science usage see convolution (computer science) . In mathematics and in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version...

int_{-infty}^{infty} h(tau) , e^{j omega (t - tau)} , d tau = e^{j omega t} int_{-infty}^{infty} h(tau) , e^{-j omega tau} , d tau = e^{j omega t} mathcal{F} { h(t)} = e^{j omega t} H(j omega)

where mathcal{F} is the Fourier transform. The Fourier transform, named after Joseph Fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions, i. ...


So, ,!e^{j omega t} is an eigenfunction of an LTI system because the system response is ,!e^{j omega t} scaled by an amount H(ω). Therefore, the eigenvalue spectrum is the Fourier transform of the operator mathbb{H}. In this transformation of the Mona Lisa, the blue vector has been rotated, but the red one has not. ... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ... The Fourier transform, named after Joseph Fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions, i. ...


See also


  Results from FactBites:
 
PlanetMath: linear time invariant system (183 words)
State-space equations, which is commonly used in modern control theory and mechanical systems.
Note that all transfer functions are LTI systems, but not all state-space equations are LTI systems.
This is version 8 of linear time invariant system, born on 2004-05-20, modified 2006-09-16.
  More results at FactBites »

 
 

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