En el análisis de sistemas , entre otros campos de estudio, un sistema lineal invariante en el tiempo (o "sistema LTI") es un sistema que produce una señal de salida a partir de cualquier señal de entrada sujeta a las limitaciones de linealidad e invariancia temporal ; estos términos se definen brevemente a continuación . Estas propiedades se aplican (exactamente o aproximadamente) a muchos sistemas físicos importantes, en cuyo caso la respuesta y (t) del sistema a una entrada arbitraria x (t) se puede encontrar directamente usando la convolución : y (t) = x (t) ∗ h (t) donde h (t) se denomina respuesta al impulso del sistemay ∗ representa la convolución (que no debe confundirse con la multiplicación, como se emplea con frecuencia con el símbolo en los lenguajes informáticos ). Además, existen métodos sistemáticos para resolver cualquier sistema de este tipo (determinando h (t) ), mientras que los sistemas que no cumplen con ambas propiedades son generalmente más difíciles (o imposibles) de resolver analíticamente. Un buen ejemplo de un sistema LTI es cualquier circuito eléctrico que consta de resistencias, condensadores, inductores y amplificadores lineales. [1]
La teoría del sistema lineal invariante en el tiempo también se utiliza en el procesamiento de imágenes , donde los sistemas tienen dimensiones espaciales en lugar de, o además de, una dimensión temporal. Estos sistemas pueden denominarse invariantes de traducción lineal para dar a la terminología el alcance más general. En el caso de sistemas genéricos de tiempo discreto (es decir, muestreados ), el término correspondiente es invariante de desplazamiento lineal . La teoría de sistemas LTI es un área de matemáticas aplicadas que tiene aplicaciones directas en el análisis y diseño de circuitos eléctricos , procesamiento de señales y diseño de filtros , teoría de control , ingeniería mecánica , procesamiento de imágenes , diseño de instrumentos de medición de muchos tipos, espectroscopía de RMN [ cita requerida ] y muchas otras áreas técnicas donde se presentan sistemas de ecuaciones diferenciales ordinarias .
Descripción general
Las propiedades definitorias de cualquier sistema LTI son la linealidad y la invariancia en el tiempo .
- Linealidad significa que la relación entre la entrada y la salida es el resultado de ecuaciones diferenciales lineales , es decir, ecuaciones diferenciales que emplean solo operadores lineales . Un sistema lineal que mapea una entrada x (t) a una salida y (t) mapeará una entrada escalada ax (t) a una salida ay (t) igualmente escalada por el mismo factor a . Y el principio de superposición se aplica a un sistema lineal: si el sistema mapea las entradas x 1 (t) y x 2 (t) a las salidas y 1 (t) y y 2 (t) respectivamente, entonces mapeará x 3 (t) = x 1 (t) + x 2 (t) a la salida y 3 (t) donde y 3 (t) = y 1 (t) + y 2 (t) .
- La invariancia de tiempo significa que si aplicamos una entrada al sistema ahora o T segundos a partir de ahora, la salida será idéntica excepto por un retraso de tiempo de T segundos. Es decir, si la salida debida a la entrada es , luego la salida debido a la entrada es . Por lo tanto, el sistema es invariante en el tiempo porque la salida no depende del momento particular en que se aplica la entrada.
El resultado fundamental de la teoría del sistema LTI es que cualquier sistema LTI puede caracterizarse por completo por una única función llamada respuesta al impulso del sistema . La salida del sistema y (t) es simplemente la convolución de la entrada al sistema x (t) con la respuesta al impulso del sistema h (t) . A esto se le llama un sistema de tiempo continuo . De manera similar, un sistema invariante en el tiempo lineal en tiempo discreto (o, más generalmente, "invariante en el desplazamiento") se define como uno que opera en tiempo discreto : y i = x i ∗ h i donde y, x y h son secuencias y la convolución, en tiempo discreto, usa una suma discreta en lugar de una integral.
Los sistemas LTI también se pueden caracterizar en el dominio de la frecuencia por la función de transferencia del sistema , que es la transformada de Laplace de la respuesta al impulso del sistema (o transformada Z en el caso de sistemas de tiempo discreto). Como resultado de las propiedades de estas transformadas, la salida del sistema en el dominio de la frecuencia es el producto de la función de transferencia y la transformada de la entrada. En otras palabras, la convolución en el dominio del tiempo es equivalente a la multiplicación en el dominio de la frecuencia.
Para todos los sistemas LTI, las funciones propias y las funciones básicas de las transformadas son exponenciales complejas . Esto es, si la entrada a un sistema es la forma de onda compleja para una amplitud compleja y frecuencia compleja , la salida será una constante compleja multiplicada por la entrada, digamos para una nueva amplitud compleja . El radio es la función de transferencia en frecuencia .
Dado que las sinusoides son una suma de exponenciales complejas con frecuencias conjugadas complejas, si la entrada al sistema es una sinusoide, entonces la salida del sistema también será una sinusoide, quizás con una amplitud diferente y una fase diferente , pero siempre con la misma frecuencia al alcanzar el estado estable. Los sistemas LTI no pueden producir componentes de frecuencia que no estén en la entrada.
La teoría del sistema LTI es buena para describir muchos sistemas importantes. La mayoría de los sistemas LTI se consideran "fáciles" de analizar, al menos en comparación con el caso no lineal o variable en el tiempo . Cualquier sistema que pueda modelarse como una ecuación diferencial lineal con coeficientes constantes es un sistema LTI. Ejemplos de tales sistemas son los circuitos eléctricos compuestos por resistencias , inductores y condensadores (circuitos RLC). Los sistemas ideales de resorte-masa-amortiguador también son sistemas LTI y son matemáticamente equivalentes a los circuitos RLC.
La mayoría de los conceptos del sistema LTI son similares entre los casos de tiempo continuo y tiempo discreto (invariante de desplazamiento lineal). En el procesamiento de imágenes, la variable de tiempo se reemplaza con dos variables de espacio, y la noción de invariancia de tiempo se reemplaza por invariancia de cambio bidimensional. Al analizar bancos de filtros y sistemas MIMO , a menudo es útil considerar vectores de señales.
A linear system that is not time-invariant can be solved using other approaches such as the Green function method. The same method must be used when the initial conditions of the problem are not null.[citation needed]
Sistemas de tiempo continuo
Impulse response and convolution
The behavior of a linear, continuous-time, time-invariant system with input signal x(t) and output signal y(t) is described by the convolution integral:[2]
(using commutativity)
where is the system's response to an impulse: is therefore proportional to a weighted average of the input function The weighting function is simply shifted by amount As changes, the weighting function emphasizes different parts of the input function. When is zero for all negative depends only on values of prior to time and the system is said to be causal.
To understand why the convolution produces the output of an LTI system, let the notation represent the function with variable and constant And let the shorter notation represent Then a continuous-time system transforms an input function, into an output function, . And in general, every value of the output can depend on every value of the input. This concept is represented by:
where is the transformation operator for time . In a typical system, depends most heavily on the values of that occurred near time Unless the transform itself changes with the output function is just constant, and the system is uninteresting.
For a linear system, must satisfy Eq.1 :
(Eq.2)
And the time-invariance requirement is:
(Eq.3)
In this notation, we can write the impulse response as
Similarly:
(using Eq.3)
Substituting this result into the convolution integral:
which has the form of the right side of Eq.2 for the case and
Eq.2 then allows this continuation:
In summary, the input function, can be represented by a continuum of time-shifted impulse functions, combined "linearly", as shown at Eq.1. The system's linearity property allows the system's response to be represented by the corresponding continuum of impulse responses, combined in the same way. And the time-invariance property allows that combination to be represented by the convolution integral.
The mathematical operations above have a simple graphical simulation.[3]
Exponentials as eigenfunctions
An eigenfunction is a function for which the output of the operator is a scaled version of the same function. That is,
where f is the eigenfunction and is the eigenvalue, a constant.
The exponential functions , where , are eigenfunctions of a linear, time-invariant operator. A simple proof illustrates this concept. Suppose the input is . The output of the system with impulse response is then
which, by the commutative property of convolution, is equivalent to
where the scalar
is dependent only on the parameter s.
So the system's response is a scaled version of the input. In particular, for any , the system output is the product of the input and the constant . Hence, is an eigenfunction of an LTI system, and the corresponding eigenvalue is .
Direct proof
It is also possible to directly derive complex exponentials as eigenfunctions of LTI systems.
Let's set some complex exponential and a time-shifted version of it.
by linearity with respect to the constant .
by time invariance of .
So . Setting and renaming we get :
i.e. that a complex exponential as input will give a complex exponential of same frequency as output.
Fourier and Laplace transforms
The eigenfunction property of exponentials is very useful for both analysis and insight into LTI systems. The one-sided Laplace transform
is exactly the way to get the eigenvalues from the impulse response. Of particular interest are pure sinusoids (i.e., exponential functions of the form where and ). The Fourier transform gives the eigenvalues for pure complex sinusoids. Both of and are called the system function, system response, or transfer function.
The Laplace transform is usually used in the context of one-sided signals, i.e. signals that are zero for all values of t less than some value. Usually, this "start time" is set to zero, for convenience and without loss of generality, with the transform integral being taken from zero to infinity (the transform shown above with lower limit of integration of negative infinity is formally known as the bilateral Laplace transform).
The Fourier transform is used for analyzing systems that process signals that are infinite in extent, such as modulated sinusoids, even though it cannot be directly applied to input and output signals that are not square integrable. The Laplace transform actually works directly for these signals if they are zero before a start time, even if they are not square integrable, for stable systems. The Fourier transform is often applied to spectra of infinite signals via the Wiener–Khinchin theorem even when Fourier transforms of the signals do not exist.
Due to the convolution property of both of these transforms, the convolution that gives the output of the system can be transformed to a multiplication in the transform domain, given signals for which the transforms exist
One can use the system response directly to determine how any particular frequency component is handled by a system with that Laplace transform. If we evaluate the system response (Laplace transform of the impulse response) at complex frequency s=jω, where ω=2πf, we obtain |H(s)| which is the system gain for frequency f. The relative phase shift between the output and input for that frequency component is likewise given by arg(H(s)).
Examples
- A simple example of an LTI operator is the derivative.
- (i.e., it is linear)
- (i.e., it is time invariant)
- When the Laplace transform of the derivative is taken, it transforms to a simple multiplication by the Laplace variable s.
- That the derivative has such a simple Laplace transform partly explains the utility of the transform.
- Another simple LTI operator is an averaging operator
- By the linearity of integration,
- it is linear. Additionally, because
- it is time invariant. In fact, can be written as a convolution with the boxcar function. That is,
- where the boxcar function
Important system properties
Some of the most important properties of a system are causality and stability. Causality is a necessity for a physical system whose independent variable is time, however this restriction is not present in other cases such as image processing.
Causality
A system is causal if the output depends only on present and past, but not future inputs. A necessary and sufficient condition for causality is
where is the impulse response. It is not possible in general to determine causality from the two-sided Laplace transform. However when working in the time domain one normally uses the one-sided Laplace transform which requires causality.
Stability
A system is bounded-input, bounded-output stable (BIBO stable) if, for every bounded input, the output is finite. Mathematically, if every input satisfying
leads to an output satisfying
(that is, a finite maximum absolute value of implies a finite maximum absolute value of ), then the system is stable. A necessary and sufficient condition is that , the impulse response, is in L1 (has a finite L1 norm):
In the frequency domain, the region of convergence must contain the imaginary axis .
As an example, the ideal low-pass filter with impulse response equal to a sinc function is not BIBO stable, because the sinc function does not have a finite L1 norm. Thus, for some bounded input, the output of the ideal low-pass filter is unbounded. In particular, if the input is zero for and equal to a sinusoid at the cut-off frequency for , then the output will be unbounded for all times other than the zero crossings.[dubious ]
Sistemas de tiempo discreto
Almost everything in continuous-time systems has a counterpart in discrete-time systems.
Discrete-time systems from continuous-time systems
In many contexts, a discrete time (DT) system is really part of a larger continuous time (CT) system. For example, a digital recording system takes an analog sound, digitizes it, possibly processes the digital signals, and plays back an analog sound for people to listen to.
In practical systems, DT signals obtained are usually uniformly sampled versions of CT signals. If is a CT signal, then the sampling circuit used before an analog-to-digital converter will transform it to a DT signal:
where T is the sampling period. Before sampling, the input signal is normally run through a so-called Nyquist filter which removes frequencies above the "folding frequency" 1/(2T); this guarantees that no information in the filtered signal will be lost. Without filtering, any frequency component above the folding frequency (or Nyquist frequency) is aliased to a different frequency (thus distorting the original signal), since a DT signal can only support frequency components lower than the folding frequency.
Impulse response and convolution
Let represent the sequence
And let the shorter notation represent
A discrete system transforms an input sequence, into an output sequence, In general, every element of the output can depend on every element of the input. Representing the transformation operator by , we can write:
Note that unless the transform itself changes with n, the output sequence is just constant, and the system is uninteresting. (Thus the subscript, n.) In a typical system, y[n] depends most heavily on the elements of x whose indices are near n.
For the special case of the Kronecker delta function, the output sequence is the impulse response:
For a linear system, must satisfy:
(Eq.4)
And the time-invariance requirement is:
(Eq.5)
In such a system, the impulse response, characterizes the system completely. I.e., for any input sequence, the output sequence can be calculated in terms of the input and the impulse response. To see how that is done, consider the identity:
which expresses in terms of a sum of weighted delta functions.
Therefore:
where we have invoked Eq.4 for the case and
And because of Eq.5, we may write:
Therefore:
(commutativity)
which is the familiar discrete convolution formula. The operator can therefore be interpreted as proportional to a weighted average of the function x[k]. The weighting function is h[-k], simply shifted by amount n. As n changes, the weighting function emphasizes different parts of the input function. Equivalently, the system's response to an impulse at n=0 is a "time" reversed copy of the unshifted weighting function. When h[k] is zero for all negative k, the system is said to be causal.
Exponentials as eigenfunctions
An eigenfunction is a function for which the output of the operator is the same function, scaled by some constant. In symbols,
- ,
where f is the eigenfunction and is the eigenvalue, a constant.
The exponential functions , where , are eigenfunctions of a linear, time-invariant operator. is the sampling interval, and . A simple proof illustrates this concept.
Suppose the input is . The output of the system with impulse response is then
which is equivalent to the following by the commutative property of convolution
where
is dependent only on the parameter z.
So is an eigenfunction of an LTI system because the system response is the same as the input times the constant .
Z and discrete-time Fourier transforms
The eigenfunction property of exponentials is very useful for both analysis and insight into LTI systems. The Z transform
is exactly the way to get the eigenvalues from the impulse response[clarification needed]. Of particular interest are pure sinusoids, i.e. exponentials of the form , where . These can also be written as with [clarification needed]. The discrete-time Fourier transform (DTFT) gives the eigenvalues of pure sinusoids[clarification needed]. Both of and are called the system function, system response, or transfer function'.
Like the one-sided Laplace transform, the Z transform is usually used in the context of one-sided signals, i.e. signals that are zero for t<0. The discrete-time Fourier transform Fourier series may be used for analyzing periodic signals.
Due to the convolution property of both of these transforms, the convolution that gives the output of the system can be transformed to a multiplication in the transform domain. That is,
Just as with the Laplace transform transfer function in continuous-time system analysis, the Z transform makes it easier to analyze systems and gain insight into their behavior.
Examples
- A simple example of an LTI operator is the delay operator .
- (i.e., it is linear)
- (i.e., it is time invariant)
- The Z transform of the delay operator is a simple multiplication by z−1. That is,
- Another simple LTI operator is the averaging operator
- Because of the linearity of sums,
- and so it is linear. Because,
- it is also time invariant.
Important system properties
The input-output characteristics of discrete-time LTI system are completely described by its impulse response . Two of the most important properties of a system are causality and stability. Non-causal (in time) systems can be defined and analyzed as above, but cannot be realized in real-time. Unstable systems can also be analyzed and built, but are only useful as part of a larger system whose overall transfer function is stable.
Causality
A discrete-time LTI system is causal if the current value of the output depends on only the current value and past values of the input.[4] A necessary and sufficient condition for causality is
where is the impulse response. It is not possible in general to determine causality from the Z transform, because the inverse transform is not unique[dubious ]. When a region of convergence is specified, then causality can be determined.
Stability
A system is bounded input, bounded output stable (BIBO stable) if, for every bounded input, the output is finite. Mathematically, if
implies that
(that is, if bounded input implies bounded output, in the sense that the maximum absolute values of and are finite), then the system is stable. A necessary and sufficient condition is that , the impulse response, satisfies
In the frequency domain, the region of convergence must contain the unit circle (i.e., the locus satisfying for complex z).
Notas
- ^ Hespanha 2009, p. 78.
- ^ Crutchfield, p. 1. Welcome!
- ^ Crutchfield, p. 1. Exercises
- ^ Phillips 2007, p. 508.
Ver también
- Circulant matrix
- Frequency response
- Impulse response
- System analysis
- Green function
- Signal-flow graph
Referencias
- Phillips, C.l., Parr, J.M., & Riskin, E.A (2007). Signals, systems and Transforms. Prentice Hall. ISBN 978-0-13-041207-2.CS1 maint: multiple names: authors list (link)
- Hespanha, J.P. (2009). Linear System Theory. Princeton university press. ISBN 978-0-691-14021-6.
- Crutchfield, Steve (October 12, 2010), "The Joy of Convolution", Johns Hopkins University, retrieved November 21, 2010
- Vaidyanathan, P. P.; Chen, T. (May 1995). "Role of anticausal inverses in multirate filter banks — Part I: system theoretic fundamentals" (PDF). IEEE Trans. Signal Process. 43 (6): 1090. Bibcode:1995ITSP...43.1090V. doi:10.1109/78.382395.
Otras lecturas
- Porat, Boaz (1997). A Course in Digital Signal Processing. New York: John Wiley. ISBN 978-0-471-14961-3.
- Vaidyanathan, P. P.; Chen, T. (May 1995). "Role of anticausal inverses in multirate filter banks — Part I: system theoretic fundamentals" (PDF). IEEE Trans. Signal Process. 43 (5): 1090. Bibcode:1995ITSP...43.1090V. doi:10.1109/78.382395.
enlaces externos
- ECE 209: Review of Circuits as LTI Systems – Short primer on the mathematical analysis of (electrical) LTI systems.
- ECE 209: Sources of Phase Shift – Gives an intuitive explanation of the source of phase shift in two common electrical LTI systems.
- JHU 520.214 Signals and Systems course notes. An encapsulated course on LTI system theory. Adequate for self teaching.
- LTI system example: RC low-pass filter. Amplitude and phase response.