Lti System Theory

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''. $y_i(t)$ is the output when the input is $x_i(t)$ . 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. Equivalently, any LTI systems can be characterized in the '' Frequency Domain '' by the system's Transfer Function , which is the transform of the system's impulse response. Different contexts use different transforms. As a result of the properties of the most common transforms, the output of the system in the frequency domain is the product of the transfer function and the transform of the input (or perhaps, the most common transforms are common precisely because of this nice property). In other words, convolution in the time domain is equivalent to multiplication in the frequency domain. For all LTI systems, the Eigenfunction s are Complex Exponentials . Because sinusoids are a linear combination of complex exponentials, 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 . LTI system theory is exceedingly good at describing many important systems. Most LTI systems are considered "easy" to analyze, at least compared to the time-varying and/or Non-linear case. Any system that can be modeled as a Differential Equation with constant coefficients is an LTI system. Examples of such systems are electrical circuits made up of resistors, inductors, and capacitors (RLC circuits). Simple spring-mass-damper systems are mathematically equivalent to RLC circuits. LTI systems need not be restricted to continuous-time systems. Discrete-time systems can also be LTI. Most of the concepts are similar between the continuous and discrete time cases. In image processing, the time variable is replaced with 2 space variables, and the notion of time-invariance is replaced by shift-invariance. When analyzing Filter Bank s and MIMO systems, it is often useful to consider Vectors of signals. CONTINUOUS TIME SYSTEMS Time invariance and linear transformation Let us start with a time-varying system whose impulse response is a 2-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 $\mathcal{H}$ represents the system operating on the input signal. The appropriate operator for this index set is a 2-dimensional function : $h(t_1, t_2) \mbox{ where } t_1, t_2 \in \mathbb{R}.$ Since $\mathcal{H}$ 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 : $y(t_1) = \int_{-\infty}^{\infty} h(t_1, t_2) \, x(t_2) \, d t_2.$ If the Linear Operator $\mathcal{H}$ is also Time-invariant , then : $h(t_1, t_2) = h(t_1 + au, t_2 + au) \qquad orall \, au \in \mathbb{R}.$ If we let : $au = -t_2, \,$ then it follows that : $h(t_1, t_2) = h(t_1 - t_2, 0). \,$ We usually drop the zero second argument to $h (t_1, t_2)$ for brevity of notation so that the superposition integral now becomes the familiar Convolution integral used in filtering 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 . Impulse response If we input a Dirac Delta Function to this system, the result of the LTI transformation is known as the Impulse Response because the delta function is an ideal impulse. We illustrate this idea as follows: \delta) (t) = \int_{-\infty}^{\infty} h(t - au) \, \delta ( au) \, d au = h(t), (by the sifting property of the Delta Function ). Note that : $h(t) = h(t, 0) \ (\mbox{with } t = t_1 - t_2)$ so that $h(t)$ is the impulse response of the system. The impulse response can be used to find the response of ''any'' input in the following way. Again using the sifting property of the $\delta(t)$ , we can write any input as a superposition of deltas: : $x(t) = \int_{-\infty}^\infty x( au) \delta(t- au) d t$ Applying the system to the input, : $\mathcal{H} x(t) = \mathcal{H} \int_{-\infty}^\infty x( au) \delta(t- au) d t$ : $\quad = \int_{-\infty}^\infty \mathcal{H} x( au) \delta(t- au) d t$ (because $\mathcal{H}$ is linear and can pass inside the integral) : $\quad = \int_{-\infty}^\infty x( au) \mathcal{H} \delta(t- au) d t$ (because $x( au)$ is constant in ''t'' and $\mathcal{H}$ is linear) : $\quad = \int_{-\infty}^\infty x( au) h(t- au) d t$ (by definition of $h(t)$ ) All information about the system is contained in the impulse response $h(t)$ . Exponentials as eigenfunctions An Eigenfunction is a function for which the output of the operator is the same function, just scaled by some amount. In symbols, : $\mathcal{H}f = \lambda f$ , where ''f'' is the eigenfunction and $\lambda$ is the Eigenvalue , a constant. The Exponential Function s $e^{s t}$ , where $s \in \mathbb{C}$ , are Eigenfunction s of a Linear , Time-invariant operator. A simple proof illustrates this concept. Suppose the input is $x(t) = e^{s t}$ . The output of the system with impulse response $h(t)$ is then : $\int_{-\infty}^{\infty} h(t - au) e^{s au} d au$ which is equivalent to the following by the commutative property of Convolution : $\int_{-\infty}^{\infty} h( au) \, e^{s (t - au)} \, d au$ : $\quad = e^{s t} \int_{-\infty}^{\infty} h( au) \, e^{-s au} \, d au$ : $\quad = e^{s t} H(s)$ , where : $H(s) = \int_{-\infty}^\infty h(t) e^{-s t} d t$ is dependent only on the parameter ''s''. So, $e^{s t}$ is an Eigenfunction of an LTI system because the system response is the same as the input times the constant $H(s)$ . Fourier and Laplace Transforms The eigenfunction property of exponentials is very useful for both analysis and insight into LTI systems. The Laplace Transform : $H(s) = \mathcal{L}\{h(t)\} = \int_{-\infty}^\infty h(t) e^{-s t} d t$ is exactly the way to get the eigenvalues from the impuse response. Of particular interest are pure sinusoids, i.e. exponentials of the form $e^{j \omega t}$ where $\omega \in \mathbb{R}$ . These are generally called complex exponentials even though the argument is purely imaginary. The Fourier Transform $H(j \omega) = \mathcal{F}\{h(t)\}$ gives the eigenvalues of pure sinusoids. Both of $H(s)$ and $H(j\omega)$ 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. The Fourier transform is used for analyzing signals that are infinite in extent. 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. x)(t) = \int_{-\infty}^\infty h(t - au) x( au) d au : $\quad = \mathcal{L}^{-1}\{H(s)X(s)\}$
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	CATEGORIES ABOUT LTI SYSTEM THEORY

	digital signal processing

	electrical engineering

	control theory

	signal processing

	fourier analysis

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Lti System Theory