Linear system

A general deterministic system can be described by an operator, , that maps an input, , as a function of to an output, , a type of black box description. A system is linear if and only if it satisfies the superposition principle, or equivalently both the additivity and homogeneity properties, without restrictions (that is, for all inputs, all scaling constants and all time.) The superposition principle means that a linear combination of inputs to the system produces a linear combination of the individual zero-state outputs (that is, outputs setting the initial conditions to zero) corresponding to the individual inputs. In a system that satisfies the homogeneity property, scaling the input always results in scaling the zero-state response by the same factor. The output versus input graph of a linear system need not be a straight line through the origin. For example, consider a system described by y(t) = k \, \frac{\mathrm dx(t)}{\mathrm dt} (such as a constant-capacitance capacitor or a constant-inductance inductor). It is linear because it satisfies the superposition principle. However, when the input is a sinusoid, the output is also a sinusoid, and so its output-input plot is an ellipse centered at the origin rather than a straight line passing through the origin. Also, the output of a linear system can contain harmonics (and have a smaller fundamental frequency than the input) even when the input is a sinusoid. For example, consider a system described by y(t) = (1.5 + \cos{(t)}) \, x(t). It is linear because it satisfies the superposition principle. However, when the input is a sinusoid of the form x(t) = \cos{(3t)}, using product-to-sum trigonometric identities it can be easily shown that the output is y(t) = 1.5 \cos{(3t)} + 0.5 \cos{(2t)} + 0.5 \cos{(4t)}, that is, the output doesn't consist only of sinusoids of same frequency as the input (), but instead also of sinusoids of frequencies and ; furthermore, taking the least common multiple of the fundamental period of the sinusoids of the output, it can be shown the fundamental angular frequency of the output is , which is different than that of the input. ==Time-varying impulse response==

The time-varying impulse response of a linear system is defined as the response of the system at time t = t2 to a single impulse applied at time In other words, if the input to a linear system is x(t) = \delta(t - t_1) where represents the Dirac delta function, and the corresponding response of the system is y(t=t_2) = h(t_2, t_1) then the function is the time-varying impulse response of the system. Since the system cannot respond before the input is applied the following causality condition must be satisfied: h(t_2, t_1) = 0, t_2 ==The convolution integral==

The output of any general continuous-time linear system is related to the input by an integral which may be written over a doubly infinite range because of the causality condition: y(t) = \int_{-\infty}^{t} h(t,t') x(t')dt' = \int_{-\infty}^{\infty} h(t,t') x(t') dt' If the properties of the system do not depend on the time at which it is operated then it is said to be time-invariant and is a function only of the time difference which is zero for (namely ). By redefinition of it is then possible to write the input-output relation equivalently in any of the ways, y(t) = \int_{-\infty}^{t} h(t-t') x(t') dt' = \int_{-\infty}^{\infty} h(t-t') x(t') dt' = \int_{-\infty}^{\infty} h(\tau) x(t-\tau) d \tau = \int_{0}^{\infty} h(\tau) x(t-\tau) d \tau Linear time-invariant systems are most commonly characterized by the Laplace transform of the impulse response function called the transfer function which is: H(s) =\int_0^\infty h(t) e^{-st}\, dt. In applications this is usually a rational algebraic function of . Because is zero for negative , the integral may equally be written over the doubly infinite range and putting follows the formula for the frequency response function: H(i\omega) = \int_{-\infty}^{\infty} h(t) e^{-i\omega t} dt ==Discrete-time systems==

The output of any discrete time linear system is related to the input by the time-varying convolution sum: y[n] = \sum_{m =-\infty}^{n} { h[n,m] x[m] } = \sum_{m =-\infty}^{\infty} { h[n,m] x[m] } or equivalently for a time-invariant system on redefining , y[n] = \sum_{k =0}^{\infty} { h[k] x[n-k] } = \sum_{k =-\infty}^{\infty} { h[k] x[n-k] } where k = n-m represents the lag time between the stimulus at time m and the response at time n. ==See also==