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Linear System Theory Introduction
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  1 Linear Systems, Superposition, and Convolution In this section we provide a brief introduction to linear systems theory. It is based on the followingdefinition:Let  L  denote a  time-invariant, linear system  with input    and output   :  L  Such systems are  additive  (the response to the sum of two inputs    and   equals the sum of theresponses to each input taken individually):       and  homogeneous  (the system can be scaled by the magnitude of the input   , where   is a scalar):    Taken together, they enjoy the  Principle of Superposition:      Several idealized input functions are of special importance in analyzing systems, the  Dirac delta function  and the  unit step function . The unit step is defined as:  if    ;  if    ;undefined   if   (1)and the Dirac delta function is its derivative:  Of course, since the step function is not continuous (notice the value at 0), one has to be careful indefining exactly what is meant by the above equation. Formally, the   is a  distribution  defined by theintegral equation:  It can be represented as the limit of a sequence of functions such as:   with   , see Fig. 1; During this limiting process we have something that resembles a physicalapproximation to the unit impulse, the first test pattern used to evaluate the lateral inhibitory network.The result is called the  impulse response  1  −20 −10 0 10 20−0.500.51t(epislon=1)−20 −10 0 10 20−0.500.511.52t(epislon=1/2)−20 −10 0 10 20−0.500.511.52t(epislon=1/4)−20 −10 0 10 20−1−0.500.511.522.5t(epislon=1/16) Figure 1:Similarly, since the step function is the integral of the impulse function, the  step response , or theresponse to a unit step function  is the integral of the impulse response function:   In general, of course, we are interested in the response of a system not to these special functions, butto an arbitrary input. The trick is to approximate this arbitrary input as a sequence of step functions (Fig.2.12 in Oppenheim and Willsky, p91), and then to use superposition to obtain the overall response. It isnecessary to  assume that the input function   is continuous , so that the approximation is meaningful.Now, suppose the input starts at   , and time is discretized into bins   seconds apart. The firststep in the approximation has height   , a constant, and the additional step at   has height  . Remembering that    , the signal   can thus be approximated by    (2)     (3)The approximate output is then given by2     (4)     (5)Now, introduce a limiting process in which the time steps become vanishingly close, so that   ,  , the sum becomes an integral and we have:   (6)    (7)Repeating this approximation procedure for a continuous input   starting at   and such that  as   yields:    (8)    (9)  (10)Thus the output can be computed as an integral function of the impulse response. Such an integral iscalled a  convolution integral , and the above expression is often abbreviated:  ReferenceAlan V. Oppenheim and Alan S. Willsky,  Signals and Systems , second edition, Prentice-Hall Inc.,1997.3
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