Signal Processing and Linear Systems-B.P.Lathi copy

A signal f t delayed by t seconds right shifted is

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Unformatted text preview: sinusoid, a nd a n e xponentially varying sinusoid. A signal t hat is symmetrical a bout t he vertical axis (t = 0) is a n e ven function of time, a nd a signal t hat is a ntisymmetrical a bout t he vertical axis is a n o dd function of time. T he p roduct of a n even function with a n o dd function results in a n o dd function. However, t he p roduct of a n even function with a n even function or a n o dd function with a n o dd function results in a n even function. T he a rea u nder a n o dd function from t = - a t o a is always zero regardless of t he value of a. O n t he o ther h and, t he a rea under an even function from t = - a t o a is two times t he a rea u nder t he same function from t = 0 t o a (or from t = - a t o 0). Every signal can be expressed as a sum of o dd a nd even function of time. A system processes i nput signals t o p roduce o utput signals (response). T he i nput is t he cause and t he o utput is its effect. In general, t he o utput is affected by two causes: t he i nternal conditions of t he s ystem (such as t he initial conditions) a nd t he e xternal input. Systems c an be classified in several ways: 1. Linear systems are characterized by the linearity property, which implies superposition; if several causes (such as various inputs a nd i nitial conditions) are acting on a linear system, t he t otal effect (response) is t he s um of t he responses from each cause, assuming t hat all t he remaining causes are absent. A system is nonlinear if it is n ot linear. 2. T ime-invariant systems are characterized by t he fact t hat s ystem parameters do not change with time. T he p arameters of time-varying p arameter s ystems change with time. 3. For memoryless (or instantaneous) systems, t he s ystem response a t any i nstant t d epends only on t he p resent value o f t he i nput (value a t t). For systems with memory (also known as dynamic systems), t he s ystem response a t any instant t d epends not only on t he p resent value of t he i nput, b ut also on t he p ast values of t he i nput (values before t ). 4. In contrast, if a system response a t t also depends on t he f uture values of t he i nput (values of i nput beyond t ), t he s ystem is noncausal. In causal systems, t he response does n ot d epend on t he f uture values of t he i nput. Because of t he dependence of t he response on t he f uture values of input, t he effect (response) 96 1 I ntroduction t o Signals a nd Systems of non causal systems occurs before cause. When t he i ndependent variable is t ime ( temporal systems), the noncausal systems are prophetic systems, a nd therefore, unrealizable, although close approximation is possible with some time delay in t he response. Noncausal systems with independent variables o ther t han t ime (e.g., space) are realizable. 5. I f t he d imensions of system elements are small compared t o t he wavelengths of t he signals, we may assume t hat each element is lumped a t a single point in space, and t he s ystem may be considered as a lumped-parameter system. T he signals u nder t his assumption are functions of time only. I f t his assumption does n ot...
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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

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