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

T he signals here are functions of space as well as

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Unformatted text preview: as discrete-time. Show t hat a s ystem described by t he e quation below is noncausal: Additional Classification o f Systems '+5 y (t) = 1 f(7') d7' 1 -5 Show t hat t his s ystem c an b e realized physically if we a ccept a delay o f 5 seconds in t he o utput. 'V T here are additional classes of systems, such as i nvertible a nd n oninvertible systems. A system S performs certain operation(s) on i nput signal(s). I f we c an obtain t he i nput f (t) b ack from t he o utput y(t) by some operation, t he s ystem 88 1 I ntroduction t o Signals a nd Systems S is said t o b e invertible. For a noninvertible system, different inputs can result in t he s ame o utput (as in a rectifier), a nd i t is impossible t o d etermine t he i nput for a given o utput. Therefore, for a n invertible system, it is essential t hat d istinct i nputs r esult in d istinct o utputs so t hat t here is one-to-one mapping between an i nput a nd t he corresponding o utput. T his ensures t hat every o utput h as a unique input. Consequently, t he s ystem is invertible. T he system t hat achieves this inverse operation [of o btaining f (t) from y(t)] is t he i nverse s ystem o f S . For instance, A system whose input and o utput are related by equation y (t) = a f (t) + b is a n invertible system. B ut a rectifier, specified by t he e quation y (t) = If(t)1 is noninvertible because t he rectification operation cannot be undone. An ideal differentiator is noninvertible because integration of its o utput c annot restore t he o riginal signal unless we know one piece of information a bout t he signal. For instance, if f (t) = 3t + 5, t he o utput of t he differentiator is y (t) = 3. I f t his o utput is a pplied t o a n i ntegrator, the o utput is 3t + c, where c is a n a rbitrary c onstant. I f we k now one piece of information a bout f (t), such as f(O) = 5, we c an determine t he i nput t o be f (t) = 3t + 5. T hus, a differentiator along with one piece of information (known as auxiliary condition) is a n invertible system.t Similarly, a system consisting of a cascade of two differentiators is invertible, if we know two independent pieces of information (auxiliary conditions) about t he i nput signal. In addition, s ystems can also be classified as s table or u nstable systems. T he c oncept of s tability is discussed in more d epth in later chapters. f:::" \l + f (t) Fig. 1 .31 Circuit for Example 1.10. • E xample 1 .10 For the series R LC circuit of Fig. 1.31, find the input-output equation relating the input voltage f (t) to the output current (loop current) y(t). Application of the Kirchhoff's voltage law around the loop yields + VR(t) + v c(t) = VL(t) -d + 3y(t) + 2 .!!.. As m entioned earlier, systems theory encompasses a variety of systems, such as electrical, mechanical, hydraulic, acoustic, electromechanical, a nd chemical, as well as social, political, economic, and biological. T he first s tep in analyzing any system is t he c onstruction of a system model, which is a m athematical expression or a rule t hat s atisfactorily approximates the dy...
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