This preview shows page 1. Sign up to view the full content.
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...
View Full Document