**Unformatted text preview: **D. Unit Impulse Function §(t)
The unit impulse ﬁmction 6(t) is one of the most important ﬁinctions in signals and systems. It is ﬁrst deﬁned as:
«505) = a $790,
°° (3)
ﬁg . Notice that 6(t) is undeﬁned at t: O. The 5(t) function can be visualized as a tall, narrow, rectangular pulse of unit area,
as shown below. The width of the rectangular pulse is a very small value c ~+ 0. its height is a very large value 1/6 —> on. For this reason, the impulse function is represented by a symbols as shown below. 56) Multilicati of 1
ﬁt) with 5(a) results in W) tilt) = MU) tilt)- (4a) lhat is, multiplication of a continuous-time function (Mt) with an impulse function located at t z 0 results in an impulse,
which is located at t = [l and has strength dam). Similarly, when the impulse is time-shifted by T, we have: 9W) 5(3 — T) I GMT) 50 - Tl (4b)
it follows from (43) that
/_ at) as) at = are) and: = no) at) d: : as). (5a) This result means that the area under the product of a function with an impulse 605) is equal to the value of that function
ﬁthe instant at which the unit impulse is located. Similarly, we have: I” at) at — T) at = as“). (51:) Generalized Function: The deﬁnition of Mt) in (3) is not mathematically rigorous since ﬁt) is undeﬁned at a! 2 0. This
problem is resolved by deﬁning the 6[t) function as a generalized function instead of a ordinary function. A generalized,
function is deﬁned by its effect on oﬂier Emotions instead of by its value at every instant of time. In the generalized sense,
we have: data}
(it C
/ an) d? = u(_t). DC- : 60:),
(6) E. Exponential Function e” Another important function is the exponential signal e“, where s is a complex number given by s = a + jw. We have: 6“ = ew'H-w)‘ = one”: = e‘”(cos wt + jsin wt), (7)
e‘ t - e(”—3"")‘ er’te‘wt = e”(cos wt — jsinwt). (8)
6% + es‘t ' est _ es't
3"" cos wt = , e“ smwt = _ (9)
2 23 where 5* z a — jw is the conjugate of s. The following functions are either special cases or can be expressed by e“: l) A constant k:
k = he“ 2) A monotonic exponential e“. w: 3) A sinusoid cos wt. and sin wt. coswt— esc+es t H ejwt+€-jwt
2e” 0:0 2 ’ E Even and Odd Functions A real function 3:66) is said to be even if
334—4) = $6“) and a real function 3:00) is said to be odd if
9:0(—t) = —:vo Even and odd functions have the following properties:
1) Even function x Even function 2 Even function. 2) Odd function x Odd function 2 Even ﬁmction. 3) Even function x Odd function = Odd function. cr=w=0 I ejwc _ e—jwt
smwt = — 4) Every signal aft) can be expressed as a sum of even and odd components as: = 930:) +2zv(—t) + 3:0) — $(—t). ‘wr G. Classiﬁcation of Systems In the analysis and design of systems, it is desirable to classify the systems according to the general properties that they
satisfy. In fact, the mathematical techniques for analyzing and designing systems depend heavily on the general characteristics
of the systems being considered. For this reason, it is necessary for us to develop a number of properties or categories that
can be used to describe the general characteristics of systems. For a system to poses a given property, the property must hold for every possible input signal to the system. If a property
holds for some input signals but not for others, the system does not possess that property. Thus, a counter example is
sufﬁcient to prove that a system does not possess a property. However; to prove that the system has some property, we must
prove that this property holds for every possible input signal. Systems may be classiﬁed in the following categories, among which two of the most important properties are linearity
and time-invariance. 1) Linear and nonlinear systems. 2) Time-invariant and time-varying systems. 3) Causal and noncausal systems. 4) Static (memoryless) and dynamic (with memory) systems. 5) Continuous-time and discrete-time systems. 6) Analog and digital systems. 7‘) Invertible and non-invertible systems. 8) Stabie and unstable systems. 9) Finite-dimensional and inﬁnite-dimensional systems. 1) Linear vs. Nonlinear Systems: The general class of systems can be subdivided into linear systems and nonlinear
systems. A linear system is one that satisﬁes the superposition principie. Simply stated, the principle of superposition
requires that the response of the system to a weighted sum of signals be equal to the corresponding weighted sum of
the response (outputs) of the system to each of the individual input signals. Deﬁnition: A relaxed system 1" is iinear Eff T[a11'1(t) + 0.232(6)] = 0,1T[$1(t)] + a2T[:r2(t)] (10) for any arbitrary input sequences 3:1(t) and 3320:), and any arbitrary constants a1 and (:2. Example:
too W9“ W7”: 17mm yo) = via-(t), \/ (Ha)
misﬁt/but)- yo) = met). \/ (11b)
‘lhen y(t) = 3.53m, X (11c)
aswz’LQd‘Xﬁ—gbp) y(t) = Amrt)+B,x_ (11d)
y(t] as“). 7( (1 1e) = Worm we) morva centre.)
=oh i4”— daggehnmr m Est) :(TEUWW) roam)
2 o(|K1(t2)tob‘£eUiz)
= oil dIWT‘J’HMi
‘e7i3nemf- _ to 33m :jidtxmr tape» 2 Ernest usnwj L ‘
s: all; \(i; + dalX-xli’lot’t‘iehhﬁ'ii bit X]: ‘R‘ 41X) Nae/Fermi =~>N0rxliuegv 35W=Iioiwtl ’t oh no»
“ dam— ex.) +13. :2 Q\ Ptiﬁ‘i’ bi.) it old Mrs) r as (Axs't‘jgtrvaeml'
@ Miiwid x3) _ X :1 = (8) We (3 \ t XL ‘+t 0&6)“ tube ‘1sz 5’ Mitten Coo “‘ap” 2) Time~invariant vs. time-variant systems: A system is called time- invariant if its input-output characteristic do not change with time. Given y(t) 2 r{3:(t)]. Suppose that the same input signal is delayed by t; to yield :r(t — t1), and
again applied to the same system. If the characteristic of the system do not change with time, the output of the relaxed
system will be y(t — t1). That is the output will be the same as the response to az[t), except that it will be delayed by
it. Deﬁnition: A relaxed system 7‘ is time-invariant (or shift-invariant) iii ac(t) —-‘ W} (13) implies that
$[t—t1)—>y(t—t1) (14) for every input signal and every time shift t1. To determine if an given system is time—invariant, we need to perform the test speciﬁed by the preceding deﬁnition.
Basic-aily we excite the system with an arbitrary input alt), which produces an output denoted by y(t). Next we delay
the input by some amount t1 and recompute the output. In general, we can write the output as w. to = Time ~ 2a)}- ' (15) Now, if this output y(t, t1) = y(t '— t}) for all possible values of t1, the system is time-invariant; On the other hand,
if y(t. t1) 72 y(t — t1) for just one value of M, the system is time—variant. Example:
we 2 w(t)—a:(t—1), \/ (Isa)
me = we). K (16b)
y(t) = :L'(—t)= K (166) = cos(wut). X (16d) Lu) %[Il)}‘lfl)= 'XlJO-tlhl)
flit-ti) = "Kit—ti) - new)
W7) W = Stitch) 9 Tim? "lawman-ti
0)) Elm l7”) '1 "l7 Xit'hl ] “Tim?” Variant"
}jit-ﬁj = but!) XUHVQ M m
to area) = Kiri-"to K ling-Widow?
flea-Jo) = x(-th)] (0‘3 80m) = new Mum) ‘ltt *ti) : new in noun») tel 300 = Kilt)
(gift-ti): the»
hush) :Xilﬁ “6) Time r Vat/Will" Tine “V0! ﬁaM' 3) Causal vs. noneausal systems: A system is said to be causal if the output of the system at any time tdepends only on 4) 5} 8) 9) present and past inputs (i.e., s:(t), 2:(t— 1), ar(f:—-2), . . .), but does not depend on ﬁiture inputs (i.e., :e(t+1), 356+?)1 - - -)-
in mathematical terms, the output of a causal system satisﬁes an equation of the form y(t) = F[x{t), $(t — l), :50 - 2), . . (17}
where FH denotes an arbitrary function. If a system does not satisfy this deﬁnition, it is noncausal.
Example:
y(t) = In) — $(t — 1), / (18a)
y(t] = Z me + k),=...,X{’b"lD) 'l’ 'Kl‘ﬁh‘imw (use)
k=-oo
y(t) 2 unit), \/ (18c)
y(t) = 30) + 3:c(t +4), 7K (tad)
y(t) = 1:62], x (lSe)
W) m 34213), X (18f)
y(t) = (18g) $(ut). X Static vs. dynamic systems: A system is called static (memoryless) if its output at any instant it depends at most on
the input at the same time, but no on past or future inputs. Otherwise, the system is said to be dynamic or to have memory. Example:
ya) = “(15); ‘/ (193)
y(t) = tx(t)+b3:3(t}, \/ (19b)
11(15) : m(t)+3at(t—1),X (19c)
yo) = Ema—k), PK (19d)
k=U
ya) = Ewe—k). X (we)
116:0 Continuous~time vs. discrete-time systems: Systems whose inputs and outputs are continuous—time signals are
continuous—time systems. On the other hand, systems whose inputs and outputs are discrete—time signals are discrete-
time systems. Analog vs. digital systems: Systems whose inputs and outputs are analog signals are analog systems. On the other
hand, systems whose inputs and outputs are digital signals are digital systems. Invertibie vs. non-invertibie systems: A system is said to be invertible if it is possible to obtain the input $(t) back
from the corresponding output y(t) by some operations. Otherwise, it is non-invenible. Stable vs. unstable systems: An arbitraiy relaxed system is said to be BlBO stable iff every bounded input produces
a bounded output. Finite-dimensional vs. inﬁnite-dimensional systems: Given a continuous—time system with input ﬁt) and output
y(t), for any nonnegative i er -i let ("l 't and mm t denote the ill“ derivative of the output y(t) and input ﬁt).
The system is said to be ﬁnite-dimensional or him ed) if for some positive integer N, the Nth derivative of the
output at time t is equal to a function of ymﬂ) and re“ (t) at time t for E] S i g N — 1. Otherwise, the system is
called inﬁnite-dimensional. - SEES-f1?“
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.H TIME-DOMAIN-ANALYSIS
i ' OF CONTINUOQS-TIME SYSTEMS In this book we consider two methods of analysis of linear time-invariant [LTD systems: the time-
domain method and the frequency-domain method. In this chapter we discuss the time—domain
analysis of linear, time-invariant, continuous-time (LTIC) systems. 2.1 INTRODUCTION For the purpose of analysis, we shall consider linear déﬁ’erential systems. This is the class of
LTIC systems introduced in Chapter 1, for which the input x0) and the output y(t) are related
by linear differential equations of the form ditty dN—ly dy
,5 F +al d,th + ' ' ' +aN—IE +4NJ’U) de dM‘lx dx
= bN_M—dtM + b~_M+[ “Yth + ' - ' +b~-1 E- + bNxU) (2.121) where all the coefﬁcients a.- and b,- are constants. Using operational notation D to represent d/dr,
we can express this equation as (DN + 01DN_E+”' + div-1D + 0ND“) = (bN—MDM + bN—M+tDM_I + ' ‘ ' + bN—ID + 52010) (21b)
01’
QtDlytr) = PtD)x(t) (2.1c)
where the polynomials Q(D) and P(D) are
Q(D) = DN+a1DN_l +.-- +a~_1D+aN (2.2a)
P(D) = main)“ + bN_M+1DM“ + - - - + bN_iD + bN (2.2s) Theoretically the powers M and N in the foregoing equations can take on any valueHowever,
practical considerations make M > N undesirable for two reasons. In Section 4.3-2, we shall Show that an LTIC system speciﬁed by . . acts as an (M ~— N )th‘order differentiator. 151 152 CHAPTER 2 TIME-DOMAIN ANALYSIS OF CONTINUOUS-TIME SYSTEMS esents an unstable system because a bounded input like the step input m I, A differentiator repr
y a differentiator. Noise is a suits in an unbounded output, 50‘). Second, the-noise is enhanced b
a wideband signal containing components of all frequencies from 0 to a very high frequency approaching oo.‘T Hence, noise contains a signiﬁcant amount of rapidly varying components. We ‘
know that the derivative of any rapidly varying signal is high. Therefore, any system speciﬁed :
by Eq. (2.1) in which M > N will magnify the high-frequency components of noise through '
differentiation. It is entirely possible for noise to be magniﬁed so much that it swamps the desired i
system output even if the noise signal at the system’s input is tolerably small. Hence, practical
systems generally use M g N. For the rest of this text we assume implicitly that M 5 N _ For
the sake of generality, we shall assume M : N in Eq. (2.1). In Chapter 1, we demonstrated that a system described by Eq. (2.1) is linear. Therefore. its
response can be expressed as the sum of two components: the zero-input component and the zero-state component (decomposition property).i Therefore,
total response = zero-input response + zero-state response (23) ' nput x (t) z 0, and thus it is the result
1 conditions) alone. It is independent
cut is the system response to the
he absence of all internal energy The zero-input component is the system response when the i
of internal system conditions (such as energy storages, initia
of the external input x0). In contrast, the zero-state compon external input x(t) when the system is in zero state, meaning t
storages: that is, all initial conditions are zero. 2.2 SYSTEM RESPONSE TO INTERNAL CONDITIONS:
THE ZERO-INPUT RESPONSE The zero-input response M30) is the solution of Eq. (2.1
QUDD’OU) = 0 ) when the input x0) = 0 so that
(2.43) 01' (1')” + aiDN"1+---+ (IN 10 + avlyott) = 0 9-4")? A solution to this equation can be obtained systematically.‘ However, we will take a shortcut by
using heuristic reasoning. Equation (2.4b) shows that a linear combination of you} and its h 1Noise is any undesirable signal, natural or man-made, that interferes with the desired signals in the system
omagnetic radiation from stars, the random motion of electrons i' Some of the sources of noise are the electr
system components, interference from nearby radio and television stations, transients produced by automobil ignition systems, and ﬂuorescent lighting.
5We can verify readin that the system described by Eq. (2.1) has zero-input response, then, by deﬁnition,
Q(Dl,‘r’0(i) = 0 If y(r) is the zero-state response, then y(r) is the solution of
QtD).r(t) = PtD)x(t) subject to zero initial conditions (zero—state). Adding these two equations, we have
QtDMyotI) + 190)] = P(D)x(t) Clearly, yea) + y(I) is the general solution of Eq. (2.1). the decomposition property. If y0(t) is Lil )1) input re-
nuator. Noise is
high frequency
:omponents. We
system speciﬁed
3f noise through
amps the desired
Hence, practical
hat M S N. For ar. Therefore, its
nponent and the (F (23) has it is the result
It is independent
1 response to the
ll internal energy )0 that
(2.43] (2.4h} take a shortcut by
If EU) and its N goals in the systern.
otion of electrons-m
luced by automobile iperty. If you ) is the 2.2 System Keeponse to Internal Conditions: The Zero-Input Response 153 successive derivatives is zero, not at some values of I, but for all 2‘. Such a result is ssible if
and on! i t its N successive derivatives are of the same form. Otherwise their Sum can never add to zero for all values of I} We know that only an exponential function 2’” has this
property. So let us assume that you} = 68’”
is a solution to Eq. (2.4b). Then
61
Dyom = 137° = cite“
d2
Dime) = Vi” = c123)"
dN ,
DNl’oO‘) = dig = cit" 8“ Substituting these results in Eq. (2.4b), we obtain
C(AN + aIAN'E + - - - + aN_[A + me“ = 0
For a nontrivial solution of this equation,
AN+a1AN"l+---+aN_1A+aN=0 (2.5a) This result means that ce“ is indeed a solution of Eq. (2.4), provided A satisﬁes Eq. (2.5a).
Note that the polynomial in Eq. (2.5a) is identical to the polynomial Q(D) in Eq. (2.4), with A
replacing D. Therefore, Eq. (2.5a) can be expressed as QUL) = 0 (2.5b) When Q (it) is expressed in factorized form, Eq. (2.5b) can be represented as
QUL) = (A - AMA - l2) - (A * AN) = 0 (2-50)-
Clearly, it has N solutions: A1, lg, ..., AN, assuming that all A,- are distinct. Consequently,
Eq. (2.4) has N possible solutions: Clem, age”, . . . , cNe’W', with cl, C2, ..._.cN as arbitrary constants. We can readily show that a general solution is given by the sum of these N solutions l
._________________________________________________________t_ lTo prove this assertion, assume that y1(t), y2(t), . . . , 37” (r) are all solutions of Eq. (2.4). Then Q(D)yi(I) = 0
Q(D)y2(1) = 0 Q(D)YN(I) = 0 Multiplying these equations by c1, c2, . . . , cN, respectively, and adding them together yields Q(D)[€t)’1(f)+ 02h“) + ' ' ' + 90:20)] = 0
This result shows that c,y.(r) + c: mm + + chnU) is also a solution of the homogeneous equation [Eq- (2.4)]. 't'cw'wrmiﬁ ‘ amasse- - j
. i 154 ® W .;
The solution of Eq. (2.4) as given in Eq. (2.6) assumes that the N characteristic roots A] , l2, ._ CHAPTER 2 TIME-DOMAIN ANALYSIS OF CONTINUOUS-TINTE SYSTEMS can at W + r-r’v are Dr “new” so that YOU) = l—‘rt'v’xIr + C2912: + ’ ' ' + 0N3”, where c1, c2 , . , C” are arbitrary constants determined by N constraints (the auxiliary CUHdi ;
on the solution. [— Observe that the polynomial QOt), which is characteristic of the system, has nothing with the input. For this reason the polynomial MA) is called the character' ' 1 1 . , a the system. The equation . ' QQ) = 0 is called the W of the system. Equation (2.5a) clearly indicates m3;-
th, . . . , AN are the roots of the characteristic equation; consequently, they are called the c 7-. tert'srr'c roots of the system. The terms chr ' r .t r —: I values and natural eye-5
are also used for characteristic rootsl The exponentials e*"‘(r' = 1, 2, .. . , n) in the zero -
response are the characteristic modes (also known as natura moes or simply as mad
Wde for each characteristic root of the system, an i
zero-input response is a linear combination of the characteristic modes of the system. .4
An LTIC system’s characteristic modes comprise its single most important attribute. it.-
acteristic modes not only determine the zero-input response but also play an importan ;
in determining the zero-state response. In other words, the entire behavior of a system '
tated primarily by its characteristic modes. In the rest of this chapter we shall See the perv
presence of characteristic modes in every aspect of system behavior. " REPEATED ROOTS AN are distinct. If there are repeated roots (same root occurring more than once), the form oi
solution is modiﬁed slightly. By direct substitution we can show that the solution of the equal (D — 90500) = 0 is given by YOU) = (C1 + €209)U In this case the root A repeats twi e. Observe that the characteristic modes in this case :7
and re“ . Continuing this pattern, we can show that for the differential equation ' (D - MUM!) = 0 M 2M
,te the characteristic modes are 2“, re , . . . , air—1e“, and that the solution is 3’00): (61+ 62! + - - - + WM)“:M Consequently, for a system with the characteristic polynomial QR) = (l - MTG - lr+1)- "(1 '- 3w) fEigenvalue is German for “characteristic value.” (2.5} tiliary conditions} has nothing to do EEC—WIQLM (2.?) indicates that A"
called the charm. mural fregtrencies n in the zero-input
rply as modes) of
e system, and the
a system. .nt attribute. Char-
an important role
)f a system is dic-
I see the pervasive C SSAlSAEI'”:
36}, the form of the
ion of the equation in this case are e"
on is ® 2.2 System Response to Internal Conditions: The Zero-Input Response 155 the characteristic modes are 6*”, tel", . . ., t"‘e"", 81”", ... , 8*“ and the solution is Yo(t)=(6'1+ ‘32? + "'+ Crtﬁl)?” + cr+i€ArHr + ‘ ' ' + 6N8”! C PLEX ROOTS Th...

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- LTI system theory, Complex number, Dirac delta function