7/21/2019 Table 3.7

http://slidepdf.com/reader/full/table-37 1/1

lhble

3.7

Form

of the

Naturat

Response

or

Dt LTt

Systems.

Rootof

Characteristl.

quation

Form

ofNaturat

Response

I(eal

and

distinct

root r

C(r)tl

Complex

or\ iugare

exp(jd)

ICr

cos(n0)

+

Crsin(nd)l(r), ,

Real

epeated

oor

r),

lKa+

Ktn+

K2n2

...

+ Kpnp)t)n

Complex,

epeated

rexp(jd)],

(C0

+

Cln + C2n2

. . .

+ Cpnp)

os(nl)

r)

+ ( D a

+ D t n + D ? n z . .

.

+ D r n p ) s i n ( n d

Table 3.8 Form of the

Forced Response or DT LTI

Systems.

Nore: lf rhe RHS nvolves

ie form

(d)tr

where

cr is alsoa

root ofthe characterkcicquadonepearedtimes, he orced

responseorm musr

be mulcipliedy n'.

Foicing fun.tlor

(RHS)

Form of forced rerponse

C

(constan0

Ct

(constan0

(d)n,(r

+ roo(

of K(d),

characteristicquation

cos ( r9+p ) f i : l cos (no )+Kzs in (nd )o rKcos (nd

(d ) cos (n0 f ) [K rcos ( r0 )+ (2s in (nd ) ] (d )

n K0+ Krn

(gcncral

irsaorder

olynomjal)

n P K o + ; i l n + K 2 n 2 + + K r n P

n(d)'

( t(o

+ Krn)(rY)

n p

d ) n

( K 0 + i : t n + K t n , + . . . + K / n r ) ( . y ) n

n c o s ( n 9 + 6 )

( K t + K r n ) c o s ( n o ) + ( , ( 3 + K r r ) s i n (