the discrete-time unit impulse sequence
DESCRIPTION
The Discrete-time Unit Impulse Sequence. 1. n. 0. Impulse Function. volt. time. Delayed Unit Impulse. The Unit Step Sequence. 1. n. 0. Relationship between Unit Impulse & Unit Step Sequences. Discrete-time unit impulse is the first difference of the discrete-time unit step. - PowerPoint PPT PresentationTRANSCRIPT
The Discrete-time Unit Impulse Sequence
1
][n
0 n
Impulse Function
time
volt
Delayed Unit Impulse
The Unit Step Sequence
.0,1][
,0,0][
nnu
nnu
1
0 n
Relationship between Unit Impulse & Unit Step Sequences
Discrete-time unit impulse is thefirst difference of the discrete-timeunit step.
].1[][][ nunun
Relationship between Unit Impulse & Unit Step Sequences
Discrete-time unit step is therunning sum of the discrete-timeunit impulse or unit sample.
n
m
mnu ].[][
1
0n
][m
0m
][m
0 mn
n
Interval of summation
Interval of summation
Alternative way of expressing the unit step as the summation (superposition) of
delayed unit impulse
0k
0
k
k].-[nu[n]or
,k]-[nu[n]
k.-nm
m,-nklet
_: variable theChanging
][ kn
0k
][ kn
0 kn<0
Interval of summation
Interval of summation
0k
k].-[nu[n]
0n
]1[ n
0n
]0[
0 n
0k
k].-[nu[n]
n][ kn
The Sampling Property of the Unit Impulse.
].[][]n-[n then x[n],nnfor
1]n-[nsincegenerally that followsIt
[n].x[0][n] x[n]
0,nfor only 1) ( nonzero is [n] Since
0.nat signal a of value thesample to
used becan sequence impulseunit The
oo
o
oo nnnx
][n
]0[x
0 n
n
]1[x
][ onn
][ onx
The Continuous-time Unit Step
.0
.0,1)(
,0,0)(
:
tat
ousdiscontinuisfunctionThis
ttu
ttu
functionstepunitofDefination
)(tu1
0 t
The Continuous-time Unit Impulse Function.
0,t,)(1(t)
0,t0,(t)
-:Defination
area
t0
Relationship between continuous-time unit step and unit impulse.
.
)()(
.)()(
dt
tdut
dtut
Integration of Unit Impulse to get Unit Stup
)(1
0t<0
0
1 )(
Interval of integration
0t
Discontinuity at t=0, poses problem of differentiation.
1
)(tu
1
0
)(lim)(,)(
)(0
ttdt
tdut
Unit and scaled Impulse
)(t1
0 t
t0
k)(tk
Scaled impulse and relationshipof unit step & impulse.
0
0
.)()(
:
),)(()()(
,1,
).()(
dttu
lyequivalentor
dtdtu
kscaleandtLetting
tkudk
t
t
Relationship of Unit and Impulse
)( t
1
0t<0
0
1 )( t
Interval of integration
0t
Interval of integration
).()()(Consider x1 ttxt
x(0)
)(t
)(t
x(t)
t
t
0
0
)()()t-(tx(t)
argument, sameby Similarly
(t).x(0)(t)x(t)
0. as )((t)
).()0()( x(t)
, intervalover constant
is x(t)small,ly sufficient For
o oo tttx
tSince
txt