Computing the output response of LTI Systems.

1
Computing the output response of LTI Systems.

• By breaking or decomposing and representing the
  input signal to the LTI system into terms of a
  linear combination of a set of basic signals.
• Using the superposition property of LTI system to
  compute the output of the system in terms of its
  response to these basic signals.

2
General Signal Representations By Basic Signal

• The basic signal - in particular the unit impulse
  can be used to decompose and represent the
  general form of any signal.
• Linear combination of delayed impulses can
  represent these general signals.

3
Response of LTI System to General Input Signal

LTI SYSTEM
General Input Signal
Output Response Signal
LTI SYSTEM
Delayed Impulse Signal 1
Response to Impulse signal N
LTI SYSTEM
Delayed Impulse Signal N
4
Representation of Discrete-time Signals in Terms
of Impulses.

• Discrete-time signals are sequences of individual
  impulses.

xn
-4
-1
2
3
n
-3
-2
0
1
4
-1
2
3
-4
n
-3
-2
0
1
4
2
3
-4
-1
n
-3
-2
0
1
4
5

• Discrete-time signals are sequences of individual
  scaled unit impulses.

xn
-4
-1
2
3
n
-3
-2
0
1
4
-1
2
3
-4
n
-3
-2
0
1
4
2
3
-4
-1
n
-3
-2
0
1
4
2
3
-4
-1
n
-3
-2
0
1
4
6

Shifted Scaled Impulses

Generally-
The arbitrary sequence is represented by a
linear combination of shifted unit impulses
dn-k , where the weights in this linear
combination are xk. The above equation is
called the sifting property of discrete-time
unit impulse.
7

As Example consider unit step signal xnun-

Generally-
The unit step sequence is represented by a
linear combination of shifted unit impulses
dn-k , where the weights in this linear
combination are ones from k0 right up to k
This is identically similar to the expression we
have derived in our previous lecture a few weeks
back when we dealt with unit step.
8
The Discrete-time Unit Impulse Responses and the
Convolution Sum Representation

• To determine the output response of an LTI system
  to an arbitrary input signal xn, we make use of
  the sifting property for input signal and the
  superposition and time-invariant properties of
  LTI system.

9
Convolution Sum Representation

• The response of a linear system to xn will be
  the superposition of the scaled responses of the
  system to each of these shifted impulses.
• From the time-invariant property, the response of
  LTI system to the time-shifted unit impulses are
  simply time-shifted responses of one another.

10

Unit Impulse Response hn
dn
hon
LTI System
0
n0

dn-k
hkn
LTI System
k
nk
11

Response to scaled unit impulse input xndn-k
x-k.dnk
x-k.h-kn
LTI System
-k
n-k
x0.dn
x0.hon
LTI System
0
n0
xk.dn-k

LTI System
k
xk.hkn
nk

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Output yn of LTI System
Thus, if we know the response of a linear system
to the set of shifted unit impulses, we can
construct the response yn to an arbitrary input
signal xn.
13

h-1n
xn
h0n
0
h1n
14

x-1h-1n
x-1dn1
0
0
x0h0n
x0dn
0
0
x1h1n
x1dn-1
0
0
xn
yn
0
0
15

• In general, the response hkn need not be
  related to each other for different values of k.
• If the linear system is also time-invariant
  system, then these responses hkn to time
  shifted unit impulse are all time-shifted
  versions of each other.
• I.e. hknh0n-k.
• For notational convenience we drop the subscript
  on h0n hn.
• hn is defined as the unit impluse (sample)
  response

16
Convolution sum or Superposition sum.
17

Convolution sum or Superposition sum.
xk
hk

18

Convolution sum or Superposition sum.
xk
hk

19

Convolution sum or Superposition sum.
xk
hk

20

Convolution sum or Superposition sum.
xk
hk
21

1
hn
ynx0hn-0x1hn-10.5hn2hn-1
1
1
x
x
x
x
1
0.5
xn
0
0.5
0.5hn
2
2
2
2hn-1
2.5
2.5
yn
2
0.5
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Modified Example 2.3
23
Modified Example 2.5
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Modified Example 2.5