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Department of Communication & Electronics Engineering Faculty of Engineering Philadelphia University. Signal Analysis & Processing Discrete -Time LTI Systems Lecture (26). Lecturer: Ibrahim Abu-Isbeih. h [ n ]. d [ n ]. LTI System. Figure 3.20. 3.6 Discrete-Time LTI Systems:.

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signal analysis processing discrete time lti systems lecture 26
Department of Communication & Electronics Engineering

Faculty of Engineering

Philadelphia University

Signal Analysis & ProcessingDiscrete -Time LTI SystemsLecture (26)

Lecturer: Ibrahim Abu-Isbeih

slide2
h[n]

d[n]

LTI

System

Figure 3.20

3.6 Discrete-Time LTI Systems:

In this section, we develop the fundamental input/output relationship for discrete-time, linear, time-invariant (LTI) systems.

It is useful to characterize an LTI system in terms of its impulse response.

The impulse response of an LTI system is defined as the output of the LTI system due to a unit impulse signal input applied at time t=0 as illustrated in Figure 3.20.

i.e., if x[n]=d [n] then y[n]=h[n].

Signal Analysis & Processing

slide3
y[n]

x[n]

LTI

System

Figure 3.21

The impulse response of the LTI system shown in Figure 3.20 is denoted by h[n], whereh[n] is the response tod[n].

Consider a discrete-time LTI system with input x[n] shown in Figure 3.21, if the impulse response of the LTI system is h[n] then the output of this LTI system is given by:

Signal Analysis & Processing

slide4
Hence, the impulse response of the LTI system h[n] is the only function needed to characterize the system completely with respect to its input and output.

The output of an LTI system with impulse response h[n] is then given by the convolution sum:

Utilizing the commutivity property of convolution, we can also write the above equation as:

Signal Analysis & Processing

slide5
The procedure of evaluating the convolution sum (the LTI system output) was discussed in session two.

For example, the output of a discrete-time LTI system with impulse response h[n]=u[n] to the input x[n]=3u[n-2] is given by:

Signal Analysis & Processing

slide6
x[n]

h[n]

3

2

1

1

1

n

n

-1

3

0

2

1

-1

3

0

2

1

Example 3.6.1:Find the output of a discrete-time LTI system with impulse response h[n]=u[n]-u[n-2] to the input

Solution:The output of the system is given by:

To find the above convolution sum:

- Graph x[n] and h[n]:

*

Signal Analysis & Processing

slide7
x[k]

h[k]

3

2

1

1

1

k

k

-1

3

0

2

1

-1

3

0

2

1

  • Replace n with kin x[n] and h[n]
  • Graph h[n-k]

h[n-k]

1

1

1

k

n-1

n

n-2

1

0

Signal Analysis & Processing

according to the convolution sum convolution can be divided into the following cases
x[k]

3

2

k

-1

3

0

2

1

According to the convolution sumconvolution can be divided into the following cases:
  • 1. For n< 0: The functions x[k] and h[n-k]
    • do not overlap, then y[n]=0

h[n-k]

1

1

1

k

n-1

n

n-2

1

0

Signal Analysis & Processing

slide9
x[k]

3

2

k

-1

3

0

2

1

  • 2. For n =0:

h[n-k]

1

1

1

k

-2

1

-1

0

Signal Analysis & Processing

slide10
x[k]

3

2

k

-1

3

0

2

1

  • 3. For n =1:

h[n-k]

1

1

1

k

-1

1

0

Signal Analysis & Processing

slide11
x[k]

3

2

k

-1

3

0

2

1

  • 4. For n =2:

h[n-k]

1

1

1

k

0

2

1

Signal Analysis & Processing

slide12
x[k]

3

2

k

-1

3

0

2

1

  • 5. For n =3:

h[n-k]

1

1

1

k

0

1

2

3

Signal Analysis & Processing

slide13
x[k]

3

2

k

-1

3

0

2

1

  • 6. For n ≥ 4:

h[n-k]

1

1

1

k

0

1

n-1

n

n-2

2

Signal Analysis & Processing

slide14
Then the output of the system is :

y[n]

5

5

3

2

n

-1

4

0

1

3

2

Signal Analysis & Processing

slide15
s[n]

u[n]

LTI

System

Figure 3.22

Step-Response:

In this lecture we introduce the step-responseof discrete-time LTI systems.

The step-response of an LTI system with impulse response h[n] is defined as the output of the LTI system due to the unit step signal input as illustrated in Figure 3.22.

i.e., if x[n]=u[n] then y[n]=s[n]=u[n]*h[n].

Signal Analysis & Processing

slide16
h[n]

5

5

5

5

………

2

2

2

2

n

0

1

2

4

-1

3

5

6

Example 3.6.2:

Find the step-response of an LTI system with impulse response h[n]=2u[n]+3u[n-4].

Signal Analysis & Processing

slide17
Solution:

To find the step-response of the given system, let x[n]=u[n] then the step-response is:

Signal Analysis & Processing

example 3 6 3 find the step response of an lti system with impulse response
h[n]

3

1

n

4

-2

-1

3

0

2

1

-1

Example 3.6.3:Find the step-response of an LTI system with impulse response

Signal Analysis & Processing

slide19
Solution:The step-response of the system is:

wherex[n]=u[n].

Using the convolution properties we have:

Signal Analysis & Processing

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