Signal Systems unit 2

1. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Signals and Systems UNIT 2
Ripal Patel
Assistant Professor,
Dr.Ambedkar Institute of Technology, Bangalore.
ripal.patel@dr-ait.org
December 1, 2020

2. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
LTI systems
• A class of systems used in signals and systems that are
both linear and time-invariant

3. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
LTI systems
• A class of systems used in signals and systems that are
both linear and time-invariant
• Linear systems are systems whose outputs for a linear
combination of inputs are the same as a linear
combination of individual responses to those inputs.

4. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
LTI systems
• A class of systems used in signals and systems that are
both linear and time-invariant
• Linear systems are systems whose outputs for a linear
combination of inputs are the same as a linear
combination of individual responses to those inputs.
• Time-invariant systems are systems where the output does
not depend on when an input was applied. These
properties make LTI systems easy to represent and
understand graphically.

5. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
LTI systems
• A class of systems used in signals and systems that are
both linear and time-invariant
• Linear systems are systems whose outputs for a linear
combination of inputs are the same as a linear
combination of individual responses to those inputs.
• Time-invariant systems are systems where the output does
not depend on when an input was applied. These
properties make LTI systems easy to represent and
understand graphically.
• Used to predict long-term behavior in a system

6. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
LTI systems
• A class of systems used in signals and systems that are
both linear and time-invariant
• Linear systems are systems whose outputs for a linear
combination of inputs are the same as a linear
combination of individual responses to those inputs.
• Time-invariant systems are systems where the output does
not depend on when an input was applied. These
properties make LTI systems easy to represent and
understand graphically.
• Used to predict long-term behavior in a system
• The behavior of an LTI system is completely defined by its
impulse response

7. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Impulse Function
The discrete version of impulse function is defined by
δ(n) =
1, n = 0
0, n 6= 0
The continuous time version of impulse function,
δ(t) =
1, t = 0
0, t 6= 0

8. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Impulse Response
• The impulse response” of a system, h[n], is the output
that it produces in response to an impulse input.
Definition: if and only if x[n] = δ[n] then y[n] = h[n]

9. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Impulse Response
• The impulse response” of a system, h[n], is the output
that it produces in response to an impulse input.
Definition: if and only if x[n] = δ[n] then y[n] = h[n]
• Given the system equation, the impulse response can be
found out just by feeding x[n] = δ[n] into the system.

10. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Impulse Response Example
• Consider the system
y[n] =
1
2
(x[n] + x[n − 1])
• Suppose we insert an impulse:
x[n] = δ[n]
• Then whatever we get at the output, by Definition, is the
impulse response. In this case it is
h[n] =
1
2
(δ[n] + δ[n − 1]) =
0.5, n = 0, 1
0, otherwise

11. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Sum
•
where, h[n]=impulse response of LTI system
x[n]=Input Signal

12. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Sum
•
where, h[n]=impulse response of LTI system
x[n]=Input Signal
•
y[n] =
∞
X
k=−∞
x[k]h[n − k]

13. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Sum
•
where, h[n]=impulse response of LTI system
x[n]=Input Signal
•
y[n] =
∞
X
k=−∞
x[k]h[n − k]
• input-excitation output-response

14. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Sum Example
Find the response y[n] of following LTI system.

15. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Sum (Graphical method)
•

16. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Sum (Graphical method)
•
•

17. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Sum (Graphical method)
•
•
•

18. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Sum (Graphical method)
•
•
•
•

19. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Sum (Graphical method)
•
•
•
•
• y(n) = [...0, 1, 4
↑
, 9, 11, 8, 2, 0, ...]

20. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Sum (Analytical method)
• Size of x(n)=A=4, Size of h(n)=B=3
Length of y(n)=A+B-1=4+3-1=6

21. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Sum (Analytical method)
• Size of x(n)=A=4, Size of h(n)=B=3
Length of y(n)=A+B-1=4+3-1=6
• x(n) is starting from 0 index n1=0
h(n) is starting from -1 index n2=-1
n1 + n2 = −1, range of n=-1 to 4

22. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Sum (Analytical method)
• Size of x(n)=A=4, Size of h(n)=B=3
Length of y(n)=A+B-1=4+3-1=6
• x(n) is starting from 0 index n1=0
h(n) is starting from -1 index n2=-1
n1 + n2 = −1, range of n=-1 to 4
• For n=-1
y[−1] =
3
X
k=0
x[k]h[−1 − k] =
x[0]h[−1] + x[1]h[−2] = (1x1) + (2x0) = 1

23. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Sum (Analytical method)
• Size of x(n)=A=4, Size of h(n)=B=3
Length of y(n)=A+B-1=4+3-1=6
• x(n) is starting from 0 index n1=0
h(n) is starting from -1 index n2=-1
n1 + n2 = −1, range of n=-1 to 4
• For n=-1
y[−1] =
3
X
k=0
x[k]h[−1 − k] =
x[0]h[−1] + x[1]h[−2] = (1x1) + (2x0) = 1
• For n=0
y[0] =
3
X
k=0
x[k]h[0 − k] =
x[0]h[0]+x[1]h[−1]+x[2]h[−2] = (1x2)+(2x1)+(3x0) = 4

24. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Sum (Analytical method)
• Size of x(n)=A=4, Size of h(n)=B=3
Length of y(n)=A+B-1=4+3-1=6
• x(n) is starting from 0 index n1=0
h(n) is starting from -1 index n2=-1
n1 + n2 = −1, range of n=-1 to 4
• For n=-1
y[−1] =
3
X
k=0
x[k]h[−1 − k] =
x[0]h[−1] + x[1]h[−2] = (1x1) + (2x0) = 1
• For n=0
y[0] =
3
X
k=0
x[k]h[0 − k] =
x[0]h[0]+x[1]h[−1]+x[2]h[−2] = (1x2)+(2x1)+(3x0) = 4
• Likewise for all the values of n
y(n) = [...0, 1, 4
↑
, 9, 11, 8, 2, 0, ...]

25. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Sum (Analytical method)
x1(n) = [1
↑
, 2, 3]
x2(n) = [2
↑
, 1, 4]
y[n] =
∞
X
k=−∞
x1[k]x2[n − k]
• Size of x1(n)=A=3, Size of x2(n)=B=3
Length of y(n)=A+B-1=3+3-1=5

26. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Sum (Analytical method)
x1(n) = [1
↑
, 2, 3]
x2(n) = [2
↑
, 1, 4]
y[n] =
∞
X
k=−∞
x1[k]x2[n − k]
• Size of x1(n)=A=3, Size of x2(n)=B=3
Length of y(n)=A+B-1=3+3-1=5
• x1(n) is starting from 0 index n1=0
x2(n) is starting from 0 index n2=0
n1 + n2 = 0, range of n=0 to 4

27. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Sum (Analytical method)
• For n=0
y[0] =
2
X
k=0
x1[k]x2[−k] =
x[0]x2[0] + x1[1]x2[−1] + x1[2]x2[−2]
= (1x2) + (2x0) + (3x0) = 2

28. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Sum (Analytical method)
• For n=0
y[0] =
2
X
k=0
x1[k]x2[−k] =
x[0]x2[0] + x1[1]x2[−1] + x1[2]x2[−2]
= (1x2) + (2x0) + (3x0) = 2
• Likewise for all the values of n y(n) = [2
↑
, 5, 12, 11, 12]

29. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Sum
Convolute the given sequences
x1[n] = αnu[n] and x2[n] = βnu[n]
• y[n] = x1[n] ∗ x2[n]

30. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Sum
Convolute the given sequences
x1[n] = αnu[n] and x2[n] = βnu[n]
• y[n] = x1[n] ∗ x2[n]
• =
P∞
k=−∞ x1[k]x2[n − k]

31. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Sum
Convolute the given sequences
x1[n] = αnu[n] and x2[n] = βnu[n]
• y[n] = x1[n] ∗ x2[n]
• =
P∞
k=−∞ x1[k]x2[n − k]
• =
P∞
k=−∞ αku[k]βn−ku[n − k]

32. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Sum
Convolute the given sequences
x1[n] = αnu[n] and x2[n] = βnu[n]
• y[n] = x1[n] ∗ x2[n]
• =
P∞
k=−∞ x1[k]x2[n − k]
• =
P∞
k=−∞ αku[k]βn−ku[n − k]
• =
P∞
k=−∞ αku[k]βnβ−ku[n − k]

33. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Sum
Convolute the given sequences
x1[n] = αnu[n] and x2[n] = βnu[n]
• y[n] = x1[n] ∗ x2[n]
• =
P∞
k=−∞ x1[k]x2[n − k]
• =
P∞
k=−∞ αku[k]βn−ku[n − k]
• =
P∞
k=−∞ αku[k]βnβ−ku[n − k]
• = βn
P∞
k=−∞ αku[k]β−ku[n − k]

34. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Sum
Convolute the given sequences
x1[n] = αnu[n] and x2[n] = βnu[n]
• y[n] = x1[n] ∗ x2[n]
• =
P∞
k=−∞ x1[k]x2[n − k]
• =
P∞
k=−∞ αku[k]βn−ku[n − k]
• =
P∞
k=−∞ αku[k]βnβ−ku[n − k]
• = βn
P∞
k=−∞ αku[k]β−ku[n − k]
• = βn
P∞
k=−∞ (α
β )k
u[k]u[n − k]

35. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Sum
•

36. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Sum
• u[k]u[n − k] = 1, 0 ≤ k ≤ n, n ≥ 0
•

37. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Sum
Convolute the given sequences
x1[n] = αnu[n] and x2[n] = βnu[n]
• = βn
Pn
k=0 (α
β )k

38. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Sum
Convolute the given sequences
x1[n] = αnu[n] and x2[n] = βnu[n]
• = βn
Pn
k=0 (α
β )k
•
= βn
[
(α
β )n+1 − 1
(α
β ) − 1
]

39. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Sum
Convolute the given sequences
x1[n] = αnu[n] and x2[n] = βnu[n]
• = βn
Pn
k=0 (α
β )k
•
= βn
[
(α
β )n+1 − 1
(α
β ) − 1
]
•
1
β − α
[βn+1
− αn+1
]

40. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Sum
The system is characterized by an impulse response
h[n] = (
3
4
)n
u[n]
Find the step response of the system. Also evaluate the output
of the system at n = ±5
• y[n] = x[n] ∗ h[n] = h[n] ∗ x[n]

41. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Sum
The system is characterized by an impulse response
h[n] = (
3
4
)n
u[n]
Find the step response of the system. Also evaluate the output
of the system at n = ±5
• y[n] = x[n] ∗ h[n] = h[n] ∗ x[n]
• =
P∞
k=−∞ h[k]x[n − k]

42. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Sum
The system is characterized by an impulse response
h[n] = (
3
4
)n
u[n]
Find the step response of the system. Also evaluate the output
of the system at n = ±5
• y[n] = x[n] ∗ h[n] = h[n] ∗ x[n]
• =
P∞
k=−∞ h[k]x[n − k]
• =
P∞
k=−∞(3
4)ku[k]u[n − k]

43. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Sum
The system is characterized by an impulse response
h[n] = (
3
4
)n
u[n]
Find the step response of the system. Also evaluate the output
of the system at n = ±5
• y[n] = x[n] ∗ h[n] = h[n] ∗ x[n]
• =
P∞
k=−∞ h[k]x[n − k]
• =
P∞
k=−∞(3
4)ku[k]u[n − k]
• =
Pn
k=0 (3
4)
k

44. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Sum
The system is characterized by an impulse response
h[n] = (
3
4
)n
u[n]
Find the step response of the system. Also evaluate the output
of the system at n = ±5
• y[n] = x[n] ∗ h[n] = h[n] ∗ x[n]
• =
P∞
k=−∞ h[k]x[n − k]
• =
P∞
k=−∞(3
4)ku[k]u[n − k]
• =
Pn
k=0 (3
4)
k
• =
( 3
4
)n+1−1
3
4
−1

45. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Integral
• Convolution Integral between two continuous signals x(t)
and h(t)
y(t) = x(t) ∗ h(t) =
Z ∞
−∞
x(τ)h(t − τ)dτ

46. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Integral
• Convolution Integral between two continuous signals x(t)
and h(t)
y(t) = x(t) ∗ h(t) =
Z ∞
−∞
x(τ)h(t − τ)dτ
• y(t)) = x(t) ∗ h(t) = h(t) ∗ x(t)

47. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Integral (Graphical Methods)
Suppose the input x(t) and impulse response h(t) of a LTI
system are given by
x(t) = 2u(t − 1) − 2u(t − 3)
h(t) = u(t + 1) − 2u(t − 1) + u(t − 3)

48. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Integral
h(t)

49. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Integral
Convolution :

50. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Integral (Analytical Methods)
x(t) = e−2t
u(t)
h(t) = u(t + 2)
• y(t) = x(t) ∗ h(t) =
R ∞
−∞ x(τ)h(t − τ)dτ

51. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Integral (Analytical Methods)
x(t) = e−2t
u(t)
h(t) = u(t + 2)
• y(t) = x(t) ∗ h(t) =
R ∞
−∞ x(τ)h(t − τ)dτ
•
R ∞
−∞ e−2τ u(τ)u(t − τ + 2)dτ

52. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Integral (Analytical Methods)
x(t) = e−2t
u(t)
h(t) = u(t + 2)
• y(t) = x(t) ∗ h(t) =
R ∞
−∞ x(τ)h(t − τ)dτ
•
R ∞
−∞ e−2τ u(τ)u(t − τ + 2)dτ
• Case 1: t + 2 0, do not overlap hence zero.

53. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Integral (Analytical Methods)
x(t) = e−2t
u(t)
h(t) = u(t + 2)
• y(t) = x(t) ∗ h(t) =
R ∞
−∞ x(τ)h(t − τ)dτ
•
R ∞
−∞ e−2τ u(τ)u(t − τ + 2)dτ
• Case 1: t + 2 0, do not overlap hence zero.
• Case 2 =
R t+2
0 e−2τ x1dτ

54. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Integral (Analytical Methods)
x(t) = e−2t
u(t)
h(t) = u(t + 2)
• y(t) = x(t) ∗ h(t) =
R ∞
−∞ x(τ)h(t − τ)dτ
•
R ∞
−∞ e−2τ u(τ)u(t − τ + 2)dτ
• Case 1: t + 2 0, do not overlap hence zero.
• Case 2 =
R t+2
0 e−2τ x1dτ
• = [e−2τ
−2 ]t+2
0 = 1
2 − 1
2e−2(t+2)

55. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Integral (Analytical Methods)
x(t) = 2u(t − 1) − 2u(t − 3)
h(t) = u(t + 1) − 2u(t − 1) + u(t − 3)
• x(t) = 1, t = 1, 2, 3

56. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Integral (Analytical Methods)
x(t) = 2u(t − 1) − 2u(t − 3)
h(t) = u(t + 1) − 2u(t − 1) + u(t − 3)
• x(t) = 1, t = 1, 2, 3
• h(t) =



1 for 1 ≤ t ≤ 3
−1 for −1 ≤ t 1
0 otherwise

57. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Integral (Analytical Methods)
x(t) = 2u(t − 1) − 2u(t − 3)
h(t) = u(t + 1) − 2u(t − 1) + u(t − 3)
• x(t) = 1, t = 1, 2, 3
• h(t) =



1 for 1 ≤ t ≤ 3
−1 for −1 ≤ t 1
0 otherwise
• y(t) = x(t) ∗ h(t) =
R ∞
−∞ x(τ)h(t − τ)dτ

58. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Integral
•

59. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Integral
•
•

60. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Integral
•
•

61. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Convolution Integral

62. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Step Response

63. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Step Response for Continuous time system

64. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Step Response Example

65. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Step Response Example

66. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Differential and Difference equation representation
of LTI systems

67. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Differential and Difference equation representation
of LTI systems

68. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Recursive evaluation of difference equation

69. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Solving Differential and Difference equations

70. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Homogeneous Solution for CT system

71. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Homogeneous Solution for DT system

72. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Homogeneous Solution Example

73. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Homogeneous Solution Example

74. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
Homogeneous Solution Example

75. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
The Particular Solution

76. 19EC34
Ripal Patel
Time domain
representation
of LTI System
Linear time-invariant
systems (LTI
systems)
Impulse Response
Convolution Sum
Convolution Sum
(Finite Sequences)
Convolution Sum
(Infinite Sequences)
Convolution
Integral
Convolution Integral
(Finite signals)
Step Response
Differential
and Difference
equation
representation
of LTI systems
The End