3. Interconnection of LTI Systems
Distributive Property for CT case:
Distributive property for DT case:
Cascade Connection of LTI systems
Prof: Sarun Soman, MIT, Manipal 3
4. Interconnection of LTI Systems
Interconnection Properties of LTI Systems
Prof: Sarun Soman, MIT, Manipal 4
5. Interconnection of LTI Systems
Relation between LTI system properties and the impulse
response.
Memory less LTI system
The o/p of a DT LTI system
To be memoryless y[n] must depend only on present value of
input
The o/p cannot depend on x[n-k] for ݇ ≠ 0
ݕ ݊ = ℎ ݇ ݊[ݔ − ݇]
ஶ
ୀିஶ
Prof: Sarun Soman, MIT, Manipal 5
6. Time Domain Representations of LTI Systems
A DT LTI system is memoryless if and only if
ℎ ݇ = ܿߜ[݇] ‘c’ is an arbitrary constant
CT system
Output
A CT LTI system is memoryless if and only if
ℎ ߬ = ܿߜ(߬) ‘c’ is an arbitrary constant
ݕ ݐ = න ℎ ߬ ݔ ݐ − ߬ ݀߬
ஶ
ିஶ
Prof: Sarun Soman, MIT, Manipal 6
7. Time Domain Representations of LTI Systems
Causal LTI systems
The o/p of a causal LTI system depends only on past or present
values of the input
DT system
ݕ ݊ = ℎ ݇ ݊[ݔ − ݇]
ஶ
ୀିஶ
future value of inputs
Prof: Sarun Soman, MIT, Manipal 7
8. Time Domain Representations of LTI Systems
For a DT causal LTI system
ℎ ݇ = 0 ݂݇ ݎ < 0
Convolution Sum in new form
For CT system
ℎ ߬ = 0 ݂߬ ݎ < 0
Convolution integral in new form
ݕ ݊ = ℎ ݇ ݊[ݔ − ݇]
ஶ
ୀ
ݕ ݐ = න ℎ ߬ ݔ ݐ − ߬ ݀߬
ஶ
Prof: Sarun Soman, MIT, Manipal 8
9. Time Domain Representations of LTI Systems
Stable LTI Systems
A system is BIBO stable if the o/p is guaranteed to be bounded
for every i/p.
Discrete time case:
Bounding convolution sum
Prof: Sarun Soman, MIT, Manipal 9
10. Time Domain Representations of LTI Systems
Assume that i/p is bounded
Condition for impulse response of a stable DT LTI system
Prof: Sarun Soman, MIT, Manipal 10
11. Time Domain Representations of LTI Systems
Continuous time case
Impulse response must be absolutely integrable.
න ℎ ߬ ݀߬ < ∞
ஶ
ିஶ
impulse response must be absolutely summable.
Prof: Sarun Soman, MIT, Manipal 11
15. Time Domain Representations of LTI Systems
Step Response
Characterizes the response to sudden changes in input.
ݏ ݊ = ℎ ݊ ∗ ]݊[ݑ
u[n-k] exist from −∞ to n
h[k] is the running sum of impulse response.
= ℎ ݇ ݊[ݑ − ݇]
ஶ
ୀିஶ
]݊[ݏ = ℎ ݇
ୀିஶ
Prof: Sarun Soman, MIT, Manipal 15
16. Time Domain Representations of LTI Systems
CT system
Relation b/w step response and impulse response.
ℎ ݐ =
݀
݀ݐ
)ݐ(ݏ
ℎ ݊ = ݏ ݊ − ݊[ݏ − 1]
Eg. Step response of an RC ckt
ݏ ݐ = න ℎ ߬ ݀߬
௧
ିஶ
Prof: Sarun Soman, MIT, Manipal 16
20. Differential and Difference equation
Representations of LTI Systems
Linear constant coefficient differential equation:
Linear constant coefficient difference equation:
Order of the equation is (N,M), representing the number of
energy storage devices in the system.
Often N>M and the order is described using only ‘N’.
ܽ
݀
݀ݐ
ݕ ݐ = ܾ
݀
݀ݐ
)ݐ(ݔ
ெ
ୀ
ே
ୀ
ܽ݊[ݕ − ݇] = ܾ݊[ݔ − ݇]
ெ
ୀ
ே
ୀ
Prof: Sarun Soman, MIT, Manipal 20
22. Differential and Difference equation
Representations of LTI Systems
Second order difference equation:
Difference equation are easily arranged to obtain recursive
formulas for computing the current o/p of the system.
(1) Shows how to obtain y[n] from present and past values of
the input.
ܽ݊[ݕ − ݇] = ܾ݊[ݔ − ݇]
ெ
ୀ
ே
ୀ
ݕ ݊ =
1
ܽ
ܾ݊[ݔ − ݇]
ெ
ୀ
−
1
ܽ
ܽݕ ݊ − ݇
ே
ୀଵ
(1)
Prof: Sarun Soman, MIT, Manipal 22
23. Differential and Difference equation
Representations of LTI Systems
Recursive evaluation of a difference equation.
Find the first two o/p values y[0] and y[1] for the system
described by ݕ ݊ = ݔ ݊ + 2ݔ ݊ − 1 − ݕ ݊ − 1 −
ଵ
ସ
݊[ݕ − 2].
Assuming that the input is ݔ ݊ = (
ଵ
ଶ
)ݑ ݊ and the initial
conditions are y[-1]=1 and y[-2]=-2.
Prof: Sarun Soman, MIT, Manipal 23
24. Solving Differential and Difference
Equations
Output of LTI system described by differential or difference
equation has two components
ݕ
- homogeneous solution
ݕ- particular solution
Complete solution
ݕ = ݕ + ݕ
Eg.
Prof: Sarun Soman, MIT, Manipal 24
25. Solving Differential and Difference
Equations
Output due to non zero initial conditions with zero input
Prof: Sarun Soman, MIT, Manipal 25
26. Solving Differential and Difference
Equations
Step response, system initially at rest.
Prof: Sarun Soman, MIT, Manipal 26
27. Solving Differential and Difference
Equations
The Homogenous Solution
CT system
Set all terms involving input to zero
Solution
ݎare the N roots of the characteristic equation
ܽ
݀
݀ݐ
ݕ ݐ = ܾ
݀
݀ݐ
)ݐ(ݔ
ெ
ୀ
ே
ୀ
ܽ
݀
݀ݐ
ݕ ݐ = 0
ே
ୀ
ݕ()ݐ = ܿ݁௧
ே
ୀଵ
Prof: Sarun Soman, MIT, Manipal 27
28. Solving Differential and Difference
Equations
DT system
Set all terms involving input to zero
Solution
ݎ are the N roots of the characteristic equation.
ܽ݊[ݕ − ݇] = ܾ݊[ݔ − ݇]
ெ
ୀ
ே
ୀ
ܽ݊[ݕ − ݇] = 0
ே
ୀ
ݕ[݊] = ܿݎ
ே
ୀଵ
Prof: Sarun Soman, MIT, Manipal 28
29. Solving Differential and Difference
Equations
If the roots are repeating ‘p’ times then there are ‘p’ distinct
terms
݁௧, ݁ݐ௧, … . ݐିଵ݁௧
and ݎ
, ݊ݎ
… . . ݊ିଵݎ
Eg.
RC ckt depicted in figure is described by the differential equation
ݕ ݐ + ܴܥ
ௗ
ௗ௧
ݕ ݐ = )ݐ(ݔ . Determine the homogenous solution.
Prof: Sarun Soman, MIT, Manipal 29
30. Solving Differential and Difference
Equations
The homogenous equation is
ݕ ݐ + ܴܥ
ௗ
ௗ௧
ݕ ݐ = 0 (1)
Solution
ݕ
ݐ = ܿଵ݁భ௧
To determine ݎଵsubstitute in (1)
ݕ ݐ + ܴܥ
݀
݀ݐ
ݕ ݐ = 0
ܿଵ݁భ௧ 1 + ܴݎܥଵ = 0
ܿଵ݁భ௧ ≠ 0
Characteristic equation
1 + ܴݎܥଵ = 0
ݎଵ = −
1
ܴܥ
Homogenous solution of the system
is
ݕ ݐ = ܿଵ݁ି
ଵ
ோ௧
Note: ܿଵ is determined later, in order
that the complete solution satisfy
the initial conditions.
ݕ
()ݐ = ܿ݁௧
ே
ୀଵ
Prof: Sarun Soman, MIT, Manipal 30
35. Solving Differential and Difference
Equations
The Particular Solution ݕ
Solution of difference or differential equation for a given input.
Assumption : output is of same general form as the input.
CT System
Input Particular solution
1 c
t ܿଵݐ + ܿଶ
݁ି௧
ܿ݁ି௧
cos(⍵ݐ + ⏀) ܿଵ cos ߱ݐ + ܿଶ sin(߱)ݐ
DT System
Input Particular solution
1 c
n ܿଵ݊ + ܿଶ
ߙ ܿߙ
cos(Ω݊ + ⏀) ܿଵ cos Ω݊ + ܿଶ sin(Ω݊)
Prof: Sarun Soman, MIT, Manipal 35
36. Example
Consider the RC ckt .Find a particular
solution for this system with an input
ݔ ݐ = cos(߱.)ݐ
Ans:
From previous example
ݕ ݐ + ܴܥ
ௗ
ௗ௧
ݕ ݐ = )ݐ(ݔ (1)
ݕ ݐ = ܿଵcos(⍵)ݐ + ܿଶ sin(⍵)ݐ
Substitute in (1)
ݕ ݐ + ܴܥ
݀
݀ݐ
ݕ
ݐ = cos(߱)ݐ
ܿଵ cos ⍵ݐ + ܿଶ sin(⍵)ݐ
− RC⍵ܿଵ sin ⍵ݐ
+ RC⍵ܿଶ cos ⍵ݐ
= cos(⍵)ݐ
Equating the coefficients of ܿଵand ܿଶ
ܿଵ + RC⍵ܿଶ = 1
−RC⍵ܿଵ + ܿଶ = 0
Solving for ܿଵand ܿଶ
ܿଵ =
1
1 + (RC⍵)ଶ
ܿଶ =
RC⍵
1 + (RC⍵)ଶ
Prof: Sarun Soman, MIT, Manipal 36
37. Example
ݕ ݐ =
1
1 + (RC⍵)ଶ
cos ⍵ݐ
+
RC⍵
1 + (RC⍵)ଶ
sin ⍵ݐ
Determine the particular solution for
the system described by the
following differential equations.
ݕ ݐ = ܿ
0 + 10ܿ = 4
ܿ =
2
5
ݕ ݐ =
2
5
b)ݔ ݐ = cos(3)ݐ
Ans:
ݕ ݐ = ܿଵcos(3)ݐ + ܿଶ sin(3)ݐ
݀
݀ݐ
ݕ
ݐ = −3ܿଵ sin(3)ݐ
+ 3 ܿଶcos(3)ݐ
−15ܿଵ sin 3ݐ + 15ܿଶ cos 3ݐ
+ 10ܿଵ cos 3ݐ
+ 10ܿଶ sin 3ݐ = 2 cos(3)ݐ
Equating coefficients
−15ܿଵ + 10ܿଶ = 0
10ܿଵ + 15ܿଶ = 2
Prof: Sarun Soman, MIT, Manipal 37
39. Solving Differential and Difference
Equations
The Complete Solution
Procedure for calculating complete solution
1. Find the form of the homogeneous solution ݕfrom the roots of the CE
equation.
2. Find a particular solution ݕby assuming that it is of the same form as
the input, yet is independent of all terms in the homogeneous solution.
3. Determine the coefficients in the homogeneous solution so that the
complete solution ݕ = ݕ
+ ݕ
satisfies the initial conditions.
Prof: Sarun Soman, MIT, Manipal 39
40. Example
Find the solution for the first order
recursive system described by the
difference equation.
ݕ ݊ −
1
4
ݕ ݊ − 1 = ]݊[ݔ
If the input ݔ ݊ = (
ଵ
ଶ
)]݊[ݑ and the
initial condition is ݕ −1 = 8.
Ans:
N=1
Homogenous solution
ݕ ݊ −
1
4
ݕ ݊ − 1 = 0
ݕ
݊ = ܿଵݎ
ݎ
−
1
4
ݎିଵ
= 0
ݎ −
1
4
= 0
ݕ ݊ = ܿଵ(
1
4
)
Particular solution
ݕ ݊ = ܿଶ(
1
2
)
ܿଶ(
1
2
)
−
1
4
ܿଶ(
1
2
)ିଵ
= (
1
2
)
ܿଶ −
1
2
ܿଶ = 1
ܿଶ = 2
Prof: Sarun Soman, MIT, Manipal 40
46. Characteristics of Systems Described by
Differential and Difference Equations
It is informative to express the o/p of a system as sum of two
components.
1) Only with initial conditions
2) Only with input
ݕ = ݕ + ݕ
Natural response ࢟ (zero i/p response)
System o/p for zero i/p
Describes the manner in which the system dissipates any stored
energy or memory.
Prof: Sarun Soman, MIT, Manipal 46
47. Characteristics of Systems Described by
Differential and Difference Equations
Procedure to calculate ݕ
1. Find the homogeneous solution.
2. From the homogeneous solution find the coefficients ܿ using initial
conditions.
Note: homogeneous solutions apply for all time, the natural
response is determined without translating initial conditions.
Eg.
A system is described by the difference equation
ݕ ݊ −
ଵ
ସ
ݕ ݊ − 1 = ݔ ݊ , ݕ −1 = 8. Find the natural response
of the system.
Prof: Sarun Soman, MIT, Manipal 47
48. Characteristics of Systems Described by
Differential and Difference Equations
Homogenous solution
ݕ
݊ = ܿଵ
1
4
Use initial conditions to find ܿଵ
Translation not required
ݕ −1 = ܿଵ
1
4
ିଵ
ܿଵ = 2
ݕ
= 2
1
4
, ݊ ≥ −1
Prof: Sarun Soman, MIT, Manipal 48
49. Characteristics of Systems Described by
Differential and Difference Equations
The Forced Response ݕ
System o/p due to the i/p signal assuming zero initial conditions.
The forced response is of the same form as the complete
solution.
Eg.
ݕ ݊ −
ଵ
ସ
ݕ ݊ − 1 = ݔ ݊ .Find the forced response of this
system if the i/p is ݔ ݊ =
ଵ
ଶ
]݊[ݑ
Ans:
Homogenous solution
Prof: Sarun Soman, MIT, Manipal 49