discrete-time systems
DESCRIPTION
Discrete-time Systems. Prof. Siripong Potisuk. Input-output Description. A DT system transforms DT inputs into DT outputs. System Interconnection. - Build more complex systems - Modify response of a system. Response of an LTI System. (Also referred to as Impulse response). - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Discrete-time Systems](https://reader035.vdocument.in/reader035/viewer/2022062323/5681680f550346895ddd9c16/html5/thumbnails/1.jpg)
Discrete-time Systems
Prof. Siripong Potisuk
![Page 2: Discrete-time Systems](https://reader035.vdocument.in/reader035/viewer/2022062323/5681680f550346895ddd9c16/html5/thumbnails/2.jpg)
Input-output Description
A DT system transforms DT inputs into DT outputs
![Page 3: Discrete-time Systems](https://reader035.vdocument.in/reader035/viewer/2022062323/5681680f550346895ddd9c16/html5/thumbnails/3.jpg)
System Interconnection
- Build more complex systems- Modify response of a system
![Page 4: Discrete-time Systems](https://reader035.vdocument.in/reader035/viewer/2022062323/5681680f550346895ddd9c16/html5/thumbnails/4.jpg)
Response of an LTI System
![Page 5: Discrete-time Systems](https://reader035.vdocument.in/reader035/viewer/2022062323/5681680f550346895ddd9c16/html5/thumbnails/5.jpg)
![Page 6: Discrete-time Systems](https://reader035.vdocument.in/reader035/viewer/2022062323/5681680f550346895ddd9c16/html5/thumbnails/6.jpg)
![Page 7: Discrete-time Systems](https://reader035.vdocument.in/reader035/viewer/2022062323/5681680f550346895ddd9c16/html5/thumbnails/7.jpg)
(Also referred to as Impulse response)
![Page 8: Discrete-time Systems](https://reader035.vdocument.in/reader035/viewer/2022062323/5681680f550346895ddd9c16/html5/thumbnails/8.jpg)
![Page 9: Discrete-time Systems](https://reader035.vdocument.in/reader035/viewer/2022062323/5681680f550346895ddd9c16/html5/thumbnails/9.jpg)
![Page 10: Discrete-time Systems](https://reader035.vdocument.in/reader035/viewer/2022062323/5681680f550346895ddd9c16/html5/thumbnails/10.jpg)
Properties of Convolution Sum A discrete-time LTI system is completely characterized by its impulse response, i.e., completely determines its input-output behavior. There is only one LTI system with a given h[n]
The role of h [n] and x [n] can be interchanged
][][
,][][][][][][
nxnh
knrrhrnxknhkxnhnxrk
Commutative Property
![Page 11: Discrete-time Systems](https://reader035.vdocument.in/reader035/viewer/2022062323/5681680f550346895ddd9c16/html5/thumbnails/11.jpg)
The Distributive Property
is equivalent to
![Page 12: Discrete-time Systems](https://reader035.vdocument.in/reader035/viewer/2022062323/5681680f550346895ddd9c16/html5/thumbnails/12.jpg)
The Associative Property
is equivalent to
![Page 13: Discrete-time Systems](https://reader035.vdocument.in/reader035/viewer/2022062323/5681680f550346895ddd9c16/html5/thumbnails/13.jpg)
- The impulse response of a causal LTI system must be zero before the impulse occurs.- Causality for a linear system is equivalent to the condition of initial rest.
0
][][][][][k
n
k
knxkhknhkxny
Stability for LTI Systems:
A necessary and sufficient condition for an LTI system tobe BIBO stable is that the impulse response is absolutelysummable.
Causality for LTI Systems:
![Page 14: Discrete-time Systems](https://reader035.vdocument.in/reader035/viewer/2022062323/5681680f550346895ddd9c16/html5/thumbnails/14.jpg)
Time-domain Description of DT LTI Systems
A general Nth-order linear constant-coefficient differenceequation
M
k
N
kkk knyaknxb
any
0 10
][][1][
Recursive equation, i.e., expresses the output at time n interms of previous values of the input and output
![Page 15: Discrete-time Systems](https://reader035.vdocument.in/reader035/viewer/2022062323/5681680f550346895ddd9c16/html5/thumbnails/15.jpg)
Solutions of LCCDE’s
- The complete solution depends on both the causal input x[n] and the initial conditions, y[-1], y[-2],……, y[-N ].
- The solution can be decomposed into a sum of two parts:
response state-zero therest) (initialonly input the todue is ][
responseinput -zero theinput) (noonly conditions initial the todue is ][
re whe][][][
1
0
10
ny
nynynyny
![Page 16: Discrete-time Systems](https://reader035.vdocument.in/reader035/viewer/2022062323/5681680f550346895ddd9c16/html5/thumbnails/16.jpg)
Finite Impulse Response (FIR) System
][)(][ 0,NFor 0 0
knxabny
M
k
k
The equation is nonrecursive, i.e., previously computedvalues of the output are not used to recursively computethe present value of the output.
The impulse response is seen to have finite duration andgiven by
otherwise0
,0][
,0
Mnnh a
nb
![Page 17: Discrete-time Systems](https://reader035.vdocument.in/reader035/viewer/2022062323/5681680f550346895ddd9c16/html5/thumbnails/17.jpg)
Infinite Impulse Response (IIR) System
recursive isequation difference the1,NFor
If the system is initially at rest, the impulse responsewill have infinite duration.
M
k
N
kkk knyaknxb
any
0 10
][][1][
![Page 18: Discrete-time Systems](https://reader035.vdocument.in/reader035/viewer/2022062323/5681680f550346895ddd9c16/html5/thumbnails/18.jpg)
Example Consider the difference equation
rest. initial assume and ][][ where][]1[21][ nKnxnxnyny
.)21(]1[
21][][
,)21(]1[
21]2[]2[
,)21(]0[
21]1[]1[
,]1[21]0[]0[
2
Knynxny
Kyxy
Kyxy
Kyxy
n
][)21( ][ 1, Setting nunhK n