Jingxian Wu wuj@uark.edu These slides are taken from Dr.. Jingxian Wu, University of Arkansas, 2020.. EE 2000 SIGNALS AND SYSTEMS... STSTEM: COMBINATION OF SYSTEMS• Combination of system
Trang 1ELEG 3124 Signals & Systems
Ch 6 Discrete-Time System
Dr Jingxian Wu
wuj@uark.edu
(These slides are taken from Dr Jingxian Wu, University of Arkansas, 2020.)
EE 2000 SIGNALS AND SYSTEMS
Trang 21 )
x
Trang 31 2
1 lim
– Energy signal: E
– Power signal: P
Trang 4) ( n x n N
3 cos( n
Trang 5SIGNAL: ELEMENTARY SIGNAL
• Unit impulse function
, 0 ,
0
,
1 )
(
n
n n
, 0 ,
1
,
0 )
(
n
n n
u
• Relation between unit impulse function and unit step
function
) 1 (
) ( )
Trang 6SIGNAL: ELEMENTARY SIGNAL
• Exponential function
) exp(
) cos(
) exp(
) ( n j 0n 0n j 0n
Trang 7• Discrete-time signals
• Discrete-time systems
• Z-transform
Trang 8SYSTEM: IMPULSE RESPONSE
• Impulse response of LTI system
– The response of the system when the input is (n )
)()
(n n
x =
System
)()
(n h n
• System response for arbitrary input
– Any signal can be decomposed as the sum of time-shifted impulses
) (
) ( )
System
)(n k
h −
– Time invariant
– Linear
)(
)(k n k x
)(k h n k x
Trang 9SYSTEM: CONVOLUTION SUM
• Convolution sum
– The convolution sum of two signals and is x (n ) h (n )
)(
)()
()
(n h n x k h n k x
• Response of LTI system
– The output of a LTI system is the convolution sum of the input and the impulse response of the system
(n h n
LTI system
Trang 10SYSTEM: CONVOLUTION SUM
( n h n x
Trang 11SYSTEM: CONVOLUTION SUM
• Example:
– Let be two
sequences, find
], 1 , 1 , 0 , 2 , 1 [ )
h
] 2 , 1 , 3 , 1 [ ) ( n = − −
x
) ( )
( n h n
Trang 12STSTEM: COMBINATION OF SYSTEMS
• Combination of systems
➔
Two systems in series
Two systems in parallel
Trang 13SYSTEM: DIFFERENCE EQUATION REPRESENTATION
• Difference equation representation of system
k
a
0 0
) (
) (
Trang 14• Discrete-time signals
• Discrete-time systems
• Z-transform
Trang 15• Bilateral Z-transform
n n
z n x z
• Unilateral Z-transform
n n
z n x z
(
• Z-transform:
– Ease of analysis– Doesn’t have any physical meaning (the frequency domain representation of discrete-time signal can be obtained through discrete-time Fourier transform)
– Counterpart for continuous-time systems: Laplace transform
Trang 17• Region of convergence (ROC)
Region of convergence
Trang 18Z-TRANSFORM: CONVERGENCE
• Convergence of causal signal
) ( )
( n u n
• Convergence of anti-causal signal
) 1 (
) ( n = u − n −
Trang 19Z-TRANSFORM: TIME SHIFTING PROPERTY
0 0
0 ( ) ( ) )
(
n
m
m n
n
z m x z
z X z n
n x Z
m n
n
z m x z
z X z n
n x Z
Trang 20Z-TRANSFORM: LTI SYSTEM
) (
– Z-domain representation
) ( )
(
0 0
z X z
b z
Y z
a
M
k
k k N
M
k
k k
z a
z b z
X
z
Y z
H
0
0
) (
)
( )
(
Trang 21Z-TRANSFORM: LTI SYSTEM
• Example
– Find the transfer function of the system described by the following
difference equation
) 1
( 2
1 ) ( )
2 (
2 ) 1 (
2 ) ( n − y n − + y n − = x n + x n −
y
Trang 22Z-TRANSFORM: STABILITY
• Stability
a z
z z
H
−
= )
– A LTI system is BIBO stable is all the poles are within the unit
circle (|a| < 1)– A LTI system is unstable is at least one pole is on or outside of the
unit circult ( )| a | 1