eleg 4603/5173l digital signal processing ch. 1 discrete
TRANSCRIPT
Department of Electrical EngineeringUniversity of Arkansas
ELEG 4603/5173L Digital Signal Processing
Ch. 1 Discrete-Time Signals and Systems
Dr. Jingxian Wu
OUTLINE
2
• Classifications of discrete-time signals
• Elementary discrete-time signals
• Linear time-invariant (LTI) discrete-time systems
• Causality and stability
• Difference equation representation of LTI systems
3
SIGNAL CLASSICIATION
• Discrete-time signal
– A signal that is defined only at discrete instants of time.
– Represented as
4cos)(
nnx
4exp
2
1)(
nnx
),(nx ,3,2,1,0,1,2,3, n
• Review: Analog v.s. digital
– Continuous-time signal
• continuous-time, continuous amplitude analog signal
– Example: speech signal
• Continuous-time, discrete amplitude
– Example: traffic light
– Discrete-time signal
• Discrete-time, discrete-amplitude digital signal
– Example: Telegraph, text, roll a dice
• Discrete-time, continuous-amplitude
– Example: samples of analog signal,
average monthly temperature
10
2
3
0
21
10
23
0
21
4
SIGNAL CLASSICIATION
),(tx
),(nx
5
SIGNAL CLASSIFICATION
• Periodic signal v.s. aperiodic signal
– Periodic signal
• The smallest value of N that satisfies this relation is the fundamental
periods.
– Is periodic?
)()( Nnxnx
– Example: )3cos( n
)cos( n
)cos( n
)4
3cos( n
is periodic if is an integer, and the smallest N
is the fundamental period.
)cos( n Nk
2
6
SIGNAL CLASSIFICATION
• Sum of two periodic signals
)(1 nx : fundamental period 1N
)(2 nx : fundamental period 2N
)()( 21 nxnx is periodic if both and are periodic.
Assume
q
p
N
N
2
1
)()( 21 nxnx
where p and q are not divisible of each other. The period is 12 qNpNN
)(1 nx )(2 nx
SIGNAL CLASSIFICATION
• Example:
– Is the signal periodic? If it is, what is the fundamental period?
7
)2
1
7
3sin()
9cos(
nn
8
SIGNAL CLASSIFICATION
• Energy signal
– Energy:
n
N
NnN
nxnxE22
)()(lim
– Review: energy of continuous-time signal
– Energy signal: E0
dttxE
2)(
SIGNAL CLASSIFICATION
• Power signal
– Power of discrete-time signal
9
N
NnN
nxN
P2
)(12
1lim
– Power signal: P0
– Review: power of continuous-time signal
T
TTdttx
TP
2)(
2
1lim
SIGNAL CLASSIFICATION
• Example
– Determine if the discrete-time exponential signal is an energy signal or
power signal
10
nnx 5.02)( 0n
OUTLINE
11
• Classifications of discrete-time signals
• Elementary discrete-time signals
• Linear time-invariant (LTI) discrete-time systems
• Causality and stability
• Difference equation representation of LTI systems
ELEMENTARY SIGNALS
• Basic signal operations
– Time shifting
• shift the signal to the right by k samples
– Reflecting
• Reflecting x(n) with respect to n = 0.
12
)( knx
)(nx )3( nx )3( nx
)( nx
)(nx )( nx
ELEMENTARY SIGNALS
• Basic signal operations
– Time scaling
• Example: Let
Find ,
13
odd ,1
even ,1)(
n
nnx
)2( nx )12( nx
ELEMENTARY SIGNALS
• Basic signal operations
– Time scaling
• Example:
the always points to x(0)
find
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]2,0,1,2,1[)( nx
3
2
3,
3),3(
nx
nxnx
ELEMENTARY SIGNALS
• Unit impulse function
– time shifting
• Unit step function
15
.0
,0
,0
,1)(
n
nn
.
,
,0
,1)(
kn
knkn
.0
,0
,1
,0)(
n
nnu
• Relation between unit impulse function and unit step
function)1()()( nunun
n
k
knu )()(
ELEMENTARY SIGNALS
• Exponential function
• Complex exponential function
– x(n) is periodic if is an integer, and the smallest integer N is the
fundamental period.
• Example
– Are the following signals periodic? If periodic, find fundamental period.
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)exp()( nnx
)sin()cos()exp()( 000 njnnjnx
Nk
0
2
njnx
9
7exp)(1
njnx
9
7exp)(2
OUTLINE
17
• Classifications of discrete-time signals
• Elementary discrete-time signals
• Linear time-invariant (LTI) discrete-time systems
• Causality and stability
• Difference equation representation of LTI systems
DISCRETE-TIME SYSTEMS
• Linear system
– Consider a system with the following input-output relationship
– The system is linear if it meets the superposition principle
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System)(1 nx )(1 ny System
)(2 nx )(2 ny
System)()( 21 nxnx )()( 21 nyny
• Time-invariant system
– Consider a system with the following input-output relationship
– The system is time-invariant if a time-shift at the input leads to the same
time-shift at the output
System)(nx )(ny
System)( knx )( kny
• Any arbitrary discrete-time signal can be decomposed as weighted
summation of the unit impulse functions
– Why?
E.g.
– Recall:
DISCRETE-TIME SYSTEM
• Impulse response of a LTI system
– The response of the system when the input is
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)(nSystem
)(nh
)(n
k
knkxnx )()()(
k
kkxx )3()()3(
dtxtx
)()()(
DISCRETE-TIME SYSTEM
• LTI response to arbitrary input
– Any arbitrary signal can be written as
– Time-invariant
– Linear
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k
knkxnx )()()(
)( knLTI
)( knh
)()( knkxk
LTI
)()( knhkxk
)(nxLTI
)()()( knhkxnyk
DISCRETE-TIME SYSTEM
• Convolution sum
– The convolution sum of two signals and is
21
)(nx )(nh
)()()()( knhkxnhnxk
• Response of LTI system
– The output of a LTI system is the convolution sum of the input and
the impulse response of the system.
)(nx)(nh
)()( nhnx
DISCRETE-TIME SYSTEM
• Examples
– 1.
– 2.
22
)()( mnnx
),()( nunx n )()( nunh n
)()( nhnx
23
DISCRETE-TIME SYSTEM
• Example
– 3. Find the step response of the system with impulse response
)(3
2cos
2
12)( nunnh
n
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DISCRETE-TIME SYSTEM
• Example:
– Let be two
sequences, find
],4,3,2,1[)( nh]2,1,0[)( nx
)()( nhnx
DISCRETE-TIME SYSTEM
• Properties: commutativity
25
)()()()( nxnhnhnx
)(nxh(n)
)()( nhnx )(nhx(n)
)()( nxnh
26
DISCRETE-TIME SYSTEM
• Properties: associativity
)()()()()()()()()( 212121 nhnhnxnhnhnxnhnhnx
)(nx)()( 21 nhnh
)(ny)(nx)(1 nh )(2 nh
)(ny
)(1 ny
)(nh
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DISCRETE-TIME SYSTEM
• Distributivity
)()()()()()()( 1121 nhnxnhnxnhnhnx
)(nx
)(1 nh
)(2 nh
)(ny+
)(nx)()( 21 nhnh
)(ny
DISCRETE-TIME SYSTSEMS
• Example
– Consider a system shown in the figure. Find
the overall impulse response.
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)1(2)()(1 nnnh )()1()(2 nunnh )(2)(3 nunh n
OUTLINE
29
• Classifications of discrete-time signals
• Elementary discrete-time signals
• Linear time-invariant (LTI) discrete-time systems
• Causality and stability
• Difference equation representation of LTI systems
CAUSALITY AND STABILITY
• Causal system
– A discrete-time system is causal if the output depends only on
values of input for
• The output does not depend on future input.
– Example:
• determine whether the following systems are causal.
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)( 0ny
0nn
)1()( 2 nxny
)(3)()( 2 nxnxny
)1()()1(3
1)( nxnxnxny
n
k
kxny )()(
1
)()(n
k
kxny
• Causality of LTI system
– An LTI system is causal if and only if h(n)=0 for n < 0
• Why?
– A signal x(n) is causal if x(n)=0 for n < 0.
– Example:
• For LTI systems with impulse responses given as follows. Find if the
systems are casual.
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CAUSALITY AND STABILITY
1
)()()()()()()(n
n
knhkxknhkxknhkxny
)2cos()( nnh
)()2cos()( nunnh
)1()()( nubnuanh nn
)1()()( nubnuanh nn
• Bounded-input bounded-output (BIBO) stable
– a system is BIBO stable if, for any bounded input x(n), the response y(n) is
also bounded.
• BIBO stability of LTI system
– An LTI discrete-time system is BIBO stable if
• Why?
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CAUSALITY AND STABILITY
)(kh
CAUSALITY AND STABILITY
• Example
– For an LTI system with impulse responses as follows. Are they BIBO
stable?
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)()5.0()( nunh n
)()5.0()( nunh n
nnh )5.0()(
)(2)( nunh n
)(2)( nunh n
OUTLINE
34
• Classifications of discrete-time signals
• Elementary discrete-time signals
• Linear time-invariant (LTI) discrete-time systems
• Causality and stability
• Difference equation representation of LTI systems
DIFFERENCE EQUATION
• Difference equation representation of LTI discrete-time system
– Any LTI discrete-time system can be represented as
– Review: any LTI continuous-time system can be represented as
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M
k
k
N
k
k knxbknya00
)()(
M
kk
k
k
N
kk
k
kdt
txdb
dt
tyda
00
)()(
DIFFERENCE EQUATION
• Simulation diagram
36
)()1()()()1()( 101 NnxbnxbnxbNnyanyany NN
DIFFERENCE EQUATION
• Example
– Draw the simulation diagram of the LTI system described by the following
difference equation
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)2(5)1()(5.0)1(3)(2 nxnxnxnyny
DIFFERENCE EQUATION
• Example
– The impulse response of an LTI system is
• Find the difference equation representation
• Draw the simulation diagram.
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]5,0,3,2[)( nh