Transcript
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DISCRETE TIME SIGNALS AND SYSTEMS
BY: DR.AZURA HAMZAH
PROF.SHOZO KOMAKI PROF.NOZOMU HAMADA
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DISCRETE TIME SIGNALS
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Discrete-time signals:
● analog signal
– t represents any physical quantity, time in sec.
● Discrete signal: discrete-time signal
– N is integer valued, represents discrete instances in times
)(txa
)(nx
}),1(),0(),1(,{)}({)( xxxnxnx
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Discrete-time signals:
Time: defined only at discrete values of time: ,
Descriptions: sequences of real or complex
numbers ,
Note A.: it take on values in the continuous
interval ,
Note B.: sampling of analogue signals:
• sampling interval, period: ,
• sampling rate: number of samples per
second,
• sampling frequency (Hz): .
( ) ( )f nT f n
T
1/Sf T
( ) ( , ) ,f n a b for a b
t nT
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SEQUENCES
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Representation by a Sequence
Concerned with processing signals that are represented by sequences.
nnxx )},({
1 2
3 4 5 6 7
8 9 10 -1 -2 -3 -4 -5 -6 -7 -8
n
x(n)
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Important Sequences
● Unit-sample sequence (n)
1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8
n
(n)
a discrete-time impulse; or
an impulse ,0,0,1,0,0,0,0
0,1)(
n
nn
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Important Sequences
● Unit-step sequence u(n)
1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8
n
u(n)
Fact:
)1()()( nunun
,1,1,1,0,0,0,0
0,1)(
n
nnu
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Important Sequences
●Real exponential sequence
nAnx )(
1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8
n
x(n)
. . .
. . .
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Important Sequences
● The exponential sequence
x[n] = Aαn
– Properties:
• If A and α are real then x[n] is real.
• Moreover if 0≤α<1 & A>0, then the sequence x[n] is positive and decreasing with increasing n.
• If -1≤α≤0, then the decreasing sequence values alternate in sign.
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Important Sequences
●Sinusoidal sequence
)cos()( 0 nAnx
n
x(n)
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– The sinusoidal sequence
x[n] = A cos(ωn+Φ)
• ω frequency of the sequence (in radians).
• A is magnitude of the sequence.
• Φ is the phase offset (in radians).
Important Sequences
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Important Sequences
●Complex exponential sequence
njenx )( 0)(
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Important Sequences
● A sequence x(n) is defined to be periodic with period N if
NNnxnx allfor )()(
Example: consider njenx 0)(
)()( 0000 )( Nnxeeeenx njNjNnjnj
kN 20 0
2
kN
0
2
must be a rational
number
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Energy of a Sequence
● Energy of a sequence is defined by
|)(| 2
n
n
nxE
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1
0
2|)(|1 N
n
x nxN
P
Power of a Sequence
● Power of a sequence is defined by
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Operations on Sequences
1)Sum
2)Product
3)Multiplication
)}()({ nynxyx
)}()({ nynxyx
)}({ nxx
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4) Time-Shift
● Delay by n0 samples
– n→n-n0
● Advance by n0 samples
– n→n+n0
● Example: δ[n] δ[n-2]
)()( 0nnxny
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● Any sequence (digital signal) can be expressed as a weighted sum of unit sample shifted in time; e.g.,:
● In General:
● On the other hand:
...]3[]2[]1[
][
]1[]2[]3[...][
321
0
123
nanana
na
nanananp
k
knkxnx ][][][
])[(][ nxTny
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Sequence Representation Using delay unit
k
knkxnx )()()(
1
2
3 4 5 6
7
8 9 10 -1 -2 -3 -4 -5 -6 -7 -8
n
x(n)
a1
a2 a7
a-3
)7()3()1()3()( 7213 nananananx
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0 0 , an integern n n n Time shifting
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5) Time Reversal
● Time reversal corresponds to reflection of the signal along n=0, the time axis (n → -n)
● Time folding
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6) Time Scaling
x[n]=u[n] n=0 y[0]=x[0]=1 y[n]=x[n/2] n=1 y[1]=x[1/2] – undefined y[1]=0 n=2 y[2]=x[1]=1 …
x[n] y[n]
0 0
1a t t t t
a
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Scaling; Signal Compression
n Kn K an integer > 1
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CLASSIFICATION OF SYSTEMS
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Types of Systems
● Causal & Non-causal
● Linear & Non Linear
● Time Variant &Time-invariant
● Stable & Unstable
● Static & Dynamic
● Recursive & Non-recursive Systems
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1) Causal vs. Non-causal Systems.
Definition.
A system is said to be causal if the output of the system at any time
n (i.e., y(n)) depends only on present and past inputs (i.e., x(n), x(n-
1), x(n-2), … ). Output does not depend on the future values of the
input. In mathematical terms, the output of a causal system satisfies
an equation of the form
where is some arbitrary function. If a system does not satisfy
this definition, it is called non-causal.
( ) ( ), ( 1), ( 2),y n F x n x n x n
[.]F
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Examples:
The causal system:
The non-causal system:
0
( ) ( ) ( )N
k
y n h k x n k
2( ) ( ) ( )y n x n bx n k
3( ) ( 1) ( 1)y n nx n bx n 10
10
( ) ( ) ( )k
y n h k x n k
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2) Linear vs. Non-linear Systems.
A discrete-time system is called linear if only if it satisfies the linear
superposition principle. In the other case, the system is called non-
linear.
Definition. A relaxed system is linear if only if
for any arbitrary input sequences and , and any
arbitrary constants and .
The multiplicative (scaling) property of a linear system:
The additivity property of a linear system:
[.]H
1 1 2 2 1 1 2 2( ) ( ) ( ) ( )H a x n a x n a H x n a H x n
1( )x n2( )x n
1a 2a
1 1 1 1( ) ( )H a x n a H x n
1 2 1 2( ) ( ) ( ) ( )H x n x n H x n H x n
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Examples:
The linear systems:
The non-linear systems:
0
( ) ( ) ( )N
k
y n h k x n k
2( ) ( ) ( )y n x n bx n k
3( ) ( ) ( 1)y n nx n bx n 0
( ) ( ) ( ) ( 1)N
k
y n h k x n k x n k
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3) Time-Invariant vs. Time-Variant Systems.
A discrete-time system is called time-invariant if its input-output
characteristics do not change with time. In the other case, the
system is called time-variant.
Definition. A relaxed system is time- or shift-invariant if
only if
implies that
for every input signal and every time shift k .
[.]H
( ) ( )Hx n y n
( ) ( )Hx n k y n k
( )x n
( ) ( )y n H x n
( ) ( )y n k H x n k
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Examples:
The time-invariant systems:
The time-variable systems:
3( ) ( ) ( )y n x n bx n
0
( ) ( ) ( )N
k
y n h k x n k
3( ) ( ) ( 1)y n nx n bx n
0
( ) ( ) ( )N
N n
k
y n h k x n k
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4) Stable vs. Unstable of Systems.
An arbitrary relaxed system is said to be bounded input - bounded
output (BIBO) stable if and only if every bounded input produces
the bounded output. It means, that there exist some finite numbers
say and , such that
for all n. If for some bounded input sequence x(n) , the output y(n)
is unbounded (infinite), the system is classified as unstable.
xM yM
( ) ( )x yx n M y n M
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Examples:
The stable systems:
The unstable system:
0
( ) ( ) ( )N
k
y n h k x n k
2( ) ( ) 3 ( )y n x n x n k
3( ) 3 ( 1)ny n x n
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5) Static vs. Dynamic Systems.
A discrete-time system is called static or memoryless if its output
at any time instant n depends on the input sample at the same time,
but not on the past or future samples of the input. In the other case,
the system is said to be dynamic or to have memory.
If the output of a system at time n is completely determined by the
input samples in the interval from n-N to n ( ), the system is
said to have memory of duration N.
If , the system is static or memoryless.
If , the system is said to have finite memory.
If , the system is said to have infinite memory.
0N
0N
0 N
N
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Examples:
The static (memoryless) systems:
The dynamic systems with finite memory:
The dynamic system with infinite memory:
3( ) ( ) ( )y n nx n bx n
0
( ) ( ) ( )N
k
y n h k x n k
0
( ) ( ) ( )k
y n h k x n k
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1. 6) Recursive vs. Non-recursive Systems.
A system whose output y(n) at time n depends on any number of the
past outputs values ( e.g. y(n-1), y(n-2), … ), is called a recursive
system. Then, the output of a causal recursive system can be
expressed in general as
where F[.] is some arbitrary function. In contrast, if y(n) at time n
depends only on the present and past inputs
then such a system is called nonrecursive.
( ) ( 1), ( 2), , ( ), ( ), ( 1), , ( )y n F y n y n y n N x n x n x n M
( ) ( ), ( 1), , ( )y n F x n x n x n M
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Examples:
The nonrecursive system:
The recursive system:
0
( ) ( ) ( )N
k
y n h k x n k
0 1
( ) ( ) ( ) ( ) ( )N N
k k
y n b k x n k a k y n k
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DISCRETE TIME SYSTEMS
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Discrete-Time System
● A discrete-time system is a device or algorithm that operates on a discrete-time signal called the input or excitation (e.g. x(n)), according to some rule (e.g. T{.}) to produce another discrete-time signal called the output or response (e.g. y(n)).
● A discrete–time system can be thought of as a transformation T(x) of an input sequence to an output sequence:
y[n] = T{x[n]}
T{●} x[n] y[n]
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Discrete-Time System
● If T{x} is restricted to have properties of
– Linearity, and
– Time invariance,
Then the system is referred to as linear time-invariant (LTI) system.
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Discrete-Time System
● Definition of LTI systems.
– x1[n] and x2[n] inputs to a discrete-time system.
– a & b, arbitrary constants, then
– The system is linear if and only if:
T{ax1[n] + bx2[n]} = aT{x1[n]} + bT{x2[n]}
Principle of superposition.
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Principle of Superposition.
T{ax1[n] + bx2[n]} = aT{x1[n]} + bT{x2[n]}
x1
x2
a
b
T{} y= T{ax1[n] + bx2[n]} ax1[n] + bx2[n]
x1
x2
T{}
T{}
a
b
bT{x2[n]}
y= aT{x1[n]}+bT{x2[n]}
aT{x1[n]}
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Properties of LTI Systems .
h1[n] h2[n] x[n] y[n]
h2[n] h1[n] x[n] y[n]
h1[n]*h2[n] x[n] y[n]
h2[n]*h1[n] x[n] y[n]
h1[n]
h2[n]
x[n] y[n]
h1[n]+h2[n] x[n] y[n]
h2[n]+h1[n] x[n] y[n]
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CONVOLUTION
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– LTI system is completely characterized by its impulse response h[n]:
• Impulse response is defined as the system’s response to a unit sample (or impulse).
• “*” denotes the convolution operator.
][*][][
][][][
nhnxny
knhkxnyk
Convolution
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.H
LTI system
unit impulse
( )n ( ) ( )h n H n
impulse response
LTI system description by convolution (convolution sum):
( ) ( ) ( ) ( ) ( ) ( )* ( ) ( )* ( )k k
y n h k x n k x k h n k h n x n x n h n
Impulse Response and Convolution
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● By applying Superposition Principle and Linear Time Invariance we can derive expression that defines convolution:
k k
kk
knhkxknTkxny
knkxTknkxTnxTny
][][])[(][][
][][)][][(])[(][
Convolution
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Convolution Sum
T [ ]
(n) h(n)
x(n) y(n)
)(*)()()()( nhnxknhkxnyk
convolution A linear shift-invariant system is completely
characterized by its impulse response.
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Properties of Convolution Math
)(*)()()()( nhnxknhkxnyk
)(*)()()()( nxnhknxkhnyk
)(*)()(*)( nxnhnhnx
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Properties of Convolution Math
h1(n) x(n) h2(n) y(n)
h2(n) x(n) h1(n) y(n)
h1(n)*h2(n) x(n) y(n)
These systems are identical.
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Properties of Convolution Math
h1(n)+h2(n) x(n) y(n)
These two systems are identical.
h1(n)
x(n)
h2(n)
y(n) +
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Properties of Convolution
Commutative Property
][*][][*][ nxnhnhnx
Distributive Property
])[*][(])[*][(
])[][(*][
21
21
nhnxnhnx
nhnhnx
Associative Property
][*])[*][(
][*])[*][(
][*][*][
12
21
21
nhnhnx
nhnhnx
nhnhnx
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Example
0 1 2 3 4 5 6
)()()( Nnununx
00
0)(
n
nanh
n
y(n)=? 0 1 2 3 4 5 6
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Example
)()()(*)()( knhkxnhnxnyk
0 1 2 3 4 5 6
k x(k)
0 1 2 3 4 5 6
k h(k)
0 1 2 3 4 5 6
k h(0k)
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Example
)()()(*)()( knhkxnhnxnyk
0 1 2 3 4 5 6
k x(k)
0 1 2 3 4 5 6
k h(0k)
0 1 2 3 4 5 6
k h(1k)
compute y(0)
compute y(1)
How to compute y(n)?
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Example
)()()(*)()( knhkxnhnxnyk
0 1 2 3 4 5 6
k x(k)
0 1 2 3 4 5 6
k h(0k)
0 1 2 3 4 5 6
k h(1k)
compute y(0)
compute y(1)
How to computer y(n)?
Two conditions have to be considered.
n<N and nN.
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Example
)()()(*)()( knhkxnhnxnyk
1
1
1
)1(
00 11
1)(
a
aa
a
aaaaany
nnn
n
k
knn
k
kn
n < N
n N
11
1
0
1
0 11
1)(
a
aa
a
aaaaany
NnnNn
N
k
knN
k
kn
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