chapter 2 discrete-time signals and systems in the time-domain yang jian [email protected] school...
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CHAPTER 2 Discrete-Time Signals and Systems
in the Time-Domain
CHAPTER 2 Discrete-Time Signals and Systems
in the Time-Domain
YANG Jian
School of Information Science and Technology
Yunnan University
云南大学滇池学院课程:现代信号处理数字信号处理 Digital Signal Processing 2
OutlineOutline
• Discrete-Time Signals
• Typical Sequences and Sequence Representation
• The Sampling Process
• Discrete-Time Systems
• Time-Domain Characterization of LTI Discrete-Time Systems
• Finite-Dimensional LTI Discrete-Time Systems
• Correlation of Signals
• Summary
云南大学滇池学院课程:现代信号处理数字信号处理 Digital Signal Processing 3
Discrete-Time SignalsDiscrete-Time Signals
• Basic signals
– Unit sample or unit impulse sequence
– Unit step sequence
– Exponential sequence
• Signal classification
– Continuous-time / discrete-time signals
– Deterministic / random signals
– Energy signals
• signals with finite energy
– Power signals
• signals with finite power
– Energy signals have zero power, and power signals have infinite energy
云南大学滇池学院课程:现代信号处理数字信号处理 Digital Signal Processing 4
Discrete-Time SignalsDiscrete-Time Signals
• Time-Domain Representation
– Sequence of numbers:
• — sequence
• — samples
• — sample value or nth samples, a real or complex value
– Figure of sequence:
• is defined only for integer value of
( )x n
n
( )x n
( ) ,0.3,0.76,0,1, 2,0.92,x n
( )x n n
云南大学滇池学院课程:现代信号处理数字信号处理 Digital Signal Processing 5
Discrete-Time SignalsDiscrete-Time Signals
• Operation on sequences
– Basic operation
• Adder / Subtraction:
• Scalar multiplication ( gain / attenuation ):
• Delay / Advance:
– Combination of Basic Operations
• Multiplier:
• Linear combination:
1 2( ) ( ) ( )x n x n y n
1 2( ) ( ) ( )x n x n y n
( ) ( )Ax n y n
0( ) ( )x n n y n
1 1 2 2( ) ( 3) ( )a x n a x n y n
云南大学滇池学院课程:现代信号处理数字信号处理 Digital Signal Processing 6
Discrete-Time SignalsDiscrete-Time Signals
• Operation on sequences
– Sampling Rate Alteration ( special operations of for discrete-time signals )
• Up-sampling:
• Down-sampling:
( / ), 0, , 2 , ,( )
0, ,
x n L n L Ly n
otherwise
( ) ( )y n x nM
云南大学滇池学院课程:现代信号处理数字信号处理 Digital Signal Processing 7
Discrete-Time SignalsDiscrete-Time Signals
• Classification of Sequences
– The number of sequences: finite / infinite
• Finite-length sequences:
– Symmetry
• conjugate-symmetric ( even ):
• conjugate-antisymmetric ( odd ):
– Periodity: periodic / aperiodic
• Periodic sequence:
1 2( ) 0, x n n N and n N
( ) ( ), , integer.x n x n kN for all n k is any
( ) ( )x n x n
( ) ( )x n x n
云南大学滇池学院课程:现代信号处理数字信号处理 Digital Signal Processing 8
Discrete-Time SignalsDiscrete-Time Signals
• Classification of Sequences
– Energy and Power Signals
2
2
: ( )
1: lim ( )
2 1
xn
K
kn K
energy x n
power P x nK
: < ,
: ,x
x
energy signals P
power signals P
云南大学滇池学院课程:现代信号处理数字信号处理 Digital Signal Processing 9
Discrete-Time SignalsDiscrete-Time Signals
• Classification of Sequences
– Other types of Classification
• Bounded:
• Absolutely summable:
• Square-summable:
( ) xx n B
( )n
x n
2( )
n
x n
云南大学滇池学院课程:现代信号处理数字信号处理 Digital Signal Processing 10
Typical Sequences and Sequence Representation
Typical Sequences and Sequence Representation
• Some Basic Sequences
– Unite sample sequence:
• An arbitrary sequence can be represented by unite sample sequence in time-domain
– Unite step sequence:
1, 0( )
0, 0
nn
n
1, 0( )
0, 0
nn
n
( ) ( ), ( ) ( ) ( 1)n
k
n k n n n
云南大学滇池学院课程:现代信号处理数字信号处理 Digital Signal Processing 11
Typical Sequences and Sequence Representation
Typical Sequences and Sequence Representation
• Sinusoidal and Exponential Sequences
– The real sinusoidal sequence:
– The exponential sequence:
• The sinusoidal sequence are periodic of period N as long as is an integer multiple of . The smallest possible N is the fundamental period of the sequence.
0( ) cos( ), x n A n n
0 0 0 0
0 0
( ) ( )
0 0
( )
cos( ) sin( )
j n n j nn
n n
x n A Ae A e e
A e n j A e n
0 N 2
云南大学滇池学院课程:现代信号处理数字信号处理 Digital Signal Processing 12
Typical Sequences and Sequence Representation
Typical Sequences and Sequence Representation
• Some Typical Sequences
– Regular window sequence:
– Real exponential sequence:
• Representation of an Arbitrary Sequence
– An arbitrary sequence can be represented as a weight sum of basic sequence and its delayed version.
1, 0 1( )
0, R
n Nw n
otherwise
( ) ( )nx n a n
( ) ( ) ( )k
x n x k n k
云南大学滇池学院课程:现代信号处理数字信号处理 Digital Signal Processing 13
The Sampling ProcessThe Sampling Process
• Uniform sampling:
– Often the discrete-time sequence is developed by uniformly sampling a continuous-time signal : ( )ax t
( ) ( )ax n x nT
• the sampling
frequency
• the sampling
angular frequency
1 ,TF T
2 ,T TF
云南大学滇池学院课程:现代信号处理数字信号处理 Digital Signal Processing 14
The Sampling ProcessThe Sampling Process
• Aliasing:
– When , a continuous-time sinusoidal signal of higher frequency would acquire the identity of a sinusoidal sequence of lower frequency after sampling.
e.g.
2T MAX
云南大学滇池学院课程:现代信号处理数字信号处理 Digital Signal Processing 15
Discrete-Time SystemDiscrete-Time System
• Discrete-time system
• Simple Discrete-Time Systems
– The accumulator
– The M-point moving-average filter
– The factor-of-L interpolator
( ) [ ( )] -y n H x n n
H [ ] Output y(n)Input x(n)
云南大学滇池学院课程:现代信号处理数字信号处理 Digital Signal Processing 16
Discrete-Time SystemDiscrete-Time System
• Classification of Discrete-Time System
– Linear system:
– Shift-Invariant System:
– LTI System:
The linear time-invariable discrete-time system satisfies both the linear and the time-invariable properties.
1 1 2 2
1 2 1 2
( ) ( ), ( ) ( ),
( ) ( ) ( ) ( )
if x n y n x n y n
then x n x n y n y n
0 0 ( ) ( ), ( ) ( )if x n y n then x n n y n n
云南大学滇池学院课程:现代信号处理数字信号处理 Digital Signal Processing 17
Discrete-Time SystemDiscrete-Time System
• Classification of Discrete-Time System
– Causal System:
In a causal discrete-time system, the th output sample
depends only on input samples for and
does notdepend on input samples for .
1 1 2 2
1 2
1 2
( ) ( ) ( ) ( )
{ ( ) ( ), }
{ ( ) ( ), }
if u n y n and u n y n
then u n u n for n N
implies also that y n y n for n N
0n
0( )y n ( )x n0n n
0n n
云南大学滇池学院课程:现代信号处理数字信号处理 Digital Signal Processing 18
Discrete-Time SystemDiscrete-Time System
• Classification of Discrete-Time System
– Stable System:
Definition of bounded-input, bounded-output ( BIBO ) stable.
• Passive and Lossless Systems
– The passivity:
– The losslessness: the same energy
x( ) ,
( ) ,
x
y
if n B n
then y n B n
2 2( ) ( )
n n
y n x n
云南大学滇池学院课程:现代信号处理数字信号处理 Digital Signal Processing 19
Discrete-Time SystemDiscrete-Time System
• Impulse and Step Responses
– Input sequence → output sequence
– Impulse response :
– Step response :
( )h n ( ) ( )n h n
( )s n ( ) ( )n s n
云南大学滇池学院课程:现代信号处理数字信号处理 Digital Signal Processing 20
Time-Domain Characterization of LTI Discrete-Time Systems
Time-Domain Characterization of LTI Discrete-Time Systems
• Input-Output Relationship
– The response y(n) of the LTI discrete-time system to x(n) will be given by the convolution sum:
– The operation
• Step 1, time-reverse:
• Step 2, shift n sampling period:
• Step 3, product:
• Step 4, summing all samples:
( ) ( ) ( ) ( ) ( ) ( ) ( )k k
y n x k h n k x n k h k x n h n
( ) ( )h k h k
( ) ( )h k h n k
( ) ( ) ( )x k h n k v k
( ) ( ) ( )k k
v k x k h n k
云南大学滇池学院课程:现代信号处理数字信号处理 Digital Signal Processing 21
Time-Domain Characterization of LTI Discrete-Time Systems
Time-Domain Characterization of LTI Discrete-Time Systems
• Some useful properties of the convolution operation
– Commutative:
– Associative for stable and single-sided sequences:
– Distributive:
1 2 2 1( ) ( ) ( ) ( )x n x n x n x n
1 2 3 1 2 3( ) [ ( ) ( )] [ ( ) ( )] ( )]x n x n x n x n x n x n
1 2 3 1 2 1 3( ) [ ( ) ( )] ( ) ( ) ( ) ( )]x n x n x n x n x n x n x n
云南大学滇池学院课程:现代信号处理数字信号处理 Digital Signal Processing 22
Time-Domain Characterization of LTI Discrete-Time Systems
Time-Domain Characterization of LTI Discrete-Time Systems
• Simple Interconnection Schemes
– Cascade Connection:
– Parallel Connection:
– Inverse System:
1 2( ) ( ) ( )h n h n h n
1 2( ) ( ) ( )h n h n h n
1 2( ) ( ) ( )h n h n n
1 2 2 1 1 2( ) ( ) ( ) ( ) ( ) ( )h n h n h n h n h n h n
1( )h n
2( )h n
1 2( ) ( )h n h n
云南大学滇池学院课程:现代信号处理数字信号处理 Digital Signal Processing 23
Time-Domain Characterization of LTI Discrete-Time Systems
Time-Domain Characterization of LTI Discrete-Time Systems
• Stability Condition in Terms of the Impulse Response
– An LTI digital filter is BIBO stable if only if its impulse response sequence is absolutely summable, i.e.:
• Causality Condition in Terms of the Impulse Response
– An LTI discrete-time system is causal if and only if its impulse response is a causal sequence satisfying the condition:
( )h n
( )n
S h n
( ) 0, 0h k for k
云南大学滇池学院课程:现代信号处理数字信号处理 Digital Signal Processing 24
Finite-Dimensional LTI Discrete-Time Systems
Finite-Dimensional LTI Discrete-Time Systems
• The difference equation:
– An important subclass of LTI discrete-time systems is characterized by a linear constant coefficient difference equation of the form:
– The order of the system is given by max( N, M )
0 0
( ) ( )N M
k kk k
d y n k p x n k
云南大学滇池学院课程:现代信号处理数字信号处理 Digital Signal Processing 25
Finite-Dimensional LTI Discrete-Time Systems
Finite-Dimensional LTI Discrete-Time Systems
• Total Solution Calculation
– The complementary solution
• The homogeneous difference equation:
• The characteristic equations:
云南大学滇池学院课程:现代信号处理数字信号处理 Digital Signal Processing 26
Finite-Dimensional LTI Discrete-Time Systems
Finite-Dimensional LTI Discrete-Time Systems
• Total Solution Calculation
– The particular solution is of the same form as specified input .
– The total solution:
( )py n
( )x n
( ) ( ) ( )c py n y n y n
云南大学滇池学院课程:现代信号处理数字信号处理 Digital Signal Processing 27
Finite-Dimensional LTI Discrete-Time Systems
Finite-Dimensional LTI Discrete-Time Systems
• Zero-Input Response and Zero-State Response
– zero-input response = complementary solution with initials;
– zero-state response = the convolution sum of x(n) and h(n).
( ) 0 ( ),
( ),
: ( ) ( )
zi
zs
zi zs
if x n the solution is y n
and if applying the specified input with
all initial conditions set to zero the solution is y n
then the total solution is y n y n
云南大学滇池学院课程:现代信号处理数字信号处理 Digital Signal Processing 28
Finite-Dimensional LTI Discrete-Time Systems
Finite-Dimensional LTI Discrete-Time Systems
• Impulse Response Calculation
云南大学滇池学院课程:现代信号处理数字信号处理 Digital Signal Processing 29
Finite-Dimensional LTI Discrete-Time Systems
Finite-Dimensional LTI Discrete-Time Systems
• Impulse Response Calculation
– The solutions
云南大学滇池学院课程:现代信号处理数字信号处理 Digital Signal Processing 30
Finite-Dimensional LTI Discrete-Time Systems
Finite-Dimensional LTI Discrete-Time Systems
• Location of Roots of Characteristic Equation for BIBO Stability
– A casual LTI system characteristic of a linear constant coefficient difference equation is BIBO stable, if the magnitude of each of the roots its characteristic equation is less than 1.
– The necessary and sufficient condition:
1k
云南大学滇池学院课程:现代信号处理数字信号处理 Digital Signal Processing 31
Finite-Dimensional LTI Discrete-Time Systems
Finite-Dimensional LTI Discrete-Time Systems
• Classification of LTI System
– Based on impulse response length
• Finite impulse response ( FIR ):
• Infinite impulse response ( IIR ):
1 2 1 2( ) 0,h n for n N and n N , with N N
2
1
( ) ( ) ( )N
k N
y n h k x n k
0
( ) ( ) ( )n
k
y n x k h n k
云南大学滇池学院课程:现代信号处理数字信号处理 Digital Signal Processing 32
Finite-Dimensional LTI Discrete-Time Systems
Finite-Dimensional LTI Discrete-Time Systems
• Classification of LTI System
– Based on the output calculation process
• Non-recursive system:
If the output sample can be calculated sequentially, knowing only the present and pass input samples.
• Recursive system:
If the computation of the output involves past output samples.
– Remarks:
• FIR — Non-recursive
• IIR — Recursive
云南大学滇池学院课程:现代信号处理数字信号处理 Digital Signal Processing 33
Correlation of SignalsCorrelation of Signals
• Definitions
– A measure of similarity between a pair of energy signals, x(n) and y(n), is given by the cross-correlation sequence defined by:
– The autocorrelation sequence of x(n) is given by:
( ) ( ) ( ), 0, 1, 2,xyn
r l x n y n l l
( ) ( ) ( ) ( ) ( ) ( )yx xyn m
r l y n x n l y m l x m r l
( ) ( ) ( )xxn
r l x n x n l
( ) ( ) [ ( )] ( ) ( )xyn
r l y n x l n y l x l
云南大学滇池学院课程:现代信号处理数字信号处理 Digital Signal Processing 34
Correlation of SignalsCorrelation of Signals
• Properties of Autocorrelation and Cross-correlation Sequences
– Set and as energies of the sequences x(n) and y(n) , then we can get
or equivalently
– If y(n) = x(n), then
• The sample value of the autocorrelation sequence has its max value at zero lag ( l = 0 ).
(0) 0xx xr (0) 0yy yr
2(0) (0) ( ) 0xx yy xyr r r l
( ) (0) (0)xy xx yy x yr l r r
( ) (0)xy xx xr l r
云南大学滇池学院课程:现代信号处理数字信号处理 Digital Signal Processing 35
Correlation of SignalsCorrelation of Signals
• Properties of Autocorrelation and Cross-correlation Sequences
– If , where N is integer and b>0 is an arbitrary number. In this case , so
( ) ( )y n bx n N
(0) ( ) (0)xx xy xxbr r l br
2y xb
云南大学滇池学院课程:现代信号处理数字信号处理 Digital Signal Processing 36
Correlation of SignalsCorrelation of Signals
• Normalized Forms of Correlation:
• Correlation Computation for Power and Periodic Signals
– Power signals:
– Periodic signals:
( )( )( ) , ( )
(0) (0) (0)xyxx
xx xyxx xx yy
r lr ll l
r r r
1 1( ) lim ( ) ( ), ( ) lim ( ) ( )
2 1 2 1
K K
xy xxK Kn K n K
r l x n y n l r l x n x n l K K
0 0
1 1( ) ( ) ( ), ( ) ( ) ( )
N N
xy xxn n
r l x n y n l r l x n x n lN N
云南大学滇池学院课程:现代信号处理数字信号处理 Digital Signal Processing 37
SummarySummary
• The LTI system has numerous applications in practice.
• The LTI system can be described by an input-output relation composed of a linear constant coefficient difference equation.
• The LTI discrete-time system is usually classified in terms of the length of its impulse response.