CHAPTER 2
Discrete-Time Signals and Systems
in the Time-Domain
YANG Jian
jianyang@ynu.edu.cn
School of Information Science and Technology
Yunnan University
Outline
• Discrete-Time Signals
• Typical Sequences and Sequence Representation
• The Sampling Process
• Discrete-Time Systems
• Time-Domain Characterization of LTI DiscreteTime Systems
• Finite-Dimensional LTI Discrete-Time Systems
• Correlation of Signals
• Summary
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Digital Signal Processing 2
Discrete-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
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Digital Signal Processing 3
Discrete-Time Signals
• Time-Domain Representation
– Sequence of numbers:
•
 x(n)
•
n
•
— sequence
— samples
x ( n) — sample value or nth samples, a real or complex
value
– Figure of sequence:
 x(n)  
,0.3,0.76,0,1, 2,0.92,

• x ( n) is defined only for integer value of n
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Digital Signal Processing 4
Discrete-Time Signals
• Operation on sequences
– Basic operation
• Adder / Subtraction:
x1 (n)  x2 (n)  y(n)
• Scalar multiplication ( gain / attenuation ):
• Delay / Advance:
Ax( n)  y( n)
x(n n0 )  y(n)
– Combination of Basic Operations
• Multiplier:
x1 (n)  x2 (n)  y(n)
• Linear combination:
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a1 x1 (n)  a2 x2 (n  3)  y(n)
Digital Signal Processing 5
Discrete-Time Signals
• Operation on sequences
– Sampling Rate Alteration ( special operations of for
discrete-time signals )
• Up-sampling:
 x( n / L),
y( n)  
 0,
n  0,  L, 2 L, ,
otherwise ,
• Down-sampling:
y( n)  x( nM )
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Digital Signal Processing 6
Discrete-Time Signals
• Classification of Sequences
– The number of sequences: finite / infinite
• Finite-length sequences:
x(n)  0, n  N1 and n  N 2
– Symmetry
• conjugate-symmetric ( even ):
• conjugate-antisymmetric ( odd ):
x ( n)  x  (  n)
x ( n)   x  (  n)
– Periodity: periodic / aperiodic
• Periodic sequence:
x(n)  x(n  kN ),
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for all n, k is any integer.
Digital Signal Processing 7
Discrete-Time Signals
• Classification of Sequences
– Energy and Power Signals

 energy :


 power :

x 

x ( n)
2
n 
K
1
2
P  lim
x
(
n
)

k  2 K  1
n  K
 energy  signals :

 power  signals :
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
 x <, P  
 x  , P  
Digital Signal Processing 8
Discrete-Time Signals
• Classification of Sequences
– Other types of Classification
• Bounded:
x ( n)  Bx  


• Absolutely summable:
n 

• Square-summable:
x ( n)  

2
x ( n)  
n 
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Digital Signal Processing 9
Typical Sequences and Sequence
Representation
• Some Basic Sequences
– Unite sample sequence:
 1,
 0,
 ( n)  
n0
n0
• An arbitrary sequence can be represented by unite sample
sequence in time-domain
– Unite step sequence:
 ( n) 
n
  (k ),
 1,
 ( n)  
 0,
n0
n0
 ( n)   ( n)   ( n  1)
k 
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Digital Signal Processing 10
Typical Sequences and Sequence
Representation
• Sinusoidal and Exponential Sequences
– The real sinusoidal sequence:
x(n)  A cos(0 n   ),
 n 
– The exponential sequence:
x( n)  A n  Ae ( 0  j0 ) n  A e 0ne j (0n )
 A e 0n cos(0 n   )  j A e 0n sin(0 n   )
• The sinusoidal sequence are periodic of period N as long
as  N is an integer multiple of 2 . The smallest
0
possible N is the fundamental period of the sequence.
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Digital Signal Processing 11
Typical Sequences and Sequence
Representation
• Some Typical Sequences
– Regular window sequence:
1,
w R ( n)  
 0,
0  n  N 1
otherwise
– Real exponential sequence:
x ( n)  a n  ( n )
• Representation of an Arbitrary Sequence
– An arbitrary sequence can be represented as a weight
sum of basic sequence and its delayed version.
x ( n) 

 x(k ) (n  k )
k 
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Digital Signal Processing 12
The Sampling Process
• Uniform sampling:
– Often the discrete-time sequence is developed by
uniformly sampling a continuous-time signal xa ( t ):
x(n)  xa (nT )
• F  1 T , the sampling
T
frequency
•   2 F , the sampling
T
T
angular frequency
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Digital Signal Processing 13
The Sampling Process
• Aliasing:
– When T  2 MAX , a continuous-time sinusoidal signal of
higher frequency would acquire the identity of a
sinusoidal sequence of lower frequency after sampling.
e.g.
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Digital Signal Processing 14
Discrete-Time System
• Discrete-time system
y( n)  H [ x( n)]
Input x(n)
-  n  
H[]
Output y(n)
• Simple Discrete-Time Systems
– The accumulator
– The M-point moving-average filter
– The factor-of-L interpolator
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Digital Signal Processing 15
Discrete-Time System
• Classification of Discrete-Time System
– Linear system:
if
x1 (n)  y1 (n), x2 (n)  y2 (n),
then
 x1 (n)   x2 (n)   y1 (n)   y2 (n)
– Shift-Invariant System:
if x(n)  y(n), then x( n  n0 )  y( n  n0 )
– LTI System:
The linear time-invariable discrete-time system satisfies
both the linear and the time-invariable properties.
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Digital Signal Processing 16
Discrete-Time System
• Classification of Discrete-Time System
– Causal System:
In a causal discrete-time system, the n0 th output sample
y( n0 ) depends only on input samples x ( n) for n  n0 and
does notdepend on input samples for n  n0 .
if u1 ( n)  y1 ( n)
then
and
u2 ( n)  y2 ( n)
{u1 ( n)  u2 ( n), for n  N }
implies also that { y1 ( n)  y2 ( n), for n  N }
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Digital Signal Processing 17
Discrete-Time System
• Classification of Discrete-Time System
– Stable System:
Definition of bounded-input, bounded-output ( BIBO ) stable.
x( n)  Bx , n
if
then
y( n)  B y , n
• Passive and Lossless Systems
– The passivity:


n 
– The losslessness:
2
y ( n) 


2
x ( n)  
n 
the same energy
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Digital Signal Processing 18
Discrete-Time System
• Impulse and Step Responses
– Input sequence → output sequence
– Impulse response h( n) :
– Step response s( n) :
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 ( n)  h( n)
 ( n)  s( n)
Digital Signal Processing 19
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:
y ( n) 


k 
k 
 x(k )h(n  k )   x(n  k )h(k )
x( n)  h( n)
– The operation
h( k )  h(  k )
h( k )  h(n  k )
Step 2, shift n sampling period:
• Step 1, time-reverse:
•
• Step 3, product:
x( k )h( n  k )  v( k )
• Step 4, summing all samples:
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

k 
k 
 v(k )   x(k )h(n  k )
Digital Signal Processing 20
Time-Domain Characterization of LTI
Discrete-Time Systems
• Some useful properties of the convolution
operation
– Commutative:
x1 (n)  x2 (n)  x2 (n)  x1 (n)
– Associative for stable and single-sided sequences:
x1 (n)  [ x2 ( n)  x3 ( n)]  [ x1 ( n)  x2 ( n)]  x3 ( n)]
– Distributive:
x1 (n)  [ x2 (n)  x3 (n)]  x1 (n)  x2 (n)  x1 (n)  x3 ( n)]
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Digital Signal Processing 21
Time-Domain Characterization of LTI
Discrete-Time Systems
• Simple Interconnection Schemes
– Cascade Connection:
h(n)  h1 (n)  h2 (n)
 h1 (n)  h2 (n)    h2 (n)  h1 (n)    h1 (n)  h2 (n) 
– Parallel Connection:
 h1 ( n)
 h2 ( n)
– Inverse System:
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h(n)  h1 (n)  h2 (n)

   h1 (n)  h2 (n) 

h1 (n)  h2 (n)   (n)
Digital Signal Processing 22
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 h( n) is absolutely summable, i.e.:
S


h( n)  
n 
• 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( k )  0,
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for k  0
Digital Signal Processing 23
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:
N
d
k 0
M
k
y( n  k )   pk x ( n  k )
k 0
– The order of the system is given by max( N, M )
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Digital Signal Processing 24
Finite-Dimensional LTI Discrete-Time
Systems
• Total Solution Calculation
– The complementary solution
• The homogeneous difference equation:
• The characteristic equations:
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Digital Signal Processing 25
Finite-Dimensional LTI Discrete-Time
Systems
• Total Solution Calculation
– The particular solution y p ( n) is of the same form as
specified input x ( n) .
– The total solution:
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y( n)  yc ( n)  y p ( n)
Digital Signal Processing 26
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).
if x( n)  0  the solution is yzi ( n),
and if applying the specified input with
all initial conditions set to zero  the solution is yzs ( n),
then the total solution is :
云南大学滇池学院课程：现代信号处理数字信号处理
y zi (n)  y zs ( n)
Digital Signal Processing 27
Finite-Dimensional LTI Discrete-Time
Systems
• Impulse Response Calculation
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Digital Signal Processing 28
Finite-Dimensional LTI Discrete-Time
Systems
• Impulse Response Calculation
– The solutions
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Digital Signal Processing 29
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:
k  1
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Digital Signal Processing 30
Finite-Dimensional LTI Discrete-Time
Systems
• Classification of LTI System
– Based on impulse response length
• Finite impulse response ( FIR ):
h(n)  0, for n  N1 and n  N 2 , with N 1  N 2
y( n) 
N2
 h(k ) x(n  k )
k  N1
• Infinite impulse response ( IIR ):
n
y( n)   x ( k )h( n  k )
k 0
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Digital Signal Processing 31
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
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Digital Signal Processing 32
Correlation 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:

rxy ( l ) 
ryx ( l ) 


 x(n) y(n  l ),
n 
y( n) x ( n  l ) 
n 
rxy ( l ) 
–
l  0, 1, 2,


m 


y(m  l ) x (m )  rxy (  l )
y( n) x[( l  n)]  y( l )  x (  l )
n 
The autocorrelation sequence of x(n) is given by:
rxx ( l ) 
云南大学滇池学院课程：现代信号处理数字信号处理

 x ( n) x ( n  l )
n 
Digital Signal Processing 33
Correlation of Signals
• Properties of Autocorrelation and Cross-correlation
Sequences
– Set rxx (0)   x  0 and ryy (0)   y  0 as energies of the
sequences x(n) and y(n) , then we can get
rxx (0)ryy (0)  rxy2 (l )  0
or equivalently
rxy (l )  rxx (0)ryy (0)   x y
– If y(n) = x(n), then
rxy (l )  rxx (0)   x
• The sample value of the autocorrelation sequence has its
max value at zero lag ( l = 0 ).
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Digital Signal Processing 34
Correlation of Signals
• Properties of Autocorrelation and Cross-correlation
Sequences
– If y( n)   bx( n  N ) , where N is integer and b>0 is an
arbitrary number. In this case  y  b2 x, so
 brxx (0)  rxy ( l )  brxx (0)
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Digital Signal Processing 35
Correlation of Signals
• Normalized Forms of Correlation:
r (l )
 xx ( l )  xx ,
rxx (0)
 xy (l ) 
rxy ( l )
rxx (0)ryy (0)
• Correlation Computation for Power and Periodic Signals
– Power signals:
K
1
rxy ( l )  lim
x( n) y( n  l ),

K  2 K  1
n  K
K
1
rxx ( l )  lim
x ( n) x ( n  l )

K  2 K  1
n  K
– Periodic signals:
1
r (l ) 
xy
N
N
 x(n) y(n  l ),
n 0
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1
r (l ) 
xx
N
N
 x ( n) x ( n  l )
n 0
Digital Signal Processing 36
Summary
• The LTI system has numerous applications in
practice.
• The LTI system can be described by an inputoutput 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.
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Digital Signal Processing 37