CHAPTER 2
Discrete-Time Signals and Systems
CHAPTER 2
Discrete-Time Signals and Systems
YANG Jian
nxryang@126.com
School of Information Science and Technology
Yunnan University
2008-03 ~ 2008-06
云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 2
OutlineOutline
• Discrete-Time Signals
• Typical Sequences and Sequence Representation
• Discrete-Time Systems
• Time-Domain Characterization of LTI Discrete-
Time Systems
• Classification of LTI Discrete-Time Systems
• Summary
• Assignment and Experiment
云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 3
Discrete-Time SignalsDiscrete-Time Signals
• Time-Domain Representation
– Sequence of numbers:
• — sequence
• — samples
• — sample value or nth samples, a real or complex value
– An example of a discrete-time signal with real-valued
samples:
• 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
云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 4
Discrete-Time SignalsDiscrete-Time Signals
• Length of a Discrete-time signal
– Finite-length or infinite-length sequence
– Length or duration of a finite-length sequence
– Appending with zeros or zero-padding
– Right-sided sequence, causal sequence
– Left-sided sequence, anticasual sequence
– Two-sided sequence
云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 5
Discrete-Time SignalsDiscrete-Time Signals
• Operation on sequences
– Basic operation
• Product / modulation:
• Addition:
• Scalar multiplication ( gain / attenuation ):
• Time-shifting, delay / advance:
• Time-reversal / folding operation:
– Combination of Basic Operations
• Linear combination
• Example 2.4, see p35
2[ ] [ ] [ ]w n x n y n= +
3[ ] [ ]w n Ax n=
4 5( ) [ 1] ( ) [ 1]w n x n w n x n= − = +
1[ ] [ ] [ ]w n x n y n= ⋅
6 ( ) [ ]w n x n= −
云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 6
Discrete-Time SignalsDiscrete-Time Signals
• Operation on sequences
– Sampling Rate Alteration
• Up-sampling, interpolation:
• Down-sampling, decimation:
[ / ], 0, , 2 , ,
[ ]
0, ,u
x n L n L L
x n
otherwise
= ± ±=
…
[ ] [ ]y n x nM=
云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 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):
• Any complex sequence x[n] can be expressed as a sum
1 2[ ] 0, x n n N and n N= < >
[ ] [ ]x n x n∗= −
[ ] [ ]x n x n∗= − −
* *
[ ] [ ] [ ]
1 1where, [ ] ( [ ] [ ]), [ ] ( [ ] [ ])
2 2
cs ca
cs ca
x n x n x n
x n x n x n x n x n x n
= +
= + − = − −
云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 8
Discrete-Time SignalsDiscrete-Time Signals
• Classification of Sequences
– Periodity: periodic / aperiodic
• Periodic sequence:
– Energy and Power Signals
2
2
: [ ]
1: lim [ ]
2 1
x
n
K
x K n K
energy x n
power P x n
K
ε ∞
=−∞
→∞ =−
= = +
∑
∑
: , 0
: ,
x x
x x
energy signals P
power signals P
ε
ε
− < ∞ = − = ∞ < ∞
[ ] [ ], for all n, k is any integer.x n x n kN= +� �
云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 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
∞
=−∞
< ∞∑
云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 10
Typical Sequences and Sequence
Representation
Typical Sequences and Sequence
Representation
• Some Basic Sequences
– Unit sample sequence:
• An arbitrary sequence can be represented by unit sample
sequence in time-domain.
– Unit step sequence:
1, 0
[ ]
0, 0
n
n
n
δ == ≠
0
1, 0
[ ]
0, 0
[ ] [ ] , [ ] [ ] [ 1]
n
k
n
n
n
n k n n n
µ
µ δ δ µ µ
=
≥= <
= = − −∑
云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 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 , i.e., . The
smallest possible N is the fundamental period of the
sequence.
0[ ] cos( ), x n A n nω φ= + −∞ < < ∞
0 0
0 0
0 0
( )
0 0
[ ] , , ,
[ ]
cos( ) sin( )
jn j
n j n
n n
x n A n e A A e
x n A e e
A e n j A e n
σ ω φ
σ ω φ
σ σ
α α
ω φ ω φ
+
+
= − ∞ < < +∞ = =
=
= + + +
0 Nω 2π 0 2N rω π=
云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 12
Typical Sequences and Sequence
Representation
Typical Sequences and Sequence
Representation
• Some Typical Sequences
– Rectangular 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 N
w n
otherwise
≤ ≤ −=
[ ] [ ]nx n a nµ=
[ ] [ ] [ ]
k
x n x k n kδ∞
=−∞
= −∑
云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 13
Discrete-Time SystemDiscrete-Time System
• Discrete-time system
• Simple Discrete-Time Systems
– The accumulator
– The M-point moving-average filter
• p52: Example 2.11, Program 2_4
– The factor-of-L interpolator
– Median filter
[ ] { [ ]} -y n T x n n= ∞ < < ∞
Discrete-time
System Output y[n]Input x[n]
云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 14
Discrete-Time SystemDiscrete-Time System
• Classification of Discrete-Time System
– Linear system
– Shift-Invariant System
– LTI System
The linear time-invariant discrete-time system satisfies
both the linearity and the time-invariance 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 nα β α β
→ →
+ → +
0 0 [ ] [ ], [ ] [ ]if x n y n then x n n y n n→ − → −
云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 15
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 not depend 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 n 0n n≤
0n n>
云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 16
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 passive (无源的) system
– The lossless system: the same energy
x[ ] ,
[ ] ,
x
y
if n B n
then y n B n
< ∀
< ∀
2 2[ ] [ ]
n n
y n x n
∞ ∞
=−∞ =−∞
≤ < ∞∑ ∑
云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 17
Discrete-Time SystemDiscrete-Time System
• Impulse and Step Responses
– Input sequence x[n] output sequence y[n]
– Unit sample response or impulse response :
– Unit step response or step response :
[ ]h n
[ ] [ ]n h nδ →
[ ]s n
[ ] [ ]n s nµ →
→
云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 18
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
∞ ∞
=−∞ =−∞
= −∑ ∑
云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 19
Time-Domain Characterization of LTI
Discrete-Time Systems
Time-Domain Characterization of LTI
Discrete-Time Systems
• Some useful properties of the convolution
operation
– Commutative (交换率):
– Associative (结合率) for stable and single-sided
sequences:
– Distributive (分配率):
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∗ + = ∗ + ∗
云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 20
• Simple Interconnection Schemes
– Cascade Connection:
– Parallel Connection:
– Inverse System:
Time-Domain Characterization of LTI
Discrete-Time Systems
Time-Domain Characterization of LTI
Discrete-Time Systems
1 2( ) ( ) ( )h n h n h n= ∗
1 2[ ] [ ] [ ]h n h n h n= +
1 2[ ] [ ] [ ]h n h n nδ∗ =
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+
→
→
→ → →↓
↑
云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 21
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= <
云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 22
Time-Domain Characterization of LTI
Discrete-Time Systems
Time-Domain Characterization of LTI
Discrete-Time Systems
• Linear Constant Coefficient 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
0
0
0
[ ] [ ]
[ ] [ ] [ ]
[ ] [ ] [ ], 1
N M
k k
k k
N M
k k
k 1 k0 0
N M
k k
k 1 k0 0
d y n k p x n k
d pthen y n y n k x n k
d d
d por y n y n k x n k if d
d d
= =
= =
= =
− = −
= − − + −
= − − + − =
∑ ∑
∑ ∑
∑ ∑
云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 23
Classification of LTI Discrete-Time
Systems
Classification of LTI Discrete-Time
Systems
• Classification 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
For a causal IIR system
[ ] [ ] [ ]
n
k
y n x k h n k
=
= −∑
云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 24
Classification of LTI Discrete-Time
Systems
Classification of LTI Discrete-Time
Systems
• Classification Based on the output calculation
process
– Non-recursive system:
If the output sample can be calculated sequentially (顺序地),
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
云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 25
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.
云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 26
Assignment and ExperimentAssignment and Experiment
• Assignment(作业)
– 第二章:2.6, 2.19, 2.21, 2.33。
– 3月19日上午10:40以前交。
• Experiment(实验)
– 实验一:离散时间信号的时域分析
– 必做内容:Q1.29, Q1.30, Q1.31, Q1.32, Q1.33。
– 选做内容: Q1.34, Q1.35。
– 3月12日完成实验,3月19日下午2:30前交实验报告。
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