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123/4/21 1Zhongguo Liu_Biomedical Engineering_Shandong
Univ.
Biomedical Signal processing
Chapter 2 Discrete-Time Signals and Systems
Zhongguo Liu
Biomedical Engineering
School of Control Science and Engineering, Shandong University
山东省精品课程山东省精品课程《《生物医学信号处理生物医学信号处理 (( 双语双语 )) 》》http://course.sdu.edu.cn/bdsp.htmlhttp://course.sdu.edu.cn/bdsp.html
http://control.sdu.edu.cn/bdsphttp://control.sdu.edu.cn/bdsp
2 04/21/232Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
Chapter 2 Discrete-Time Signals and Systems
2.0 Introduction2.1 Discrete-Time Signals: Sequences2.2 Discrete-Time Systems2.3 Linear Time-Invariant (LTI)
Systems2.4 Properties of LTI Systems2.5 Linear Constant-Coefficient
Difference Equations
3 04/21/233Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
Chapter 2 Discrete-Time Signals and Systems
2.6 Frequency-Domain Representation of Discrete-Time Signals and systems
2.7 Representation of Sequences by Fourier Transforms
2.8 Symmetry Properties of the Fourier Transform
2.9 Fourier Transform Theorems2.10 Discrete-Time Random Signals2.11 Summary
4 04/21/234Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
2.0 IntroductionSignal: something conveys information,
represented mathematically as functions of one or more independent variables. Classified as:
Continuous-time (analog) signals, discrete-time signals, digital signals
Signal-processing systems are classified along the same lines as signals: Continuous-time (analog) systems, discrete-time systems, digital systems
5 04/21/235Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
2.1 Discrete-Time Signals: Sequences
Discrete-Time signals are represented as
In sampling,
1/T (reciprocal of T) : sampling frequency
integer:,, nnnxx
periodsamplingTnTxnx a :,
Cumbersome, so just use
x n
6 04/21/236Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
Figure 2.1 Graphical representation of a discrete-
time signal
Abscissa: continuous line : is defined only at discrete instants x n
7 Figure 2.2
EXAMPLE Sampling the analog waveform
)(|)(][ nTxtxnx anTta
8 04/21/238Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
Sum of two sequences
Product of two sequences
Multiplication of a sequence by a number α
Delay (shift) of a sequence
Basic Sequence Operations
][][ nynx
integer:][][ 00 nnnxny
][][ nynx
][nx
9 04/21/239Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
Basic sequences
Unit sample sequence (discrete-time impulse, impulse)
01
00
n
nn
10 04/21/2310Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
Basic sequences
k
knkxnx ][][][ arbitrary sequence
7213 7213 nananananp
A sum of scaled, delayed impulses
11 04/21/2311Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
Basic sequences
Unit step sequence
00
01][
n
nnu
n
k
knu ][
0
][]2[]1[][][k
knnnnnu
]1[][][ nunun First backward difference
0, 0 ,1, 00 01 0since
n
k
when nk when nkk k
12 04/21/2312Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
Basic SequencesExponential sequences nAnx ][A and α are real: x[n] is realA is positive and 0<α<1, x[n] is positive
and decrease with increasing n-1<α<0, x[n] alternate in sign, but
decrease in magnitude with increasing n : x[n] grows in magnitude as n
increases1
13 04/21/2313Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
EX. 2.1 Combining Basic sequences
00
0][
n
nAnx
n
If we want an exponential sequences that is zero for n <0, then
][][ nuAnx n
Cumbersome
simpler
14 04/21/2314Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
Basic sequences
Sinusoidal sequence
nallfornwAnx 0cos][
15 04/21/2315Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
Exponential Sequences
0jwe jeAA
nwAjnwA
eAeeAAnxnn
nwjnnjwnjn
00 sincos
][ 00
1
11
Complex Exponential Sequences
Exponentially weighted sinusoidsExponentially growing
envelopeExponentially decreasing envelope
0[ ] jw nx n Ae is refered to
16 04/21/2316Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
Frequency difference between continuous-time and discrete-time complex exponentials or
sinusoids
njwnjnjwnwj AeeAeAenx 000 22][
: frequency of the complex sinusoid or complex exponential
: phase
0w
0 0[ ] cos 2 cosx n A w r n A w n
17 04/21/2317Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
Periodic Sequences
A periodic sequence with integer period N
nallforNnxnx ][][
NwnwAnwA 000 coscos
integer,20 iskwherekNw
02 / , integerN k w where k is
18 04/21/2318Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
EX. 2.2 Examples of Periodic Sequences
Suppose it is periodic sequence with period N ][][ 11 Nnxnx
4/cos][1 nnx
4/cos4/cos Nnn integer:,4/4/24/ kNnkn
01, 8 2 /k N w
2 / ( / 4) 8N k k
19 04/21/2319Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
Suppose it is periodic sequence with period N ][][ 11 Nnxnx
8/3cos8/3cos Nnn
integer:,8/38/328/3 kNnkn 16,3 Nk
EX. 2.2 Examples of Periodic Sequences
8/3cos][1 nnx 8
3
8
2
02 / 2 / (3 / 8)N k w k
0 02 3 / 2 / ( continuous signal)N w w for
20 04/21/2320Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
EX. 2.2 Non-Periodic Sequences
Suppose it is periodic sequence with period N
][][ 22 Nnxnx
nnx cos][2
)cos(cos Nnn
2 , : integer,
integer
for n k n N k
there is no N
21 04/21/2321Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
High and Low Frequencies in Discrete-time signal
0[ ] cos( )x n A w n
(b) w0 = /8 or 15/8
(c) w0 = /4 or 7/4
(d) w0 =
Frequency: The rate at which a repeating event occurs.
(a) w0 = 0 or 2
22 04/21/2322Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
2.2 Discrete-Time System
Discrete-Time System is a trasformation or operator that maps input sequence x[n] into a unique y[n]
y[n]=T{x[n]}, x[n], y[n]: discrete-time signal
T{ }‧x[n] y[n]
Discrete-Time System
23 04/21/2323Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
EX. 2.3 The Ideal Delay System
nnnxny d ],[][
If is a positive integer: the delay of the system, Shift the input sequence to the right by samples to form the output .
dn
dn
If is a negative integer: the system will shift the input to the left by samples, corresponding to a time advance.
dn
dn
24 04/21/2324Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
EX. 2.4 Moving Average
2
11 2
1 1 21 2
1
11
1 ... 1 ...1
M
k M
y n x n kM M
x n M x n M x n x n x n MM M
x[m]
mnn-5
dummy index m
y[n]for n=7, M1=0, M2=5
25 04/21/2325Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
Properties of Discrete-time
systems
2.2.1 Memoryless (memory)
system
Memoryless systems:
the output y[n] at every value of n depends
only on the input x[n] at the same value of n
2][nxny
26 04/21/2326Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
Properties of Discrete-time systems
2.2.2 Linear SystemsIf ny1T{ }‧ nx1
ny2 nx2 T{ }‧
nay nax T{ }‧
nbxnaxnx 213 nbynayny 213 T{ }‧
nyny 21 nxnx 21 T{ }‧ additivity property
homogeneity or scaling同 ( 齐 ) 次性
propertyprinciple of superposition
and only If:
27 04/21/2327Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
Example of Linear System
Ex. 2.6 Accumulator system
n
k
kxny
nbynaykxbkxa
kbxkaxkxny
n
k
n
k
n
k
n
k
2121
2133
n
k
kxny 11
n
k
kxny 22
nxandnx 21
for arbitrary
nbxnaxnx 213 when
28 04/21/2328Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
Example 2.7 Nonlinear Systems
Method: find one counterexample
222 1111 counterexample
2][nxny For
][log10 nxny
110log1log10 1010
counterexample
For
29 04/21/2329Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
Properties of Discrete-time systems
2.2.3 Time-Invariant SystemsShift-Invariant Systems
012 nnxnx 012 nnyny
ny1T{ }‧ nx1
T{ }‧
30 04/21/2330Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
Example of Time-Invariant System
Ex. 2.8 Accumulator system
n
k
kxny
01 nnxx
01011
0
1
nnykxnkxkxnynn
k
n
k
n
k
31 04/21/2331Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
Ex. 2.9 The compressor system
nMnxny ,T{ }‧ x n
0
T{ }‧
0
n 2n 2y n x n
1 0x n x n n
0
1n T{ }‧ 2 1n
0
0
y n n0
1 1y n n 2( 1)x n
1 1 0- y n x Mn x Mn n 0 0 y n n x M n n
32 04/21/2332Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
Properties of Discrete-time
systems
2.2.4 Causality
A system is causal if, for every
choice of , the output
sequence value at the index
depends only on the input
sequence value for
0n
0nn
0nn
33 04/21/2333Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
Ex. 2.10 Example for Causal System
Forward difference system is not Causal
Backward difference system is Causal
nxnxny 1
1 nxnxny
34 04/21/2334Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
Properties of Discrete-time
systems
2.2.5 Stability
Bounded-Input Bounded-Output (BIBO) Stability: every bounded input sequence produces a bounded output sequence.
nallforBnx x ,
nallforBny y ,
if
then
35 04/21/2335Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
Ex. 2.11 Testing for Stability or Instability
2][nxny
nallforBnx x ,
nallforBBny xy ,2
if
then
is stable
36 04/21/2336Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
Accumulator system
n
k
kxny
boundedn
nnunx :
01
00
boundednotnn
nkxkxny
n
k
n
k
:01
00
Accumulator system is not stable
Ex. 2.11 Testing for Stability or Instability
37 04/21/2337Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
2.3 Linear Time-Invariant (LTI) Systems
Impulse response
0nn
nh n
0nnh
T{ }‧
T{ }‧
38 04/21/2338Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
LTI Systems: Convolution
k
knkxnx
k k
y n T x k n k x k T n k
Representation of general sequence as a linear combination of delayed impulse
principle of superposition
An Illustration Example ( interpretation 1 )
nh n
n k h n k
k
x k h n k x n h n
39 04/21/2339Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
k
y n x k h n k
40 04/21/2340Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
Computation of the Convolution
reflecting h[k] about the origion to obtain h[-k]
Shifting the origin of the reflected sequence to k=n
( interpretation 2 )
k
knhkxny
nkhknh kh kh
41
Convolution can be realized by
–Reflecting(reversing) h[k] about the origin to obtain h[-k].–Shifting the origin of the reflected sequences to k=n.–Computing the weighted moving average of x[k] by using the weights given by h[n-k].
k
y n x k h n k
42 04/21/2342Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
Ex. 2.13 Analytical Evaluation of the
Convolution
otherwise
NnNnununh
0
101
For system with impulse response
h(k)
0
k
y n x k h n k
Find the output at index n
nuanx ninput
43 04/21/2343Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
00y n n
otherwise
Nnnh
0
101 nuanx n
h(k)
0
0
h(n-k) x(k)
h(-k)
0
44 04/21/2344Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
1 00, 0 1n n N n N
a
aankhkxny
nn
k
kn
k
1
1 1
00
h(-k)
0
h(k)
0
45 04/21/2345Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
a
aa
a
aa
ankhkxny
NNn
nNn
n
Nnk
kn
Nnk
1
1
11
11
11
h(-k)
0
h(k)
0
1 0 1n N n N
46 04/21/2346Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
nNa
aa
Nna
an
ny
NNn
n
1,1
1
10,1
10,0
1
1
47 04/21/2347Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
2.4 Properties of LTI Systems
Convolution is commutative(可交换的 )
nxnhnhnx
h[n]x[n] y[n]
x[n]h[n] y[n]
nhnxnhnxnhnhnx 2121
Convolution is distributed over addition
48 04/21/2348Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
Cascade connection of systems
nhnhnh 21
x [n] h1[n] h2[n] y [n]
x [n] h2[n] h1[n] y [n]
x [n] h1[n] ]h2[n]
y [n]
49 04/21/2349Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
Parallel connection of systems
nhnhnh 21
50 04/21/2350Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
Stability of LTI SystemsLTI system is stable if the impulse
response is absolutely summable .
k
khS
kk
knxkhknxkhny
xBnx xk
y n B h k
Causality of LTI systems 0,0 nnhHW: proof, Problem
2.62
51 04/21/2351Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
Impulse response of LTI systems
Impulse response of Ideal Delay systems
,d dh n n n n a positive fixed integer Impulse response of Accumulator
nun
nknh
n
k 0,0
0,1
52 04/21/2352Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
Impulse response of Moving Average systems
2
11 2
1 21 2
1 21 2
1
1
1,
1
0 ,
1 1
1
M
k M
h n n kM M
M n MM M
otherwise
u n M u n MM M
1M 2M
53 04/21/2353Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
Impulse response of Moving Average systems
1 21 2
11
1h n u n M u n M
M M
1M2M
1u n M
2 1u n M
2 1M
1M
54
Impulse response of Forward Difference
nnnh 1
1 nnnh
Impulse response of Backward Difference
55
Finite-duration impulse response (FIR) systems
The impulse response of the system has only a finite number of nonzero samples.
The FIR systems always are stable.
otherwise,
MnM,MM
knMM
nhM
Mk
01
1
1
1
2121
21
2
1
n
S h n
such as:
56
Infinite-duration impulse response (IIR)
The impulse response of the system is infinite in duration
nun
nknh
n
k 0,0
0,1
nuanh n
n
S h n
Stable IIR System:
1a
Unstable system
n
S u n
57
Output of the ideal delay system
d dy n x n n n x n n
The convolution of a shifted impulse sequence with any signal x[n] is easily evaluated by simply shifting x[n] by the displacement of the impulse.
Any noncausal FIR system can be made causal by cascading it with a sufficiently long delay.
Useful ideal delay system
58
Equivalent systems
1 1h n n n n
1 1 1n n n n n
59
Inverse system
nnunu
nnnunh
1
1
nnhnhnhnh ii
60
2.5 Linear Constant-Coefficient Difference Equations
M
mm
N
kk mnxbknya
00
An important subclass of linear time-invariant systems consist of those system for which the input x[n] and output y[n] satisfy an Nth-order linear constant-coefficient difference equation.
61
Ex. 2.14 Difference Equation Representation of the
Accumulator
,n
k
y n x k
1
1k
n
y n x k
11
nynxkxnxnyn
k
nxnyny 1
62
Block diagram of a recursive ( 递推 ) difference equation
representing an accumulator
1y n y n x n
63
Ex. 2.15 Difference Equation Representation of the Moving-
Average System with 01 M
11
12
2
MnunuM
nh
2
02 1
1 M
k
knxM
nyrepresentation 1
another representation 1
nuMnnM
nh
11
12
2
64
nuMnnM
nh
11
12
2
11
12
21
Mnxnx
Mnx
nxnyny 11
11
11 2
2
MnxnxM
nyny
M
mm
N
kk mnxbknya
00
65
Difference Equation Representation of the
System
In Chapter 6, we will see that many
(unlimited number of ) distinct
difference equations can be used to
represent a given linear time-
invariant input-output relation.
66
Solving the difference equation
Without additional constraints or information, a linear constant-coefficient difference equation for discrete-time systems does not provide a unique specification of the output for a given input.
M
mm
N
kk mnxbknya
00
67
Solving the difference equation
Output:
nynyny hp Particular solution: one output
sequence for the given input ny p nxp
Homogenous solution: solution for the homogenous equation( ):
nyh
00
N
khk knya
N
m
nmmh zAny
1
where is the roots ofmz 00
N
k
kk za
0x n
M
mm
N
kk mnxbknya
00
68
Solving the difference equation recursively
If the input and a set of auxiliary value
nx
1 10 0
0, ,1, 2,3,N M
k k
k k
a by n y n k x n nk
a a
1
0 1
1, 2, 3,
,N M
k k
k kN N
a by n N y n
n N
k x k
N
n
N
a a
N
1 , 2 ,y y y N are specified.
y(n) can be written in a recurrence formula:
69
Example 2.16 Recursive Computation of Difference Equation
1 , , 1y n ay n x n x n K n y c
Kacy 0
aKcaKacaayy 2001
KacaaKcaaayy 232012
KacaKacaaayy 3423023
1 0n ny n a c a foK r n
70
Example 2.16 Recursive Computation of Difference Equation
nxnyany 11
caxyay 11 112
cacaaxyay 2111 223
1 1ny n a c for n cacaaxyay 3211 334
1n ny n a c Ka for ln l nu a
1for n cynKnxnxnayny 11
1 0n ny n a c a foK r n
71 04/21/2371Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
Impulse response of Moving Average systems
2
11 2
1 21 2
1 21 2
1
1
1,
1
0 ,
1 1
1
M
k M
h n n kM M
M n MM M
otherwise
u n M u n MM M
1M 2M
Review
72 04/21/2372Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
Impulse response of Moving Average systems
1 21 2
11
1h n u n M u n M
M M
1M2M
1u n M
2 1u n M
2 1M
1M
Review
73
Example 2.16 Recursive Computation of Difference Equation
The system is noncausal.The system is not linear.
is not time invariant.
0nx n K n
00
1 n nny n a c Ka u n n for all n
1n ny n a c Ka for ln l nu a nxnyany 11
0, 0, 0when K x n K n should y n
1 1y n ay n x n x n K n y c
When
74
Summary
The system for which the input and
output satisfy a linear constant-
coefficient difference equation:
The output for a given input is not
uniquely specified. Auxiliary
conditions are required.
M
mm
N
kk mnxbknya
00
75
SummaryIf the auxiliary conditions are in the form
of N sequential values of the output,
1 10 0
0, ,1,2,3,N M
k k
k k
a by n y n k x n nk
a a
later value can be obtained by rearranging the difference equation as a recursive relation running forward in n,
1 , 2 , ,y y y N
M
mm
N
kk mnxbknya
00
76
Summary
and prior values can be obtained by rearranging the difference equation as a recursive relation running backward in n.
1
0 1
1, 2, 3,
,N M
k k
k kN N
a by n N y n
n N
k x k
N
n
N
a a
N
1 , 2 , ,y y y N
M
mm
N
kk mnxbknya
00
77
Difference Equation Representation of the
SystemIf a system is characterized by a linear
constant-coefficient difference equation and is further specified to be linear, time invariant, and causal, the solution is unique.
In this case, the auxiliary conditions are stated as initial-rest conditions( 初始松弛条件 ).
nx ny
0nn 0nn
The auxiliary information is that if the input
is zero for ,then the output, is constrained to be zero for
78
Summary
Linearity, time invariance, and causality of the system will depend on the auxiliary conditions.
If an additional condition is that the system is initially at rest, then the system will be linear, time invariant, and causal.
79
Example 2.16 with initial-rest conditions
1 1 0y n ay n x n x n K n y
since 0, 0x n n
ny n Ka u n
If the input is , again with initial-rest conditions, then the recursive solution is carried out using the initial condition
0nnK
00 nn,ny
00
n ny n Ka u n n
80
Discussion
If the input is , with initial-rest conditions,
0nnK 00 nn,ny
Note that for , initial rest implies that
00 n 01 y
It does mean that if .
01 00 Nnyny 00 nn,nx
Initial rest does not always means
1 0y y N
00
n ny n Ka u n n
81
2.6 Frequency-Domain Representation of Discrete-Time Signals and systems
2.6.1 Eigenfunction and Eigenvalue for LTI
is called as the eigenfunction of the system , and the associated eigenvalue is
nx
nxH
T
nxnxHnxTny If
82
Eigenfunction and Eigenvalue
Complex exponentials is the eigenfunction for LTI discrete-time systems:
,jwnx n ne
k
k k
y n h n x n h k x n k
jw n k jwk jwnh k h k
jw jwnH
e e e
e e
frequency responseeigenvalue
eigenfunction
83
Frequency response
is called as frequency response of the system.
jwH e
jw jw jwR IH e H e jH e
jwj H ejw jwH e H e e
Magnitude, phase
Real part, imaginary part
k
jwjwk jwn jwny n h k He e e e
84
Example 2.17 Frequency response of the ideal Delay
dnnxny jwnx n e
jwnjwnnnjw eeeny dd djwnjwH e e
dh n n n jw jwn
n
H e h n e
From defination(2.109):
jwnd
n
jwndn n e e
85
Example 2.17 Frequency response of the ideal Delay
cos sin
1
d
jw
d
jwnjw
d d
jw jwR I
j H ejwn jw
H e e
wn j wn
H e jH e
e H e e
86
Linear combination of complex exponential
k
njwk
kenx
k kjw jw nk
k
y n H e e
87
Example 2.18 Sinusoidal response of LTI systems
00 0cos
2 2
jw n jw nj jA Ax n A w n e e e e
0 0 0 0
2 2jw jw n jw jw nj jA A
y n H He e e e e e
00 0 0*
,jwH ejw jw jw
if h n is real then
H e H e H e e
0 00cos ,jw jwy n A H e w n H e
88
Sinusoidal response of the ideal Delay
djw nw,eH 01
0cosx n A w n
0 0
0
coscos
d
d
y n A w n w nA w n n
0 00cos ,jw jwy n A H e w n H e
0jw jw ndH e e
89
Periodic Frequency Response
The frequency response of discrete-time LTI systems is always a periodic function of the frequency variable with period
w 2
2 2j w j w n
n
H e h n e
2 2j w jwn j n jwne e e e
2j w jwH e H e
2 ,j w r jwH e H e for r an integer
jwn
n
h n e
90
Periodic Frequency Response
The “low frequencies” are frequencies close to zero
The “high frequencies” are frequencies close to
More generally, modify the frequency with
, r is integer. 2 r
jweHor w
0 2w We need only specify over
91
Example 2.19 Ideal Frequency-Selective
Filters
Frequency Response of Ideal Low-pass Filter
92
Frequency Response of Ideal High-pass Filter
93
Frequency Response of Ideal Band-stop Filter
Ideal Low-pass Filter
94
Frequency Response of Ideal Band-pass Filter
95
Example 2.20 Frequency Response of the Moving-Average
System
otherwise,
MnM,MMnh
01
121
21
1
1 2
1 2
2
1
12
1
1
1
1 1
jwM
n M
jw M
jwnH eM M
jwM
jwM M
e
e ee
96
1
12
11 2
1
1jw
jw MjwM
jwM MH e
e ee
1 11 2 1 22 12 2
22 211 2
1
jw M M jw M Mjw M M
jw jwM M
e e ee e
1 11 2 1 212 12 2
211 2
1
1
jw M M jw M Mjw M M
jwM M
e e ee
1 2 22 111 2
sin 1 21
sin 2jw M M
M M
w M M
we
97
Frequency Response of the Moving-Average System
45
45
1 2 22 111 2
sin 1 21
sin 2jw jw M M
M M
w M MH e
we
M1 = 0 and M2 = 4
211 2
sin 5 21
sin 2jw j w
M M
wH e
we
相位也取决于符号,不仅与指数相关
98
2.6.2 Suddenly applied Complex Exponential Inputs
In practice, we may not apply the complex exponential inputs e
jwn to a system, but the more practical-appearing inputs of the form
x[n] = ejwn u[n]
0, 0 0, 0y n n for x n n
i.e., x[n] suddenly applied at an arbitrary time, which for convenience we choose n=0.
For causal LTI system:
99
2.6.2 Suddenly applied Complex Exponential Inputs nuenx jwn
0
0, 0
, 0jwk jwn
n
k
n
y n x n h nh k e e n
0 1
jwk jwn jwk jwn
k k n
h k e e h k e e
For n≥0
For causal LTI system
1
jw jwn jwk jwnss t
k n
H e e h k e e y n y n
[ ]jw n k
k
h k e u n k
0
[ ]jw n k
k
h k e u n k
100
2.6.2 Suddenly applied Complex Exponential Inputs
Steady-state Response
0
jw jwn jwk jwnss
k
y n H e e h k e e
1nk
jwnjwkt eekhny
Transient response
101
2.6.2 Suddenly Applied Complex Exponential Inputs
(continue)
1 1 0
jwk jwnt
k n k n k
y n h k e e h k h k
For infinite-duration impulse response (IIR)
For stable system, transient response must become increasingly smaller as n ,
Illustration of a real part of suddenly applied complex exponential Input with IIR
102
If h[n] = 0 except for 0 n M (FIR), then the transient response yt[n] = 0 for n+1 >
M. For n M, only the steady-state
response exists
2.6.2 Suddenly Applied Complex Exponential Inputs
(continue)
Illustration of a real part of suddenly applied complex exponential Input with FIR
103
2.7 Representation of Sequences by Fourier
Transforms(Discrete-Time) Fourier Transform, DTFT, analyzing
n
jw jwnX x ne e
dweeXnx jwnjw
2
1
If is absolutely summable, i.e. then exists. (Stability)
nx n
x n
jweX
Inverse Fourier Transform, synthesis
104
Fourier Transform
jweXjjwjwI
jwR
jw eeXejXeXeX
X : , ,jwe magnitude magnitude spectrum
amplitude spectrum
spectrum,spectrumFourier
,transformFourier:eX jw
spectrumphase,phase:eX jw
rectangular form polar form
105
Principal Value (主值) is not unique because any
may be added to without affecting the result of the complex exponentiation.
jweX 2 r jweX
Principle value: is restricted to the range of values between . It is denoted as
jweX and
jwAR X eG
: phase function is referred as a continuous function of for
w w0 arg jwX e
jwj X ee
106
Impulse response and Frequency response
The frequency response of a LTI system is the Fourier transform of the impulse response.
jw jwn
n
H e h n e
dweeHnh jwnjw
2
1
107
Example 2.21: Absolute Summability
0 0
njw n jwn jw
n n
X e a e ae
The Fourier transform
nuanx nLet
0
1, 1
1n
n
a if aa
n+1
n
1 1lim 1
1 1
jwjw
jw jw
aeif ae
ae ae
( )
1or if a
108
2.6.2 Suddenly applied Complex Exponential Inputs nuenx jwn
0
0, 0
, 0jwk jwn
n
k
n
y n x n h nh k e e n
For n≥0
For causal LTI system
Illustration of a real part of suddenly applied complex exponential Input with IIR
0 1
jwk jwn jwk jwn
k k n
h k e e h k e e
ssy n ty n
Review
109
2.7 Representation of Sequences by Fourier
Transforms(Discrete-Time) Fourier Transform, DTFT, analyzing
n
jw jwnX x ne e
dweeXnx jwnjw
2
1
If is absolutely summable, i.e. then exists. (Stability)
nx n
x n
jweX
Inverse Fourier Transform, synthesis
Review
110
Discussion of convergence
2
n
x n
Absolute summability is a sufficient condition for the existence of a Fourier transform representation, and it also guarantees uniform convergence.
Some sequences are not absolutely summable, but are square summable, i.e.,
111
Discussion of convergence
Sequences which are square
summable, can be represented by a
Fourier transform, if we are willing to
relax the condition of uniform
convergence of the infinite sum
defining .
jweX
Then we have Mean-square Convergence.
n
jw jwnX x ne e
2
n
x n
112
Discussion of convergence
n
jwnjw enxeX
M
Mn
jwnjwM enxeX
0
2dweXeXlim jw
Mjw
M
The error may not approach zero at each value of as , but total “energy” in the error does.
jwM
jw eXeX w M
Mean-square convergence
113
Example 2.22 : Square-summability for the ideal
Lowpass Filter
ww,
ww,eH
c
cjwlp 0
1
1 1
2 2p
cc
c c
ww jwn jwnl w w
h n dwjn
e e
Since is nonzero for , the ideal lowpass filter is noncausal.
nhlp 0n
sin1,
2cjw n jw nc c w n
njn n
e e
114
Example 2.22 Square-summability for the ideal
Lowpass Filter
M
Mn
jwncjwM e
n
nwsineH
Define
is not absolutely summable. nhlp
jwn
n
c en
nwsin
does not converge uniformly for all w.
sinlp
cw nh n
n approaches zero as
,
but only as .
n
n1
115
Example 2.22 Square-summability for the ideal
Lowpass Filter
sinMjw j nc
Mn
w
M
w nH e e
n
Define
sin1,
2p
c c
c
w j nl w
w nh n d n
ne
1 sin[ ) / 2]
2 sin[ ) / 2]jw
Mc
c
w
w
wH e d
w
(2M+1)(
(
116
Gibbs Phenomenon
jwMH e
M=1 M=3
M=7 M=19
1 sin[ ) / 2]
2 sin[ ) / 2]c
c
w
w
wd
w
(2M+1)(
(
117
Example 2.22 continued
cwwAs M increases, oscillatory
behavior at
is more rapid, but the size of the ripple does not decrease. (Gibbs Phenomenon)
M
cww
As , the maximum amplitude of the oscillation does not approach zero, but the oscillations converge in location toward the point .
118
Example 2.22 continued
02
dweHeHlim jw
Mjw
lpM
nhlp
jwM eH
jwlp eH
However, is square summable, and converges in the mean-square sense to
does not converge
uniformly to the discontinuous
function .
jwn
n
c en
nwsin
jwlp eH
119
Example 2.23 Fourier Transform of a constant
The sequence is neither absolutely summable nor square summable.
nallfornx 1
rweXr
jw 22
The impulses are functions of a continuous variable and therefore are of “infinite height, zero width, and unit area.”
nxDefine the Fourier transform of :
120
Example 2.23 Fourier Transform of a constant:
proof
1
2
12 2
2
jw jwn
jwn
r
x n X e e dw
w r e dw
2 jwn
r
w r e dw
jwnw e dw
0 1j ne w dw
121
Example 2.24 Fourier Transform of Complex Exponential
Sequences
njwenx 0
r
jw rwweX 22 0
00
jw njwnw w e dw e
0
1
2
12 2
2
jw jwn
jwn
r
x n X e e dw
w w r e dw
1 2 2r
w r
Proof
122
Example: Fourier Transform of Complex Exponential
Sequences
n,eanxk
njwk
k
2 2jwk k
k r
X e a w w r
r k
kkjw rwwaeX 22
123
Example: Fourier Transform of unit step
sequence
nunx
rjw
jw rwe
eU 21
1
124
2.8 Symmetry Properties of the Fourier Transform
Conjugate-symmetric sequence
Conjugate-antisymmetric sequence
nxnx ee
nxnx oo
nxnxnx oe
1
2e ex n x n x n x n
nxnxnxnx oo
2
1
125
Symmetry Properties of real sequence
even sequence: a real sequence that is Conjugate-symmetric
odd sequence: real, Conjugate-antisymmetric
nxnx ee
nxnx oo
nxnxnx oe
nxnxnxnx ee 2
1
nxnxnxnx oo 2
1
real sequence:
126
jwo
jwe
jw eXeXeX
jwe
jwjwjwe eXeXeXeX
2
1
jwo
jwjwjwo eXeXeXeX
2
1
Decomposition of a Fourier transform
Conjugate-antisymmetricConjugate-symmetric
127
x[n] is complex jweXnx
jweXnx jweXnx
jwo eXnxImj
jwjwIo eXImjejXnx
1Re
2x n x n x n 1
2jw jw jwX X X
ee e e
1
2jw jw jw
RX e X e X e 1
2ex n x n x n
128
x[n] is real
jw jwx n x n X e X e
jwjw eXeX
jwRe eXnx jw
Io ejXnx
jwI
jwI eXeX
jwjw eXeX
jw jw jw jwR I R IX e jX e X e jX e
jw jwR RX e X e
Conjugate-symmetric
129
Ex. 2.25 illustration of Symmetry Properties
nuanx n 11
1
aif
aeeX
jwjw
1
1jw jw
jwX e X e
ae
jwR
jwR eX
wcosaa
wcosaeX
21
12
jwI
jwI eX
wcosaa
wsinaeX
21 2
jwjw eXwcosaa
eX
212 21
1
jwjw eXwcosa
wsinataneX
1
1
x[n] , a is real
130
Ex. 2.25 illustration of Symmetry Properties
a=0.75(solid curve) and a=0.5(dashed curve)
Real part
Imaginary part
2
1 cos
1 2 cos
a w
a a w
2
sin
1 2 cos
a w
a a w
131
Its magnitude is an even function, and phase is odd.
Ex. 2.25 illustration of Symmetry Properties
1 22
1
1 2 cos
jwX ea a w
1 sintan
1 cosjw a w
X ea w
a=0.75(solid curve) and a=0.5(dashed curve)
132
2.9 Fourier Transform Theorems
2.9.1 Linearity
1 1jwx n X e
F 2 2jwx n X e
F
1 2 1 2jw jwax n bx n aX e bX e F
{ [ ]}jwX e x nF 1[ ] { }jwx n X eF
[ ] jwx n X eF
133
Fourier Transform Theorems
2.9.2 Time shifting and frequency shifting
jweXnx
jwd
jwndx n n X ee
00 wwjnjw eXnxe
134
Fourier Transform Theorems2.9.3 Time reversal
jweXnx
jweXnx
jweXnx If is real, nx
jw jwx n x n X e X e
jw jwx n x n X e X e
If is real, even, is real, even.
nx jwX e
135
Fourier Transform Theorems2.9.4 Differentiation in Frequency
jw
n
jwnx n X e x n e
dw
edXjnnx
jw
jw
n
jwndX e dj j x n
dw dw
e
n
jwnnx n e
136
Fourier Transform Theorems
jweXnx
dweXnxE jw
n
22
2
1
is called the energy density spectrum 2jweX
2.9.5 Parseval’s Theorem
[ ]1
2jw jwn
n n
E x n x n X e e dw x n
jwX e 1
2jw jwn
n
X e x n e dw
137
Fourier Transform Theorems
2.9.6 Convolution Theorem
jweXnx jweHnh
k
y n x k h n k x n h n
jwjwjw eHeXeY
if
HW: proof
138
Fourier Transform Theorems
2.9.7 Modulation or Windowing Theorem
jweXnx jweWnw
nwnxny
deWeXeY wjjjw
2
1
HW: proof
139
Fourier transform pairs
1n 00
jwnenn
k
kwn 221
11
1n
jwa u n a
ae
12
1 jwk
u n w ke
2
11 1
1
n
jwn a u n a
ae
140
Fourier transform pairs wjjw
pp
pn
erewcosrrnu
wsin
nwsinr2221
11
1
ww,
ww,eX
n
nwsin
c
cjwc
0
1
2sin 1 21, 0
0, sin 2jwMw Mn M
x notherwise w
e
1
( ) 1 1p p p p
jw jw
jw jw jw jwjw jw
e e
r e e re e re e
1( )pjw nre
1( )pjw nre
141
Fourier transform pairs
00 2 2
k
jw n w w ke
00 01
cos ( )2
jw n j jw n jw n e e
0 02 2j j
k
e w w k e w w k
142
Ex. 2.26 Determine the Fourier Transform of sequence 5 nuanx n
jw
jwn
aeeXnuanx
1
111
jw
wjjwwjjw
n
ae
eeXeeX
nuanxnx
1
555
15
2
512
jw
wjjwjw
ae
eaeXaeXnxanx
1
55
25
25
143
Ex. 2.27 Determine an inverse
Fourier Transform of jwjwjw
beaeeX
11
1
jwjw
jw
be
bab
ae
baaeX
11
nubba
bnua
ba
anx nn
144
Ex. 2.28 Determine the impulse response from the frequency respone:
0,
,
cjwhp
cjwnd
w wH e
w we
1jw jw jwlp
jwn jwn jwnd d dhp lpH e H e H ee e e
1,
0,cjw
lpc
w wH e
w w
sin c dd d d
dhp lp
w n nh n n n h n n n n
n n
145
Ex. 2.29 Determine the impulse response for a difference equation:
Impulse response nnx
14
11
2
1 nxnxnyny
14
11
2
1 nnnhnh
jwjwjwjw eeHeeH 4
11
2
1
1 11 14 41 1 11 1 12 2 2
jw
jw jw
jw jw jwH e
e ee e e
146
12
1
4
1
2
11
nununhnn
Ex. 2.29 Determine the impulse response for a difference equation:
1
1 41 11 12 2
jw
jw
jw jwH e
ee e
12
nu n
11 1 14 2
nu n
147
2.10 Discrete-Time Random Signals
Deterministic: each value of a
sequence is uniquely determined by a
mathematically expression, a table of
data, or a rule of some type.
Stochastic signal: a member of an
ensemble of discrete-time signals that
is characterized by a set of probability
density function.
148
2.10 Discrete-Time Random Signals
For a particular signal at a particular time,
the amplitude of the signal sample at that
time is assumed to have been determined
by an underlying scheme of probability.
That is, is an outcome of some
random variable nx nx
149
2.10 Discrete-Time Random Signals
is an outcome of some random variable
( not distinguished in notation).nX
nx
The collection of random variables is called a random process(随机过程 ).
The stochastic signals do not directly have Fourier transform, but the Fourier transform of the autocorrelation and autocovariance sequece often exist.
150
Fourier transform in stochastic signals
The Fourier transform of autocorrelation sequence has a useful interpretation in terms of the frequency distribution of the power in the signal.
The effect of processing stochastic signals with a discrete-time LTI system can be described in terms of the effect of the system on the autocorrelation sequence.
151
Stochastic signal as inputLet be a real-valued sequence
that is a sample sequence of a wide-sense stationary discrete-time random process (随机过程 ).
nx
nh nx ny
kk
knxnhkxknhny
If the input is stationary, then so is the output
Consider a stable LTI system with real h[n].
152
Stochastic signal as input
The mean of output process
0
yk
jx x
k
m E y n h k E x n k
m h k H e m
mXn = E{Xn }, mYn= E(Yn}, can be written
as mx[n] = E{x[n]}, my[n] =E(y[n]}.
In our discussion, no necessary to distinguish between the random variables Xn andYn and their specific values x[n] and y[n].
wide-sense stationary
153
Stochastic signal as inputThe autocorrelation function of output
yy m E y n y n m
is called a deterministic autocorrelation sequence or autocorrelation sequence of
nh nchh
hhk
where l h k h l kc
ll kl
( )xx xx hhk r l
h k h r m r k m l lc
k r
E h k h r x n k x n m r
k r
h k x n k h r x n m r
154
Stochastic signal as input
jwxx
jwhh
jwyy eeCe
2jwjwjwjwhh eHeHeHeC
2jw jw jw
yy xxe H e e
yy m E y n y n m xx hhl
m l lc
*hhk
where l h k h l k h l h lc
DTFT of the autocorrelation function of output
power (density) spectrum
real h[n]
155
Total average power in output
2
2
t
10
21
2otal average power in output
jwyy yy
jw jwxx
E y n e dw
H e e dw
2jw jw jw
yy xxe H e e provides the motivation for the term
power density spectrum.
能量无限
dweXnxE jw
n
22
2
1Parseval’s Theorem
能量有限
0jwe
For Ideal bandpass system
1 1
2 2
b b
a a
jw jwxx xxe dw e dw
Since is a real, even, its FT
is also real and even, i.e., jw jwxx xxe e
jwxx e xx m
jwxx
jwjwyy eeHe
2so is
Suppose that H(e jw ) is an ideal bandpass filter, as shown in Figure .
210
2jw jw
yy xxH e e dw
157
For Ideal bandpass system
1 10
2 2
b b
a a
jw jwyy xx xxe dw e dw
( ) 0
lim 0 0b a
yy
0jwxx e for all w
能量非负
the power density function of a real signal is real, even, and nonnegative.
the area under for
can be taken to represent the mean
square
value of the input in that frequency
band.
jwxx e a b
158
Ex. 2.30 White Noise
2xx xm m
2jw jwmxx xx x
m
e m e for all w
2 21 10
2 2jw
xx xx x xe dw dw
The average power of a white noise is
A white-noise signal is a signal for which
Assume the signal has zero mean. The power spectrum of a white noise is
159
A noise signal with power spectrum
can be assumed to be the
output of a LTI system with white-
noise input.
A noise signal whose power spectrum is not
constant with frequency.
Color Noise
jwyy e
22
xjwjw
yy eHe
160
Suppose
Color Noise
2
22 2
2
2
1
1
1 2 cos
jw jwyy x xjw
x
e H eae
a a w
nh n a u n
1
1jw
jwH e
ae
161
Cross-correlation between the input and output
xy
k
m E x n y n m
E x n h k x n m k
jwxx
jwjwxy eeHe
xxk
xx
h k m k
h k k
162
Cross-correlation between the input and
outputIf 2
xx xm m
2 2xy xx x x
k
m h k k h k m k h m
That is, for a zero mean white-noise
input, the cross-correlation between
input and output of a LTI system is
proportional to the impulse response
of the system.
163
Cross power spectrum between the input and
output
The cross power spectrum is proportional
to the frequency response of the system.
w,e xjw
xx2
jwx
jwxy eHe 2
164
2.11 Summary
Define a set of basic sequence.Define and represent the LTI
systems in terms of the convolution, stability and causality.
Introduce the linear constant-coefficient difference equation with initial rest conditions for LTI , causal system.
Recursive solution of linear constant-coefficient difference equations.
165
2.11 Summary
Define FIR and IIR systems
Define frequency response of the LTI system.
Define Fourier transform.
Introduce the properties and theorems of Fourier transform. (Symmetry)
Introduce the discrete-time random signals.
166 23/4/21166Zhongguo Liu_Biomedical Engineering_Shandong U
niv.
Chapter 2 HW
2.1, 2.2, 2.4, 2.5, 2.7, 2.11, 2.12,2.15, 2.20, 2.62
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