12015-10-161zhongguo liu_biomedical engineering_shandong univ. biomedical signal processing chapter...

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.


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




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



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



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



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



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



Basic sequences

Unit sample sequence (discrete-time impulse, impulse)

01

00

n

nn



Basic sequences

k

knkxnx ][][][ arbitrary sequence

7213 7213 nananananp

A sum of scaled, delayed impulses



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



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



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



Basic sequences

Sinusoidal sequence

nallfornwAnx 0cos][



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



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



Periodic Sequences

A periodic sequence with integer period N

nallforNnxnx ][][

NwnwAnwA 000 coscos

integer,20 iskwherekNw

02 / , integerN k w where k is



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



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



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



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



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



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



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



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



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:



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



Example 2.7 Nonlinear Systems

Method: find one counterexample

222 1111 counterexample

2][nxny For

][log10 nxny

110log1log10 1010

counterexample

For



Properties of Discrete-time systems

2.2.3 Time-Invariant SystemsShift-Invariant Systems

012 nnxnx 012 nnyny

ny1T{ }‧ nx1

T{ }‧



Example of Time-Invariant System

Ex. 2.8 Accumulator system

n

k

kxny

01 nnxx

01011

0

1

nnykxnkxkxnynn

k

n

k

n

k



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




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



Ex. 2.10 Example for Causal System

Forward difference system is not Causal

Backward difference system is Causal

nxnxny 1

1 nxnxny




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



Ex. 2.11 Testing for Stability or Instability

2][nxny

nallforBnx x ,

nallforBBny xy ,2

if

then

is stable



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



2.3 Linear Time-Invariant (LTI) Systems

Impulse response

0nn

nh n

0nnh

T{ }‧

T{ }‧



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



k

y n x k h n k



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



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



00y n n

otherwise

Nnnh

0

101 nuanx n

h(k)

0

0

h(n-k) x(k)

h(-k)

0



1 00, 0 1n n N n N

a

aankhkxny

nn

k

kn

k

1

1 1

00

h(-k)

0

h(k)

0



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



nNa

aa

Nna

an

ny

NNn

n

1,1

1

10,1

10,0

1

1



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



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]



Parallel connection of systems

nhnhnh 21



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



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



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




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


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




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




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


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


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


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


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.


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



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


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


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


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


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


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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