discrete-time signals and systems 主講人：虞台文. content introduction discrete-time...

Discrete-Time Signals and Systems

主講人：虞台文

Content Introduction Discrete-Time Signals---Sequences Linear Shift-Invariant Systems Stability and Causality Linear Constant-Coefficient Difference Equations Frequency-Domain Representation of Discrete-Time Signals and

Systems Representation of Sequences by Fourier Transform Symmetry Properties of Fourier Transform Fourier Transform Theorems The Existence of Fourier Transform Important Transform Pairs

Introduction


The Taxonomy of Signals

Signal: A function that conveys information

Time

Amplitude

analog signalscontinuous-time

signalsdiscrete-time

signalsdigital signals

Continuous

Continuous

Discrete

Discrete

Signal Process Systems

SignalProcessing

System

SignalProcessing

Systemsignal output

Facilitate the extraction of desired information e.g.,

• Filters• Parameter estimation


analogsystem

analogsystemsignal output

continuous-time signal continuous-time signal

discrete-time

system

discrete-time

systemsignal output

discrete-time signal discrete-time signal

digitalsystem

digitalsystemsignal output

digital signal digital signal

A important class of systems


Linear Shift-Invariant Systems.Linear Shift-Invariant Systems.

Linear Shift-Invariant Discrete-Time Systems.Linear Shift-Invariant Discrete-Time Systems.

In particular, we’ll discuss

Discrete-Time Signals---

Sequences


Representation by a Sequence

Discrete-time system theory– Concerned with processing signals that are

represented by sequences.

nnxx )},({ nnxx )},({

1 2

3 4 5 6 7

8 9 10-1-2-3-4-5-6-7-8

n

x(n)

Important Sequences

Unit-sample sequence (n)

00

01)(

n

nn

00

01)(

n

nn

1 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8

n

(n)

Sometime call (n) a discrete-time impulse; oran impulse

Important Sequences

Unit-step sequence u(n)

00

01)(

n

nnu

00

01)(

n

nnu

1 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8

n

u(n)

Fact:

)1()()( nunun )1()()( nunun

Important Sequences

Real exponential sequence

nanx )(nanx )(

1 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8

n

x(n)

. . .

. . .

Important Sequences

Sinusoidal sequence

)cos()( 0 nAnx )cos()( 0 nAnx

n

x(n)

Important Sequences

Complex exponential sequence

njenx )( 0)( njenx )( 0)(

Important Sequences

A sequence x(n) is defined to be periodic with period N if

NNnxnx allfor )()( NNnxnx allfor )()( Example: consider

njenx 0)( )()( 0000 )( Nnxeeeenx njNjNnjnj

kN 20 0

2

k

N0

2

must be a rational

number

Energy of a Sequence

Energy of a sequence is defined by

|)(| 2

n

n

nxE |)(| 2

n

n

nxE

Operations on Sequences

Sum

Product

Multiplication

Shift

)}()({ nynxyx )}()({ nynxyx

)}()({ nynxyx )}()({ nynxyx

)}({ nxx )}({ nxx

)()( 0nnxny )()( 0nnxny

Sequence RepresentationUsing delay unit

() ( ) )(k

n kx n x k

() ( ) )(k

n kx n x k

1

2

3 4 5 6

7

8 9 10-1-2-3-4-5-6-7-8

n

x(n)

a1

a2 a7

a-3

)7()3()1()3()( 7213 nananananx )7()3()1()3()( 7213 nananananx

Linear Shift-Invariant Systems


Systems

T [ ]T [ ]x(n) y(n)=T[x(n)]

Mathematically modeled as a unique transformation or

operator.

Linear Systems

T [ ]T [ ]x(n) y(n)=T[x(n)]

)]([)]([)]()([ 2121 nxbTnxaTnbxnaxT )]([)]([)]()([ 2121 nxbTnxaTnbxnaxT

Examples:

Ideal Delay System )()( dnnxny

Accumulator

n

k

kxny )()(

Moving Average2

11 2

1( ) ( )

1

k M

k M

y n x n kM M

T [ ]T [ ]x(n) y(n)=T[x(n)]

Examples:


Accumulator

n

k

kxny )()(

Moving Average

Mk

Mk

knxMM

ny1

)(1

1)(

21

T [ ]T [ ]x(n) y(n)=T[x(n)]Are these system linear?Are these system linear?

Examples:

A Memoryless System 2)]([)( nxny

T [ ]T [ ]x(n) y(n)=T[x(n)]

Is this system linear?Is this system linear?

T [ ]T [ ]x(n) y(n)=T[x(n)]

Linear Systems

)()()( knkxnxk

)()()( knkxnxk

)()()( knkxTnyk

)()()( knkxTnyk

)]([)()( knTkxnyk

)()( nhkx kk

時間 k 之 impulse

於時間 n 時之輸出值

Shift-Invariant Systems

x(n) y(n)=T[x(n)]T [ ]T [ ]

x(nk) y(nk)

x(n) y(n)

x(n-1) y(n-1)

x(n-2) y(n-2)

Shift-Invariant Systems

x(n) y(n)=T[x(n)]T [ ]T [ ]

x(n-k) y(n-k)

x(n) y(n)

x(n-1) y(n-1)

x(n-2) y(n-2)

輸入 /輸出關係僅與時間差有關輸入 /輸出關係僅與時間差有關

Linear Shift-Invariant Systems

T [ ]T [ ]x(n) y(n)=T[x(n)])()()( knkxnxk

)()()( knkxnxk

)()()( knkxTnyk

)()()( knkxTnyk

)]([)()( knTkxnyk

)()( knhkxk

時間 k 之 impulse

於時間 n 時之輸出值僅與時間差有關

Impulse Response

T [ ]T [ ]

x(n)=(n) h(n)=T[(n)]

0 0

00

Convolution Sum

T [ ]T [ ]

(n) h(n)

x(n) y(n)

)(*)()()()( nhnxknhkxnyk

convolution

A linear shift-invariant system is completely characterized by its impulse response.

A linear shift-invariant system is completely characterized by its impulse response.

Characterize a System

h(n)h(n)x(n) x(n)*h(n)

Properties of Convolution Math

)(*)()()()( nhnxknhkxnyk

)(*)()()()( nxnhknxkhnyk

)(*)()(*)( nxnhnhnx )(*)()(*)( nxnhnhnx


h1(n)h1(n)x(n) h2(n)h2(n) y(n)

h2(n)h2(n)x(n) h1(n)h1(n) y(n)

h1(n)*h2(n)h1(n)*h2(n)x(n) y(n)

These systems are identical.These systems are identical.


h1(n)+h2(n)h1(n)+h2(n)x(n) y(n)

These two systems are identical.These two systems are identical.

h1(n)h1(n)

x(n)

h2(n)h2(n)

y(n)+

Example

0 1 2 3 4 5 6

)()()( Nnununx

00

0)(

n

nanh

n

y(n)=?0 1 2 3 4 5 6

Example

)()()(*)()( knhkxnhnxnyk

0 1 2 3 4 5 6k

x(k)

0 1 2 3 4 5 6kh(k)

0 1 2 3 4 5 6kh(0k)

Example


0 1 2 3 4 5 6k

x(k)

0 1 2 3 4 5 6kh(0k)

0 1 2 3 4 5 6kh(1k)

compute y(0)

compute y(1)

How to computer y(n)?

Example


0 1 2 3 4 5 6k

x(k)

0 1 2 3 4 5 6kh(0k)

0 1 2 3 4 5 6kh(1k)

compute y(0)

compute y(1)

How to computer y(n)?

Two conditions have to be considered.Two conditions have to be considered.

n<N and nN.n<N and nN.

Example


1

1

1

)1(

00 11

1)(

a

aa

a

aaaaany

nnn

n

k

knn

k

kn

n < N

n N

11

1

0

1

0 11

1)(

a

aa

a

aaaaany

NnnNn

N

k

knN

k

kn

Example


1

1

1

)1(

00 11

1)(

a

aa

a

aaaaany

nnn

n

k

knn

k

kn

n < N

n N

11

1

0

1

0 11

1)(

a

aa

a

aaaaany

NnnNn

N

k

knN

k

kn

012

345

0 5 10 15 20 25 30 35 40 45 50

Impulse Response ofthe Ideal Delay System


)()( dnnnh )()( dnnnh

By letting x(n)=(n) and y(n)=h(n),

(n nd)(n nd)0 1 2 3 4 5 6 nd

Impulse Response ofthe Ideal Delay System

你必須知道

(n nd)(n nd)0 1 2 3 4 5 6 nd

)()(*)( dd nnxnnnx

(n nd)扮演如下功能：• Shift; or• Copy

(n nd)扮演如下功能：• Shift; or• Copy

Impulse Response ofthe Moving Average

Moving Average2

11 2

1( ) ( )

1

k M

k M

y n x n kM M

2

11 2

1( ) ( )

1

M

k M

h n n kM M

2

11 2

1( ) ( )

1

M

k M

h n n kM M

otherwise

MnMMMnh

01

1)( 21

21

otherwise

MnMMMnh

01

1)( 21

21

M1 0 M2

. . . . . .你能以 (n k)解釋嗎 ?你能以 (n k)解釋嗎 ?

Impulse Response ofthe Accumulator

)()()( nuknhn

k

)()()( nuknhn

k

你能解釋嗎 ?你能解釋嗎 ?

Accumulator

n

k

kxny )()(

0

. . .

Stability and Causality


Stability Stable systems --- every bounded input

produce a bounded output (BIBO) Necessary and sufficient condition for a BIBO

k

khS |)(|

k

khS |)(|

ProveNecessary Condition for Stability

Show that if x is bounded and S < , then y is bounded.

kk

khMknxkhny |)(|)()(|)(|

where M = max x(n)

ProveSufficient Condition for Stablility

Show that if S = , then one can find a bounded sequence x such that y is unbounded.

*( )( ) 0

( ) | ( ) |

0 ( ) 0

h nh n

x n h n

h n

Define

Skh

khkhkxy

kk

|)(|

|)(|)()()0(

2

Example: Show that the linear shift-invariant system with

impulse response h(n)=anu(n) where |a|<1 is stable.

1

1|)(|

00 k

k

k aakhS

1

1|)(|

00 k

k

k aakhS

Causality Causal systems --- output for y(n0) depends only

on x(n) with n n0. A causal system whose impulse response h(n)

satisfies

0for 0)( nnh 0for 0)( nnh

Linear Constant-Coefficient Difference Equations


N-th Order Difference Equations

M

kk

N

kk knxbknya

00

)()(

M

kk

N

kk knxbknya

00

)()(

Examples:


Accumulator )()1()( nxnyny

Moving Average

Mk

k

knxM

ny0

)(1

1)(

Compute y(n)

M

kk

N

kk knxbknya

00

)()(

M

kk

N

kk knxbknya

00

)()(

M

k

kN

k

k knxa

bkny

a

any

0 01 0

)()()(

M

k

kN

k

k knxa

bkny

a

any

0 01 0

)()()(

The Ideal Delay System

)()( dnnxny )()( dnnxny

Delay Delay Delay. . .x(n) y(n)

nd sample delaysx(n) y(n)

)()( dnnnh )()( dnnnh

The Moving Average

0

1( ) ( )

1

M

k

y n x n kM

0

1( ) ( )

1

M

k

y n x n kM

0

1( ) ( )

1

M

k

h n n kM

)1()(1

1

Mnunu

M

)(*)1()(1

1nuMnn

M

The Moving Average

0

1( ) ( )

1

M

k

y n x n kM

0

1( ) ( )

1

M

k

y n x n kM

)(*)1()(1

1)( nuMnn

Mnh

)(*)1()(

1

1)( nuMnn

Mnh

Attenuator

1

1

M+

M+1 sampledelay

Accumulatorsystem

+_

Frequency-Domain Representation of



Sinusoidal and Complex Exponential Sequences

Play an important role in DSP

LTI

h(n)

LTI

h(n)

njenx )(

k

knxkhny )()()(

k

knjekh )()(

jn

k

jk eekh )(

jnj eeH )(

Frequency Response

nje jnj eeH )()( jeH

eigenvalueeigenfunction

k

jkj ekheH )()(

k

jkj ekheH )()(

Frequency Response

k

jkj ekheH )()(

k

jkj ekheH )()(

( ) ( ) ( )j j jR IH e H e jH e ( ) ( ) ( )j j j

R IH e H e jH e

)(|)(|)(

jeHjj eeHeH)(|)(|)(

jeHjj eeHeH

magnitude

phase

Example:The Ideal Delay System

)()( dnnxny )()( dnnxny )()( dnnnh )()( dnnnh

( ) ( ) ( ) dj nj j k j kd

k k

H e h k e k n e e

1|)(| jeH

dj neH )(

magnitude

phase

Example:The Ideal Delay System

dnjj eeH )(

)cos()( 0 nAnx

njjnjj eeA

eeA

nx 00

22)( dd njnjjnjnjj eee

Aeee

Any 0000

22)(

)()( 00

22dd nnjjnnjj ee

Aee

A

])(cos[)( 0 dnnAny

Periodic Nature ofFrequency Response

k

jkj ekheH )()(

k

jkj ekheH )()(

k

jkj ekheH )2()2( )()(

k

jkekh )(

)( jeH

,2,1,0

)()( )2(

m

eHeH mjj

,2,1,0

)()( )2(

m

eHeH mjj


k

jkj ekheH )()(

k

jkj ekheH )()(,2,1,0

)()( )2(

m

eHeH mjj

,2,1,0

)()( )2(

m

eHeH mjj

|)(| jeH

234 2 3 4


k

jkj ekheH )()(

k


)()( )2(

m

eHeH mjj

,2,1,0

)()( )2(

m

eHeH mjj

|)(| jeH

234 2 3 4

Generally, we choose

To represent one period in frequency domain.


k

jkj ekheH )()(

k


)()( )2(

m

eHeH mjj

,2,1,0

)()( )2(

m

eHeH mjj

|)(| jeH

Low

FrequencyHigh

FrequencyHigh

Frequency

|)(| jeH

cc

1

|)(| jeH

aa

1

bb

|)(| jeH

cc

1

Ideal Frequency-Selective Filters

Lowpass Filter

Bandstop Filter

Highpass Filter

Moving Average

Mk

k

knxM

ny0

)(1

1)(

Mk

k

knxM

ny0

)(1

1)(

M

k

knM

nh0

)(1

1)(

M

k

knM

nh0

)(1

1)(

k

jkj ekheH )()(

h(n)

0 0 M

M

k

jkeM 01

1

j

Mj

e

e

M 1

1

1

1 )1(

)(

)(

1

12/2/2/

2/)1(2/)1(2/)1(

jjj

MjMjMj

eee

eee

M

)(

)(

1

12/2/

2/)1(2/)1(2/

jj

MjMjMj

ee

eee

M

)2/sin(

]2/)1(sin[

1

1 2/ Me

MMj

Moving Average

)2/sin(

]2/)1(sin[

1

1)( 2/ M

eM

eH Mjj

)2/sin(

]2/)1(sin[

1

1|)(|

M

MeH j

Moving Average

)2/sin(

]2/)1(sin[

1

1)( 2/ M

eM

eH Mjj

)2/sin(

]2/)1(sin[

1

1|)(|

M

MeH j

-0.5

0

0.5

1

1.5

-4 -3 -2 -1 0 1 2 3 4

-1

-0.5

0

0.5

1

-4 -3 -2 -1 0 1 2 3 4

• M=4• Lowpass• Try larger M

Representation of Sequences by

Fourier Transform


Fourier Transform Pair

n

n

njj enxeX )()(

n

n

njj enxeX )()(

Analysis

deeXnx njj )(2

1)(

deeXnx njj )(2

1)(

Synthesis

Inverse Fourier Transform(IFT)

Fourier Transform(FT)

Prove

deeXnx njj )(2

1)(

( ) 2j n me d

n = m

n

n

njj enxeX )()(

n

n

njj enxeX )()(

deeXnx njj )(2

1)(

deemx njm

m

mj)(2

1

deemx njmjm

m

)(2

1

demx mnjm

m

)()(2

1

)()(

1 )( mndjemnj

mnj

)(

)(

1 mnjemnj

)()(

)(

1 mnjmnj eemnj

)(sin2)(

1mnj

mnj

0)(

)(sin2

mn

mn

de mnj )(n m

Prove

n

n

njj enxeX )()(

n

n

njj enxeX )()(

deeXnx njj )(2

1)(

deemx njm

m

mj)(2

1

deemx njmjm

m

)(2

1

demx mnjm

m

)()(2

1

deeXnx njj )(2

1)(

= x(n)

Prove

n

n

njj enxeX )()(

n

n

njj enxeX )()(

deeXnx njj )(2

1)(

deemx njm

m

mj)(2

1

deemx njmjm

m

)(2

1

demx mnjm

m

)()(2

1

deeXnx njj )(2

1)(

Notations

n

n

njj enxeX )()(

n

n

njj enxeX )()(

Analysis

deeXnx njj )(2

1)(

deeXnx njj )(2

1)(

SynthesisInverse Fourier Transform

(IFT)

Fourier Transform(FT)

)]([)( nxeX j F

)]([)( 1 j- eXnx F

)()( jeXnx F )()( jeXnx F

Real and Imaginary Parts

( ) [ ]j j n

n

X e x n e

( ) [ ]j j n

n

X e x n e

Fourier Transform (FT)

is a complex-valued function

)()()( jI

jR

j ejXeXeX

Magnitude and Phase

)(|)(|)(

jeXjjj eeXeX

)()()( jI

jR

j ejXeXeX

)( jeX

)( jR eX

)( jI eX

|)(| jeX

)( jeX

magnitudemagnitude phasephase


Symmetry Properties of Fourier Transform

Conjugate-Symmetric andConjugate-Antisymmetric Sequences

Conjugate-Symmetric Sequence

Conjugate-Antisymmetric Sequence

)()( * nxnx ee )()( * nxnx ee

)()( * nxnx oo )()( * nxnx oo

an even sequence if it is real.

an odd sequence if it is real.

Sequence Decomposition

Any sequence can be expressed as the sum of a conjugate-symmetric one and a conjugate-antisymmetric one, i.e.,

)()()( nxnxnx oe )()()( nxnxnx oe ConjugateSymmetric

ConjugateAntisymmetric

)](*)([)( 21 nxnxnxe )](*)([)( 2

1 nxnxnxe )](*)([)( 21 nxnxnxo )](*)([)( 2

1 nxnxnxo

Function Decomposition

Any function can be expressed as the sum of a conjugate-symmetric one and a conjugate-antisymmetric one, i.e.,

)()()( jo

je

j eXeXeX )()()( jo

je

j eXeXeX

ConjugateSymmetric

ConjugateAntiymmetric

)](*)([)( 21 jjj

e eXeXeX )](*)([)( 21 jjj

e eXeXeX )](*)([)( 21 jjj

o eXeXeX )](*)([)( 21 jjj

o eXeXeX

Conjugate-Symmetric andConjugate-Antiymmetric Functions

Conjugate-Symmetric Function

Conjugate-Antisymmetric Function

)()( * je

je eXeX )()( * j

ej

e eXeX

)()( * jo

jo eXeX )()( * j

oj

o eXeX

an even function if it is real.

an odd function if it is real.

Symmetric Properties


n

jnjnj

n

eXenxenx )()()(


magnitude

phase

magnitude

phase


)()(* * jeXnx F )()(* * jeXnx F)()( jeXnx F )()( jeXnx F

magnitude

phase

magnitude

phase

n

njnj

n

enxenx*

)()(*

*

)(

n

njenx )(* jeX



magnitude

phase

magnitude

phase


)()}(Re{ je eXnx F )()}(Re{ j

e eXnx F


)()}(Im{ jo eXnxj F )()}(Im{ j

o eXnxj F

)](*)([)}(Re{ 21 nxnxnx

)]()([)](*)([ *21

21 jj eXeXnxnx F

)](*)([)}(Im{ 21 nxnxnxj

)]()([)](*)([ *21

21 jj eXeXnxnx F


)()( jRe eXnx F )()( j

Re eXnx F


)()( jIo ejXnx F )()( j

Io ejXnx F

)](*)([)( 21 nxnxnxe

)]()([)](*)([ *21

21 jj eXeXnxnx F

)](*)([)( 21 nxnxnxo

)]()([)](*)([ *21

21 jj eXeXnxnx F

Symmetric Properties for Real Sequence x(n)


magnitude

phase

Facts:

1. real part is even

2. Img. part is odd

3. Magnitude is even

4. Phase is odd

)()( jR

jR eXeX

)()( jI

jI eXeX

|)(||)(| jI

jI eXeX

)()( jj eXeX


Fourier Transform Theorems

Linearity

)()()()( jj ebYeaXnbynax F )()()()( jj ebYeaXnbynax F

11

)()(])()([n

nj

n

njnj

n

enybenxaenbynax

)()( jj ebYeaX

Time Shifting Phase Change

)()( jnjd eXennx dF )()( jnj

d eXennx dF

nj

ndd ennxnnx

)()]([F

)( jnj eXe d

)()( dnnj

n

enx

nj

n

nj enxe d

)(

Frequency Shifting Signal Modulation

)()( )( 00 jnj eXnxe F )()( )( 00 jnj eXnxe F

n

njnjnj enxenxe )()]([ 00F

n

njenx )( 0)(

)( )( 0 jeX

Time Reversal


n

njenxnx )()]([F

n

njenx )()(

)( jeX

Differentiation in Frequency

)()(

jeX

d

djnnx F )()(

jeX

d

djnnx F

n

njennxnnx )()]([F

n

nj

d

denx

j)(

1

n

njenxd

dj )( )( njeX

d

dj

The Convolution Theorem

)()()()()()(

jjj

k

eHeXeYknhkxny F )()()()()()(

jjj

k

eHeXeYknhkxny F

n

njenyny )()]([F

n

nj

k

eknhkx )()(

k n

njeknhkx )()(

k n

knjenhkx )()()(

k n

njkj enhekx )()(

)()( jj eHeX

The Modulation or Window Theorem

deWeXeYnwnxny jjj )()(2

1)()()()( )(F


1)()()()( )(F

n

njj enxnweY )()()(

n

njnjj edeeXnw )()(2

1

deeXnwn

njj )()()(2

1

denweXn

njj )()()(2

1

deWeX jj )()(2

1 )(

Parseval’s Theorem

n

jj deYeXnynx )()(2

1)(*)( *

n

jj deYeXnynx )()(2

1)(*)( *

( )1( ) *( ) ( ) ( )

2j n j j

n

x n y n e X e Y e d

)()(*

)()(*

j

j

eYny

eXnxF

F

)()(*

)()(*

j

j

eYny

eXnxF

F


1)()()()( )(F


1)()()()( )(F

Facts:

Letting =0, then proven.

Parseval’s TheoremEnergy Preserving

n

j deXnx 22 |)(|2

1|)(|

n

j deXnx 22 |)(|2

1|)(|

nn

nxnxnx )(*)(|)(| 2

deXeX jj )()(2

1 *

deX j 2|)(|2

1

Example: Ideal Lowpass Filter

cc

)( jeH

c

cjeH0

||1)(

deeHnh njjc

c

)(2

1)(

c

c

de nj

2

1

c

c

njdenj

nj )(2

1

c

c

njenj

2

1

n

nc

sin

-60 -40 -20 0 20 40 60-0.2

0

0.2

0.4

0.6


,2,1,0 sin

)(

nn

nnh c ,2,1,0

sin)(

nn

nnh c The ideal lowpass fileter

Is noncausal.


-60 -40 -20 0 20 40 60-0.2

0

0.2

0.4

0.6

,2,1,0 sin

)(

nn

nnh c ,2,1,0

sin)(

nn

nnh c The ideal lowpass fileter

Is noncausal.To approximate the ideal lowpass filter usin

g a window.

njM

Mn

cj en

neH

sin

)(nj

M

Mn

cj en

neH

sin

)(

-4 -3 -2 -1 0 1 2 3 4-1

0

1

2

M=3

-4 -3 -2 -1 0 1 2 3 4-1

0

1

2M=5

-4 -3 -2 -1 0 1 2 3 4-1

0

1

2M=19


njM

Mn

cj en

neH

sin

)(nj

M

Mn

cj en

neH

sin

)(


The Existence of Fourier Transform

Key Issue

n

njj enxeX )()(

n

njj enxeX )()(

Analysis

deeXnx njj )(2

1)(

deeXnx njj )(2

1)(

Synthesis

Does X(ej) exist for all ?

We need that |X(ej)| < for all

Sufficient Condition for Convergence

n

nx |)(| allfor |)(| jeX

|)(|)( |)(|

n

nj

n

njj enxenxeX

n

njenx |||)(|

n

nx |)(|

More On Convergence

Define

M

Mn

njjM enxeX )()(

Uniform Convergence

0|)()(|lim

jM

j

MeXeX

Mean-Square Convergence

0|)()(|lim 2

jM

j

MeXeX


Important Transform Pairs

Fourier Transform Pairs

Sequence Fourier Transform

)(n 1

)( dnn dnje

)1|(| )( anuan jae1

1

)(nu

kj

kae

)2(1

1

)()1( nuan n 2)1(

1 jae



)1|(| )(sin

)1(sin

rnunr

p

pn

jjp erer 22cos21

1

n

nc

sin

||0

||1)(

c

cjeX

otherwise

Mnnx

0

01)( 2/

)2/sin(

]2/)1(sin[ MjeM



nje 0

k

k)2(2 0

)cos( 0 n

k

jj keke )]2()2([ 00

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