Discrete-Time Signals and Systems 主主主 主主主 :
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
Discrete-Time Signals and Systems
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
Signal Process Systems
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
Signal Process 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
Discrete-Time Signals and Systems
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
Discrete-Time Signals and 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:
Ideal Delay System )()( dnnxny
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
Properties of Convolution Math
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.
Properties of Convolution Math
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
)()()(*)()( knhkxnhnxnyk
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
)()()(*)()( knhkxnhnxnyk
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
)()()(*)()( knhkxnhnxnyk
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
)()()(*)()( knhkxnhnxnyk
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
Ideal Delay System )()( dnnxny
)()( 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
Discrete-Time Signals and Systems
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
Discrete-Time Signals and Systems
N-th Order Difference Equations
M
kk
N
kk knxbknya
00
)()(
M
kk
N
kk knxbknya
00
)()(
Examples:
Ideal Delay System )()( dnnxny
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
Discrete-Time Signals and Systems
Discrete-Time Signals and Systems
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
Periodic Nature ofFrequency Response
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
Periodic Nature ofFrequency Response
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
Generally, we choose
To represent one period in frequency domain.
Periodic Nature ofFrequency Response
k
jkj ekheH )()(
k
jkj ekheH )()(,2,1,0
)()( )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
Discrete-Time Signals and Systems
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
Discrete-Time Signals and Systems
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
)()( jeXnx F )()( jeXnx F
n
jnjnj
n
eXenxenx )()()(
)()( jeXnx F )()( jeXnx F
magnitude
phase
magnitude
phase
Symmetric Properties
)()(* * jeXnx F )()(* * jeXnx F)()( jeXnx F )()( jeXnx F
magnitude
phase
magnitude
phase
n
njnj
n
enxenx*
)()(*
*
)(
n
njenx )(* jeX
Symmetric Properties
)()(* * jeXnx F )()(* * jeXnx F)()( jeXnx F )()( jeXnx F
magnitude
phase
magnitude
phase
Symmetric Properties
)()}(Re{ je eXnx F )()}(Re{ j
e eXnx F
)()( jeXnx F )()( jeXnx 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
Symmetric Properties
)()( jRe eXnx F )()( j
Re eXnx F
)()( jeXnx F )()( jeXnx 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)
)()(* * jeXnx F )()(* * jeXnx F)()( jeXnx F )()( jeXnx F
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
Discrete-Time Signals and Systems
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
)()( jeXnx F )()( jeXnx F
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
deWeXeYnwnxny jjj )()(2
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
deWeXeYnwnxny jjj )()(2
1)()()()( )(F
deWeXeYnwnxny jjj )()(2
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
Example: Ideal Lowpass Filter
,2,1,0 sin
)(
nn
nnh c ,2,1,0
sin)(
nn
nnh c The ideal lowpass fileter
Is noncausal.
Example: Ideal Lowpass Filter
-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
Example: Ideal Lowpass Filter
njM
Mn
cj en
neH
sin
)(nj
M
Mn
cj en
neH
sin
)(
Discrete-Time Signals and Systems
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
Discrete-Time Signals and Systems
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
Fourier Transform Pairs
Sequence Fourier Transform
)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
Fourier Transform Pairs
Sequence Fourier Transform
nje 0
k
k)2(2 0
)cos( 0 n
k
jj keke )]2()2([ 00