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Digital Signal Processing, II, Zheng-Hua Tan, 20061
Digital Signal Processing, Fall 2006- E-Study -
Zheng-Hua Tan
Department of Electronic SystemsAalborg University, Denmark
Lecture 2: Fourier transforms and frequency response
Digital Signal Processing, II, Zheng-Hua Tan, 20062
Course at a glance
Discrete-time signals and systems
Fourier-domain representation
DFT/FFT
Systemstructures
Filter structures Filter design
Filter
z-transform
MM1
MM2
MM9,MM10MM3
MM6
MM4
MM7 MM8
Sampling andreconstruction MM5
Systemanalysis
System
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Digital Signal Processing, II, Zheng-Hua Tan, 20063
Part I: System impulse response
System impulse responseLinear constant-coefficient difference equations Fourier transforms and frequency response
Digital Signal Processing, II, Zheng-Hua Tan, 20064
FIR systems – reflected in the h[n]Ideal delay
Forward difference
Backward difference
Finite-duration impulse response (FIR) systemThe impulse response has only a finite number of nonzero samples.
integer. positive a ],[][ ],[][
dd
d
nnnnhnnnxny
−=∞<<∞−−=
δ
][]1[][][]1[][
nnnhnxnxny
δδ −+=−+=
]1[][][]1[][][
−−=−−=
nnnhnxnxny
δδ
nd
0
-10
3
Digital Signal Processing, II, Zheng-Hua Tan, 20065
IIR systems – reflected in the h[n]
Accumulator
Infinite-duration impulse response (IIR) systemThe impulse response is infinitive in duration.
StabilityFIR systems always are stable, if each of h[n] values is finite in magnitude.IIR systems can be stable, e.g.
][][][
][][
nuknh
kxny
n
k
n
k
==
=
∑
∑
−∞=
−∞=
δ 0
…
∞<−==
<=
∑∞ |)|1(1||
1|| with ][][
0aaS
anuanhn
n
∞<= ∑∞
−∞=
?|][|n nhS
Digital Signal Processing, II, Zheng-Hua Tan, 20066
Cascading systems
Causality Ideal delay
Any noncausal FIR system can be made cause by cascading it with a sufficiently long delay!
0 ,0][?
<= nnh
][][ dnnnh −= δ
][]1[][difference Forward
nnnh δδ −+=
]1[][][difference Backward
−−= nnnh δδ
]1[][delay sameple-One
−= nnh δ][nx
][nx ][ny
][ny
4
Digital Signal Processing, II, Zheng-Hua Tan, 20067
Cascading systems
Accumulator + Backward difference system
Inverse system:
][][systemr Accumulato
nunh =
][][ nnh δ=
]1[][][difference Backward
−−= nnnh δδ][nx
][nx ][nx
][nx
][][*][][*][ nnhnhnhnh ii δ==
Digital Signal Processing, II, Zheng-Hua Tan, 20068
Part II: LCCD equations
System impulse responseLinear constant-coefficient difference equations Fourier transforms and frequency response
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Digital Signal Processing, II, Zheng-Hua Tan, 20069
LCCD equations
An important class of LTI systems: input and output satisfy an Nth-order LCCD equations
Difference equation representation of the accumulator
][]1[][
]1[][][][][
][]1[
][][
1
1
nxnyny
nynxkxnxny
kxny
kxny
n
k
n
k
n
k
=−−
−+=+=
=−
=
∑
∑
∑
−
−∞=
−
−∞=
−∞=
∑∑==
−=−M
mm
N
kk mnxbknya
00][][
One-sampledelay
x[n]
y[n-1]
y[n]
Recursive representation
Digital Signal Processing, II, Zheng-Hua Tan, 200610
Part III: Fourier transforms
System impulse responseLinear constant-coefficient difference equationsFourier transforms and frequency response
Frequency-domain representation of discrete-time signals and systemsSymmetry properties of the Fourier transformFourier transform theorems
6
Digital Signal Processing, II, Zheng-Hua Tan, 200611
Signal representations
A sum of scaled, delayed impulse
Sinusoidal and complex exponential sequencesSinusoidal input sinusoidal response with the same frequency and with amplitude and phase determined by the systemComplex exponential sequences are eigenfunctions of LTI systems.signal representation based on sinusoids or complex exponentials
)cos(][ 0 φω += nAnx
∑∞
−∞=
−=k
knkxnx ][][][ δ
njenx ω=][
∑∞
−∞=
−=k
knhkxny ][][][
Digital Signal Processing, II, Zheng-Hua Tan, 200612
Eigenfunctions
Complex exponentials as input to system h[n]
Define Then
nj
k
kj
k
knjnj
nj
eekh
ekheTny
nenx
ωω
ωω
ω
)][(
][}{][
,][
)(
∑
∑∞
−∞=
−
∞
−∞=
−
=
==
∞<<∞−=
njjk
kjj
eeHny
ekheH
ωω
ωω
)(][
][)(
=
= ∑∞
−∞=
−
Eigenvalue Eigenfunction
A – linear operator
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Digital Signal Processing, II, Zheng-Hua Tan, 200613
Eigenvalue – called frequency response
Frequency response is generally complex
Frequency response of the ideal delay system
Method 1: using the eigenfunction
Method 2: using the impulse response
.)(,1|)(|
).sin()(
),cos()(
dj
j
dj
I
dj
R
neHeH
neH
neH
ω
ω
ω
ω
ω
ω
ω
−=∠
=
−=
=
dnj
n
njd
j eenneH ωωω δ −∞
−∞=
−∑ =−= ][)(
d
dd
njj
njnjnnjnj
eeH
eeeeTnyωω
ωωωω
−
−−
=
===
)(
}{][ )(
)(|)(|
)()()(ωω
ωωω
jeHjj
jI
jR
j
eeH
ejHeHeH∠=
+=
describes changes in magnitude and phase
][][ ][][ dd nnnhnnxny −=→−= δ
Digital Signal Processing, II, Zheng-Hua Tan, 200614
Frequency response
The frequency response of discrete-time LTI systems is always a periodic function of the frequency variable w with period .
Only specify over the intervalThe ‘low frequencies’ are close to 0The ‘high frequencies’ are close to
)(][][)( 2)2()2( ωπωπωπω j
n
njnj
n
njj eHeenhenheH === ∑∑∞
−∞=
−−∞
−∞=
+−+
πωπ ≤<−
π±
π2
8
Digital Signal Processing, II, Zheng-Hua Tan, 200615
Ideal frequency-selective filters
For which the frequency response is unity over a certain range of frequencies, and is zero at the remaining frequencies.
Ideal low-pass filter: passes only low and rejects high
Digital Signal Processing, II, Zheng-Hua Tan, 200616
Ideal frequency-selective filters
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Digital Signal Processing, II, Zheng-Hua Tan, 200617
Sinusoidal response of LTI systems
Sinusoidal input sinusoidal response with the same frequency and with amplitude and phase determined by the system
)cos(|)(| ][ then
)(2
)(2
][
22
)cos(][
0
0
0
0000
00
θφω
φω
ω
ωωφωωφ
ωφωφ
++=
+=
+=
+=
−−−
−−
neHAny
eeHeAeeHeAny
eeAeeAnAnx
j
njjjnjjj
njjnjj
θωωωω ω −∠−− === eeHeeHeHeH
nhjeHjjj j
|)(||)(|)()(
thenreal, is ][ if0
0000 )(*
Digital Signal Processing, II, Zheng-Hua Tan, 200618
Signal representation
More than sinusoids, a broad class of signals can be represented as a linear combination of complex exponentials:
If x[n] can be represented as a superposition of complex exponentials, output y[n] can be computed by using the frequency response, which is similar to the function of impulse response.
∑=k
njk
kenx ωα][
∑=∴k
njjk
kk eeHny ωωα )(][
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Digital Signal Processing, II, Zheng-Hua Tan, 200619
Frequency-domain representation of x[n]
By Fourier transforms Fourier representation:
In general, Fourier transform is complex
)(|)(|
)()()(ωω
ωωω
jeXjj
jI
jR
j
eeX
ejXeXeX∠=
+=
ωπ
ωω deeX
x[n]
njj )(21
sinusoidscomplex smallmally infinitesi ofion superposit a asrepresent
njj
njj
enxeX
deeXnx
ωω
π
πωω ω
π−
∞
∞−
−
∑
∫
=
=
][)( where
)(21][ Inverse Fourier transform
w is continuous-time variable
Fourier transformn is discrete-time variable
Fourier spectrumSpectrumMagnitude spectrumAmplitude spectrumPhase spectrum
Digital Signal Processing, II, Zheng-Hua Tan, 200620
Frequency and impulse responses
Are a Fourier transform pair
Fourier transform is periodic with period
njj enxeX ωω −∞
∞−∑= ][)(
ωπ
π
πωω
ωω
deeHnh
enheH
njj
n
njj
∫
∑
−
∞
−∞=
−
=∴
=
)(21][
][)( Recall
π2
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Digital Signal Processing, II, Zheng-Hua Tan, 200621
Sufficient condition for Fourier transform
Condition for the convergence of the infinite sum
X[n] is absolutely summable, then its Fourier transform exists (sufficient condition).
∞<≤
≤
=
∑
∑
∑
∞
∞−
−∞
∞−
−∞
∞−
|][|
|||][|
|][| |)(|
nx
enx
enxeX
nj
njj
ω
ωω
Digital Signal Processing, II, Zheng-Hua Tan, 200622
Example: ideal lowpass filter
Frequency response
Noncausal.Not absolutely summable.
⎩⎨⎧
≤<<
=πωω
ωωω
|| ,0|| ,1
)(c
cjlp eH
∞<<∞−== ∫− nn
ndenh cnj
lpc
c,
sin21][
πω
ωπ
ω
ωω
∑∞
−∞=
−=n
njcjlp e
nn
eH ωω
πωsin
)(
∑−=
−=M
Mn
njcjM e
nn
eH ωω
πωsin
)(
12
Digital Signal Processing, II, Zheng-Hua Tan, 200623
Example: Fourier transform of a constant
Constant sequence
Its Fourier transform is defined as the periodic impulse train
1][ =nx
)2(2)( reXr
j πωπδω ∑∞
−∞=
+=
njj
njj
enxeX
deeXnx
ωω
π
πωω ω
π−
∞
∞−
−
∑
∫
=
=
][)( where
)(21][
Digital Signal Processing, II, Zheng-Hua Tan, 200624
Part III: Fourier transforms
System impulse responseLinear constant-coefficient difference equationsFourier transforms and frequency response
Frequency-domain representation of discrete-time signals and systemsSymmetry properties of the Fourier transformFourier transform theorems
13
Digital Signal Processing, II, Zheng-Hua Tan, 200625
Symmetry properties of Fourier transform
Conjugate-symmetric sequenceConjugate-antisymmetric sequenceAny , ][nx
][][ * nxnx ee −=
][][ * nxnx oo −−=
][][][
][])[][(21][
][])[][(21][ define
])[][(21])[][(
21][
**
**
**
nxnxnx
nxnxnxnx
nxnxnxnx
nxnxnxnxnx
oe
oo
ee
+=∴
−−=−−=
−=−+=
−−+−+=
)())()((21)(
)())()((21)(
where)()()(
**
**
ωωωω
ωωωω
ωωω
jo
jjjo
je
jjje
jo
je
j
eXeXeXeX
eXeXeXeX
eXeXeX
−−
−−
−=−=
=+=
+=
Digital Signal Processing, II, Zheng-Hua Tan, 200626
Symmetry properties of Fourier transform
Table 2.1
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Digital Signal Processing, II, Zheng-Hua Tan, 200627
Part III: Fourier transforms
System impulse responseLinear constant-coefficient difference equationsFourier transforms and frequency response
Frequency-domain representation of discrete-time signals and systemsSymmetry properties of the Fourier transformFourier transform theorems
Digital Signal Processing, II, Zheng-Hua Tan, 200628
Linearity of the Fourier Transform
)()(][][
)(][
)(][
2121
22
11
ωω
ω
ω
jjF
jF
jF
ebXeaXnbxnax
eXnx
eXnx
+↔+
↔
↔
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Digital Signal Processing, II, Zheng-Hua Tan, 200629
Time shifting and frequency shifting
)(][
)(][
)(][
)( 00 ωωω
ωω
ω
−
−
↔
↔−
↔
jF
nj
jnjF
d
jF
eXnxe
eXennx
eXnx
d
Digital Signal Processing, II, Zheng-Hua Tan, 200630
Time reversal
)(][
real. is ][ if)(][
)(][
* ω
ω
ω
jF
jF
jF
eXnx
nxeXnx
eXnx
↔−
↔−
↔
−
16
Digital Signal Processing, II, Zheng-Hua Tan, 200631
Differentiation in frequency
ω
ω
ω
dedXjnnx
eXnxjF
jF
)(][
)(][
↔
↔
Digital Signal Processing, II, Zheng-Hua Tan, 200632
Parseval’s theorem
ωπ
π
π
ω
ω
∫∑ −
∞
−∞=
==
↔
deXnxE
eXnx
j
n
jF
22 |)(|21|][|
)(][
17
Digital Signal Processing, II, Zheng-Hua Tan, 200633
The convolution theorem
)()()(
][*][][][][
)(][
)(][
ωωω
ω
ω
jjjk
jF
jF
eXeHeY
nhnxknhkxny
eHnh
eXnx
=
=−=
↔
↔
∑∞
−∞=
Digital Signal Processing, II, Zheng-Hua Tan, 200634
The modulation or Windowing theorem
∫−−=
=↔
↔
π
π
θωθω
ω
ω
θπ
deWeXeY
nwnxnyeWnw
eXnx
jjj
jF
jF
)()(21)(
][][][)(][
)(][
)(
18
Digital Signal Processing, II, Zheng-Hua Tan, 200635
Fourier transform theorems
Table 2.2
Digital Signal Processing, II, Zheng-Hua Tan, 200636
Fourier transform pairs
Table 2.3
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Digital Signal Processing, II, Zheng-Hua Tan, 200637
Example
Determining a Fourier transform using Tables 2.2 and 2.3
ω
ωω
ω
ωωωω
ωω
j
jj
n
j
jjjj
n
jj
Fn
aeeaeX
nuanxanxae
eeXeeX
nuanxnxae
eXnuanx
−
−
−
−−
−
−
−=
−==−
==
−=−=−
=↔=
1)(
])5[(i.e. ][][1
)()(
])5[(i.e. ]5[][1
1)(][][
552
5
5
15
2
512
11
]5[][ −= nuanx n
Digital Signal Processing, II, Zheng-Hua Tan, 200638
Summary
System impulse responseLinear constant-coefficient difference equations Fourier transforms and frequency response
Frequency-domain representation of discrete-time signals and systemsSymmetry properties of the Fourier transformFourier transform theorems
20
Digital Signal Processing, II, Zheng-Hua Tan, 200639
Course at a glance
Discrete-time signals and systems
Fourier-domain representation
DFT/FFT
Systemstructures
Filter structures Filter design
Filter
z-transform
MM1
MM2
MM9,MM10MM3
MM6
MM4
MM7 MM8
Sampling andreconstruction MM5
Systemanalysis
System