signal & linear system analysis
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Principles of Communications I (Fall, 2002) Signal & Linear System Analysis
NCTU EE 1
Signal & Linear System Analysis
Signal Model and Classifications
Deterministic vs. Random
Deterministic signals: completely specified function of time. Predictable, no uncertainty
e.g. tAtx 0cos)( ω= , ∞<<−∞ t ; where A and 0ω are constants
Random signals (stochastic signals): take on random values at any given time instant and characterized by pdf (probability density function)
“Not completely predictable”, “with uncertainty”
e.g. nindex at time tossedshown when valued)( icenx =
Model: A (large, maybe ∞ ) set of waveforms each associated with a prob-ability measure
e.g. pdf characterizing the out-of-band radio noise
Principles of Communications I (Fall, 2002) Signal & Linear System Analysis
NCTU EE 2
Periodic vs. Aperiodic
Periodic signal: A signal )(tx is periodic iff ∃ a constant 0T , such that )()( 0 txTtx =+ , t∀
The smallest such 0T is called “fundamental period” or “simply period”. Aperiodic signal: cannot find a finite 0T such that )()( 0 txTtx =+ , t∀
Phasor signal & Spectra
A special periodic function tjjtj eAeAetx 00 )()(~ ωθθω ⋅== +
phasor rotating)(~ ≡tx ; phasor≡θjAe ; numbers real , ≡θA
Why this complex signal? 1. Key part of modulation theory 2. Construction signal for almost any signal 3. Easy mathematical analysis for signal 4. Phase carries time delay information
More on Phasor Signal:
1. Information is contained in A and θ (given a fixed ))(or 00 ωf .
2. The related real sinusoidal function:
)}(~Re{)cos()( 0 txtAtx =+= θω or )(*~21)(~
21)cos()( 0 txtxtAtx +=+= θω
3. In vector form graphically:
Principles of Communications I (Fall, 2002) Signal & Linear System Analysis
NCTU EE 3
4. Frequency-domain representation
Single-sided (SS) amp. & phasor vs. double-sided (DS):
Line spectra:
E.g. find SS and DS spectra of )sin()( 0 θω += tAtx
Ans: put into cosine form first
Singularity Functions: opposed to regular functions
Unit impulse function )(tδ :
1. Defined by
)0()()0()()()()(0
0
0
0xdttxdtttxdtttx === ∫∫∫
+
−
+
−
∞
∞−δδδ ; 1)(
0
0=∫
+
−
dttδ
2. It defines a precise sample point of x(t) at time t (or 0t if )( 0tt −δ )
dttttxtx )()()( 00 −= ∫∞
∞−δ
3. Basic function for linearly constructing a time signal
ττδτ dtxtx )()()( −= ∫∞
∞−
4. Properties: (Z & T, pp. 25-26 (don’t bother properties 5 and 6))
)(||
1)( ta
at δδ = ; )()( tt −= δδ : even function
Principles of Communications I (Fall, 2002) Signal & Linear System Analysis
NCTU EE 4
5. What is )(tδ precisely? some of intuitive ways of realizing it:
E.g. 1 ⎪⎩
⎪⎨⎧ <= →
elsewhere)(or otherwise,0
||,21lim)( 0
εεδ ε
tt
E.g. 2 2
0sin1lim)( ⎟
⎠⎞
⎜⎝⎛=
→ επ
πεδ
ε
tt
t
Unit step function u(t):
dttdutdtu
t )()( ;)()( == ∫ ∞−δλλδ
Signal types classified by energy & power This classification will be needed for the later analysis of deterministic and random signals
Energy: dttxET
TT ∫−→∞≡ 2|)(|lim
Power: dttxT
PT
TT ∫−→∞≡ 2|)(|
21lim
Energy Signals: iff 0)(P 0 =∞<< E
Power Signals: iff )(E 0 ∞=∞<< P
Principles of Communications I (Fall, 2002) Signal & Linear System Analysis
NCTU EE 5
Example-1 )()(1 tuAetx tα−=
Example-2 )()(2 tAutx =
Example-3 )cos()( 03 θω += tAtx
Note: 1. If x(t) is periodic, then it is meaningless to find its energy, we only need
to check its power
dttxT
PT
Tt∫+≡0
0
2
0|)(|1
2. Noise is often persistent and is often a power signal 3. Deterministic and aperiodic signals are often energy signals 4. A realizable LTI system can be represented by a signal and mostly is a
energy signal 5. Power measure is useful for signal and noise analysis 6. The energy and power classifications of signals are mutually exclusive
(cannot be both at the same time). But a signal can be neither energy nor power signal
Signal Space & Orthogonal Basis
Applying the Sophomore’s Linear Algebra
Basis vectors (for vector space): (essential in DSP & communication theory)
N-dimensional basis vectors: Nbbb ,,, 21 L
Degree of freedom and independence:
E.g.: In geometry, any 2-D vector ⎥⎦
⎤⎢⎣
⎡=
qp
x can be decomposed into com-
ponents along two orthogonal basis vectors, (or expanded by these two vectors)
2211 bxbxx +=
Meaning of “linear” in linear algebra:
222111 )()( byxbyxyx +++=+
Principles of Communications I (Fall, 2002) Signal & Linear System Analysis
NCTU EE 6
Basis functions (for function space): (indispensable for general signal analysis) A general function can also be expanded by a set of basis functions (in an approximation sense)
∑∞
−∞=
≈n
nn tXtx )()( φ
or more feasibly
∑=
≈N
nnn tXtx
1)()( φ
Define the matching (or correlation) operation as
∑ ∫∫∞
−∞=
∞
∞−
∞
∞−
≈n
mnnm dtttXdtttx )()()()( φφφ
If we define orthogonality as
⎩⎨⎧ =
≡−≡∫∞
∞− ..,0,1
)()()(womn
mndttt mn δφφ
then
mm Xdtttx =∫∞
∞−
)()( φ
E.g. )cos()( 0tmtm ωφ = for periodic even x(t); set of )(tmφ orthogonal?
E.g. Taylor’s expansion for cosine function, basis functions? orthogonal? Remarks:
1. Using Freshmen calculus can show that function approximation ex-pansion by orthogonal basis functions is an optimal LSE approxima-tion
2. Is there a very good set of orthogonal basis functions? 3. Concept and relationship of spectrum, bandwidth and infinite continu-
ous basis functions
Principles of Communications I (Fall, 2002) Signal & Linear System Analysis
NCTU EE 7
Fourier Series & Fourier Transform
Fourier Series: ∑∞
−∞=
=n
tnfjneXtx 02)( π
∫+ −= 00
0
02
0
)(1 Tt
t
tnfjn dtetx
TX π
Sinusoidal Representation
[ ]∑∞
=
−−++=
1
)(220
00)(ˆn
tfnjn
tnfjn eXeXXtx ππ
If x(t) is real,
nXjnn eXX ∠= , nXj
nn eXX −∠−− = nXj
n eX ∠−=
][)(ˆ )2()2(
10
00 XntnfjXntnfj
nn eeXXtx ∠+−∠+
∞
=
++= ∑ ππ
)2cos(2 01
0 nn
n XtnfXX ∠++= ∑∞
=
π Cosine FS
Note: Index starts from 1 (not ∞ )
Trigonometric FS:
[ ])2sin()sin()2cos()cos()(ˆ 001
0 tnfXtnfXXXtx nnn
n ππ ∠−∠+= ∑∞
=
∑∑∞
=
∞
=
++=1
001
0 )2sin()2cos(n
nn
n tnfbtnfaX ππ
where )cos(2 nnn XXa ∠= , )sin(2 nnn XXb ∠=
Or, ∫+
=00
0
)2cos()(20
0
Tt
tn dttnftxT
a π
∫+
= 00
0
)2sin()(20
0
Tt
tn dttnftxT
b π
∑ ∑∞
=
∞
=
++=1 1
000 )2sin()2cos(
2)(ˆ
n nnn tnfbtnfaatx ππ
Principles of Communications I (Fall, 2002) Signal & Linear System Analysis
NCTU EE 8
Properties of Fourier series
∫+ −= 00
0
02)(1 Tt
t
tnfj
nn dtetxX π
λ
“DC” coefficient: ∫+ −= 00
0
0)0(2
00 )(1 Tt
t
tfj dtetxT
X π
∫+
= 00
0
)(1
0
Tt
tdttx
T = average value of x(t)
“AC” coefficients: ∫+
−=00
0
)]2sin()2)[cos((100
0
Tt
tn dttnfjtnftxT
X ππ
∫ ∫+ +
−= 00
0
00
0
)2sin()(1)2cos()(10
00
0
Tt
t
Tt
tdttnftx
Tjdttnftx
Tππ
If x(t) is real, then, Xnjnnnn eXXjXX ∠=+= ]Im[]Re[
where ∫+
= 00
0
)2cos()(1]Re[ 00
Tt
tn dttnftxT
X π
dttnftxT
XTt
tn ∫+
= 00
0
)2sin()(1]Im[ 00
π
Hence,
]Re[]Re[ nn XX =− ]Im[]Im[ nn XX −=− *nn XX =− nn XX =− (even function) and
nn XX −∠=∠ − (odd function)
Linearity x(t) ↔ ak
y(t) ↔ bk
Ax(t)+By(t) ↔ Aak + Bbk
Time Reversal
x(t) ↔ ak
x(-t) ↔ a-k
Time Shifting
ktfjk2
0 ae )tx(t 00π−↔−
Principles of Communications I (Fall, 2002) Signal & Linear System Analysis
NCTU EE 9
Time Scaling x(αt) ↔ ak But the fundamental frequency changes
Multiplication x(t) ↔ ak
y(t) ↔ bk
x(t)y(t) ↔
Conjugation and Conjugate Symmetry
x(t) ↔ ak
x*(t) ↔ a*-k
If x(t) is real ⇒ a-k = ak*
Parseval’s Theorem
Power in time domain = power in frequency domain
dttxT
P Ttt
ox
201
1)(1
∫+=
∑∑∞
−∞=
∞
−∞==⎥⎦
⎤⎢⎣⎡=
nn
nno XXT
TPx 22
0
1
∑∞
−∞=−
llklba
Principles of Communications I (Fall, 2002) Signal & Linear System Analysis
NCTU EE 10
Example: half-rectified sinewave
Example:
Principles of Communications I (Fall, 2002) Signal & Linear System Analysis
NCTU EE 11
Fourier Series Good orthogonal basis functions for a periodic function:
1. Intuitively, basis functions should be also periodic
2. Intuitively, periods of the basis functions should be equal to the period or integer fractions of the target signal
3. Fourier found that sinusoidal functions are good and smooth functions to expand a periodic function
Synthesis & analysis: (reconstruction & projection) Given periodic )(tx with period
00
1f
T = , 00 2 fπω = , it can be synthe-
sized as
∑∞
−∞=
≈n
tjnneXtx 0)( ω
nX : Spectra coefficient, spectra amplitude response
To synthesize it we must first analyze it and find out nX . By orthogonality
∫+ −= 00
0
0)(1
0
Tt
t
tjnn dtetx
TX ω
Fourier Transform
Good orthogonal basis functions for a aperiodic function:
1. Already know sinusoidal func-tions are good choice
2. Sinusoidal components should not be in a “fundamental & har-monic” relationship
3. Aperiodic signals are mostly fi-nite duration
4. Consider the aperiodic function as a special case of periodic func-tion with infinite period
Synthesis & analysis: (reconstruction & projection) Given aperiodic )(tx with period
∞→=df
T 10 , dfd πωω 20 == , we can
synthesize it as
∫
∫∑∞
∞−
∞
∞−
∞
−∞=>−
=
=≈
dfefX
deXeXtx
ftj
tj
n
tjndnd
π
ωω
ωωω
π
2
0
)(
)(21lim)(
By orthogonality
)( )(
)()( 2
txofresponsefrequencytxofTransformFourier
dtetxfX ftj
≡≡
= ∫∞
∞−
− π
Hence,
FT Inverse)()( 2 ≡= ∫∞
∞−
dfefXtx ftj π
Principles of Communications I (Fall, 2002) Signal & Linear System Analysis
NCTU EE 12
Frequency components: 1. Decompose a periodic signal into
countable frequency components 2. Has a fundamental freq. and many
other harmonics
|| of phase:amplitude|:|
|| 00
nn
n
tjnXjn
tjnn
XXX
eeXeX n
∠
= ∠ ωω
3. Discrete line spectra
Power Spectral Density: 2|| nX
and (by Parseval’s equality)
∑∫∞
−∞=
+==
nn
Tt
tXdttx
TP 22
0
|||)(|1 00
0
In real basis functions:
∑
∑
∑
∞
=
∞
=
∞
=
−−
+
+=
∠++=
∠+=+
10
100
100
0
sin
cos
)cos(||2)(
)cos(||2
00
nn
nn
nnn
nn
tjnn
tjnn
tnB
tnAX
XtnXXtx
XtnXeXeX
ω
ω
ω
ω
ωω
note that *nn XX =− for real x(t)
Frequency components: 1. Decompose an aperiodic signal into
uncountable frequency components 2. No fundamental freq. and contain all
possible freq.
∞<<∞−= ∠
feefXefX ftjfXjftj ππ 2)(2 |)(|)(
3. Continuous spectral density
Energy Spectral Density: 2|)(|)( fXfG ≡
and
∫∫∞
∞−
∞
∞−== dffXdttxE 22 |)(||)(|
In real basis functions: As your exercise example!
Principles of Communications I (Fall, 2002) Signal & Linear System Analysis
NCTU EE 13
What x(t) has Fourier series? 1. Expansion by orthogonal basis func-
tions can be shown is equivalent to finding nX using the LSE (or MSE) cost function:
}])({[})]({[ 22 0∑−=
−=N
Nn
tjnnN eXtxEteE ω
2. Would 0})]({[ 2 →teE N as ∞→N ?3. This requires square integrable condi-
tion (for the power signal):
∫ ∞<0
2|)(|T
dttx
and not necessarily 0|)(| →teN
4. Dirichlet’s conditions: (a) finite no. of finite discontinuities (b) finite no. of finite max & min. (c) absolute integrable
∫ ∞<0
|)(|T
dttx
Dirichlet’s condition implies conver-gence almost everywhere, except at some discontinuities.
What x(t) has Fourier Transform? 1. Expansion by orthogonal basis func-
tions can be shown is equivalent to finding )( fX using the LSE (or MSE) cost function:
}])()({[})]({[ 222 dfefXtxEteET
T
ftjT ∫−−= π
2. Would 0})]({[ 2 →teE T as ∞→T ?3. This requires square integrable condi-
tion (for the energy signal):
∫∞
∞−∞<dttx 2|)(|
4. Dirichlet’s conditions: (a) finite no. of finite discontinuities (b) finite no. of finite max & min. (c) absolute integrable
∫∞
∞−∞<dttx |)(|
Dirichlet’s condition implies conver-gence almost everywhere, except at some discontinuities.
Some General Symmetry Definitions
(a) Symmetric (even): real )( ),()( txtxtx −=
(b) Conjugate –symmetric: )(*)( txtx −=
(c) Anti-symmetric (odd): real )( ),()( txtxtx −−=
(d) Conjugate –anti-symmetric: )(*)( txtx −−=
Principles of Communications I (Fall, 2002) Signal & Linear System Analysis
NCTU EE 14
Symmetry Properties of x(t) and Its Fourier Function
For real periodic x(t), *nn XX =− , or nnnn XXXX −∠=∠= −− |,||| .
For real aperiodic x(t), )()( * fXfX −= or |)(||)(| fXfX −= , )()( fXfX −−∠=∠
Example:
Fourier Transform of Singular Functions
)(tδ is not an energy signal (hence doesn’t satisfy Dirichlet condition).
However, its FT can be obtained by formal definition.
,1)( ⎯→⎯FTtδ ),(1 fFT δ⎯→⎯
,)( 020
fjFT AettA πδ −⎯→⎯− ),( 00 ffAAe FTtfj −⎯→⎯ δπ
Example: ∑∞
−∞=
−n
nTt )( 0δ
Fourier Transform of Periodic Signals—Periodic signals are not energy signals (don’t satisfy Dirichlet’s conditions)
Given a periodic )(tx , ∑∞
−∞=
=n
tjnneXtx 0)( ω
⇒ ∑∞
−∞=
−=n
n nffXfX )()( 0δ
Example-1 tf02cos π
Example-2 ∑∞
−∞=
−n
nTt )( 0δ
Principles of Communications I (Fall, 2002) Signal & Linear System Analysis
NCTU EE 15
Properties of Fourier Transform (p.223 of O.W.Y)
Principles of Communications I (Fall, 2002) Signal & Linear System Analysis
NCTU EE 16
Fourier Transform Pairs (p.225 of O.W.Y)
Principles of Communications I (Fall, 2002) Signal & Linear System Analysis
NCTU EE 17
Relationship Between FT of an Aperiodic Signals & Its Periodic Extension
Let FT of an aperiodic pulse signal p(t) be )()}({ fPtp =ℑ ,
We can generate a periodic signal )(tx by duplicating )(tp at every sT in-
terval , then
∑∑∞
−∞=
∞
−∞=
−=−=n
sn
s nTtptpnTttx )()(*])([)( δ
From convolution theorem,
∑∑
∑∞
−∞=
∞
−∞=
∞
−∞=
−=×−=
×−ℑ=
nsss
nss
ns
nffnfPffPnfff
fPnTtfX
)()()()(
)(]})({[)(
δδ
δ
Principles of Communications I (Fall, 2002) Signal & Linear System Analysis
NCTU EE 18
Poisson sum formula: By taking inverse FT of above eq.
∑∑
∑∑∞
−∞=
∞
−∞=
−
∞
−∞=
−∞
−∞=
−
=−ℑ=
−ℑ=−==ℑ
n
tnfjss
nsss
nsss
ns
senfPfnffnfPf
nffnfPfnTtptxfX
πδ
δ
21
11
)(})({)(
})()({)()()}({
∑∑∞
−∞=
∞
−∞=
=−n
tnfjss
ns
senfPfnTtp π2)()( --> Poisson sum formula
Power Spectral Density & Correlation Energy Signal
∫∫ −∞→
∞
∞−
−−−−
+=+=−∗=
ℑ∗ℑ=ℑ=ℑ≡T
TTdxxdxxxx
fXfXfXfXfG
λτλλλτλλττ
τφ
)()(lim)()()()(
)](*[)]([)](*)([)}({)( 1111
Time-average autocorrelation function
φ(0) = E signal energy
Time average autocorrelation function of power signals
⎪⎪⎩
⎪⎪⎨
⎧
+
+=
+≡
∫∫−∞→
signalpower periodic ifsignalpower aperiodic if
,)()(1
,)()(21lim
)()()(
0
*
0
*
*
T
T
TT
dttxtxT
dttxtxT
txxR
τ
τ
τττ
∫∞
∞−= dffSR )()0(
)}({)( τRfS ℑ= Power spectral density
For periodic power signal
∑∞
−∞=
−=ℑ=n
n nffXRfS )(||)}({)( 02 δτ
Principles of Communications I (Fall, 2002) Signal & Linear System Analysis
NCTU EE 19
Interpretation
1. )(τφ and )(τR measure the similarity between the signal at time t
and τ+t
2. )( fG and )( fS represents the signal energy or power per unit fre-
quency at freq. f.
Properties of )(τR :
1. ττ ∀≥== ,)()()0( 2 RtxpowerR , )0()}(max{ RR =τ
2. )(τR is even for real signal: )()()()( * τττ RtxtxR =−≡−
3. If x(t) does not contain a periodic component
2
||)()(lim txR =
→∞τ
τ
4. If x(t) is periodic with period 0T , then )(τR is periodic in τ with
the same period
5. fRfS ∀≥ℑ= ,0)}({)( τ
Crosscorrelation of two power signals )(),( tytx
∫−→∞−=
+=−≡
T
TT
xy
dttytxT
tytxtytxR
)()(21lim
)()()()()(
*
**
τ
τττ
Crosscorrelation of two energy signals )(),( tytx
∫∞
∞−−≡ dttytxxy )()()( * ττφ
Remarks: )()( * ττ −= yxxy RR , )()( * τφτφ −= yxxy
Principles of Communications I (Fall, 2002) Signal & Linear System Analysis
NCTU EE 20
Signals & Linear Systems
1. )}({)( txHty = ,
2. H: an operator representing some mechanism and operation on x(t) and/or y(t) to produce y(t); generally it can be approximated by a differential equation.
3. More specifically, the linear constant-coefficient differential eq. suits for most applications and makes life a lot easier!
Linear & Time-Invariant (LTI) System
Linear systems:
Satisfies superposition principle
)()()}({)}({)}()({)( 2122112211 tytytxHtxHtxtxHty +=+=+= αααα
Time-invariant:
Delayed input produces a delayed output (due to the non-delayed input)
)}({)( txHty = --> )()}({ 00 ttyttxH −=−
Complete characterization of LTI systems:
The unit impulse function is key to the characterization.
)}({)( tHth δ≡
λλδλ dtxtx )()()( −= ∫∞
∞−
λλδλλλδλ dtHxdtxHtxHty )}({)(})()({)}({)( −=−== ∫∫∞
∞−
∞
∞−
)(tx )(tyΗ
Principles of Communications I (Fall, 2002) Signal & Linear System Analysis
NCTU EE 21
If TI,
)()()()}({
)(*)()()()}({)(
000 ttydtthxttxH
thtxdthxtxHty
−=−−=−
≡−==
∫
∫
∞
∞−
∞
∞−
λλλ
λλλ
Principles of Communications I (Fall, 2002) Signal & Linear System Analysis
NCTU EE 22
Convolution form holds iff LTI
1. Duality of signal x(t) & system h(t)
λλλλλλ dtxhdthxty )()()()()( −=−= ∫∫∞
∞−
∞
∞−
Convolution theorem:
)()(})()({)()}({ fXfHdtxhfYty =−ℑ=ℑ ∫∞
∞−λλλ
Key application: generally )()( fXfH is easier than )(*)( thtx ….
BIBO stability:
A system is BIBO if output is bounded, given any bounded input
conditionDirichlet ofelement main |)(|
|)(||})(max{||})()(max{||})(max{|
⇒∞<⇒
∞<=−=
∫∫∫
∞
∞−
∞
∞−
∞
∞−
λλ
λλλλλ
dh
dhtxdtxhty
Causality:
1. A system is causal if: current output does not depend on future input; or current input does not contribute to the output in the past A truth in reality.
2. All the systems in nature world are causal. Causality generally is not a problem in most circumstance. However, it matters frequently when we want to counteract the effect of hostile communication channel, such that we need an ideal equalizer, which is non-causal.
0for ,0)(
)()()()()(0
<=⇒
−=−= ∫∫∞∞
∞−
tth
dtxhdtxhty λλλλλλ
Principles of Communications I (Fall, 2002) Signal & Linear System Analysis
NCTU EE 23
Paley-Wiener Condition:
If (1) ∞<∫∞
∞−dffH 2)(
(2) ∞<= tth ,0)(
⇒ ∞<+∫
∞
∞−df
fH(f)
21ln
Remarks: |H(f)| cannot grow too fast. |H(f)| cannot be exactly zero over a finite band of frequency
If (1) ∞<∫∞
∞−dffH 2)(
(2) ∞<+∫
∞
∞−df
fH(f)
21ln
⇒ ) ,0)(( causal is that H(f)such H(f) ∞<=∠∃ tth
Eigenfunctions of LTI System
If )()}({ tgtg α=Η , where α is a constant w.r.t time parameter, then
⎩⎨⎧
function eigen :)(eigenvalue :
tgα
numbercomplex arbitrary an : ,)( sAetx st=
λλα
αλλλλ
λ
λλ
dAeh
txAedehdAehty
s
ststs
−∞
∞−
−∞
∞−
−∞
∞−
∫∫∫
=
===
)(
)(])([)()( )(
Let tfjs iπ2= , then )(])([)( 22 txAedehty tfjfj ii αλλ πλπ == −∞
∞−∫
Principles of Communications I (Fall, 2002) Signal & Linear System Analysis
NCTU EE 24
Correlation functions related by LTI systems:
1. ∫∞
∞−−=∗= λλτλτττ dRhRhR xxyx )()()()()(
2. )()()()( * ττττ xy RhhR ∗∗−=
3. )()()( fSfHfS xyx ⋅=
4. )(|)(|)( 2 fSfHfS xy ⋅=
pf) (1) & (2) from definitions. (3) & (4) from )()}({ fHh −=−ℑ τ and
)()}({ ** fHh −=ℑ τ
System Transmission Distortion & System Frequency Response (a) Since almost any input x(t) can be represented by a linear combination of
orthogonal sinusoidal basis functions ftje π2 , we only need to input
ftjAe π2 to the system to characterize the system’s properties, and the
eigenvalue
)()( 2 fHdteth ftj == −∞
∞−∫πα
carries all the system information responding to ftjAe π2 .
(b) In communication theory, transmission distortion is of primary concern in high-quality transmission of data. Hence, the system representing transmission channel is the key investigation target.
Three types of distortion of a transmission channel:
1. Amplitude distortion: linear system but the amplitude response is not constant.
2. Phase (delay) distortion: linear system but the phase shift is not a linear function of frequency.
3. Nonlinear distortion: nonlinear system
Principles of Communications I (Fall, 2002) Signal & Linear System Analysis
NCTU EE 25
Group delay:
)()( ,)(21)( fHf
dffdfTg ∠=−= θθ
π
For a linear phase system, 00 2)( ftHf πθ −∠=
0)( tfTg = , a constant
If Tg(f) is not a constant, sinusoidal inputs of different frequencies have different delays.
Cf. Phase delay: fffTp π
θ2
)()( −=
Ideal general filters:
Principles of Communications I (Fall, 2002) Signal & Linear System Analysis
NCTU EE 26
Realizable filters approximating the ideal filters:
1. Butterworth filter: simple 2. Chebyshev filter: smaller maximum deviation 3. Bessel filter: approximately linear phase in passband
Principles of Communications I (Fall, 2002) Signal & Linear System Analysis
NCTU EE 27
Time-Bandwidth Product—uncertainty principle It can be argued that a narrow time signal has a wide (frequency) bandwidth, and vice versa:
constant)()( ≥× bandwidthduration
1. Definition of T: equal areas
0|)()(|)(|)0( =
∞
∞−
∞
∞−=≥= ∫∫ ffXdttxdttxTx
2. Definition of W:
)0()(|)(|)0(2 xdffXdffXWX =≥= ∫∫∞
∞−
∞
∞−
3. Combine 1 and 2:
211
)0()0(2 ≥⇒≥≥ TW
TXxW
Principles of Communications I (Fall, 2002) Signal & Linear System Analysis
NCTU EE 28
Sampling Theory Ideal sampling
Ideal sampling signal: impulse train (an analog signal)
( ) ( )∑∞
−∞=−=
nnTtts δ , T: sampling period
Analog (continuous-time) signal: )(tx
Sampled (continuous-time) signal: )(txδ
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )∑∑
∑∞
−∞=
∞
−∞=
∞
−∞=
−=−=
−==
nss
ns
ns
nTtnTxnTttx
nTttxtstxtx
δδ
δδ
( ) ( ) ( ) ( )
( ) ( ) ( )∑ ∑
∑∞
−∞=
∞
−∞=
∞
−∞=
−=−∗=
−∗=∗=
k kssss
kss
kffXfkfffXf
kffffXfSfXfX
δ
δδ
][)(
Two Cases:
(1) no aliasing: Wf s 2> , and
(2) aliasing: Wf s 2< , where W is the highest nonzero frequency
component of )( fX .
After sampling, the replicas of )( fX overlap in frequency domain.
That is, the higher frequency components of )( fX overlap with the
lower frequency components of ( )sffX − .
Principles of Communications I (Fall, 2002) Signal & Linear System Analysis
NCTU EE 29
Nyquist Sampling Theorem:
Let x(t) be a bandlimited signal with 0)( =fX for Wf ≥|| . (i.e., no com-
ponents at frequencies greater than W ) Then x(t) is uniquely determined
by its samples K,2,1,0),(][ ±±== nnTxnx s , if WT
fs
s 21≥= .
-- Undersampling: Wf s 2<
-- Overdampling: Wf s 2>
Reconstruction:
Ideal reconstruction filter:
WfBWeBfHfH s
ftj −≤≤Π= − ,)2
()( 020
π
)()()()(
00
20
0
ttxHftyefXHffY
s
ftjs
−=⇒=⇒ − π
Alternative expression
∑∑∞
−∞=
∞
−∞=
−−=−=n
ssn
ss nTttBnTxBHnTthnTxty )](2[sinc)(2)()()( 00
Two types of reconstruction errors
Principles of Communications I (Fall, 2002) Signal & Linear System Analysis
NCTU EE 30
∑∑−
=
−
=
==1
0
1
0
)()(1)(1)(N
k
nkN
N
k
nkN WkXkH
NWkY
Nny
DFT & FFT
DFT
1,,1,0 ,
1,,1,0 ,1
1
0
2
1
0
2
−==
−==
∑
∑−
=
−
−
=
NkexX
NneXN
x
N
n
Nnkj
nk
N
k
Nnkj
kn
L
L
π
π
Fast convolution by FFT
∑∑−
=
−
=
−
=
=
===
=1
0
1
0
/22 )()(|)()(
|)()(N
n
nkN
N
n
Nnkj
Nk
j
ezj
WnhenheHkH
zHeH j
ππω
ω
ωω
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