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Lecture 2Signals and Systems (I)
Principles of Communications
Fall 2008
NCTU EE Tzu-Hsien Sang
Outlines
• Signal Models & Classifications
• Signal Space & Orthogonal Basis
• Fourier Series &Transform
• Power Spectral Density & Correlation
• Signals & Linear Systems
• Sampling Theory
• DFT & FFT
2
Signal Models and Classifications
• The first step to knowledge: classify things.
• What is a signal?
• Usually we think of one-dimensional signals; can our scheme extend to higher dimensions?
• How about representing something uncertain, say, a noise?
• Random variables/processes – mathematical models for signals
3
• Deterministic signals: completely specified functions of time. Predictable, no uncertainty, e.g. , with A and are fixed.
• 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. x(n) = dice value at the n-th toss.
4
)cos()( 0tAtx 0
5
• Periodic vs. Aperiodic signals
• Phasors and why are we obsessed with sinusoids?
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.)(~ 00 )( tjjtj eAeAetx
Singularity functions (they are not functions at all!!!)
• Unit impulse function :
• Defined by
• It defines a precise sample point of x(t) at the incidence t0:
• Basic function for linearly constructing a time signal
• Properties: ;
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)0()()0()()()()(0
0
0
0xdttxdtttxdtttx
dttttxtx )()()( 00
dtxtx )()()(
)(||
1)( t
aat )()( tt
• What is precisely? some intuitive ways of imaging it:
• Unit step funcgtion:8
elsewhere)(or otherwise,0
||,2
1lim)( 0
tt
2
0sin
1lim)(
t
tt
)(t
dt
tdutdtu
t )()( ;)()(
Energy Signals & Power Signals
• Energy:
• Power:
• Energy signals: iff
• Power signals: iff
• Examples:
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dttxET
TT 2|)(|lim
dttxT
PT
TT 2|)(|
2
1lim
0)( 0 PE)( 0 EP
)()(1 tuAetx t
)()(2 tAutx
)cos()( 03 tAtx
• If x(t) is periodic, then it is meaningless to find its energy; we only need to check its power.
• Noise is often persistent and is often a power signal.• Deterministic and aperiodic signals are often energy
signals.• A realizable LTI system can be represented by a
signal and mostly is a energy signal.• Power measure is useful for signal and noise analysis.• 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.
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Signal Spaces & Orthogonal Basis
• The consequence of linearity: N-dimensional basis vectors:
• Degree of freedom and independence: For example, in geometry, any 2-D vector can be decomposed into components along two orthogonal basis vectors, (or expanded by these two vectors)
• Meaning of “linear” in linear algebra:
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Nbbb ,,, 21
2211 bxbxx
222111 )()( byxbyxyx
• A general function can also be expanded by a set of basis functions (in an approximation sense)
or more feasibly
• Define the inner product as (“arbitrarily”)
and the basis is orthogonal
then12
n
nn tXtx )()(
N
nnn tXtx
1
)()(
.)()()(),(
dttytxtytx
..,0
,1)()()(
wo
mnmndttt mn
.)()(
dtttxX mm
• Examples: cosine waves
What good are they?
• Taylor’s expansion: orthogonal basis?
• Using calculus can show that function approximation expansion by orthogonal basis functions is an optimal LSE approximation.
• Is there a very good set of orthogonal basis functions?
• Concept and relationship of spectrum, bandwidth and infinite continuous basis functions.
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)cos()( 0tmtm
Fourier Series & Fourier Transform
• Fourier Series:
• Sinusoids (when?):
• If x(t) is real,
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n
tnfjneXtx 02)(
00
0
02
0
)(1 Tt
t
tnfjn dtetx
TX
1
)(220
00)(ˆn
tfnjn
tnfjn eXeXXtx
*n
Xjn
Xjnn XeXeXX nn
][)(ˆ )2()2(
10
00 XntnfjXntnfj
nn eeXXtx
)2cos(2 01
0 nn
n XtnfXX
Notice the integral bounds.
• Or, use both cosine and sine:
with
Yet another formulation:
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)2sin()sin()2cos()cos()(ˆ 001
0 tnfXtnfXXXtx nnn
n
1
001
0 )2sin()2cos(n
nn
n tnfbtnfaX
)cos(2 nnn XXa )sin(2 nnn XXb
00
0
)2cos()(2
00
Tt
tn dttnftxT
a
00
0
)2sin()(2
00
Tt
tn dttnftxT
b
1 1
000 )2sin()2cos(
2)(ˆ
n nnn tnfbtnfa
atx
• Some Properties
• Linearity If x(t) ak and y(t) bk
then Ax(t)+By(t) Aak + Bbk
• Time Reversal
If x(t) ak then x(-t) a-k
• Time Shifting• Time Scaling
x(t) ak But the fundamental frequency changes
• Multiplication x(t)y(t)
• Conjugation and Conjugate Symmetry
x(t) ak and x*(t) a*-k
If x(t) is real a-k = ak*16
ktπfjk a e) tx(t 002
0
llklba
• Parseval’s Theorem
Power in time domain = power in frequency domain
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dttxT
PTt
to
x
201
1
)(1
nn
nnox XXT
TP
22
0
1
Some Examples
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Extension to Aperiodic Signals
• Aperiodic signals can be viewed as having periods that are “infinitely” long.
• Rigorous treatments are way beyond our abilities. Let’s use our “intuition.”
• If the period is infinitely long. What can we say about the “fundamental frequency.”
• The number of basis functions would leap from countably infinite to uncountably infinite.
• The synthesis is now an integration..
• Remember, both cases are purely mathematical construction.
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“The wisdom is to tell the minute differences between similar-looking things and to find the common features
of seemingly-unrelated ones…”
Fourier Series Fourier Transform
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.
Good orthogonal basis functions for a aperiodic function:1.Already know sinusoidal functions are good choice.2.Sinusoidal components should not be in a “fundamental & harmonic” relationship.3.Aperiodic signals are mostly finite duration.4.Consider the aperiodic function as a special case of periodic function with infinite period
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Synthesis & analysis: (reconstruction & projection)Given periodic with period , and , it can be synthesized as
: Spectra coefficient, spectra amplitude response Before synthesizing it, we must first analyze it first and find out .By orthogonality
Synthesis & analysis: (reconstruction & projection)Given aperiodic with period , and , we can
synthesize it as
By orthogonality
Hence,
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)(tx0
0
1
fT
00 2 f
n
tjnneXtx 0)(
nX
nX
00
0
0)(1
0
Tt
t
tjnn dtetx
TX
)(tx
df
T1
0dfd 20
dfefX
deXeXtx
ftj
tj
n
tjndn
d
2
0
)(
)(2
1lim)(
)(
)(
)()( 2
txofresponsefrequency
txofTransformFourier
dtetxfX ftj
FT Inverse)()( 2
dfefXtx ftj
22
Frequency components:1.Decompose a periodic signal into countable frequency components.2.Has a fundamental freq. and many other harmonics.
3.Discrete line spectra Power Spectral Density:
and (by Parseval’s equality)
Frequency components:1.Decompose an aperiodic signal into uncountable frequency components2.No fundamental freq. and contain all possible freq.
3.Continuous spectral density Energy Spectral Density:
and
|| of phase:
amplitude|:|
|| 00
nn
n
tjnXjn
tjnn
XX
X
eeXeX n
2|| nX
nn
Tt
tXdttx
TP 22
0
|||)(|1 00
0
f
eefXefX ftjfXjftj 2)(2 |)(|)(
2|)(|)( fXfG
dffXdttxE 22 |)(||)(|
23
In real basis functions:
note that
for real x(t).
Exercises!
10
100
100
0
sin
cos
)cos(||2)(
)cos(||2
00
nn
nn
nnn
nn
tjnn
tjnn
tnB
tnAX
XtnXXtx
XtnX
eXeX
*nn XX
Conditions of Existence
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1. Expansion by orthogonal basis functions can be shown is equivalent to finding using the LSE (or MSE) cost function:
2. Would as ?3. This requires square integrable
condition (for the power signal):
and not necessarily 4. Dirichlet’s conditions:finite no. of finite discontinuities;finite no. of finite max & min.;absolute integrable:
Dirichlet’s condition implies convergence almost everywhere, except at some discontinuities.
1. Expansion by orthogonal basis functions can be shown is equivalent to finding using the LSE (or MSE) cost function:
2. Would as ?3. This requires square integrable
condition (for the energy signal):
4. Dirichlet’s conditions:finite no. of finite discontinuities;finite no. of finite max & min.;absolute integrable:
Dirichlet’s condition implies convergence almost everywhere, except at some discontinuities.
}])({[})]({[ 22 0
N
Nn
tjnnN eXtxEteE
0})]({[ 2 teE N N
0
2|)(|T
dttx
0|)(| teN
0
|)(|T
dttx
}])()({[})]({[ 222 dfefXtxEteET
T
ftjT
0})]({[ 2 teE T T
dttx 2|)(|
dttx |)(|
