신호및시스템 Chapter 1.
신호및시스템 [ Chapter 1. ]
Prof. Sung-Il Chien, School of EE, KNU 2
Introduction
1.1 What Is a Signal?
SignalsSpeech signalsImages or visual signalsInternetStock informationetc.
A signal is formally defined as a function of one or more variables, which
conveys information on the nature of a physical phenomenon.
variablesOne-dimensional – e.g. Speech
Multidimensional – e.g. Image
신호및시스템 [ Chapter 1. ]
Prof. Sung-Il Chien, School of EE, KNU 3
Introduction
1.2 What Is a System?
A system is formally defined as an entity that manipulates one or more
signals to accomplish a function, thereby yielding new signals.
SystemInput
signal
Output
signal
신호및시스템 [ Chapter 1. ]
Prof. Sung-Il Chien, School of EE, KNU 4
Introduction
Cf. modulation – the process of converting the message signal into a form that is
compatible with the transmission characteristics of the channel.
Analog communication
Digital communication
sampling, quantization, coding
Two modes of communications
Broadcasting
Point-to-point communication
pathfinder
신호및시스템 [ Chapter 1. ]
Prof. Sung-Il Chien, School of EE, KNU 5
Introduction
1.3 Overview of Specific Systems
Communication system
, distortion
noise introduced
interference
At the some time, high speed in communication system is required to cover
recent requirement of massive data.
Transmitter Channel Receiver
Messagesignal
Transmittedsignal
Receivedsignal
Estimateof message
signal
In channel
신호및시스템 [ Chapter 1. ]
Prof. Sung-Il Chien, School of EE, KNU 6
Introduction
신호및시스템 [ Chapter 1. ]
Prof. Sung-Il Chien, School of EE, KNU 7
Introduction
Control systems
FeedbackDigital control system
Controller Plant∑ ∑)(teReference
input )(tx
)(tv
Disturbance)(tv
Output)(ty
Sensor(s)Feedback signal)(tr
신호및시스템 [ Chapter 1. ]
Prof. Sung-Il Chien, School of EE, KNU 8
Introduction
Remote sensing
The process of acquiring information about an object of interest without
being in physical contact with it.
Usually, multiple sensors covering a large part of the electromagnetic
spectrum is required.
Radar sensors, Infrared sensors, X-ray
SAR satisfactory operation day and night and under all weather
conditions
still achieving high-resolution imaging capability instead requiring sophisticated signal-processing operation
e.g. Fourier transform (FT) (FFT)
신호및시스템 [ Chapter 1. ]
Prof. Sung-Il Chien, School of EE, KNU 9
Introduction
Biomedical signal processing
Biological signals – ECG (electrocardiogram)
Neurons EEG (electroencephalogram)
A record of fluctuations in the electrical activity of large
groups of neurons in the brain
신호및시스템 [ Chapter 1. ]
Prof. Sung-Il Chien, School of EE, KNU 10
Introduction
Auditory System
Analog Versus Digital Signal Processing
신호및시스템 [ Chapter 1. ]
Prof. Sung-Il Chien, School of EE, KNU 11
Introduction
1.4 Classification of Signals
* real-valued signal, complex-valued signal
Continuous-time and discrete-time signals
A discrete-time signal can be derived from a continuous-time signal by
sampling it at a uniform rate.
: sampling period
)(tx defined at discrete instants of time
K,2,1,0),(][ ±±== nnxnx
sampling quantization
Streams of bits
(to the world of computer)
신호및시스템 [ Chapter 1. ]
Prof. Sung-Il Chien, School of EE, KNU 12
Introduction
Even and odd signals
Even signalOdd signal
ttxtx allfor)()( =−ttxtx allfor)()( −=−
Ex. 1. 1 Is the signal and even or an odd function of time t ?
Solution: Replace t with –t yields
Hence, is an odd signal.
)(tx
( )
≤≤−
=otherwise,0
,sin)(
TtTtx T
tπ
( )
( ))(
otherwise,0
,sin
otherwise,0
,sin)(
txTtT
TtTtx
Tt
Tt
−=
≤≤−−
=
≤≤−−
=−
π
π
)(tx
신호및시스템 [ Chapter 1. ]
Prof. Sung-Il Chien, School of EE, KNU 13
Introduction
Arbitrary signal x(t)Even-odd decomposition
show that
)()()( txtxtx oe +=
)()(
)()(
txtx
txtx
oo
ee
−=−=−
[ ]
[ ])()(2
1)(
)()(2
1)(
txtxtx
txtxtx
o
e
−−=
−+=
tetx t cos)( 2−=Ex. 1.2
Solution:
tt
tetetx
tt
tetetx
tetetx
tto
tte
tt
cos)2sinh(
)coscos(21
)(
cos)2cosh(
)coscos(21
)(
)cos()cos()(
22
22
22
−=
−=
=
+=
=−=−
−
−
신호및시스템 [ Chapter 1. ]
Prof. Sung-Il Chien, School of EE, KNU 14
Introduction
A complex-valued signal is said to be conjugate symmetric if
Let , Then
)(tx
)()( * txtx =−
)()()( tjbtatx += )()()(* tjbtatx −= )()()( tjbtatx −+−=−)()( tata =− )()( tbtb −=−
신호및시스템 [ Chapter 1. ]
Prof. Sung-Il Chien, School of EE, KNU 15
Introduction
Periodic signals, nonperiodic signals
The smallest value of T that satisfied the above equation is called the
fundamental period of .
Fundamental frequency
Angular frequency
As for periodic discrete-time signals
The smallest N fundamental period
Fundamental(angular) frequency
)0(allfor)()( >+= TtTtxtx
)(tx
)in(1
HzT
f =
)secondperradianin(2
T
πω =
nNnxnx integerallfor][][ +=
N
π2=Ω
신호및시스템 [ Chapter 1. ]
Prof. Sung-Il Chien, School of EE, KNU 16
Introduction
Deterministic signals, random signals
Energy signals, power signals
The instantaneous power dissipated in the resistor
in the signal analysis,
Now, the instantaneous power of the signal is expressed as
e.g. noise generated in the amplifier
)()(or)(
)( 22
tRitpR
tvtp ==
ohm1=R
)(or)()( 22 titvtp =
)()( 2 txtp =
신호및시스템 [ Chapter 1. ]
Prof. Sung-Il Chien, School of EE, KNU 17
Introduction
Total energy of
and its average power
Average power of a periodic signal
Similarly,
for a periodic signal
Usually
)(tx
∫ ∫−
∞
∞−∞→==
2/
2/
22 )()(limT
TTdttxdttxE
∫−∞→=
2
2
2 )(1
limT
TTdttx
TP
∫−=2/
2/
2 )(1 T
Tdttx
TP
∑
∑
−=∞→
∞
−∞=
=
=N
NnN
n
nxN
P
nxnxE
][2
1lim
])[ofenergytotal(][
2
2
∑−
=
=1
0
2 ][1 N
n
nxN
P
deterministic & nonperiodic signal energy signal
random and periodic power
)0( ∞<< E
)0( ∞<< P
신호및시스템 [ Chapter 1. ]
Prof. Sung-Il Chien, School of EE, KNU 18
Introduction
1.5 Basic Operations on Signals
Operations performed on dependent variables
Amplitude scaling
Addition
Multiplication
Differentiation
integration
][][or)()( ncxnytcxty ==
)()()( 21 txtxty +=
)()()( 21 txtxty =
)()( txdt
dty =
∫ ∞−=
tdxty ττ )()(
신호및시스템 [ Chapter 1. ]
Prof. Sung-Il Chien, School of EE, KNU 19
Introduction
Operations performed on the independent variableTime scaling: continuous-time signal
Time scaling: discrete-time signal
)()( atxty =
0][][ >= kknxny
expandedis)(10compressedis)(1
tyatya
<<>
2=k
신호및시스템 [ Chapter 1. ]
Prof. Sung-Il Chien, School of EE, KNU 20
Introduction
Reflection
Time shifting
)()( txty −=
Ex. 1.3
Cf. even signals an even signal is the same as its reflected version.)()( txtx =−
leftthetoshiftedis)(0rightthetoshiftedis)(0
0
0
txttxt
<>
)()( 0ttxty −=
Ex. 1.4
신호및시스템 [ Chapter 1. ]
Prof. Sung-Il Chien, School of EE, KNU 21
Introduction
Precedence rule for time shifting and time scaling
Correct way: is shifted to right b then is compressed as much as a
)()( batxty −=
)(tx )()( btxtv −=)(tv )()( atvty =
Ex. 1.5 )32()( += txty
)3()( += txtv )2()( tvty =The incorrect applying of the precedence leads to the result shown in below.
)(tx
( ))3(2)( += txty
신호및시스템 [ Chapter 1. ]
Prof. Sung-Il Chien, School of EE, KNU 22
Introduction
]32[][ += nxny
]3[][first += nxnv
Ex. 1.6
0]6[]3[1]4[]2[
1]2[]1[0]0[]0[
]2[][
=−=−−=−=−
=−=−==
=
vyvyvy
vy
nvny
Original signal at are lost2&2 −== nn
신호및시스템 [ Chapter 1. ]
Prof. Sung-Il Chien, School of EE, KNU 23
Introduction
1.6 Elementary Signals
Exponential signals
Discrete-time case
atBetx =)(
lexponentiadecaying0lexponentiagrowing0
<>
aa
nBrnx =][
galternatin0growing1decaying10
⇒<⇒>⇒<<
rr
r
신호및시스템 [ Chapter 1. ]
Prof. Sung-Il Chien, School of EE, KNU 24
Introduction
Sinusoidal signals
Discrete version
N is the number of samples contained in a cycle
ωπφω 2
)cos()( =+= TtAtx
( )
)()cos()2cos()cos(
)(cos)(
txtAtATtA
TtATtx
=+=++=++=
++=+
φωφπωφωω
φω
periodic
)cos(][ φ+Ω= nAnx
( ))cos(
)(cos][φφ
+Ω+Ω=++Ω=+
NnANnANnx
)cleradians/cy(2
radians)(2
N
m
mN
ππ
=Ω
=Ω
신호및시스템 [ Chapter 1. ]
Prof. Sung-Il Chien, School of EE, KNU 25
Introduction
Ex. 1.7
]5cos[3][
]5sin[][
2
1
nnx
nnx
ππ
=
=
(a) Find common fundamental period.
Solution:Angular freq. of x1[n] and x2[n] is Ω = 5π radians/cycle
525
2
2
m
m
mN
=
=
Ω=
ππ
π
For x1[n] and x2[n] to be periodic, N must be an integer
N = 2, 4, 6, …
신호및시스템 [ Chapter 1. ]
Prof. Sung-Il Chien, School of EE, KNU 26
Introduction
(b) Express the composite sinusoidal signal ][][][ 21 nxnxny +=
in the form of y[n] = Acos(Ωn + φ) and evaluate the amplitude A and phase φ.
)sin()sin()cos()cos()cos( φφφ nAnAnA Ω−Ω=+ΩSolution:
Letting Ω = 5πthe right-hand side of identity is of the same form as x1[n] + x2[n]
Ex. 1.7
]5cos[3][
]5sin[][
2
1
nnx
nnx
ππ
=
=
1)sin( −=φA 3)cos( =φA
3
1
][ofamplitude
][ofamplitude
)cos(
)sin()tan(
2
1 −===nx
nx
φφφ radians3/πφ −=
1)3/sin()sin( −=−= πφ AA 2=A
−=∴
35cos2][
ππnny
신호및시스템 [ Chapter 1. ]
Prof. Sung-Il Chien, School of EE, KNU 27
Introduction
Use the following identity:
)cos(sincos 022 φθθθ −+=+ BABA
A
B10 tan−=φ
1,3 == BA
33
1tan2 0
22 πφ ===+∴ BA
신호및시스템 [ Chapter 1. ]
Prof. Sung-Il Chien, School of EE, KNU 28
Introduction
The sum of two periodic signals is also periodic.
The product of two periodic signals is also periodic.
N1: a period of x1(n) and N2: a period of x2(n)
The fundamental period is
)()()( 21 nxnxnx +=
),gcd( 21
21
NN
NNN =
)()()( 21 nxnxnx =
The period is also),gcd( 21
21
NN
NNN =
However, the fundamental period may be smaller by the following properties.
)sin()sin(cossin
)cos()cos(coscos
)cos()cos(sinsin
21
21
21
BABABA
BABABA
BABABA
++−=
++−=
+−−=
신호및시스템 [ Chapter 1. ]
Prof. Sung-Il Chien, School of EE, KNU 29
Introduction
Relation between sinusoidal and complex exponential signals
)identitysEuler'(sincos θθθ je j +=
tjtjj
tj
BeeAeAetAtx
ωωφ
φωφωReRe
Re)cos()( )(
===+= +
φjAeB =
)sin()cos()( φωφωφωω +++== + tjAtAAeBe tjtj
Similarly Re)cos( njBenA Ω=+Ω φIm)sin( njBenA Ω=+Ω φ
K,2,1,0&4
for ==ΩΩ ne nj π
신호및시스템 [ Chapter 1. ]
Prof. Sung-Il Chien, School of EE, KNU 30
Introduction
Exponentially damped sinusoidal signals
Step function
0)sin()( >+= − αφωα tAetx t
)()( tutx =
<≥
=00
01][
n
nnu
신호및시스템 [ Chapter 1. ]
Prof. Sung-Il Chien, School of EE, KNU 31
Introduction
Impulse function
≠==
0001
)(nn
nδ
0for0)( ≠= ttδ ∫∞
∞−=1)( dttδ
Dirac delta functions
One way to visualize is to view it as the limiting form of a rectangular pulse of unit area.)(tδ
)(lim)(0
tgt TT→
=δ sifting property : )()()( 00 txtttx =−∫∞
∞−δ
0)(1
)( >= ata
at δδ (Proof is included in the text.)
)62.1()()( tudt
dt =δ
Conversely, )63.1()()( ∫ ∞−= tdtu ττδ
신호및시스템 [ Chapter 1. ]
Prof. Sung-Il Chien, School of EE, KNU 32
Introduction
Ex. 1.9The capacitor C is assumed to be initially uncharged.
At t = 0, the switch connecting the dc voltage
source V0 to the RC circuit is closed.
Find the voltage v(t) across the capacitor for t ≥ 0.
Solution:The switching operation is represented by a step function V0u(t)
as shown in the equivalence circuit.
The capacitor cannot charge suddenly, so, with it being initially uncharged, we have
(equivalence circuit)
.0)0( =v
For t = ∞, the capacitor becomes fully charged; hence,
.)( 0Vv =∞Recognizing that the voltage across the capacitor increases exponentially with
a time constant RC, we may thus express v(t) as
).()1()( )/(0 tueVtv RCt−−=
신호및시스템 [ Chapter 1. ]
Prof. Sung-Il Chien, School of EE, KNU 33
Introduction
Ex. 1.10
The capacitor is initially uncharged
and the switch connecting it to the dc voltage
source V0 is suddenly closed at time t = 0.
Determine the current i(t) that flows through the capacitor for t ≥ 0.
)()( 0 tuVtv =
Solution:
The voltage across the capacitor is expressed as
By definition, the current flowing through the capacitor is
.)(
)(dt
tdvCti =
Hence, for the problem at hand, we have
)(
)()(
0
0
tCVdt
tduCVti
δ=
=
신호및시스템 [ Chapter 1. ]
Prof. Sung-Il Chien, School of EE, KNU 34
Introduction
Derivatives of Impulse
doublet:
)()( )1( ttdt
d δδ =
( ) )70.1()2/()2/(lim)( 1
0
)1( ∆−−∆+= ∆→∆ttt δδδ
∫∞
∞−= )71.1(;0)()1( dttδ
∫∞
∞− ==− )72.1(.|)()()(00
)1(tttf
dt
ddttttf δ
Generally, ∫∞
∞− ==− .|)()()(00
)(ttn
nn tf
dt
ddttttf δ
신호및시스템 [ Chapter 1. ]
Prof. Sung-Il Chien, School of EE, KNU 35
Introduction
Ramp function
<≥
=00
0)(
t
tttr
)()( ttutr =
<≥
=00
0][
n
nnnr
][][ nnunr =
신호및시스템 [ Chapter 1. ]
Prof. Sung-Il Chien, School of EE, KNU 36
Introduction
1.7 Systems Viewed as Interconnections of Operations
)()( txHty = ][][ nxHny =
e.g. discrete-time shift operator kS
Ex. 1.8 Moving average system
)1(3
1
]2[]1[][3
1][
2SSH
nxnxnxny
++=
−+−+=
Cascade form Parallel form
kS][nx ][ knx −
신호및시스템 [ Chapter 1. ]
Prof. Sung-Il Chien, School of EE, KNU 37
Introduction
1.8 Properties of Systems
Stability
A system is said to be bounded input-bounded output (BIBO) stable if and
only if every bounded inputs results in a bounded output.
where
tMty y allfor|)(| ∞<≤
∞<≤ xMtx |)(|
Ex. 1.14 1ifunstableis][][ >= rnxrny n
Solution: nMnx x allfor][ ∞<≤
xnnn Mrnxrnxrny ≤== ][][][Then
unstable1if →∞→> nrr
신호및시스템 [ Chapter 1. ]
Prof. Sung-Il Chien, School of EE, KNU 38
Introduction
Memory
Causality
A system is said to be causal if the present value of the output signal depends
only on the present and/or past value of the input signal.
memory)(1
)(
memoryless)(1
)(
→=
→=
∫ ∞−
tdv
Lti
tvR
ti
ττ
noncausal]1[][]1[3
1][
causal]2[]1[][3
1][
→−+++=
→−+−+=
nxnxnxny
nxnxnxny
신호및시스템 [ Chapter 1. ]
Prof. Sung-Il Chien, School of EE, KNU 39
Introduction
Invertibility
not invertible
Time invariance
Two operators commute with each other !
H)(tx )(ty invH)(tx
[ ] )()()( txHHtxHHtyH invinvinv ==operatoridentity →= IHH inv
)(2 txy =
)()( txHty = )()( 00 txSttx t=−
0tS)(tx )( 0ttx −
H)(tyi H
)(tx0tS
)(tyo
)()()( 00 txHSttxHty t
i =−= )()()( 00 txHStySty tto ==
invarianttime00 −→= HSHS tt
신호및시스템 [ Chapter 1. ]
Prof. Sung-Il Chien, School of EE, KNU 40
Introduction
Assume a system is time invariant
and
Question? Find for input
H)(tx )(ty
)(ty )( 0ttx −
H)( 0ttx − )(ty
)()(
)(
)(
)()(
0
0
0
0
0
ttytyS
txHS
txHS
ttxHty
t
t
t
−==
=
=
−=
신호및시스템 [ Chapter 1. ]
Prof. Sung-Il Chien, School of EE, KNU 41
Introduction
Linearity
A system is said to be linear if it satisfies the principles of superpositionnonlinear
==
=
∑
∑
=
=N
iii
N
iii
txaHtHxty
txatx
1
1
)()()(
)()(
if the system is linear )()(where)()(1
txHtytyaty ii
N
iii ==∑
=
∑=
=N
iii txHaty
1
)()(
Ex. 1.19
Sol.
linear?][][ nnxny =
∑=
=N
iii nxanx
1
][][if
linear][][][][y111
→=== ∑∑∑===
n
iii
n
iii
N
iii nyannxanxann
신호및시스템 [ Chapter 1. ]
Prof. Sung-Il Chien, School of EE, KNU 42
Introduction
1.9 NoiseExternal sources of noiseInternal sources of noise
신호및시스템 [ Chapter 1. ]
Prof. Sung-Il Chien, School of EE, KNU 43
Introduction
Thermal NoiseTime-average-squared value (v(t): thermal noise voltage)
k: Boltzman’s constant
Tabs: absolute temperature
R: resistance
∆f: width of the frequency band in Hertz over which v(t) is measured
∫−∞→=
T
TTdttv
Tv )95.1(.)(
2
1lim 22
22 vlots4 fRkTv abs ∆=
where,
신호및시스템 [ Chapter 1. ]
Prof. Sung-Il Chien, School of EE, KNU 44
Introduction
1.10 Theme ExampleMoving-Average Systems
∑−
=−=
1
0
][1
][N
k
knxN
ny
신호및시스템 [ Chapter 1. ]
Prof. Sung-Il Chien, School of EE, KNU 45
Introduction
신호및시스템 [ Chapter 1. ]
Prof. Sung-Il Chien, School of EE, KNU 46
Introduction