Signal & Systems - KOCWcontents.kocw.net/KOCW/document/2015/chungnam/kimjin... · 2016. 9. 9. ·...
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Kim, J. Y. IC & DSP Research Group Signal & Systems Chonnam National University Dept. of Electronics and Computer Eng. IC&DSP Research Group Kim, Jin Young
Transcript of Signal & Systems - KOCWcontents.kocw.net/KOCW/document/2015/chungnam/kimjin... · 2016. 9. 9. ·...
Kim, J. Y.
IC & DSP
Research
Group
Signal & Systems
Chonnam National University
Dept. of Electronics and Computer Eng.
IC&DSP Research Group
Kim, Jin Young
Kim, J. Y.
IC & DSP
Research
Group
Kim, J. Y.
IC & DSP
Research
Group
What is a Signal?
(DEF) Signal : A signal is formally defined as a
function of one or more variables, which
conveys information on the nature of physical
phenomenon.
Message??
Kim, J. Y.
IC & DSP
Research
Group
Kim, J. Y.
IC & DSP
Research
Group
What is a System?
(DEF) System : A system is formally defined as
an entity that manipulates one or more signals
to accomplish a function, thereby yielding new
signals.
system output
signal input
signal
Kim, J. Y.
IC & DSP
Research
Group
Some Interesting Systems
Communication system
Control systems
Remote sensing system
Biomedical system(biomedical signal
processing)
Auditory system
Kim, J. Y.
IC & DSP
Research
Group
Some Interesting Systems
Communication system
Kim, J. Y.
IC & DSP
Research
Group
Some Interesting Systems
Control systems
Kim, J. Y.
IC & DSP
Research
Group
Some Interesting Systems
Remote sensing system
Perspective view of Mount Shasta
(California), derived from a pair of stereo
radar images acquired from orbit with the
shuttle Imaging Radar (SIR-B). (Courtesy
of Jet Propulsion Laboratory.)
Kim, J. Y.
IC & DSP
Research
Group
SAR: Synthetic Aperture
Radar 합성개구 레이더에서 사용하는 전파 빔은 비교적 펄스폭도 넓고 안테나
직경도 작아서 빔의 각도 범위도 넓은 편이다. 그 대신 레이더를 비행기나
인공위성에 싣고서 빠르게 이동을 하면서 레이더 반사파를 연속적으로
수신한다. 이렇게 하면 전파가 반사되어 돌아오는 동안 이동한 거리만큼
마치 레이더 안테나의 직경이 길어지는 효과가 나타나므로 보다 예리하게
반사파를 수신할 수 있게 된다. 그러므로 공중에서 넓은 범위의 지상의
고해상도의 영상을 획득하는 데 아주 효과적이다.
Kim, J. Y.
IC & DSP
Research
Group
Some Interesting Systems
Biomedical system(biomedical signal
processing)
Kim, J. Y.
IC & DSP
Research
Group
Some Interesting Systems
Auditory system
Dancing outer hair cell
Kim, J. Y.
IC & DSP
Research
Group
Classification of Signals
Continuous and discrete-time signals
Continuous and discrete-valued signals
Even and odd signals
Periodic signals, non-periodic signals
Deterministic signals, random signals
Causal and anticausal signals
Right-handed and left-handed signals
Finite and infinite length
Kim, J. Y.
IC & DSP
Research
Group
Continuous and discrete-time
signals
Continuous signal - It is defined for all time t : x(t)
- t : real number
Discrete-time signal
- It is defined only at discrete instants of time:
x[n]=x(nT)
- n : integer
Kim, J. Y.
IC & DSP
Research
Group
Continuous and Discrete valued
singals
CV corresponds to a continuous y-axis
DV corresponds to a discrete y-axis
Digital signal
Kim, J. Y.
IC & DSP
Research
Group
Even and odd signals
Even signals : x(-t)=x(t)
Odd signals : x(-t)=-x(t)
Even and odd signal decomposition
xe(t)= 1/2·(x(t)+x(-t)) xo(t)= 1/2·(x(t)-x(-t))
Kim, J. Y.
IC & DSP
Research
Group
왼쪽 대칭 오른쪽 대칭
에든버러대: 얼굴만 딱 봐도 어릴 적 고생했는지
척 알 수 있다? (영국 데일리 텔레그래프)
Kim, J. Y.
IC & DSP
Research
Group
Periodic signals, non-periodic
signals
Periodic signals
- A function that satisfies the condition
x(t)=x(t+T) for all t
- Fundamental frequency : f=1/T
- Angular frequency : = 2 f=2/T
Non-periodic signals
Kim, J. Y.
IC & DSP
Research
Group
Kim, J. Y.
IC & DSP
Research
Group
Angular Frequency
Angular frequency is a measure of how fast an object is rotating.
Kim, J. Y.
IC & DSP
Research
Group
Determinism is the philosophical proposition
that every event, including human cognition
and behaviour, decision and action, is
causally determined by an unbroken chain of
prior occurrences.[1]
*Determinism
생년, 월, 일, 시
육십갑자로 구성되어
모두 8글자
사주팔자는 태어난 순간
우주로부터 받은 기운
Kim, J. Y.
IC & DSP
Research
Group
Time-wave zero
Kim, J. Y.
IC & DSP
Research
Group
Deterministic signals, random
signals
Deterministic signals -There is no uncertainty with respect to its value at any
time. (ex) sin(3t)
Random signals - There is uncertainty before its actual occurrence.
Kim, J. Y.
IC & DSP
Research
Group
Random Performance
Kim, J. Y.
IC & DSP
Research
Group
Causal and anticausal Signals
Causal signals : zero for all negative time
Anticausal signals : zero for all positive time
Noncausal : nozero values in both positive and
negative time
causal
signal
anticausal
signal noncausal
signal
Kim, J. Y.
IC & DSP
Research
Group
Right-handed and left-handed
Signals
Right-handed and left handed-signal :
zero between a given variable and
positive or negative infinity
Kim, J. Y.
IC & DSP
Research
Group
Finite and infinite length
Finite-length signal : nonzero over a
finite interval tmin< t< tmax
Infinite-length singal : nonzero over all
real numbers
Kim, J. Y.
IC & DSP
Research
Group
Basic Operations on Signals
Operations performed on dependent
signals
Operations performed on the
independent signals
Kim, J. Y.
IC & DSP
Research
Group
Operations performed on
dependent signals
Amplitude scaling
Addition
Multiplication
Differentiation
Integration
( ) ( )y t cx t
1 2( ) ( ) ( )y t x t x t
1 2( ) ( ) ( )y t x t x t
( ) ( )d
y t x tdx
( ) ( )
t
y t x d
Kim, J. Y.
IC & DSP
Research
Group
Kim, J. Y.
IC & DSP
Research
Group
( ) ( )y t cx t
1 2( ) ( ) ( )y t x t x t
Kim, J. Y.
IC & DSP
Research
Group
Operations performed on the
independent signals
Time scaling
a>1 : compressed
0<a<1 : expanded
( ) ( )y t x at
Spatial scaling
Kim, J. Y.
IC & DSP
Research
Group
Operations performed on the
independent signals
Reflection ( ) ( )y t x t
Kim, J. Y.
IC & DSP
Research
Group
Operations performed on the
independent signals
Time shifting
- Precedence Rule for time shifting
& time scaling
0( ) ( )y t x t t
( ) ( ) ( ( ))b
y t x at b x a ta
Kim, J. Y.
IC & DSP
Research
Group
The incorrect way of applying the precedence rule. (a) Signal x(t).
(b) Time-scaled signal v(t) = x(2t). (c) Signal y(t) obtained by shifting
v(t) = x(2t) by 3 time units, which yields y(t) = x(2(t + 3)).
The proper order in which the operations of time scaling and time
shifting (a) Rectangular pulse x(t) of amplitude 1.0 and duration 2.0,
symmetric about the origin. (b) Intermediate pulse v(t), representing
a time-shifted version of x(t). (c) Desired signal y(t), resulting from
the compression of v(t) by a factor of 2.
Kim, J. Y.
IC & DSP
Research
Group
Elementary Signals
Exponential signals
Sinusoidal signals
Exponentially damped sinusoidal signals
( ) atx t Be
( ) cos( )x t A t
( ) cos( )atx t Ae t
Kim, J. Y.
IC & DSP
Research
Group
Elementary Signals
Step function ( ) ( )x t u t
Kim, J. Y.
IC & DSP
Research
Group
(a) Rectangular pulse x(t) of amplitude A and duration of 1 s, symmetric
about the origin. (b) Representation of x(t) as the difference of two step
functions of amplitude A, with one step function shifted to the left by ½
and the other shifted to the right by ½ ; the two shifted signals are denoted
by x1(t) and x2(t), respectively. Note that x(t) = x1(t) – x2(t).
Kim, J. Y.
IC & DSP
Research
Group
Kim, J. Y.
IC & DSP
Research
Group
Elementary Signals
Impulse function ( ) ( )x t t
(a) Evolution of a rectangular pulse of unit area into an impulse of unit
strength (i.e., unit impulse). (b) Graphical symbol for unit impulse.
(c) Representation of an impulse of strength a that results from allowing
the duration Δ of a rectangular pulse of area a to approach zero.
Kim, J. Y.
IC & DSP
Research
Group
Elementary Signals
Ramp function
Integral of step function
( ) ( )x t r t
Kim, J. Y.
IC & DSP
Research
Group
Systems Viewed as
Interconnection of Operations
system output
signal input
signal
Digital system
Kim, J. Y.
IC & DSP
Research
Group
Passive filter:
RLC bandpass filter
Analogue system
Kim, J. Y.
IC & DSP
Research
Group
Properties of Systems
Stability
Memory
Invertibility
Time Invariance
Linearity
Kim, J. Y.
IC & DSP
Research
Group
Kim, J. Y.
IC & DSP
Research
Group
Stability(1)
BIBO stable : A system is said to be
bounded-input bounded-output stable iff
every bounded input results in a bounded
output.
Its Importance : the collapse of Tacoma
Narrows suspension bridge, pp.45
| ( ) | | ( ) |x yt x t M t y t M
Kim, J. Y.
IC & DSP
Research
Group
Dramatic photographs
showing the collapse of the
Tacoma Narrows
suspension bridge on
November 7, 1940. (a)
Photograph showing the
twisting motion of the
bridge’s center span just
before failure.
(b) A few minutes after the
first piece of concrete fell,
this second photograph
shows a 600-ft section of
the bridge breaking out of
the suspension span and
turning upside down as it
crashed in Puget Sound,
Washington. Note the car in
the top right-hand corner of
the photograph.
Kim, J. Y.
IC & DSP
Research
Group
Kim, J. Y.
IC & DSP
Research
Group
Stability(2)
Example pp.46
- y[n]=1/3(x[n]+x[n-1]+x[n-2])
- y[n]=rnx[n], where r>1
1[ ] [ ] [ 1] [ 2]
3
1(| [ ] | | [ 1] | | [ 2] |)
3
1( )
3x x x x
y n x n x n x n
x n x n x n
M M M M
Kim, J. Y.
IC & DSP
Research
Group
Memory
Memory system : A system is said to possess
memory if its output signal depends on past
values of the input signal
Memoryless system
(example)
1( ) ( )
1( ) ( )
[ ] [ ] [ 1]
t
i t v tR
i t v dL
y n x n x n
Kim, J. Y.
IC & DSP
Research
Group
Causality
Causal system : A system is said to be causal if
the present value of the output signal depends
only on the present and/or past values of the
input signal.
Non-causal system
(example)
y[n]=x[n]+1/2x[n-1]
y[n]=x[n+1]+1/2x[n-1]
Kim, J. Y.
IC & DSP
Research
Group
What’s your answer?
Hey guys - can someone tell me why i get two contradicting asnwers...
i am asked to determine whether the following system is causal or not:
y[k+1] + 0.64y[k-1] = x[k] + 2x[k-1]
method 1:
put k = k-1 (delay the signal by one unit)
therefore , y[k] = -0.64y[k-2] +x[k-1] +2x[k-2]
it is cleary that this signal is casual because a present output value is dependent
on present and past input/output values.
method 2:
put k = k+1
therefore, 0.64y[k] = y[k+2] +x[k+1] +2x[k]
now the system is clearly not causal!!!! whys is this happening!!
i can't see any error in my maths. Also , shifting should not alter the signal
characteristics!!!
please someone help!!!
Kim, J. Y.
IC & DSP
Research
Group
Invertiblity(1)
Invertible system : A system is said to be
invertible if the input of the system can be
recovered from the system output.
H:xy, H-1:yx
H-1{y(t)}= H-1{H{x(t)}}, H-1H=I
H H-1
x(t) x(t) y(t)
Kim, J. Y.
IC & DSP
Research
Group
Invertiblity(2)
(Example)
-
-
1( ) ( ) ( ) ( )
td
y t x d x t L y tL dt
2( ) ( )y t x t
Kim, J. Y.
IC & DSP
Research
Group
Time Invariance (1)
Time invariant system : A system is said to be
time invariant if a time delay or time advance of
the input signal leads to a identical time shift in
the output signal.
0
0 0
( ) { ( )}
{ { ( )}} { ( )}
i
t t
y t H x t t
H S x t HS x t
0
0
0 0
( ) { ( )}
{ { ( )}} { ( )}
t
t t
y t S y t
S H x t S H x t
Kim, J. Y.
IC & DSP
Research
Group
Time Invariance (2)
St0 H
x(t) yi(t) x(t-t0)
H St0
x(t) y0(t)
Are following two systems equivalent?
Kim, J. Y.
IC & DSP
Research
Group
Time Invariance (3)
Examples 1( ) ( )
( )( )
( )
t
y t x dL
x ty t
R t
Kim, J. Y.
IC & DSP
Research
Group
Linearity(1)
Linear system : A system is said to be linear if it
satisfies the principle of superposition.
1
1
?
1 1
( ) ( )
( ) { ( )} { ( )}
{ ( )} ( )
N
i i
i
N
i i
i
N N
i i i i
i i
x t a x t
y t H x t H a x t
a H x t a y t
Kim, J. Y.
IC & DSP
Research
Group
Linearity(2)
a1
a2
aN
.
.
.
. . .
H
x1(t)
x2(t)
xN(t)
.
.
. y(t)
H
H
H
.
.
.
a1
a2
aN
. . .
.
.
.
x1(t)
x2(t)
xN(t)
y(t)
Kim, J. Y.
IC & DSP
Research
Group
Linearity(3)
Examples
-
-
Check superposition with simple two
inputs.
[ ] [ ]y n nx n
( ) ( ) ( 1)y t x t x t
1 1 2 2( ) ( ) ( )x t a x t a x t
Kim, J. Y.
IC & DSP
Research
Group
Invertible, time invariant and
linear?
Show an example of a system which is
invertible, time invariant and nonlinear.
x
y
Kim, J. Y.
IC & DSP
Research
Group
Theme Examples Example of multiple propagation paths in a wireless
communication environment.
Kim, J. Y.
IC & DSP
Research
Group
Kim, J. Y.
IC & DSP
Research
Group
Tapped-delay-line model of a linear communication
channel, assumed to be time-invariant
Kim, J. Y.
IC & DSP
Research
Group
Stock Price : filtering
(a) Fluctuations in the closing stock price of Intel over a three-year
period.
(b) Output of a four-point moving-average system.
Kim, J. Y.
IC & DSP
Research
Group
LTI System
Linear time-invariant
Impulse response and convolution
Exponentials as eigenfunctions
Fourier and Laplace Transform
(z-transforma and discrete-time transforms)
Causality and Stability
