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Signals and Systems
Introduction
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Textbook
Simon Haykin and Barry Van Veenand Systems, Second Edition, Joh
Sons, 2003.
40%,60%,
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Introductions to Signals
Signal:
a function of one or more independentvariables; typically contains information about
the behavior or nature of some physicalphenomena.
e.g. Voice (audio), TV (audio + video), light, voltage,
current, stock price, etc.
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Introductions to System
System:
responds to a particular signal input byproducing other signal.
e.g. Biological sensory system, electronic circuits,automobile, etc.
SystemInput signal Output signal
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Objectives
System characterization
how it responds to input signal
(e.g. human auditory system)
System design
to process signal in a particular way
(e.g. signal restoration, signal identification,image processing)
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Examples of Signals
Audio (intensity vs. time)
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Examples of Signals
TV signal (voltage vs. time)modulated picture signal + audio signal; carrier signal;
system involves :antenna, tuner, CRT
Horizontal
Sync Pulse
Active Video
Signal
Color Burst
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Examples of Signals (cont.)
Biomedical signal (voltage vs. time)
e.g. Electrocardiogram
Traffic flow (quantity vs. time) volume, composition, pattern, mobility;
system involved: traffic lights, roads, junctions
Network Throughput
# of packets/sec., pattern
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Examples of Signals (cont.)
Stock price (index vs. time; $ vs. time)
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Examples of Signals (cont.)
Picture (intensity vs. space)
on film -- continuous space, continuous tone
on computer -- discrete, quantized
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Continuous vs. Discrete Signal
x(t)
t
x[n]
n
Discrete-time Signal(independent variable: n)
Continuous-time Signal(independent variable: t)
x[n] is also referred as a sequence. Any particularone in x[n] is called a sample.
(a).
(b).
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Transformation of
independent variable
(1). Reflection
x(t) x(-t)
x[n] x[-n] x(t)
t
x[n]
n
x[-n]
n
t
x(-t)
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Transformation of
independent variable (cont.)
(2). Scaling
x(t) x(ct)
x(t)
t
x(t/2)
t
x(2t)
t
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Transformation of
independent variable (cont.)
(3). Time-shift
x[n] x[n-n0]
x[n]
n
x[n-2]
n
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Transformation of
independent variable (cont.)
x(t)
1 2 t
x(t+1)
-1 1 2 t
-2/3 2/3
x(1.5t+1)
1 2 t
To perform transformation x(t)
x(a t+b)
You have to do time-shiftingthenscaling.
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Even and Odd functions
A signal is called an even signal (function) if
A signal is called an odd signal (function) if
x(t) = - x(-t)x[n] = - x[-n]
x(t) = x(-t)x[n] = x[-n]
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Even and Odd functions (cont.)
Any signal can be broken into sum of one evenand one odd signal.
x(t)= Ev{x(t)} + Od {x(t)}
Q: How to find the even (odd) part of a signal.
A: Ev{x(t)} = (1/2) [x(t) + x(-t)]
Od {x(t)} = (1/2) [x(t) - x(-t)]
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Even and Odd functions (cont.)
Ex:
x[n]
n-3 -2 -1 0 1 2 3
11/2
x[n]
n-3 -2 -1 0 1 2 3
1
Ev{x[n]} = (1/2) [x[n] + x[-n]] Od {x[n]} = (1/2) [x[n] - x[-n]]
x[n]
n0 1 2 3
1/2
-1/2
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Periodicand AperiodicSignal
IF x(t) is periodic with period T, then
x(t) = x(t+T)
IF x[n] is periodic with period N, then
x[n] = x[n+mN] ; m an integerFundamental period (T0 or N0) :
the smallest positive value of (T or N) for which
the above equation holds.Aperiodicis also called Non-periodic
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Unit Step and Unit Impulse
function
Unit Step function:
u(t) = 0, t0
Unit Impulse function:
d(t) = d u(t) / dt
also
* A system can be characterized by its unit step response
or unit impulse response.
1
t
1
t'dt)(t'u(t) t
d
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Discrete Time Unit Step and
Unit Impulse Sequence
Unit Step function:
u[n] = 0, n
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Discrete Time Unit Step and
Unit Impulse Sequence (cont.)
d[n] = u[n]-u[n-1] : first difference of unit step
u[n]= : running sum of unit sample
or equivalently u[n]=
n
m
m][d
0
][k
knd
x[n]
n-3 -2 -1 0 1 2 3
1
......x[n]
n-3 -2 -1 0 1 2 3
x[n-1]
n-3 -2 -1 0 1 2 3
= +
n
x[n-2]
-3 -2 -1 0 1 2 3
+ + ...
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Overview of System
System: any process that results in thetransformation of signal.
x(t) y(t)
Continuous-time
x[n] y[n]
Discrete-time
A system is continuous-time if both input and outputare continuous-time.A system is discrete-time if both input and outputare discrete-time.
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Interaction of Multiple Systems
(a) Cascade (Series)
x(t) y(t) z(t)
System #1 System #2
#1
#2
(b) Parallel
I t ti f M lti l S t
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Interaction of Multiple Systems
(cont.)
(c) Series/Parallel
#2
#3
#1
x2
( )2( )2
+
-
Ex. y[n]=(2x[n]-x[n]2)2
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Properties of System
Memory/Memoryless
Invertibility and Inverse System
CausalityStability
Time-invariance
Linearity
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(1). Memory and Memoryless
A system is memoryless if the output onlydepends on input at the same time
Ex. (i) Resister is a memoryless component.
y(t)=R x(t) where y(t) : voltage
x(t) : current
R: resistance
k
kxny ][][(ii) With memory : e.g.
Capacitor is also with memory.
(2)
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(2). Invertibility and Inverse
System
A system is invertible if distinct inputs lead todistinct outputs.
x(t) y(t) z(t) = x(t)
System #1 System #2
If z(t)=x(t), then system #2 is the inverse system
of system #1.E.g. y(t)=2 x(t) then z(t) =0.5 y(t)
]1[][][][][
n-ynynzthenkxnyn
k
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(3). Causality
A system is causal if the output at anytime only dependson the input at the present time and before.
E.g. y[n]=x[n]-x[n-1] : causal
y(t)=x(t+1) : non-causal
Causal property is more important for real-timeprocessing.
But for some applications, such as image-processing, no
need to process the data causally.E.g.
M
Mk
knxM
ny ][12
1][
Q:
Memoryless Causal ?
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(4). Stability
A system is stable if bounded input givesbounded output.
--- BIBO stable
Eg. x(t) : the horizontal force; y(t) : vertical displacement
x(t)
y(t)
Stable System
x(t)y(t)
Unstable System
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E.g. ][)1(][][ nunkunyn
k
][)1( nun
Input: u[n], boundedOutput: (n+1)u[n], non-bounded as
][lim nyn
(4). Stability (cont.)
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(5). Time-invariance
A system is time-invariant if a time shift in theinput only causes a time shift in the output.
i.e. If x[n] y[n], then x[n-n0] y[n-n0]
Ex.1 y(t)=sin(x(t))
Let y1(t)= sin(x1(t)), x2(t)=x1(t-t0)
Then y2(t)=sin(x2(t))=sin(x1(t-t0))
= y1(t- t0)
time-invariant (T.I.)
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(5). Time-invariance (cont.)
Ex.2 y[n]=n x[n]
Let y1[n]= n x1[n] & x2[n]=x1[n-n0]
Then y2[n]=n x2[n]=n x1[n-n0]
However y1[n-n0]=(n-n0) x1[n-n0]
y2[n]
not time-invariant (T.I.)
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(6). Linearity
A system is linear if
1. the response to x1(t)+x2(t) is y1(t) +y2(t)
---additivity
2. the response to ax1
(t) is ay1
(t), where a isany complex constant.
---scaling
Combine the above two properties, we can conclude
ax1(t)+ bx2(t) ay1(t) + by2(t)---superposition property
For discrete-time :
ax1[n]+ bx2[n] ay1[n] + by2[n]
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