signals and systems 1 lecture 11 dr. ali. a. jalali september 13, 2002
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Signals and Systems 1 Lecture 11 Dr. Ali. A. Jalali September 13, 2002. Signals and Systems 1. Lecture # 11 First TEST Review. EE 327 fall 2002. Signals and Systems. (Signals & Systems). Sequences. Signals. Systems. Electrocardiogram (ECG or EKG). EE 327 fall 2002. - PowerPoint PPT PresentationTRANSCRIPT
Signals and Systems 1Lecture 11
Dr. Ali. A. JalaliSeptember 13, 2002
Signals and Systems 1
Lecture # 11First TEST ReviewFirst TEST Review
EE 327 fall 2002
Signals and Systems
Signals Sequences Systems
(Signals & Systems)
Electrocardiogram (ECG or EKG)
EE 327 fall 2002
Signals and Systems
1. A signal is the physical form of a waveform, like a sound wave or a radio wave.
2. Time is often the independent variable for signal.3. The independent variable can be 1-D or 2-D (space x, y
in image), 3-D or N-D4. A system is an object or channel that changes a signal
that passes through it.5. Amplifiers are systems that increase the amplitude of
signals passing through them. 6. Attenuators are systems that decrease the amplitude of
signals passing through them.
EE 327 fall 2002
Signals and Systems1. Signals
1. physical form of a waveform
2. e.g. a sound, electrical current, radio wave
2. Systems1. a channel that changes a signal that passes through it
2. e.g. a telephone connection, a room, a vocal tract
SystemInput Signal Output Signal
EE 327 fall 2002
Signals:
1. Classification of Signals1. Deterministic and Stochastic signals
2. Periodic and Aperiodic signals
3. Continuous time (CT) and Discrete time (DT)
4. Causal and anti-causal signals
5. Right and left sided signals
6. Bounded and unbounded signals
7. Even and odd signals
EE 327 fall 2002
Building –block signals Unit impulse – definition1. The unit impulse (t), is an important signal of CT systems. The Dirac delta function,
is not a function in the ordinary sense. It is defined by the integral relation
And is called a generalized function.
1. The unit impulse is not defined in terms of its values, but is defined by how it acts inside an integral when multiplied by a smooth function f(t). To see that the area of the unit impulse is 1, choose f(t) = 1 in the definition. We represent the unit impulse schematically as shown below; the number next to the impulse is its area.
.)()()( dtftf
EE 327 fall 2002
).()()()0( dtff
Unit impulse- narrow pulse approximation1. To obtain an intuitive feeling for the unit impulse, it is often
helpful to imagine a set of rectangular pulses where each pulse has width and height 1/ so that its area is 1.
The unit impulse is the quintessential tall and narrow pulse!
EE 327 fall 2002
Unit impulse- intuiting the definition1. To obtain some intuition about the meaning of the
integral definition of the impulse, we will use a tall rectangular pulse of unit area as an approximation to the unit impulse.
2. As the rectangular pulse gets taller and narrower,
)0()0(
)()(lim0
ff
dtttf
EE 327 fall 2002
Unit Step Function1. Integration of the unit impulse yields the unit
step function
which is defined as
.01
00)(
tif
tiftu
EE 327 fall 2002
dtut
)()(
%F1_7b Signal g(t) multiplied by a pulse functions
EE 327 fall 2002
Successive integration of the unit impulse 1. Successive integration of the unit impulse yields a
family of functions.
2. Later we will talk about the successive derivatives of (t), but these are too horrible to contemplate in the first lecture.
EE 327 fall 2002
Building-block signals can be combined to make a
rich population of signals
1. Unit steps and ramps can he combined to produce pulse signals.
EE 327 fall 2002
Example:1. Describe analytically the signals shown in
Solution: Signal is (A/2)t at , turn on this signal at t = 0 and turn it off again at t = 2. This gives,
)].2()([2
)( tututA
tf
EE 327 fall 2002
20 t
A
0 2t
f(t)
See page 9 of text for more examples.
Continuous Systems1. Preview2. A system is transforms input signals into output signals.
3. A continuous-time system receives an input signal x(t) and generates an output signals y(t).
4. y(t)=h(t)x(t) means the system h(t) acts on input signal x(t) to produce output signal y(t).
5. We concentrate on systems with one input and one output signal, i.e., Single-input, single output (SISO) systems.
6. Systems often denoted by block diagram.
7. Lines with arrows denote signals (not wires). Arrows show inputs and outputs
EE 327 fall 2002 Signals and Systems 1
Continuous-timeSystem
h(t)
x(t)Input
y(t)Output
Continuous Systems
Example: Electric Network
Systems
Classifications of systems:1. Linear and nonlinear systems.
2. Time Invariant and time varying systems.
3. Causal, noncausal and anticausal systems.
4. Stable and unstable systems.
5. Memoryless systems and systems with memory.
6. Continuous and Discrete time systems.EE 327 fall 2002 Signals and Systems 1
Continuous SystemsLinearity:
principle of homogeneity (c is real constant).
principle of additively
homogeneity and additively (Principle of superposition)
EE 327 fall 2002 Signals and Systems 1
x(t)=C 1 x1(t) y(t)=C 1 y1(t) LSx1(t) y1(t) LS
x1(t) y1(t)
x2(t) y2(t)
x(t)=x1(t)+x2(t) y(t)=y1(t)+y2(t) LS
LS LS
x1(t) y1(t)
x2(t) y2(t)
LS
LS LS
X(t)=C1x1(t)+C2x2(t) y(t)=C1y1(t)+C2y2(t)
21 , cc
Continuous SystemsLinearity example: Let the response of a linear system at rest due to the
system input be given by and let the response of the same system at rest due to another system input be
Then the response of the same system at rest due to input given by
Is simply obtained as:
,0,1)(1 ttf
EE 327 fall 2002 Signals and Systems 1
,0,5.05.0)( 21 teety tt
,0),sin()(2 tttf
;0),cos()10/10(2.05.0)( 22 tteety tt
,0),sin(32)(3)(2 21 tttftf
.0),cos(10
1034.05.01)(3)(2)( 2
21 tteetytyty tt
Continuous Systems
EE 327 fall 2002 Signals and Systems 1
LS
LSTT
t t
t t
x(t) y(t)
x(t-T) y(t-T)
x(t-T) y(t-T)
x(t) y(t)
1
1 1 4
31
Time Invariance:
Shifted inputShifted outputFor All value of t and T.
Continuous SystemsTime invariant example: Let the response of a time-invariant linear system at rest
due to be given by Then, the system response due to the shifted system input defined by
Is
,0),(1 ttf
EE 327 fall 2002 Signals and Systems 1
,0,43)( 21 teety tt
6,0
6),6()( 1
2 t
ttftf
6,0
6,43)(
)6(2)6(
2t
teety
tt
Differential Equation Model
1. Many physical systems are described by linear differential equations.
2. Reducing differential equations to algebraic equations
3. Homogeneous solution, exponential solution and natural frequencies.
4. Particular solution, system function and poles-zeros
5. Total solution, initial condition and steady-state
6. Conclusion
The Nth-order Differential Equation Model
X(t) system input,Y(t) system output and our practical restriction order
1. The general linear constant-coefficient Nth-Order DE for SISO systems are:
a y t ady t
dta
d y t
dtb x tN
N
N0 1 0
bdx t
dta
d x t
dtL
L
L1 .OR:
a
d y t
dtb
d x t
dtk
k
kk
N
k
k
kk
L
0 0
.
LN
Initial Condition Solution of Differential Equation
This is Characteristic Equation
Can be written as:
Or as factored form:
Characteristic roots are:
Where may be real or complex (conjugate pairs).
011
00 N
N sasasa
.00
N
k
kk Sa
NN rsrsrs ,,, 2211
0210
NN
N
k
kk rsrsrsasa
Nrrr ,,, 21
Example Solution
ttIC eCeCt 2
21)(
1- Find CE?The homogeneous differential equation is:Yielding the CE as:
2- CR?Using the quadratic formula, CR are:
3- Is this system stable?Both roots are negative real so the system is stable.
4- Algebraic form of IC response?
5- Constant IC solution? and Solving gives: and Thus
.2,12,1 r
023 ttt
211)0( CCIC .20]/)([ 210
CCdttdtIC
.0232 ss
21 C .12 C .0,2)( 2 teet ttIC
Unit Impulse Response The response of an LTI system to an input of unit impulse function is called the unit impulse response.
Important: When determining the unit impulse response h(t) of an LTI system, it is necessary to make all initial conditions zero. (output due to input not energy stored in system)
EE 327 fall 2002 Signals and Systems 1
LTIx(t)=(t) y(t)= h(t)
Convolution Integral
EE 327 fall 2002 Signals and Systems 1
dthxtx )()()(
1. The convolution integral is one of the most important results used in the study of the response of linear systems.
2. If we know the unit impulse response h(t) for a linear system, by using the convolution integral we can compute the system output for any known input x(t).
3. In the following integration integral h(t) is the system’s unit impulse response.
EE 327 fall 2002 Signals and Systems 1
RC y(t)
+
-
dfC
tvtRfty
dfC
dfC
tRfty
dfC
tRfty
t
c
t
t
)(1
)()()(
)(1
)(1
)()(
)(1
)()(
00
0
0
Total response = ZIR + ZSR
Example for total response of system
f(t)
IC response, Force response and Steady state response
t
2u(t+1)
-u(t-2)
3u(t)
f(t)
t
2u(t+1)
-u(t-2)
3u(t)f(t)
1
2
0
3
2
-2
-1
f(t)= -2u(t+1)+3u(t)-u(t-2)EXAMPLE:
-2
1
2
t
f(t)
t
(t+2)u(t+2)
(t-2)u(t-2)
f(t)
2
0
4
2-2
f(t)= -(t+4)u(t+4)+(t+2)u(t+2)+(t-2)u(t-2)-(t-4)u(t-4)EXAMPLE:
-2
-4 40
-4 4
-2
-4
2-2
8
-(t-4)u(t-4)
-(t+4)u(t+4)
Sketch the following sequence?4u(n-3)-2(n-6)u(n-6)+2(n-8)u(n-8)
0 1 2 3 4 5 6 7 8 9
4
0 1 2 3 4 5 6 7 8 9
4
Sketch the following sequence?2(n+5)u(n+5)-3nu(n )+(n-10)u(n-10)
0 1 2 3 4 5 6 7 8 9 10-5
10
n
Slop=2
Slop=-3
2(n+5)u(n+5)
-3nu(n)
(n-10)u(n-10)
Slop=1
After n = 10 the total slop is zero.
Graphical Solution
elsewhere0,
011,
10,1
)(f2 t
tt
t
elswhere
tt
tt
t
,0
01,
10,
)(f1
EXAMPLE:
)(1 tf )(2 tf
0 0 11-1 -1
1 1
t t
*
t-
2121 ,)d-(t)f(f(t)f*(t)fy(t)
elswhere,0
01,
10,
)(f1
elsewhere0,
011,
10),(1
)(f2
t
tt
t
2-2
)(2 tf
0 1-1
1
t-2+t
2-2
)(2 tf
0 1-1
1
t-2+t -1+t
-2<t<-1
-1 < t < 0
t
d1
)1)((
t
t
tddtdt
0
0
1
1
1)1)(()1)]((1[))]((1[
2-20 1-1
1
t-2+t -1+t
2-20 1-1
1
t-2+t -1+t
1 < t < 2
1
2)1)((
td
0 < t < 1
1
1
1
0
0
2))(1())}((1[))}((1[
t
t
tddtdt
)(2 tf
)(2 tf
Convolution Plane
t-
elsewhere0,
011,
10,1
)(f2 t
tt
t
elswhere
tt
tt
t
,0
01,
10,
)(f1
EXAMPLE:
)(1 tf )(2 tf
0 0 11-1 -1
1 1
t t
*
t-
2121 ,)d()f-(tf(t)f*(t)fy(t)
0)(2 f
a bc
d
e0
1
1t
0
-1
1
1t
1)(2 f
0)(2 f
1)(2f
Regions:
,t ,1t 1t ,1
ttf )(1
elswhere
tt
tt
t
,0
01),(
10,
)(f1
elsewhere0,
011,
10,1
)(f2
1
-1 1f
g h
t 1t
ttf )(1 0)(1 tf0)(1 tf
2-2
Integral arrangement:Integral arrangement:
dtt
)1)((y(t)1
1
y(t)
-2 < t <-1
-1 < t < 0
1 < t < 2
Region
a
b c d
h
+
Integral
+ dtdtdtt
t
t)1)(()1)(()1)((y(t)
1
0
0
1
0 < t < 1 e f g+ + dtdtdtt
t
t)1)(()1)(()1)((y(t)
1
0
0
1
dtt
)1)((y(t)1
1
t < -2 0y(t)
0y(t) t > 2
t-
2121 ,)d()f-(tf(t)f*(t)fy(t)