Signals and Systems 1 Lecture 11 Dr. Ali. A. Jalali September 13, 2002

Signals and Systems 1Lecture 11

Dr. Ali. A. JalaliSeptember 13, 2002

Signals and Systems 1

Lecture # 11First TEST ReviewFirst TEST Review

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Signals and Systems

Signals Sequences Systems

(Signals & Systems)

Electrocardiogram (ECG or EKG)

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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.

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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

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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

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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

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).()()()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!

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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

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Unit Step Function1. Integration of the unit impulse yields the unit

step function

which is defined as

.01

00)(

tif

tiftu

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dtut

)()(

%F1_7b Signal g(t) multiplied by a pulse functions

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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.

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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.

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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

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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

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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)


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


,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


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


,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)


LTIx(t)=(t) y(t)= h(t)

Convolution Integral


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.


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)

Signals and Systems 1 Lecture 11 Dr. Ali. A. Jalali September 13, 2002

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Transcript of Signals and Systems 1 Lecture 11 Dr. Ali. A. Jalali September 13, 2002