Signals and Systems 1Lecture 3

Dr. Ali. A. JalaliAugust 23, 2002

Signals and Systems 1

Lecture # 3Introduction to Signals

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Example 1

1. Example 1

2. Interpret and sketch the generalized function x(t) where

3.

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).4()( 4/ tetx t

Example 1:Solution:1. To determine the meaning of x(t) we place it in an

integral

2. Let = t+4 so that

1. From the definition of the unit impulse,

The integral equals e-1 . Therefore

dttedttx t

)4()( 4/

dttet

)4(4/

de )(4/)4(

)0()()( fdtttf

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).4()( 1 tetx

Example 1: Graphical solution1. The result can also be seen graphically.

The left panel shows both and (t+4), and the right panel shows their

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4/te

t

Unit impulse- what do we need it for?

1. The unit impulse is a valuable idealization and is used widely in science and engineering. Impulses in time are useful idealizations.

2. Impulse of current in time delivers a unit charge instantaneously to network.

3. Impulse of force in time delivers an instantaneous momentum to a mechanical system.

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Unit impulse- what do we need it for?1. Impulses in space are also useful.

2. Impulse of mass density in space represents a point mass.

3. Impulse of charge density in space represents a point charge.

4. Impulse of light intensity in space represents a point of light.

5. We can imagine impulses in space and time

6. Impulse of light intensity in space and time represents a brief flash of light at a point in space.

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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

)()(

Unit Step Function (Figure 1.7a text)

%F1_7a Unit step functiont=-2:0.01:5; % make t a vector of 701 pointsq=size(t);f=zeros(q(1),q(2)); % set f = a vector of zerosq=size(t(201:701));f(201:701)=ones(q(1),q(2));% set final 500 points of f to 1plot(t,f),title('Fig.1.7a Unit step function');axis([-2,5,-1,2]); % sets limits on axesxlabel('time, t');ylabel(' u(t)');grid;

%F1_7a Unit step functiont=-2:0.01:5; % make t a vector of 701 pointsq=size(t);f=zeros(q(1),q(2)); % set f = a vector of zerosq=size(t(201:701));f(201:701)=ones(q(1),q(2));% set final 500 points of f to 1plot(t,f),title('Fig.1.7a Unit step function');axis([-2,5,-1,2]); % sets limits on axesxlabel('time, t');ylabel(' u(t)');grid;

.0

)( 00

ottif

ttifBttBu

Generic stepfunction EE 327 fall 2002

%F1_7b Signal g(t) multiplied f(101:501)=2.5-

cos(5*t(101:501) by a pulse functions ([u(t+1)-u(t-3)]

%F1_7b Signal g(t) multiplied by a pulse functionst= -2:0.01:5;q=size(t);f=zeros(q(1),q(2));f(101:501)=2.5-cos(5*t(101:501));plot(t,f),title('Fig.1.7b Signal g(t) multiplied by a pulse

functions');axis([-2,5,-1,4]);xlabel('time, t');ylabel(' g(t)[u(t+1)-u(t-3)]');grid;

%F1_7b Signal g(t) multiplied by a pulse functionst= -2:0.01:5;q=size(t);f=zeros(q(1),q(2));f(101:501)=2.5-cos(5*t(101:501));plot(t,f),title('Fig.1.7b Signal g(t) multiplied by a pulse

functions');axis([-2,5,-1,4]);xlabel('time, t');ylabel(' g(t)[u(t+1)-u(t-3)]');grid;

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

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Unit impulse as the derivative of the unit step1. As an example of the method for dealing with generalized

functions consider the generalized function

2. Since u(t) is discontinuous, its derivative does not exist as an ordinary function, but it does as a generalized function. To see what x(t) means, put it in an integral with a smooth testing function

3. 4. And apply the usual integration-by-parts theorem

5. to obtain

).()( tudt

dtx

).0()()()(0

fdttfdt

dfty

,)()()()( dttfdt

dtututf

,)()()( dttudt

dtfty

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Unit impulse as the derivative of the unit step. Cont’d

1. The result is that

1. which, from the definition of the unit impulse, implies that

2. That is, the unit impulse is the derivative of the unit step in a generalized function sense.

),0()()( fdttudt

dtf

).()( tudt

dt

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Real Exponential Functions1. Exponential signals are characterized by exponential

function

Where e is the Naperian constant 2.718… and a and A are real constants.

atAetf )(

f(t)

Time, t

atAetf )(

Ramp Functions1. A shifted ram function with slop B is defined as

Unit ramp function being at t=0 by making B=1 and t0=0 and multiplying by u(t), giving

)()( 0ttBtg

.0,

0,0)()(

tt

tttutr

Time, tf(

t)

r(t)=tu(t)

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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Sinusoidal Functions1. A sinusoidal function

frequency in hertz or cycles per second, the phase shift in radians, the radian frequency is rad/s and the period

is sExponential functions, as in

Where B is the amplitude, is angular frequency in radians/second, and is the phase shift in radians.

0

2cos)cos()2cos()(

T

tAtAftAtf

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)cos(22

)( )()( tBeB

eB

te tjtj

f.2 fT /10

f.2

Exponentialy Modulated Sinusoidal Functions1. If s sinusoid is multiplied by a real exponential, we have an

exponentially modulated sinusoid that also can arise as a sum of complex exponentials, as in

)2cos()( ftAetf t

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)cos(22

)( )()( tBeeB

eeB

te tjttjt

Example:

)1cos(3)( 2.0 tetf t Turned on at t = +1by multipling shiftedunit stepu(t-1)

Building-block signals can be combined to make a

rich population of signals

1. Eternal complex exponentials and unit steps can be combined to produce causal and anti-causal decaying exponentials.

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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.

Sequences1. Unite Sample Sequence2. The unit sample sequence is the discrete-time version of the unit impulse in

CT situations.3. Definition of unit sample sequence:

4. Thus it is possible to represent an arbitrary sequence as the weighted sum of unit sample sequences.

)(n

.0,0

0,1)(

m

mm

)3( n1

n

… …

1

n

… …0

m = n m = n-3Plots of Unite Sample Sequence

0 1 2 3

Sequences1. Unite Step Sequence2. The unit step sequence is the discrete-time version of the unit

step in CT situations.3. Definition of unit step sequence:

4. The unit step sequence u(n) is related to unit sample sequence by

.0,0

0,1)(

n

nnu

1

n

…

Plots of Unite Step Sequence

0 1 2 3

.)()(

n

m

mnu U(n)

0

00 ,0

,)(

nn

nnBnnBu

Generic step sequence

Sequences1. Ramp Sequence2. A shifted ramp sequence with slop of B is defined by:

3. The unit ramp sequence and shifted ramp sequences

4. Example: g(t) = 2(n-10).

)()( 0nnBng

MATLAB Code:n=-10:1:20;f=2*(n-10);stem(n,f);

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Real Exponential Sequences1. Real exponential sequence is defined as: Example for A = 10 and a = 0.9, as n goes to infinity the sequence approaches

zero and as n goes to minus infinity the sequence approaches plus infinity.

naAnf )()(

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)()()( nuaAnp nComposite sequence:

Multiplying point byPoint by the step sequence

MATLAB Code:n=-10:1:10;f =10*(.9).^n;stem(n,f);axis([-10 10 0 30]);

Sinusoidal Sequence1. A sinusoidal sequence may be described as:

2. Where A is positve real number (amplitude), N is the period, and a is the phase.

3. Example: 4. A = 5, N = 16 5. And

6. MATLAB Code:7. n=-20:1:20;

8. f=5*[cos(n*pi/8+pi/4)];

9. stem(n,f);

N

nAnf

2cos)(

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.4/a

Exponentialy Modulated Sinusoidal Sequence

1. By multiplying an exponential sequence by sinusoidal sequence, we obtain an exponentially modulated sequence described by:

2. Example: 3. A = 10, N = 16, a = 0.94. And5. MATLAB Code:6. n=-20:1:20;

7. f=10*[0.9 .^n];

8. g=[cos(2*n*pi/16+pi/4)];

9. h=f .*g;

10. stem(n,h);

11. axis([-20 20 -30 70]);

N

naAng n 2

cos)()(

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.4/

Example:1. Use the Sequence definition, describe analytically

the following sequence.

1

n

A Pulse Sequence

0 1 2 3

f(n)

-1-2

Solution: This pulse sequence can be describe by

f(t) = u(n) – u(n-3).

The first step sequence turn on the pulse at n = 0,and second step turns it off at n = 3.

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See page 13 of textFor more examples.

A Road MapSignals and Systems

Mathematical Model, usually difference or differential equations

LP FilterSpring Mass Dc MotorAny phisicalsystems

SolutionsFind

x(t) or

x(n)(.))(dt

dtx or x(n+1)

Ordinary DELaplace transformZ transformConvolutionSimulationHock upFourier transformFourier seriesTransfer functionState space modelSignal flow graphBlock diagramUnit sample responseFrequency response

Time Domain and Frequency Domain solutions.Continuous and Discrete-time Systems.

Time Domain and Frequency Domain solutions.Continuous and Discrete-time Systems.

Imp

lemen

tation

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Conclusions1. Introduction to signals and systems,

2. Signals, definitions and classifications,

3. Building block signals- eternal complex exponentials and impulse,

4. Mathematical description of signals,

5. MATLAB examples,

6. A road map.

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