signals and systems 1

7/23/2019 Signals and Systems 1

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Ertem Tuncel

EE 110A Signals and Systems

Introduction to

Signals and Systems



What is a signal?

• Definition: A signal is a function of one ormore independent variables.

time (t )

space ( x) or ( x, y)

spatiotemporal ( x,t ) or ( x, y,t )



Examples

• Speech signal. Time is the independent variable

• Amplitude = acoustic pressure



Examples

• Unemployment rate. Also a time signal.

• Amplitude = % of unemployment among people

over 16 years old.



Examples

• Image signal. Space variables ( x, y)

x

y

• Amplitude = brightness



Examples

• Video signal. Space variables ( x, y) and timevariable t

• Amplitude = brightness of RGB components



• We will focus on one-dimensional signals.

In this course

• Time will be the independent variable

• Discrete-time signals (110B)

• Continuous-time signals (110A)

unemployment rate

stock market data

sampled signals

audio signals

voltage/current in a circuit with AC power



What is a system?

• Definition: A system is a relationship betweenits input signal, typically x(t ), and its output

signal, typically y(t ).

SYSTEM y(t ) x(t )

• Any legitimate relation between x(t ) and y(t )

forms a system.



Examples

• x(t ) = force applied on the car at time t

• y(t ) = displacement of the car at time t



• = voltage applied on the circuit at time t

Examples

• = voltage on the capacitor at time t



Examples

• x(t ) = solar radiation at time t

• y(t ) = temperature at a location at time t



• We will study important properties of systems.

In this course

• The focus will be on linear time-invariant (LTI)systems

Linearity

Time-invarianceInvertibility

Memory

Causality

Continuous-time input and output in EE110A

Stability

Discrete-time input and output in EE110B



Some important signals

• The unit step signal:

t

1

A very simple signal.



• The impulse signal:


t

What if we took the derivative of u(t )?

At t = 0, it seems that u(t ) has infinite slope.





For consistency, though, we must remember

What a strange function!!

integrate it from to 0.000000001and you get zero

integrate it from to 0.000000001

and you get one





To keep this consistency, one can think of

the impulse function as a limit.

t

Area =1

Area =1

Area =1

Area =1



• The unit ramp signal:


What if we integrate u(t )?

t

r (t )

Slope =1



• The unit ramp signal:


Conversely, we also have

• Summary:

integrate integratedifferentiate differentiate




• The exponential signal:

t 1

t

1

t

1

• Three cases:





• As (positive) changes,

t

1

x(t )

increasing





• As (negative) changes,

t

1

x(t )

increasing



Periodic signals

• A signal is said to have a period T if

• Example: square wave

• If T is a period, so are 2T , 3T , 4T , ...

t

x(t )

T 2T 0-T



Periodic signals

• An important class of examples is sinusoidals:

• is called the amplitude

• is called the frequency

• is called the phase

• Convention:



Periodic signals

• To find T , we need to solve

• Towards that end, we need to use the identity

which is true for any integer k .

• In other words, T must satisfy forsome k .

• Smallest such T :



Periodic signals• Another way to understand this behavior is to

look at complex exponentials

• For this signal to have a period T , we need

implying that

or that



Digression: Complex algebra

• In rectangular coordinates,

Real Imaginary

Re

Im

x

y z




• In polar coordinates,

Re

Im

z

Magnitude

Angle

r




• In rectangular coordinates,




• In polar coordinates,



Digression: Complex algebra• Two important identities:

• Proof:

• Add and subtract the two to find the desired

result.



Simple signal transformations

• Why does this cause a right shift?

• Time shift: Let

for some .

t

y(t )

t

x(t )

• The key is to see that the signal y copies at

time instant t the "old value" of x at




• So this is resulting in the "mirror image"

of the signal around the y-axis.

t

x(t )

• Time reversal: Let

t

y(t )




• Scaling:

for some .

t

x(t )

1

t

y(t )

t

y(t )



Transformation combos• What if we have a transformation such as

• Looks like a combo of time shift, time reversal,

and scaling. But with what order?

x(t )

• Option 1:

TIME

REVERSAL

x(-t )RIGHT SHIFT

3 UNITS

x(3-t )SCALE BY 2

x(3-2t )

• Option 2:

x(t ) LEFT SHIFT

3 UNITS

x(t+3)TIME

REVERSAL

x(3-t )SCALE BY 2

x(3-2t )

?



Transformation combos x(t )

TIME

REVERSAL

x(-t )RIGHT SHIFT

3 UNITS

x(3-t )SCALE BY 2

x(3-2t )

x(t ) LEFT SHIFT

3 UNITS

x(t+3)TIME

REVERSAL

x(3-t )SCALE BY 2

x(3-2t )

• Option 3: x(t ) TIME

REVERSAL

x(-t )SCALE BY 2

x(-2t ) RIGHT SHIFT

1.5 UNITS

x(3-2t )

• Option 4: x(t ) LEFT SHIFT

3 UNITS

x(t+3)SCALE BY 2

x(3+2t ) TIME

REVERSAL

x(3-2t )




TIME

REVERSAL

x(-t )RIGHT SHIFT

3 UNITS

x(3-t )SCALE BY 2

x(3-2t )

x(t ) LEFT SHIFT

3 UNITS

x(t+3)TIME

REVERSAL

x(3-t )SCALE BY 2

x(3-2t )

x(t ) TIME

REVERSAL

x(-t )SCALE BY 2

x(-2t ) RIGHT SHIFT

1.5 UNITS

x(3-2t )

x(t ) LEFT SHIFT

3 UNITS

x(t+3)SCALE BY 2

x(3+2t ) TIME

REVERSAL

x(3-2t )

• Option 5: x(t )SCALE BY 2

x(2t ) TIME

REVERSAL

x(-2t ) RIGHT SHIFT

1.5 UNITS

x(3-2t )




TIME

REVERSAL

x(-t )RIGHT SHIFT

3 UNITS

x(3-t )SCALE BY 2

x(3-2t )

x(t ) LEFT SHIFT

3 UNITS

x(t+3)TIME

REVERSAL

x(3-t )SCALE BY 2

x(3-2t )

x(t ) TIME

REVERSAL

x(-t )SCALE BY 2

x(-2t ) RIGHT SHIFT

1.5 UNITS

x(3-2t )

x(t ) LEFT SHIFT

3 UNITS

x(t+3)SCALE BY 2

x(3+2t ) TIME

REVERSAL

x(3-2t )

x(t )SCALE BY 2 x(2t ) TIME

REVERSAL x(-2t ) RIGHT SHIFT1.5 UNITS

x(3-2t )

• Option 6: x(t )

SCALE BY 2 x(2t ) LEFT SHIFT

1.5 UNITS

x(2t+3)TIME

REVERSAL

x(3-2t )



Transformation combos x(t ) LEFT SHIFT

3 UNITS

x(t+3)SCALE BY 2

x(3+2t ) TIME

REVERSAL

x(3-2t )

t

x(t )

1

LEFT SHIFT3 UNITS

t

x(t +3)

-2-3

SCALE BY 2

t

x(3+2t )

-1-1.5

TIMEREVERSAL

t

x(3-2t )

1 1.5



Transformation combos

t

x(t )

1t

x(3-2t )

1 1.5

• Could we have computed this without goingthrough the transformations one by one?

3-2t t

0 1.5

1 1

• Yes. Pick important time instants in the

original signal, and find out what t needs to be for 3-2t to correspond to those instants.



Linarity

• A system is linear if

SYSTEM

SYSTEM

and

impliesSYSTEM

for any



Linearity

• Problem: Is a linear system?



Linearity




Linearity


• But what about the case a = 1 and b = 0?

• Remember that the condition needs to be

satisfied FOR ANY

NONLINEAR



Linearity


• To see this, just take any a and b NOT

satisfying a+b = 1.

NONLINEAR



Time invariance

• A system is time-invariant if

implies

SYSTEM y(t ) x(t )

SYSTEM

for any x(t ) and .



Time invariance

• Problem: Is a time-invariantsystem?

x(t )



Time invariance


x(t )

TIME VARIANT



Time invariance

• Problem: Is a time-invariant system?

x(t )

i i i



Time invariance


x(t )

TIME VARIANT

If i d b hi



If in doubt, try this out• You can try to find an example to prove non-

linearity or time-variance

• For the last example, , try this:

t

x(t )

1t

1/2

t

x(t -1)

1 2

SYSTEM

SYSTEM

t 1/2 1

M d C li



Memory and Causality

• A system is memoryless if at time instant t ,the value of y(t ) depends only on the current

value of x(t ), and not on any past or future

value of it.

• A system is causal if at time instant t , the

value of y(t ) depends only on the current and

past value of x(t ), and not on any future valueof it.

• Obviously, memorylessness implies causality,

but not vice versa.

M d C li




• Problem: Is a memorylesssystem? If not memoryless, is it causal?

• Solution:

MEMORYLESS

CAUSAL

M d C lit




• Problem: Is amemoryless system? If not memoryless, is it

causal?

• Solution:

HAS MEMORY

CAUSAL

M d C lit




• Problem: Is a memorylesssystem? If not memoryless, is it causal?

• Solution:

HAS MEMORY

NON-CAUSAL

M d C lit



Memory and Causality• Problem: Is

a memoryless system? If not memoryless, is it

causal?

• Solution: y(t ) clearly depends on all values of

x() between the time instants and t .

HAS MEMORY CAUSAL

I tibilit



Invertibility

• A system is invertible if there exists anothersystem which outputs x(t ) when its input is

y(t ).

SYSTEM x(t ) y(t ) INVERSE

SYSTEM x(t )

• This should be true for ALL x(t ).

I tibilit



Invertibility

• But this definition seems to require that youactually find the inverse system.

• Alternative definition: A system is invertible

if no two distinct input signals yield the sameoutput.

NOT

INVERTIBLE

SYSTEM

y(t )

SYSTEM y(t )

I tibilit



Invertibility

• Problem: Is an invertible system?

NOT

INVERTIBLE

I tibilit



Invertibility

• Problem: Is an invertible system?• Solution: Yes, because you can reverse the

operation by

INVERTIBLE

I tibilit



Invertibility• Problem: Is an

invertible system?

NOT

INVERTIBLE

t

SYSTEM

t

t

SYSTEM

t 1 2-1

I tibilit



Invertibility• Problem: Is

an invertible system?

• Solution: Yes, because you can reverse the

operation by

INVERTIBLE

I tibilit



Invertibility

and

• Alternatively, if the system were not

invertible, there would exist two inputsand yielding the same output.

• But that would mean that for all t and ,

I tibilit



Invertibility

• Letting , this is the same as

• Contradiction! No such can exist.

INVERTIBLE

St bilit



Stability

• A system is stable if bounded inputs yield bounded outputs.

• Mathematically speaking, a system is stable if

for some B implies

for some C.

• This should be true for ALL x(t ).

Stabilit



Stability• A familiar example of a stable system:

• Obviously no bounded source can create an

infinite voltage on the capacitor.

Stability



Stability• A familiar example of an unstable system:

L

C

• If with , theoutput will blow up.

Stability



Stability• A frightening example of an unstable system:

http://youtu.be/j-zczJXSxnw

Stability

http://youtu.be/j-zczJXSxnw



Stability• Here is another unstable system:

OBJECT OF

UNIT MASS

F (t )

Stability



Stability• Here is another unstable system:

OBJECT OF

UNIT MASS

F (t )

x(t )

• Newton's Law:

• This implies that even with a very small force,

you can obtain infinite displacement.

Stability



Stability

• Problem: Is a stable system?

• Taking C = B in the definition then leads to

the conclusion that the system is...

STABLE

• Solution: If , then certainly

Stability



Stability• Problem: Is a stable

system?

STABLE

• Solution: If , then

• Take

signals and systems 1

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