Chapter 2

Updated 04/18/23

Time reversal:

Time Reversal

X(t) Y=X(-t)

Time scaling

Example: Given x(t), find y(t) = x(2t). This SPEEDS UP x(t) (the graph is shrinking)

The period decreases!

What happens to the period T?

The period of x(t) is 2 and the period of y(t) is 1,

TimeScaling

X(t) Y=X(at)

a>1 Speeds up Smaller period Graph shrinks!a<1 slows down Larger period Graph expandsa>1 Speeds up Smaller period Graph shrinks!a<1 slows down Larger period Graph expands

Example of Time Scaling (tt/a)

http://tintoretto.ucsd.edu/jorge/teaching/mae143a/lectures/1analogsignals.pdf

Time scaling

• Given y(t), – find w(t) = y(3t) – v(t) = y(t/3).

Time Shifting• The original signal g(t) is

shifted by an amount t0 .Time Shift: y(t)=g(t-to)

• g(t)g(t-to) // to>0 Signal Delayed Shift to the right

TimeShifting

X(t) Y=X(t-to)

Delay

Time Shifting

• The original signal x(t) is shifted by an amount t0 .

Time Shift: y(t)=x(t-to)

• X(t)X(t-to) // to>0 Signal Delayed Shift to the right

• X(t)X(t+to) // to<0 Signal Advanced Shift to the left

TimeShifting

X(t) Y=X(t-to)

Time Shifting Example

• Given x(t) = u(t+2) -u(t-2), – find

• x(t-t0)=• x(t+t0)=

Answer:• x(t-t0)= u(t-to+2) -u(t-to-2), • x(t+t0)= u(t+to+2) -u(t+to-2),

But how can we draw this function?

Note: Unit Step Functiona discontinuous continuous-time signal

Draw

• x(t) = u(t+1)- u(t-2)

u(t+1)- u(t-2)

t=0 t

Time Shifting

• Determine x(t) + x(2-t) , where x(t) = u(t+1)- u(t-2)– Which is x(t):

– find x(2-t): Reverse and then advance in time First find y(t)=x[-(t-2)];

u(t+1)- u(t-2)

t=0

First we reverse x(t) Then, we delay it by 2 unites as shown below:

Add the two functions: x(t) + x(2-t)

Summary

Or rewrite as: X[-(t+1)]Hence, reverse the signal in time. Then shift to the left of t=0by one unit!

Or rewrite as: X[-(t-2)]Hence, reverse the signal in time. Then shift to the right of t=0by two units!

shift to the left of t=0 by two units!Shifting to the right; increasing in time Delaying the signal!

Delayed/ Moved right

Advanced/Moved left

Reversed &Delayed

This is really: X(-(t+1))

SeeNotes

Amplitude Operations

In general:

y(t)=Ax(t)+B

B>0 Shift up

B<0 Shift down

|A|>1 Gain

|A|<1 Attenuation

A>0NO reversal

A<0 reversal

Reversal

Scaling

Scaling

Amplitude Operations

Given x2(t), find 1 - x2(t).

Signals can be added or multiplied

Multiplication of two signals:x2(t)u(t)

Ans.

Ans.

Step unit function

Remember: This is y(t) =1

Amplitude Operations

Given x2(t), find 1 - x2(t).

Signals can be added or multiplied e.g., we can filter parts of a signal!

Multiplication of two signals:x2(t)u(t)

Step unit function

Remember: This is y(t) =1

Note: You can also think of it as X2(t) being amplitude revered and then shifted by 1.

Signal Characteristics

• Even and odd signals1. X(t) = Xe(t) + Xo(t)

2. X(-t) = X(t) Even

3. X(-t) = -X(t) Odd

• PropertiesXe + Ye = ZeXo + Yo = ZoXe + Yo = Ze + Zo

Xe * Ye = ZeXo * Yo = ZeXe * Yo = Zo

Represent Xo(t) in terms of X(t) only!

Represent Xe(t) in terms of X(t) only!

Signal Characteristics

• Even and odd signals1. X(t) = Xe(t) + Xo(t)

2. X(-t) = X(t) Even

3. X(-t) = -X(t) Odd

• PropertiesXe + Ye = ZeXo + Yo = ZoXe + Yo = Ze + Zo

Xe * Ye = ZeXo * Yo = ZeXe * Yo = Zo

KnowThese!

Proof Examples

• Prove that product of two even signals is even.

• Prove that product of two odd signals is odd.

• What is the product of an even signal and an odd signal? Prove it!

)()()(

)()()(

)()()(

21

21

21

txtxtx

txtxtx

txtxtx

Eventx

txtxtx

txtxtx

txtxtx

)(

)()()(

)()()(

)()()(

21

21

21

Change t -t

Given:

Signal Characteristics:Find uo(t) and ue(t)

Remember:

Signal Characteristics

Symmetric across the vertical axis Anti-symmetric across the vertical axis

Example

• Given x(t) find xe(t) and xo(t)

5

4___

5

2___

5

2___

Example

• Given x(t) find xe(t) and xo(t)

5

4___

5

2___

5

2___

-5-5

Example

• Given x(t) find xe(t) and xo(t)4___

5

2___

5

2___

4e-0.5t

2___2e-0.5t

-2___

2___2e-0.5t

2e+0.5t

Example

• Given x(t) find xe(t) and xo(t)4___

5

2___

5

2___

4e-0.5t

2___2e-0.5t

-2___

2___2e-0.5t

2e+0.5t-2e+0.5t

Periodic and Aperiodic Signals

• Given x(t) is a continuous-time signal • X (t) is periodic iff X(t) = x(t+nT) for any T

and any integer n• Example

– Is X(t) = A cos(t) periodic?• X(t+nT) = A cos(t+Tn)) = • A cos(t+2n)= A cos(t)

– Note: f0=1/T0; = Angular freq.– T0 is fundamental period; T0 is the minimum

value of T that satisfies X(t) = x(t+T)

Periodic signalsExamples:

• =>Show that sin(t) is in fact a periodic signal.

• Use a graph • Show it mathematically

– What is the period?– Is this an even or odd signal?

• =>Is tesin(t) periodic?– X(t) = x(t+T)?

Sum of periodic Signals

• X(t) = x1(t) + X2(t)

• X(t+nT) = x1(t+m1T1) + X2(t+m2T2)

• m1T1=m2T2 = To = Fundamental period

• Example: – cos(t/3)+sin(t/4)– T1=(2)/(/3)=6; T2 =(2)/(/4)=8; – T1/T2=6/8 = ¾ = (rational number) = m2/m1

– m1T1=m2T2 Find m1 and m2

– 6.4 = 3.8 = 24 = To (n=1)

Sum of periodic Signals

• X(t) = x1(t) + X2(t)

• X(t+nT) = x1(t+m1T1) + X2(t+m2T2)

• m1T1=m2T2 = To=Fundamental period

• Example: – cos(t/3)+sin(t/4)– T1=(2)/(/3)=6; T2 =(2)/(/4)=8; – T1/T2=6/8 = ¾

– m1T1=m2T2 T1/T2= m2/m16/8=3/46.4 = 3.8 24 = To

Note that T1/T2 must be RATIONAL (ratio of integers)

An irrational number is any real number which cannot be expressed as a fraction a/b, where a and b are integers, with b non-zero.

Read about Irrational Numbers: http://www.mathsisfun.com/irrational-numbers.html

Product of periodic SignalsX(t) = xa(t) * Xb(t) =

= 2sin[t(7/24)]* cos[t(/24)];

find the period of x(t)•We know: = 2sin[t(7/12)/2]* cos[t(/12)/2];

– Using Trig. Itentity:

•x(t) = sin(t/3)+sin(t/4)•Thus, To=24 , as before!

http://www.sosmath.com/trig/Trig5/trig5/pdf/pdf.html

Sum of periodic Signals – may not always be periodic!

– T1=(2)/()= 2; – T2 =(2)/(sqrt(2)); – T1/T2= sqrt(2); – Note: T1/T2 = sqrt(2)

is an irrational number– X(t) is aperiodic

tttxtxtx 2sincos)()()( 21

Note that T1/T2 is NOT RATIONAL (ratio of integers) In this case!

Sum of periodic Several Signals

• Example– X1(t) = cos(3.5t)– X2(t) = sin(2t) – X3(t) = 2cos(t7/6)– Is v(t) = x1(t) + x2(t) + x3(t) periodic? – What is the fundamental period of v(t)? – Find even and odd parts of v(t).

Do it on your own!

Sum of periodic Signals (cont.)

– X1(t) = cos(3.5t) f1 = 3.5/2 T1 = 2 /3.5

– X2(t) = sin(2t) f2 = 2/2 T2 = 2 /2

– X3(t) = 2cos(t7/6) f3 = (7/6)/2 T3 = 2 /(7/6) T1/T2 = 4/7 Ratio or two integers T1/T3 = 1/3 Ratio or two integers Summation is periodic

– m1T1 = m2T2 = m3T3 = To ; Hence we find To

– The question is how to choose m1, m2, m3 such that the above relationship holds

– We know: 7(T1) = 4(T2) & 3(T1) = 1(T3) ; m1(T1)=m2(T2)

– Hence: 21(T1) = 12(T2)= 7(T3);

Thus, fundamental period: To = 21(T1) = 21(2 /3.5)=12(T2)=12

Find even and odd parts of v(t).

T1 T2 T32 2 12

m1 m2 m37.3=21 2.6=12 7.1=7

m1xT1 m2xT2 m3xT3 12 12

Important Engineering Signals

• Euler’s Formula (polar form or complex exponential form)– Remember: ej = 1|_ and arg [ ej ] = can you prove these?)

• Unit Step Function (Singularity Function)

• Use Unit Step Function to express a block function (window)

-T/t T/t

notes

notes

notes

What is sin(A+B)? Or cos(A+B)?

Important Engineering Signals

• Euler’s Formula

• Unit Step Function (Singularity Function)– Can you draw x(t) = cos(t)[u(t) – u(t-2)]?

• Use Unit Step Function to express a block function (window)

-T/2 T/2

notes

notes

Next

Unit Step Function Properties

kttuttuttu )]([)]([)( 02

00

0),/()( 00 aattutatu

Examples:

Note: U(-t+3)=1-u(t-3)

Unit Step Function Applications: Creating Block Function (window)

• rect(t/T)

• Can be expressed as u(T/2-t)-u(-T/2-t) – Draw u(t+T/2) first; then reverse it!

• Can be expressed as u(t+T/2)-u(t-T/2)

• Can be expressed as u(t+T/2).u(T/2-t)

-T/2 T/2

1

-T/2 T/2

1

-T/2 T/2

1

-T/2 T/2

Note: Period is T; & symmetric

Unit Step Function Applications: Creating Unit Ramp Function

)()()()( 00

0

0

0

ttuttddtutft

t

t

Unit ramp function can be achieved by:

to to+1

1

notes

Non-zero only for t>t0

Example: Using Mathematica: (t-2)*unitstep[t-2] – click here:

http://www.wolframalpha.com/input/?i=%28t-2%29*unitstep%5Bt-2%5D

Unit Step Function Applications: Example

)2(5)1(]1[)()(3)( tututttututf

Unit Step Function Applications:Example

• Plot

• t<-2 f(t)=0• -2<t<-1

f(t)=3[t+2]• -1<t<1 f(t)=-3t• 1<t<3 f(t)=-3• 3<t< f(t)=0

)3(3)1(]1[3)1(]1[6)2(]2[3)( tututtuttuttf

http://www.wolframalpha.com/input/?i=3*%28t%2B2%29*UnitStep%28t%2B2%29-6*%28t%2B1%29*UnitStep%28t%2B1%29%2B3*%28t-1%29*UnitStep%28t-1%29%2B3*UnitStep%28t-3%29

Using Mathematica - click here:

Unit Impulse Function (t)• Not real (does not exist in nature – similar to i=sqrt(-1) • Also known as Dirac delta function

– Generalized function or testing function

• The Dirac delta can be loosely thought of as a function of the real line which is zero everywhere except at the origin, where it is infinite

• Note that impulse function is not a true function – it is not defined for all values

– It is a generalized function

0

0 to

(t)

(t-to)

= (0)Mathematical definition

Mathematical definition

Unit Impulse Function (t)

• Also note that

• Also

1. Scaling– K(t) Area (or weight) under = K

2. Multiplication– X(t) (t) X(0) (t) = Area (or weight) under

3. Time Shift – X(t) (t-to) X(to) (t-to)

• Example: Draw 3x(t-1) (t-3/2) where x(t)=sin(t)1.Using x(t) (t-to) X(to) (t-to) ; 2.3x(3/2-1) (t-3/2)=3sin(1/2) (t-3/2)

Unit Impulse Properties

Unit Impulse Properties

• Integration of a test function

– Example

• Other properties:

Make sure you can understand why!

Unit Impulse Properties

• Example: Verify

• Evaluate the following

Schaum’s p38

1

1

2 ?)()13( tt

Schaum’s p40

Remember:

• A system is an operation for which cause-and-effect relationship exists– Can be described by block diagrams – Denoted using transformation T[.]

• System behavior described by mathematical model

Continuous-Time Systems

T [.]X(t) y(t)

System - Example

• Consider an RL series circuit– Using a first order equation:

dt

tdiLRtitVVtV

dt

tdiLtV

LR

L

)()()()(

)()(

LV(t)

R

Interconnected Systems

notes

• Parallel

• Serial (cascaded)

• Feedback

LV(t)

R

L

R

Interconnected System Example• Consider the following systems with 4 subsystem• Each subsystem transforms it input signal• The result will be:

– y3(t)=y1(t)+y2(t)=T1[x(t)]+T2[x(t)]– y4(t)=T3[y3(t)]= T3(T1[x(t)]+T2[x(t)])– y(t)= y4(t)* y5(t)= T3(T1[x(t)]+T2[x(t)])* T4[x(t)]

Feedback System

• Used in automatic control

• Example: The following system has 3 subsystems. Express the equation denoting interconnection for this system - (mathematical model will depend on each individual subsystem)

– e(t)=x(t)-y3(t)= x(t)-T3[y(t)]=– y(t)= T2[m(t)]=T2(T1[e(t)]) y(t)=T2(T1[x(t)-y3(t)])= T2(T1( [x(t)] - T3[y(t)] ) ) =– =T2(T1([x(t)] –T3[y(t)]))

Find this first Then,

calculate this

System Properties

Continuous-Time Systems - Properties

• Systems with Memory:

• Examples – Memoryless/Memory?

Has memory if output depends on inputs other than the one defined at current time

Continuous-Time Systems - Properties

Inverse of a System (invertibility)

Examples

Noninvertible Systems

Each distinct input distinct output

Thermostat Example!(notes)

Continuous-Time Systems - Invertible

• If a system is invertible it has an Inverse System

• Example: y(t)=2x(t)– System is invertible: For any x(t) we get a distinct output y(t)– Thus, the system must have an Inverse

• x(t)=1/2 y(t)=z(t)

y(t)System

Inverse System

x(t) x(t)

y(t)=2x(t)System(multiplier)

Inverse System(divider)

x(t) x(t)

If the system is not invertible it does not have an INVERSE!

Continuous-Time Systems - CausalityCausality (non-anticipatory system)- A System can be causal with non-causal components!

Examples

??

Remember: Reverse is not TRUE!

?? What it t<0?

Depends on cause-and-effect no future dependency

Continuous-Time Systems - Stability

• Many different definitions • Bounded-input-bounded-output

• Example: – An ideal amplifier y(t) = 10 x(t) B2=10 B1– Square system: y(t)=x^2 B2=B1^2

• A system can be unstable or marginally stable• More examples

Continuous-Time Systems – Time Invariance

Time-shift in input results in time-shift in output system always acts the same way (Fix System)

To test: y(t)|t-to y(t)|x(t-to)Example: y(t) = e^x(t):y(t)|t=t-to e^x(t-to)y(t)|x(t-to) e^x(t-to) Time Invariance

Testing for Time Invariance

What if the system is time reversal? (next slide)

Continuous-Time Systems – Time Invariance

Example of a system:U(1-t)

Draw the First output!

Draw the Second output!

Are they the same?

Continuous-Time Systems – Time Invariance

Example of a system:U(1-t)

Time reversal operation is NOT time invariant!

Pay attention!Due to time - reversal

Continuous-Time Systems – Time Invariance

• Examples: – Do these yourself!

notes

Continuous-Time Systems – Linearity

• A linear system must satisfy superposition condition (additive and homogeneity hoh-muh-juh-nee-i-tee)

• Note: for a linear system a zero input always generates an output zero

• Example notes

We need to prove: aT[x1]+bT[x2]=T[a.x1+b.x2]

More…

• Example

• Summary

notes

Practice Problems

• Schaum’s Outlines:– 1.1(p19), 1.5(p23). 1.6-1.7(p24)1.16(p30),

1.22(p35)– There will be more….(check back)

• Textbook: Chapter 2- Solved Problems– 1, 3, 5-10, 13, 14, 16, 17, 21, 22, 24, 25-30,

33-35, 48, 51, 53-61