# Chapter 2 Updated 4/11/2017.

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Chapter 2 Updated 4/11/2017

Time reversal: X(t) Y=X(-t) Time Reversal
Example: Playing a tape recorder backward Beatle Revolution 9 is said be played backward!

Time scaling Example: Given x(t), find y(t) = x(2t). This SPEEDS UP x(t) (the graph is shrinking) The period decreases! X(t) Y=X(at) Time Scaling Example: fast forwarding / slow playing What happens to the period T? The period of x(t) is 2 and the period of y(t) is 1, a>1  Speeds up  Smaller period  Graph shrinks! a<1  slows down  Larger period  Graph expands

Example of Time Scaling (tt/a)

Time scaling Given y(t), find w(t) = y(3t) v(t) = y(t/3).

Time Shift: y(t)=g(t-to)
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 X(t) Y=X(t-to) Time Shifting Delay

Time Shift: y(t)=x(t-to)
Time Shifting X(t) Y=X(t-to) 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 Example Delay – at the air port / radio stations / LRC circuits Time advancing is not physically realizble! Very important: x(-t) is not the same is x(-(t-2))***********

Time Shifting Example Given x(t) = u(t+2) -u(t-2), find 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 Function a 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) 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 See This is really: Notes X(-(t+1)) Delayed/ Moved right
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 Or rewrite as: X[-(t+1)] Hence, reverse the signal in time. Then shift to the left of t=0 by one unit! Or rewrite as: X[-(t-2)] Hence, reverse the signal in time. Then shift to the right of t=0 by two units! See Notes This is really: X(-(t+1))

Amplitude Operations In general: y(t)=Ax(t)+B B>0  Shift up
B<0  Shift down |A|>1 Gain |A|<1 Attenuation A>0NO reversal A<0 reversal Reversal Scaling Scaling

Signals can be added or multiplied
Amplitude Operations Given x2(t), find 1 - x2(t). Ans. Remember: This is y(t) =1 Multiplication of two signals:x2(t)u(t) Ans. Step unit function Signals can be added or multiplied

Amplitude Operations Given x2(t), find 1 - x2(t).
Note: You can also think of it as X2(t) being amplitude revered and then shifted by 1. Given x2(t), find 1 - x2(t). Remember: This is y(t) =1 Multiplication of two signals:x2(t)u(t) Step unit function Signals can be added or multiplied  e.g., we can filter parts of a signal!

Signal Characteristics
Even and odd signals X(t) = Xe(t) + Xo(t) X(-t) = X(t)  Even X(-t) = -X(t)  Odd Properties Represent Xo(t) in terms of X(t) only! Represent Xe(t) in terms of X(t) only! Xe * Ye = Ze Xo * Yo = Ze Xe * Yo = Zo Xe + Ye = Ze Xo + Yo = Zo Xe + Yo = Ze + Zo

Signal Characteristics
Even and odd signals X(t) = Xe(t) + Xo(t) X(-t) = X(t)  Even X(-t) = -X(t)  Odd Properties Xe * Ye = Ze Xo * Yo = Ze Xe * Yo = Zo Xe + Ye = Ze Xo + Yo = Zo Xe + Yo = Ze + Zo Know These!

Proof Examples Prove that product of two even signals is even.
Change t -t 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!

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

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

Example Given x(t) find xe(t) and xo(t) 4___ 5 2___ 2___ 5 5

Example Given x(t) find xe(t) and xo(t) 4___ 5 2___ 2___ -5 5 -5 5

Example Given x(t) find xe(t) and xo(t) 4___ 4e-0.5t 2___ 2___ 2___
-2___

Example Given x(t) find xe(t) and xo(t) 4___ 4e-0.5t 2___ 2___ 2___
-2___

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(wt) periodic? X(t+nT) = A cos(w(t+Tn)) = A cos(wt+w2np)= A cos(wt) Note: f0=1/T0; wo=2p/To <= Angular freq. T0 is fundamental period; T0 is the minimum value of T that satisfies X(t) = x(t+T)

Periodic signals Examples:
=>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(tp/3)+sin(tp/4) T1=(2p)/(p/3)=6; T2 =(2p)/(p/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
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. X(t) = x1(t) + X2(t) X(t+nT) = x1(t+m1T1) + X2(t+m2T2) m1T1=m2T2 = To=Fundamental period Example: cos(tp/3)+sin(tp/4) T1=(2p)/(p/3)=6; T2 =(2p)/(p/4)=8; T1/T2=6/8 = ¾ m1T1=m2T2  T1/T2= m2/m16/8=3/46.4 = 3.8  24 = To Read about Irrational Numbers:

Product of periodic Signals
X(t) = xa(t) * Xb(t) = = 2sin[t(7p/24)]* cos[t(p/24)]; find the period of x(t) We know: = 2sin[t(7p/12)/2]* cos[t(p/12)/2]; Using Trig. Itentity: x(t) = sin(tp/3)+sin(tp/4) Thus, To=24 , as before!

Sum of periodic Signals – may not always be periodic!
T1=(2p)/(1)= 2p; T2 =(2p)/(sqrt(2)); T1/T2= sqrt(2); Note: T1/T2 = sqrt(2) is an irrational number X(t) is aperiodic 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 = 3.5/2p  T1 = 2p /3.5 X2(t) = sin(2t)  f2 = 2/2p  T2 = 2p /2 X3(t) = 2cos(t7/6)  f3 = (7/6)/2p  T3 = 2p /(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(2p /3.5)=12(T2)=12p T1 T2 T3 2p/3.5 2p/2 12p/7 m1 m2 m3 7.3=21 2.6=12 7.1=7 m1xT1 m2xT2 m3xT3 12p Find even and odd parts of v(t).

Important Engineering Signals
Euler’s Formula (polar form or complex exponential form) Remember: ejF = 1|_F and arg [ ejF ] = F (can you prove these?) Unit Step Function (Singularity Function) Use Unit Step Function to express a block function (window) notes What is sin(A+B)? Or cos(A+B)? notes notes -T/t T/t

Important Engineering Signals
Euler’s Formula Unit Step Function (Singularity Function) Can you draw x(t) = cos(t)[u(t) – u(t-2p)]? Use Unit Step Function to express a block function (window) notes notes Next -T/2 T/2

Unit Step Function Properties
Examples: Note: U(-t+3)=1-u(t-3)

Unit Step Function Applications: Creating Block Function (window)
1 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) Note: Period is T; & symmetric -T/2 T/2 1 -T/2 T/2 1 -T/2 T/2 -T/2 T/2

Unit Step Function Applications: Creating Unit Ramp Function
Unit ramp function can be achieved by: 1 notes to to+1 Non-zero only for t>t0 Example: Using Mathematica: (t-2)*unitstep[t-2] – click here:

Unit Step Function Applications: Example
Use this on Mathematica: 3*UnitStep(t)+t*UnitStep(t)-(t-1)*UnitStep(t-1)-5*UnitStep(t-2)

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*(t+2)*UnitStep(t+2)-6*(t+1)*UnitStep(t+1)+3*(t-1)*UnitStep(t-1)+3*UnitStep(t-3) Using Mathematica - click here:

Unit Impulse Function d(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 = f(0) Mathematical definition Mathematical definition d(t) d(t-to) to

Unit Impulse Function d(t)
Also note that Also

Unit Impulse Properties
Scaling Kd(t) Area (or weight) under = K Multiplication X(t) d(t) X(0) d(t) = Area (or weight) under Time Shift X(t) d(t-to) X(to) d(t-to) Example: Draw 3x(t-1) d(t-3/2) where x(t)=sin(t) Using x(t) d(t-to) X(to) d(t-to) ; 3x(3/2-1) d(t-3/2)=3sin(1/2) d(t-3/2)

Unit Impulse Properties
Integration of a test function Example Other properties: Sin^2(a/b) Make sure you can understand why!

Unit Impulse Properties
Example: Verify Evaluate the following Schaum’s p38 Schaum’s p40 Remember:

Continuous-Time Systems
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 X(t) y(t) T [.]

System - Example Consider an RL series circuit
Using a first order equation: R L V(t)

Interconnected Systems
Parallel Serial (cascaded) Feedback notes R R L L V(t)

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 Thermostat Example! (notes) Each distinct input  distinct output

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) System Inverse System x(t) y(t) x(t) System (multiplier) Inverse System (divider) x(t) y(t)=2x(t) x(t) If the system is not invertible it does not have an INVERSE!

Continuous-Time Systems - Causality
Causality (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
Testing for Time Invariance 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 What if the system is time reversal? (next slide) Time-shift in input results in time-shift in output  system always acts the same way (Fix System)

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) Pay attention! Due to time - reversal Time reversal operation is NOT time invariant!

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

More… notes Example Summary

Practice Problems Schaum’s Outlines:
1.1(p19), 1.5(p23) (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