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Chapter 2 Updated 4/13/2015

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Time reversal: Time Reversal X(t)Y=X(-t)

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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, Time Scaling X(t)Y=X(at) a>1 Speeds up Smaller period Graph shrinks! a<1 slows down Larger period Graph expands a>1 Speeds up Smaller period Graph shrinks! a<1 slows down Larger period Graph expands

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Example of Time Scaling (t t/a)

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

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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 Time Shifting X(t)Y=X(t-to) Delay

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

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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?

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Note: Unit Step Function a discontinuous continuous-time signal

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Draw x(t) = u(t+1)- u(t-2) u(t+1)- u(t-2) t=0t

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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)

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Summary 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! 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)) See Notes

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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

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Amplitude Operations Given x 2 (t), find 1 - x 2 (t). Signals can be added or multiplied Multiplication of two signals:x 2 (t)u(t) Ans. Step unit function Remember: This is y(t) =1

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Amplitude Operations Given x 2 (t), find 1 - x 2 (t). Signals can be added or multiplied e.g., we can filter parts of a signal! Multiplication of two signals:x 2 (t)u(t) Step unit function Remember: This is y(t) =1 Note: You can also think of it as X 2 (t) being amplitude revered and then shifted by 1.

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Signal Characteristics Even and odd signals 1.X(t) = Xe(t) + Xo(t) 2.X(-t) = X(t) Even 3.X(-t) = -X(t) Odd Properties Xe + Ye = Ze Xo + Yo = Zo Xe + Yo = Ze + Zo Xe * Ye = Ze Xo * Yo = Ze Xe * Yo = Zo Represent Xo(t) in terms of X(t) only! Represent Xe(t) in terms of X(t) only!

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Signal Characteristics Even and odd signals 1.X(t) = Xe(t) + Xo(t) 2.X(-t) = X(t) Even 3.X(-t) = -X(t) Odd Properties Xe + Ye = Ze Xo + Yo = Zo Xe + Yo = Ze + Zo Xe * Ye = Ze Xo * Yo = Ze Xe * Yo = Zo Know These!

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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! Change t -t

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Given: Signal Characteristics: Find u o (t) and u e (t) Remember:

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

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

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

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Example Given x(t) find xe(t) and xo(t) 4___ 5 2___ 5 4e -0.5t 2___ 2e -0.5t -2___ 2___ 2e -0.5t 2e +0.5t

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Example Given x(t) find xe(t) and xo(t) 4___ 5 2___ 5 4e -0.5t 2___ 2e -0.5t -2___ 2___ 2e -0.5t 2e +0.5t -2e +0.5t

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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)

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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 te sin(t) periodic? –X(t) = x(t+T)?

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Sum of periodic Signals X(t) = x1(t) + X2(t) X(t+nT) = x1(t+m 1 T 1 ) + X2(t+m 2 T 2 ) m 1 T 1 =m 2 T 2 = 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 –m 1 T 1 =m 2 T 2 Find m1 and m2 –6.4 = 3.8 = 24 = To (n=1)

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Sum of periodic Signals X(t) = x1(t) + X2(t) X(t+nT) = x1(t+m 1 T 1 ) + X2(t+m 2 T 2 ) m 1 T 1 =m 2 T 2 = To=Fundamental period Example: –cos(t /3)+sin(t /4) –T1=(2 )/( /3)=6; T2 =(2 )/( /4)=8; –T1/T2=6/8 = ¾ –m 1 T 1 =m 2 T 2 T1/T2= m 2 /m 1 6/8=3/4 6.4 = 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:

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Product of periodic Signals X(t) = x a (t) * X b (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!

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

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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!

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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). T1T2T3 2 2 12 m1m2m3 7.3=212.6=127.1=7 m1xT1m2xT2m3xT3 12

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

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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/2T/2 notes Next

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

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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/2T/2 1 -T/2T/2 1 -T/2T/2 1 -T/2T/2 Note: Period is T; & symmetric

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

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Unit Step Function Applications: Example

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Plot t<-2 f(t)=0 -2

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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 0to (t) (t-to) = (0) Mathematical definition

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Unit Impulse Function (t) Also note that Also

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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) –Using x(t) (t-to) X(to) (t-to) ; –3x(3/2-1) (t-3/2)=3sin(1/2) (t-3/2) Unit Impulse Properties

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Integration of a test function –Example Other properties: Make sure you can understand why!

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

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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)

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

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

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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)]

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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

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System Properties

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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

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Continuous-Time Systems - Properties Inverse of a System (invertibility) Examples Noninvertible Systems Each distinct input distinct output Thermostat Example! (notes)

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

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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

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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

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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)

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Continuous-Time Systems – Time Invariance Example of a system: U(1-t) Draw the First output! Draw the Second output! Are they the same?

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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

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Continuous-Time Systems – Time Invariance Examples: –Do these yourself! notes

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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[x 1 ]+bT[x 2 ]=T[a.x 1 +b.x 2 ]

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More… Example Summary notes

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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

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