2Time reversal: X(t) Y=X(-t) Time Reversal Example: Playing a tape recorder backwardBeatle Revolution 9 is said be played backward!
3Time scalingExample: 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)TimeScalingExample: fast forwarding / slow playingWhat 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
6Time Shift: y(t)=g(t-to) Time ShiftingThe 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 rightX(t)Y=X(t-to)TimeShiftingDelay
7Time Shift: y(t)=x(t-to) Time ShiftingX(t)Y=X(t-to)TimeShiftingThe 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 rightX(t)X(t+to) // to<0 Signal Advanced Shift to the leftExample Delay – at the air port / radio stations / LRC circuitsTime advancing is not physically realizble!Very important: x(-t) is not the same is x(-(t-2))***********
8Time 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?
9Note: Unit Step Function a discontinuous continuous-time signal
11Time 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)
12Summary 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 rightAdvanced/Moved leftReversed &DelayedOr 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!SeeNotesThis is really:X(-(t+1))
13Amplitude Operations In general: y(t)=Ax(t)+B B>0 Shift up B<0 Shift down|A|>1 Gain|A|<1 AttenuationA>0NO reversalA<0 reversalReversalScalingScaling
14Signals can be added or multiplied Amplitude OperationsGiven x2(t), find 1 - x2(t).Ans.Remember:This is y(t) =1Multiplication of two signals:x2(t)u(t)Ans.Step unit functionSignals can be added or multiplied
15Amplitude 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) =1Multiplication of two signals:x2(t)u(t)Step unit functionSignals can be added or multiplied e.g., we can filter parts of a signal!
16Signal Characteristics Even and odd signalsX(t) = Xe(t) + Xo(t)X(-t) = X(t) EvenX(-t) = -X(t) OddPropertiesRepresent Xo(t) in terms of X(t) only!Represent Xe(t) in terms of X(t) only!Xe * Ye = ZeXo * Yo = ZeXe * Yo = ZoXe + Ye = ZeXo + Yo = ZoXe + Yo = Ze + Zo
17Signal Characteristics Even and odd signalsX(t) = Xe(t) + Xo(t)X(-t) = X(t) EvenX(-t) = -X(t) OddPropertiesXe * Ye = ZeXo * Yo = ZeXe * Yo = ZoXe + Ye = ZeXo + Yo = ZoXe + Yo = Ze + ZoKnowThese!
18Proof Examples Prove that product of two even signals is even. Change t -tProve 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!
19Signal Characteristics: Find uo(t) and ue(t) Remember:Given:
20Signal Characteristics Anti-symmetricacross the vertical axisSymmetric across the vertical axis
21ExampleGiven x(t) find xe(t) and xo(t)4___52___2___55
22ExampleGiven x(t) find xe(t) and xo(t)4___52___2___-55-55
23Example Given x(t) find xe(t) and xo(t) 4___ 4e-0.5t 2___ 2___ 2___ -2___
24Example Given x(t) find xe(t) and xo(t) 4___ 4e-0.5t 2___ 2___ 2___ -2___
25Periodic and Aperiodic Signals Given x(t) is a continuous-time signalX (t) is periodic iff X(t) = x(t+nT) for any T and any integer nExampleIs 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)
26Periodic signals Examples: =>Show that sin(t) is in fact a periodic signal.Use a graphShow it mathematicallyWhat is the period?Is this an even or odd signal?=>Is tesin(t) periodic?X(t) = x(t+T)?
27Sum of periodic Signals X(t) = x1(t) + X2(t)X(t+nT) = x1(t+m1T1) + X2(t+m2T2)m1T1=m2T2 = To = Fundamental periodExample:cos(tp/3)+sin(tp/4)T1=(2p)/(p/3)=6; T2 =(2p)/(p/4)=8;T1/T2=6/8 = ¾ = (rational number) = m2/m1m1T1=m2T2 Find m1 and m26.4 = 3.8 = 24 = To (n=1)
28Sum 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 periodExample:cos(tp/3)+sin(tp/4)T1=(2p)/(p/3)=6; T2 =(2p)/(p/4)=8;T1/T2=6/8 = ¾m1T1=m2T2 T1/T2= m2/m16/8=3/46.4 = 3.8 24 = ToRead about Irrational Numbers:
29Product 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!
30Sum 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 numberX(t) is aperiodicNote that T1/T2 is NOT RATIONAL (ratio of integers) In this case!
31Sum of periodic Several Signals ExampleX1(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!
32Sum of periodic Signals (cont.) X1(t) = cos(3.5t) f1 = 3.5/2p T1 = 2p /3.5X2(t) = sin(2t) f2 = 2/2p T2 = 2p /2X3(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 periodicm1T1 = m2T2 = m3T3 = To ; Hence we find ToThe question is how to choose m1, m2, m3 such that the above relationship holdsWe 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)=12pT1T2T32p/3.52p/212p/7m1m2m37.3=212.6=127.1=7m1xT1m2xT2m3xT312pFind even and odd parts of v(t).
33Important 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)notesWhat is sin(A+B)? Or cos(A+B)?notesnotes-T/tT/t
34Important Engineering Signals Euler’s FormulaUnit 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)notesnotesNext-T/2T/2
35Unit Step Function Properties Examples:Note:U(-t+3)=1-u(t-3)
36Unit Step Function Applications: Creating Block Function (window) 1rect(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/2T/21-T/2T/21-T/2T/2-T/2T/2
37Unit Step Function Applications: Creating Unit Ramp Function Unit ramp function can be achieved by:1notestoto+1Non-zero only for t>t0Example: Using Mathematica:(t-2)*unitstep[t-2] – click here:
38Unit Step Function Applications: Example Use this on Mathematica:3*UnitStep(t)+t*UnitStep(t)-(t-1)*UnitStep(t-1)-5*UnitStep(t-2)
39Unit Step Function Applications: Example Plott<-2 f(t)=0-2<t<-1 f(t)=3[t+2]-1<t<1 f(t)=-3t1<t<3 f(t)=-33<t< f(t)=03*(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:
40Unit Impulse Function d(t) Not real (does not exist in nature – similar to i=sqrt(-1)Also known as Dirac delta functionGeneralized function or testing functionThe 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 infiniteNote that impulse function is not a true function – it is not defined for all valuesIt is a generalized function= f(0)Mathematical definitionMathematical definitiond(t)d(t-to)to
42Unit Impulse Properties ScalingKd(t) Area (or weight) under = KMultiplicationX(t) d(t) X(0) d(t) = Area (or weight) underTime ShiftX(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)
43Unit Impulse Properties Integration of a test functionExampleOther properties:Sin^2(a/b)Make sure you can understand why!
44Unit Impulse Properties Example: VerifyEvaluate the followingSchaum’s p38Schaum’s p40Remember:
45Continuous-Time Systems A system is an operation for which cause-and-effect relationship existsCan be described by block diagramsDenoted using transformation T[.]System behavior described by mathematical modelX(t)y(t)T [.]
46System - Example Consider an RL series circuit Using a first order equation:RLV(t)
47Interconnected Systems ParallelSerial (cascaded)FeedbacknotesRRLLV(t)
48Interconnected System Example Consider the following systems with 4 subsystemEach subsystem transforms it input signalThe 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)]
49Feedback 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 firstThen, calculate this
51Continuous-Time Systems - Properties Systems with Memory:Examples – Memoryless/Memory?Has memory if output depends on inputs other than the one defined at current time
52Continuous-Time Systems - Properties Inverse of a System (invertibility)ExamplesNoninvertible SystemsThermostatExample!(notes)Each distinct input distinct output
53Continuous-Time Systems - Invertible If a system is invertible it has an Inverse SystemExample: y(t)=2x(t)System is invertible: For any x(t) we get a distinct output y(t)Thus, the system must have an Inversex(t)=1/2 y(t)=z(t)SystemInverseSystemx(t)y(t)x(t)System(multiplier)InverseSystem(divider)x(t)y(t)=2x(t)x(t)If the system is not invertible it does not have an INVERSE!
54Continuous-Time Systems - Causality Causality (non-anticipatory system)- A System can be causal with non-causal components!ExamplesRemember: Reverse is not TRUE!???? What it t<0?Depends on cause-and-effect no future dependency
55Continuous-Time Systems - Stability Many different definitionsBounded-input-bounded-outputExample:An ideal amplifier y(t) = 10 x(t) B2=10 B1Square system: y(t)=x^2 B2=B1^2A system can be unstable or marginally stableMore examples
56Continuous-Time Systems – Time Invariance Testing for Time InvarianceTo 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 InvarianceWhat 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)
57Continuous-Time Systems – Time Invariance Example of a system:U(1-t)Draw the First output!Draw the Second output!Are they the same?
58Continuous-Time Systems – Time Invariance Example of a system:U(1-t)Pay attention!Due to time - reversalTime reversal operation is NOT time invariant!
59Continuous-Time Systems – Time Invariance Examples:Do these yourself!notes
60Continuous-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 zeroExampleWe need to prove: aT[x1]+bT[x2]=T[a.x1+b.x2]notes