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Chapter 2

Chapter 2. Updated 6/6/2014. Time reversal:. X(t). Y=X(-t). Time Reversal. Time scaling. Example: Given x ( t ), find y ( t ) = x (2 t ). This SPEEDS UP x ( t ) (the graph is shrinking ) The period decreases!. X(t). Y=X(at). Time Scaling. What happens to the period T ? .

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Chapter 2

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  1. Chapter 2 Updated 6/6/2014

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

  3. 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 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

  4. Example of Time Scaling (tt/a) http://tintoretto.ucsd.edu/jorge/teaching/mae143a/lectures/1analogsignals.pdf

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

  6. 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

  7. 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

  8. 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?

  9. Note: Unit Step Functiona discontinuous continuous-time signal

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

  11. Time Shifting t=0 • 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)

  12. Summary 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-2)] Hence, reverse the signal in time. Then shift to the right of t=0 by two units! Or rewrite as: X[-(t+1)] Hence, reverse the signal in time. Then shift to the left of t=0 by one unit! See Notes This is really: X(-(t+1))

  13. 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

  14. 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

  15. Amplitude Operations 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!

  16. 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

  17. 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!

  18. Proof Examples 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!

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

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

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

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

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

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

  25. Periodic and Aperiodic Signals • Given x(t) is a continuous-time signal • X (t) is periodic iffX(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)

  26. 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)?

  27. 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)

  28. 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: http://www.mathsisfun.com/irrational-numbers.html

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

  30. 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!

  31. 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!

  32. 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 Find even and odd parts of v(t).

  33. 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

  34. 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

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

  36. 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

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

  38. Unit Step Function Applications: Example

  39. 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 Using Mathematica - click here: 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

  40. 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) 0 d(t-to) 0 to

  41. Unit Impulse Function d(t) • Also note that • Also

  42. 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)

  43. Unit Impulse Properties • Integration of a test function • Example • Other properties: Make sure you can understand why!

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

  45. 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 [.]

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

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

  48. 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)]

  49. 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

  50. System Properties

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