1. Chapter 2
Updated 4/11/2017

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

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
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

4. Example of Time Scaling (tt/a)

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

6. 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

7. 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))***********

8. 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?

9. Note: Unit Step Function a 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 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 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))

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. 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

15. 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!

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 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!

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___

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

25. 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)

26. 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)?

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:

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!

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 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).

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:

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

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
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:

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)
d(t-to) 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:
Sin^2(a/b)
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

51. 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

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

53. 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!

54. 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

55. 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

56. 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)

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

58. 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!

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

60. 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

61. More…
notes
Example
Summary

62. 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