1. 1.1 Continuous-time and discrete-time signals
1 Signal and System
1. Signals and Systems
1.1 Continuous-time and discrete-time signals
1.1.1 Examples and Mathematical Representation
A. Examples
(1) A simple RC circuit
Source voltage Vs and
Capacitor voltage Vc

2. Retarding frictional force ρV Velocity V
1 Signal and System
(2) An automobile
Force f from engine
Retarding frictional force ρV
Velocity V

3. 1 Signal and System
(3) A Speech Signal

4. 1 Signal and System
(4) A Picture

5. (5) Vertical Wind Profile
1 Signal and System
(5) Vertical Wind Profile

6. (1) Continuous-time Signal
1 Signal and System
B. Types of Signals
(1) Continuous-time Signal

7. (2) Discrete-time Signal
1 Signal and System
(2) Discrete-time Signal

8. (1) Function Representation Example: x(t) = cos0t x(t) = ej 0t
1 Signal and System
C. Representation
(1) Function Representation
Example: x(t) = cos0t
x(t) = ej 0t
(2) Graphical Representation
Example: ( See page before )

9. 1.1.2 Signal Energy and Power
1 Signal and System
1.1.2 Signal Energy and Power
A. Energy (Continuous-time)
Instantaneous power:
Let R=1Ω, so
p(t)=i2(t)=v2(t)=x2(t)

10. Energy over t1 t  t2: Total Energy: Average Power:
1 Signal and System
Energy over t1 t  t2:
Total Energy:
Average Power:

11. B. Energy (Discrete-time)
1 Signal and System
B. Energy (Discrete-time)
Instantaneous power:
Energy over n1 n  n2:
Total Energy :
Average Power:

12. C. Finite Energy and Finite Power Signal
1 Signal and System
C. Finite Energy and Finite Power Signal
Finite Energy Signal :
( P  0 )
Finite Power Signal :
( E   )

13. 1.2 Transformations of the Independent Variable
1 Signal and System
1.2 Transformations of the Independent Variable
1.2.1 Examples of Transformations
A. Time Shift
Right shift : x(t-t0)
x[n-n0] (Delay)
Left shift : x(t+t0)
x[n+n0] (Advance)

14. 1 Signal and System
Examples

15. x(-t) or x[-n] : Reflection of x(t) or x[n]
1 Signal and System
B. Time Reversal
x(-t) or x[-n] : Reflection of x(t) or x[n]

16. C. Time Scaling x(at) ( a>0 ) Stretch if a<1
1 Signal and System
C. Time Scaling
x(at) ( a>0 )
Stretch if a<1
Compressed if a>1
Example 1.1

17. Definition: There is a posotive value of T which :
1 Signal and System
1.2.2 Periodic Signals
Definition: There is a posotive value of T which :
x(t)=x(t+T) , for all t
x(t) is periodic with period T .
T  Fundamental Period
For Discrete-time period signal:
x[n]=x[n+N] for all n
N  Fundamental Period

18. 1 Signal and System
Examples of periodic signal

19. Even signal: x(-t) = x(t) or x[-n]= x[n]
1 Signal and System
1.2.3 Even and Odd Signals
Even signal: x(-t) = x(t) or x[-n]= x[n]
Odd signal : x(-t)= -x(t) or x[-n]= -x[n]
Even-Odd Decomposition:
or:

20. 1 Signal and System
Examples

21. 1.3 Exponential and Sinusoidal signal
1 Signal and System
1.3 Exponential and Sinusoidal signal
1.3.1 Continuous-time Complex Exponential
and Sinusoidal Signals
A. Real Exponential Signals
x(t)= C eat ( C, a are real value)

22. B. Periodic Complex Exponential and Sinusoidal Signals
1 Signal and System
B. Periodic Complex Exponential and
Sinusoidal Signals
(1) x(t) = e j0t
(2) x(t) = Acos(0t+)
(3) x(t) = e jk0t
All x(t) satisfy for x(t) = x(t+T) , and T=2/ 0
So x(t) is periodic.

23. and cos0t = (e j0t + e -j0t ) / 2 sin0t = (e j0t - e -j0t ) / 2
1 Signal and System
Euler’s Relation:
e j0t = cos0t + sin 0t
and cos0t = (e j0t + e -j0t ) / 2
sin0t = (e j0t - e -j0t ) / 2
We also have

24. C. General Complex Exponential Signals x(t) = C e jat ,
1 Signal and System
C. General Complex Exponential Signals
x(t) = C e jat ,
in which C = |C| ej , a = r + j 0 So
x(t) = |C| ej eat ej0t
= |C| eat ej(0t+  )
= |C| eat cos(0t+ ) + j |C| eat sin(0t+ )

25. 1 Signal and System
Signal waves

26. 1.3.2 Discrete-time Complex Exponential and Sinusoidal Signals
1 Signal and System
1.3.2 Discrete-time Complex Exponential
and Sinusoidal Signals
Complex Exponential Signal (sequence) :
x[n] = C n or x[n] = C en

27. A. Real Exponential Signal
1 Signal and System
A. Real Exponential Signal
Real Exponential Signal
x[n] = C n
(a) >1
(b) 0<<1
(c) -1<<0
(d) <-1

28. B. Sinusoidal Signals = cos 0n + jsin0n 1 Signal and System
Complex exponential:
x[n] = e j0n
= cos 0n + jsin0n
Sinusoidal signal:
x[n] = cos(0n+)

29. C. General Complex Exponential Signals
1 Signal and System
C. General Complex Exponential Signals
Complex Exponential Signal:
x[n] = C n
in which C = |C| ej ,  = ||ej0 (polar form)
then
x[n]=|C| ||ncos(0n + )+j|C| ||nsin(0n + )

30. 1 Signal and System
Real or Imaginary of Signal

31. 1.3.3 Periodicity Properties of Discrete-time Complex Exponentials
1 Signal and System
1.3.3 Periodicity Properties of Discrete-time
Complex Exponentials
Continuous-time: e j0t , T=2/0
Discrete-time: e j0n , N=?
Calculate period:
By definition: e j0n = e j0(n+N)
thus e j0N = 1 or 0N = 2 m
So N = 2m/0
Condition of periodicity: 2/0 is rational

32. 1 Signal and System
Periodicity Properties

33. 1.4 The Unit Impulse and Unit Step Functions
1 Signal and System
1.4 The Unit Impulse and Unit Step Functions
1.4.1 The Discrete-time Unit Impulse and Unit
Step Sequences
(1) Unit Sample(Impulse):

34. (2) Relation Between Unit Sample and Unit Step
1 Signal and System
Unit Step Function:
(2) Relation Between Unit Sample and Unit Step or

35. (3) Sampling Property of Unit Sample
1 Signal and System
(3) Sampling Property of Unit Sample

36. 1 Signal and System
Illustration of Sampling

37. 1.4.2 The Continuous-time Unit Step and Unit Impulse Functions
1 Signal and System
1.4.2 The Continuous-time Unit Step and Unit
Impulse Functions
(1) Unit Step Function:

38. Unit Impulse Function:
1 Signal and System
Unit Impulse Function:

39. (2) Relation Between Unit Impulse and Unit Step
1 Signal and System
(2) Relation Between Unit Impulse and Unit Step

40. (3) Sampling Property of (t)
1 Signal and System
(3) Sampling Property of (t)
Example 1.7

41. 1.5 Continuous-time and Discrete-time System
1 Signal and System
1.5 Continuous-time and Discrete-time System
Definition:
(1) Interconnection of Component,device,
subsystem…. (Broadest sense)
(2) A process in which signals can be
transformed. (Narrow sense)
Representation of System:
(1) Relation by the notation

42. Continuous-time system
1 Signal and System
(2) Pictorial Representation
Continuous-time system
x(t)
y(t)
Discrete-time system
x[n]
y[n]

43. 1.5.1 Simple Example of systems
1 Signal and System
1.5.1 Simple Example of systems
Example 1.8:
RC Circuit in Figure 1.1 : Vc(t)  Vs(t)
RC Circuit (system)
vs(t)
vc(t)

44. Balance in bank (system)
1 Signal and System
Example 1.10:
Balance in a bank account from month
to month:
balance y[n]
net deposit --- x[n]
interest %
so y[n]=y[n-1]+1%y[n-1]+x[n]
or y[n]-1.01y[n-1]=x[n]
Balance in bank (system)
x[n]
y[n]

45. 1.5.2 Interconnections of System
1 Signal and System
1.5.2 Interconnections of System
(1) Series(cascade) interconnection

46. (2) Parallel interconnection
1 Signal and System
(2) Parallel interconnection
Series-Parallel interconnection

47. (3) Feed-back interconnection
1 Signal and System
(3) Feed-back interconnection

48. 1 Signal and System
Example of Feed-back interconnection

49. 1.6 Basic System Properties
1 Signal and System
1.6 Basic System Properties
1.6.1 Systems with and without Memory
Memoryless system: It’s output is dependent only on the input at the same time.
Features: No capacitor, no conductor, no delayer.
Examples of memoryless system:
y(t) = C x(t) or y[n] = C x[n]
Examples of memory system:
or y[n]-0.5y[n-1]=2x[n]

50. 1.6.1 Invertibility and Inverse Systems
1 Signal and System
1.6.1 Invertibility and Inverse Systems
Definition:
(1) If system is invertibility,then an inverse system
exists.
(2) An inverse system cascaded with the original
system,yields an output equal to the input.

51. 1 Signal and System

52. Memoryless systems are causal.
1 Signal and System
1.6.3 Causality
Definition:
A system is causal If the output at any time depends only on values of the input at the present time and in the past.
For causal system, if x(t)=0 for t<t0, there must be y(t)=0 for t<t0.
( nonanticipative )
Memoryless systems are causal.

53. 1 Signal and System
x(t)
y(t) t1 t2

54. Small inputs lead to responses that don not diverge.
1 Signal and System
1.6.4 Stability
Definition:
Small inputs lead to responses that don not diverge.
Finite input lead to finite output:
if |x(t)|<M, then |y(t)|<N .
Examples:
Stable pendulum
Motion of automobile

55. 1 Signal and System
Example 1.13

56. Characteristics of the system are fixed over time.
1 Signal and System
1.6.5 Time Invariance
Definition:
Characteristics of the system are fixed over time.
Time invariant system:
If x(t)  y(t), then x(t-t0)  y(t-t0) .
Example

57. 1 Signal and System
x(t)
y(t)
x(t-t0)
y(t-t0)

58. The system possesses the important property of superposition:
1 Signal and System
1.6.6 Linearity
Definition:
The system possesses the important property of superposition:
(1) Additivity property:
The response to x1(t)+x2(t) is y1(t)+y2(t) .
(2) Scaling or homogeneity property:
The response to ax1(t) is ay1(t) .
(where a is any complex constant, a0 .)

59. L Represented in block-diagram: x1(t) x2(t) y1(t) y2(t) a x1(t)
1 Signal and System
Represented in block-diagram: L
x1(t)
x2(t)
y1(t)
y2(t)
a x1(t)
x1(t) +x2(t)
ax1(t) +bx2(t)
a y1(t)
y1(t) +y2(t)
ay1(t) +by2(t)
Example

60. LTI LTI System Linear and Time-invariant system x(t) y(t) x(t-t0)
1 Signal and System
LTI System
Linear and Time-invariant system
LTI
x(t)
y(t)
x(t-t0)
ax(t) +bx(t-t0)
y(t-t0)
ay(t) +by(t-t0)
Problems: