1. SIGNALS & SYSTEMS

2. Contents of the Lecture
Signal & System?
Time-domain representation of LTI system
Fourier transform and its application
Z transform and its application
Digital Filter & Its Application

3. Can you believe it?

4. Examples of System

5. 1. INTRODUCTION

6. What is a Signal?
(DEF) Signal : A signal is formally defined as a function of one or more variables, which conveys information on the nature of physical phenomenon.
나는 무엇을 생각할까요?

7. What is a System?
(DEF) System : A system is formally defined as an entity that manipulates one or more signals to accomplish a function, thereby yielding new signals.
system
output signal
input signal

8. Some Interesting Systems
Communication system
Control systems
Remote sensing system
Biomedical system(biomedical signal processing)
Auditory system

9. Some Interesting Systems
Communication system

10. Some Interesting Systems
Control systems

11. Some Interesting Systems
Papero

12. Some Interesting Systems
Remote sensing system
Perspectival view of Mount Shasta (California), derived from a pair of stereo radar images acquired from orbit with the shuttle Imaging Radar (SIR-B). (Courtesy of Jet Propulsion Laboratory.)

13. Some Interesting Systems
Biomedical system(biomedical signal processing)

14. Some Interesting Systems
Auditory system

15. Classification of Signals
Continuous and discrete-time signals
Continuous and discrete-valued signals
Even and odd signals
Periodic signals, non-periodic signals
Deterministic signals, random signals
Causal and anticausal signals
Right-handed and left-handed signals
Finite and infinite length

16. Continuous and discrete-time signals
Continuous signal
- It is defined for all time t : x(t)
Discrete-time signal
- It is defined only at discrete instants of time : x[n]=x(nT)

17. Continuous and Discrete valued singals
CV corresponds to a continuous y-axis
DV corresponds to a discrete y-axis
Digital signal

18. Even and odd signals Even signals : x(-t)=x(t)
Odd signals : x(-t)=-x(t)
Even and odd signal decomposition
xe(t)= 1/2·(x(t)+x(-t)) xo(t)= 1/2·(x(t)-x(-t))

20. Periodic signals, non-periodic signals
- A function that satisfies the condition
x(t)=x(t+T) for all t
- Fundamental frequency : f=1/T
- Angular frequency : = 2/T
Non-periodic signals

21. Deterministic signals, random signals
-There is no uncertainty with respect to its value at any time. (ex) sin(3t)
Random signals
- There is uncertainty before its actual occurrence.

22. Causal and anticausal Signals
Causal signals : zero for all negative time
Anticausal signals : zero for all positive time
Noncausal : nozero values in both positive and negative time
causal signal
anticausal signal
noncausal signal

23. Right-handed and left-handed Signals
Right-handed and left handed-signal : zero between a given variable and positive or negative infinity

24. Finite and infinite length
Finite-length signal : nonzero over a finite interval tmin< t< tmax
Infinite-length singal : nonzero over all real numbers

25. Basic Operations on Signals
Operations performed on dependent signals
Operations performed on the independent signals

26. Operations performed on dependent signals
Amplitude scaling
Addition
Multiplication
Differentiation
Integration

27. Operations performed on the independent signals
Time scaling
a>1 : compressed
0<a<1 : expanded

28. Operations performed on the independent signals
Reflection

29. Operations performed on the independent signals
Time shifting
- Precedence Rule for time shifting & time scaling

30. The incorrect way of applying the precedence rule. (a) Signal x(t)
The incorrect way of applying the precedence rule. (a) Signal x(t). (b) Time-scaled signal v(t) = x(2t). (c) Signal y(t) obtained by shifting v(t) = x(2t) by 3 time units, which yields y(t) = x(2(t + 3)).
The proper order in which the operations of time scaling and time shifting (a) Rectangular pulse x(t) of amplitude 1.0 and duration 2.0, symmetric about the origin. (b) Intermediate pulse v(t), representing a time-shifted version of x(t). (c) Desired signal y(t), resulting from the compression of v(t) by a factor of 2.

31. Elementary Signals Exponential signals Sinusoidal signals
Exponentially damped sinusoidal signals

32. Elementary Signals
Step function

33. (a) Rectangular pulse x(t) of amplitude A and duration of 1 s, symmetric about the origin. (b) Representation of x(t) as the difference of two step functions of amplitude A, with one step function shifted to the left by ½ and the other shifted to the right by ½; the two shifted signals are denoted by x1(t) and x2(t), respectively. Note that x(t) = x1(t) – x2(t).

34. Elementary Signals Impulse function
(a) Evolution of a rectangular pulse of unit area into an impulse of unit strength (i.e., unit impulse). (b) Graphical symbol for unit impulse. (c) Representation of an impulse of strength a that results from allowing the duration Δ of a rectangular pulse of area a to approach zero.

35. Elementary Signals
Ramp function

36. Systems Viewed as Interconnection of Operations
output signal
input signal

37. Properties of Systems Stability Memory Invertibility Time Invariance
Linearity

38. Stability(1)
BIBO stable : A system is said to be bounded-input bounded-output stable iff every bounded input results in a bounded output.
Its Importance : the collapse of Tacoma Narrows suspension bridge, pp.45

39. Dramatic photographs showing the collapse of the Tacoma Narrows suspension bridge on November 7, (a) Photograph showing the twisting motion of the bridge’s center span just before failure. (b) A few minutes after the first piece of concrete fell, this second photograph shows a 600-ft section of the bridge breaking out of the suspension span and turning upside down as it crashed in Puget Sound, Washington. Note the car in the top right-hand corner of the photograph.

40. Stability(2) Example pp.46 - y[n]=1/3(x[n]+x[n-1]+x[n-2])
- y[n]=rnx[n], where r>1

41. Memory
Memory system : A system is said to possess memory if its output signal depends on past values of the input signal
Memoryless system
(example)

42. Memory or memoryless?

43. Causality
Causal system : A system is said to be causal if the present value of the output signal depends only on the present and/or past values of the input signal.
Non-causal system
(example)
y[n]=x[n]+1/2x[n-1]
y[n]=x[n+1]+1/2x[n-1]

44. Invertiblity(1)
Invertible system : A system is said to be invertible if the input of the system can be recovered from the system output.
H:xy, H-1:yx
H-1{y(t)}= H-1{H{x(t)}}, H-1H=I H
H-1
x(t)
y(t)

45. Invertiblity(2)
(Example) -

46. Time Invariance (1)
Time invariant system : A system is said to be time invariant if a time delay or time advance of the input signal leads to a identical time shift in the output signal.

47. Time Invariance (2) Are following two systems equivalent? St0 H x(t)
yi(t)
x(t-t0)
y0(t)

48. Time Invariance (3)
Examples

49. Linearity(1)
Linear system : A system is said to be linear if it satisfies the principle of superposition.

50. Linearity(2) a1 a2 aN .  H x1(t) x2(t) xN(t) y(t) H . a1 a2 aN 

51. Linearity(3)
Examples -
Check superposition with simple two inputs.

52. Theme Examples
Example of multiple propagation paths in a wireless communication environment.

53. Tapped-delay-line model of a linear communication channel, assumed to be time-invariant

54. Stock Price : filtering
(a) Fluctuations in the closing stock price of Intel over a three-year period.
(b) Output of a four-point moving-average system.

55. References
S. Haykin and B. Van Veen, Signals and Systems, 3rd ed. Wiley and Sons, Inc, 2003.
Kim Jin Young, “Handout”, IC & DSP Research, EE Dept. Chonnam National University, 2005.