1. Chapter 1. -Signals and Systems
Media Network Lab.

2. Signals and Systems Why signals and systems? 1. 1 Signal?
Signal processing : from signal to signal
(communication, control, multimedia)
Pattern recognition : from signal to information
Synthesis : from information to signal
1. 1 Signal?
f(t): 1D
f(x, y): 2D
f(x, y, z) or f(x, y, t): 3D
4D?
Signals and systems
Cont.

3. 1. 4 Classification of Signals
Continuous-time vs discrete-time
Even vs odd signal
Conjugate symmetric signal
Periodic signal
Deterministic and random signal
Power/energy signal
Communication system
Cont.

4. 1. 4 Classification of Signals
a. Continuous-time -> sampling -> discrete-time
b. Even and odd signals
Odd signal (1.3)
Example 1.1 Even and Odd Signal
Fig 1.12 (a) Continuous-time signal x(t).
(b) discrete-time signal x[n].
->
(1.5절 먼저하고 올 것)
Cont.

5. 1.4.b Odd signals and even signals
Theorem: All signals can be represented by sum of odd signal and even signal.
Problem 1.1 Odd signal + even signal
(a)
(b)
(c)
(d)
Cont.

6. 1.4.c symmetric 1.4.d periodic c. Conjugate symmetric signal
(1. 6)
d. Periodic vs aperiodic signal
Definition : A signal x(t) is periodic if and only if there exists
non-zero T which satisfies
Frequency: (1. 8)
angular frequency: (1.9)
Cont.

7. 1.4.d Periodic signals Periodic discrete signal (1. 11) Cont.
Figure1.14 (a) Square wave with amplitude A = 1 and
period T = 0.2s.
(b) Rectangular pulse of amplitude A and
duration T1.
Figure 1.16 Discrete-time square wave alternative between –1 and +1.
Figure 1.17 Aperiodic discrete-time signal consisting of three nonzero samples.
Cont.

8. 1.4.d periodic 1.4.e Deterministic
Problem 1.5 periodic or aperiodic?
(a)
(b)
(c)
(d)
(e)
(f)
(g)
How about
e. Deterministic and random signals random signals
=> 확률 및 랜덤변수
Cont.

9. 1.4.f Energy signal and power signals
Power [Watt or J/sec] 가 유한하면, power signal.
(ex. 집에 들어오는 전력)
Energy [J or W∙sec]가 유한하면, energy signal.
(ex. ‘아’하는 소리)
(1. 12) (1. 13)
normalized power (R=1Ω) -> (1.14)
 (1. 15) (1. 16)
(1. 18) (1. 19)
Cont.

10. 1.4.f Energy signal and power signals
Problem 1.9 Power or energy signal
(a) (b)
(c) 

11. 1.5 Basic operations 1.5.1 basic operations amplitude scaling (노래방 볼륨)
(1. 21)
addition
(1. 22), (노래방 마이크 2개)
multiplication
(modulation, sampling)
differentiation
 (1. 24)
Cont.

12. 1.5.2.a time scaling a. time scaling Prob 1.10 Let -> Cont.
Up-sampling and down-sampling (decimation)
Cont.

13. 1.5.2.b Reflection
b. reflection (reverse) : Michael Jackson, 서태지 ‘피가 모자라’
Example 1.13
Problem The discrete-time signal
-> Find
Cont.

14. 1.5.2.c Time shifting c. Time shifting
Problem The discrete-time signal
-> Find
Cont.

15. 1.5.3 Shifting and Time Scaling
Example 1.5 Precedence Rule
incorrect!
Cont.

16. 1.5.3 Shifting and Time Scaling
Problem 1. 14
(a) (b)
(c) (d)
(e) (f)
Answer:
Cont.

17. 1.5.3 Shifting and Time Scaling
Ex 1.16 Precedence Rule for Discrete-Time Signal -> Find Problem 1.15 consider a discrete-time signal Answer:
Cont.

18. 1.6 Elementary Signals 1.6.1 Exponential signals Exponential
Sinusoidal
Complex exponential
Step function and impulse => ramp signals
Exponential signals
(1. 31)  (1. 32)
Figure 1.28 (a) a<0, B>0. (b) a>0, B>0
Cont.

19. 1.6.2 Sinusoidal Signals 1.6.2 Sinusoidal Signals (1. 35)
Periodic with period of
Periodic? ->
Figure (a) a<0, B>0. (b) a>0, B>0
Cont.

20. 1.6.2 Sinusoidal Signals  Periodic if  Angular frequency Cont.
Figure 1.31
(a) A cos(ωt + Φ)
with phase Φ = +/6 .
(b) A sin (ωt + Φ) with phase Φ = +/6
Figure 1.33 Discrete-time sinusoidal signal.
Cont.

21. 1.6.2 Sinusoidal Signals Example 1.17 Discrete-Time sinusoidal signals
Problem The fundamental period?
Cont.

22. 1.6.3 Complex Exponential signals
Problem 1.17 Consider the following sinusoidal signals:
(a) (b)
(c) (d)
1.6.3 Complex Exponential signals
->
->
->
Cont.

23. 1.6.4 Exponential Damped Sinusoidal signals
(1. 48) Problem 1.20 Complex-valued exponential signal Evaluate and for the following cases: (a) real, (b) imaginary, (c) complex,
Figure 1.35 Exponentially damped sinusoidal signal Ae-at sin(ωt), with A = 60 and α = 6.
Cont.

24. Step Function
Example 1.8 Rectangular Pulse Problem 1.22 Represent x[n] by using step functions
-> 

25. 1.6.6 Impulse Function Continuous time impulse functions (1. 61)
(1. 68)

26. 1.6.6 Impulse Function
Discrete time impulse function
Cont.

27. Continuous time: δ(t)  integration  u(t)
1.6.8 Ramp Function
Continuous time: δ(t)  integration  u(t)
δ(t)  differentiation  u(t)
Discrete time: δ[n]  summation  u[n]
δ[n]  difference  u[n]
Cont.

28. Signals and Systems
1.7 Systems viewed as interconnections of operations
Example 1.12 Moving average system
1.8 Properties of Systems (p. 55)
stable
memory
causal
inverse
time invariance
linear
Cont.

29. BIBO (bounded input, bounded output)
1.8.1 STABILITY  BIBO
BIBO (bounded input, bounded output)
For BI for all t
if BO for all t, then Stable!
Ex Stable?
Sol) Let
 stable

30. 1.8.1 STABILITY  BIBO
Fig 1.52 Collapse of the Tacoma Narrows suspension bridge on November 7, 1940.
Photograph showing the twisting motion of the bridge’s center span just before failure.
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. (Courtesy of the Smithsonian Institution.)
Cont.

31. 1.8.1 STABILITY  BIBO
Ex , where r>1. Stable? Sol) for all n Feedback (recursive) system 
(1. 117)
The feedback coefficient ρ determines the stability of the filter. Stable if |ρ|<1
…………………………

32. 1.8.2 Memory or Memory-less 1.8.3 Causal
1.8.3 CAUSALITY
출력이 현재 또는 과거의 입력에 의해 결정된다.
 no memory

33. 1.8.4 INVERTIBILITY (P.59)
Ex) 2 2
Cont.

34. 1.8.4 INVERTIBILITY (P.59)
Example 1.16 Invertible? Multipath Communication Channels
Fig. 1.67
Example of multiple paths in a wireless communication environment.
time-invariant.
Cont.

35. 1.8.5 TIME INVARIANCE (P.60) 언제나 같은 대답 (always the same response)
입력을 time-shift하면 출력도 time-shift
(Time shifted input -> time shifted output)
Problem 1.33
Let z[n]=x[n-2],
then not T. I.
Cont.

36. 1.8.6 LINEARITY (P.63)
The same amplitude scaling, addition in both input and output signals.
Definition) For and , linear if
where
Superposition of the input -> superposition of output
Cont.

37. 1.8.6 LINEARITY (P.63)
Example 1.19 Linear Discrete-Time System (P.64) linear? Solution) Let where
 Linear!
Yes!!
Cont.

38. 1.8.6 LINEARITY (P.63) Example 1.20 Linear? (p. 65) Solution) Let
Problem linear?  Linear!!
Problem Is it possible for a linear system to be non-causal?
Problem Linear?
No But,
Ex) y[n]=x[n]+1 Linear? -> Non-linear!
non-linear.
Cont.

39. Linear Time-Invariant Systems
LTI
System