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1

Signal & Systems

Hsin-chia Lu

https://ceiba.ntu.edu.tw/941s_and_s_vlsi

2

Forward

Signal System Response

Input voltage, current

Circuit Output voltage,current

Depression of accelerator padel

AutomobileAutomobile

speed

3

Signal: characterization, enhancement, noise suppressionexamples: image restoration, speech enhancement

System: characterization, synthesisexamples: chemical processing plants

Analysis tool: Fourier analysis, Fourier series, Fourier transformSignal type:

continuous: voltage in a circuit, temperature,…discrete: closing stock market average, …

4

• Quantization of continuous signals into discrete signals.

• DSP ( digital signal process) is possible due to advance in computer and digital signal processing power.– Audio (CD, MP3) , Video (VCD, DVD, DVB) , …

• Continuous and discrete formulations are presented is parallel in this book. They are similar but not identical.

5

What is a signal?

6

7

Other Signal Types: photos

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Other Signal Types

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Continuous Time vs. Discrete Time

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Signal Energy & Power

• For a resistor with voltage v(t) and current of i(t), the power is

with total energy of

and power of

• Our definition is

Energy:

11

Signal Energy & Power (cont.)

• For , the energy is

• Power is

12

X(t)=t

X[n]=4

Finite duration signals

Examples

InfiniteInfiniteThird

FiniteInfiniteSecond

ZeroFiniteFirst

P ∞E∞Class of signals

13

Transformations of Signal (Time shift)

14

Transformations of Signal (Time shift)

15

Transformations of Signal

(Reflection)

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Transformations of Signal

(Reflection)

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Transformations of Signal (Scaling)

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Transformations of Signal

(Examples)

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Periodic Signals

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Periodic Signals

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Even and Odd Signals

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Even and Odd Decomposition

23

Real Exponential Signals

x(t) = Ceat

24

Periodic Sinusoidal Signals (Imaginary Exponential)

if

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Relationship betweenFrequency and period

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Periodic Sinusoidal Signals (cont.)

• Euler’s relation

• We obtain

• Energy per period

28

Harmonically Related Complex Exponentials

• All complex exponentials with period of T0

or

• Define

• Sets of

with period of

29

General Complex Exponential Signals

x(t) = Ceat

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Discrete-Time Complex Exponential

• Complex exponential signal or sequence

• Real complex exponential:

C and a are real

• Sinusoidal Signal

x[n] = Can

x[n] = Ceβn a = eβ

or

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32

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General Complex Exponential Signalsx(t) = Ceat

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Periodicity of Discrete-Time Complex Exponentials

• Continuous time

– Rate of oscillation increases with ω0

– Periodic for any value of ω0

• Discrete-time

Meaningful in

• For ω0 = π, the highest oscillation is

35

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Periodicity of Complex Exponentials (cont.)

• For a period of N

we need

or

• Fundamental period

37

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Harmonically Related Periodic Exponentials(A common period of N)

• Frequencies are multiple of 2π/N

• N distinct periodic exponentials

because

φ0[n] = 1, φ1[n] = ej2πn/N , φ2[n] = ej4πn/N, …, φΝ−1[n] = ej2π(Ν−1)n/N

39

Unit Impulse Function

⎩⎨⎧

=≠

=01

00][

n

nnδ

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Unit Step Function

⎩⎨⎧

≥<

=01

00][

n

nnu

]1[][][ −−= nununδ

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Unit Step Function (cont.)

∑+∞

−∞==

mmnu ][][ δ

42

Unit Step Function (cont.)

=

∞=

−=

−=

0

0

][

][][

k

k

kn

knnu

δ

δ

m to k = n-m

Some properties:][][][][

][]0[][][

000 nnnxnnnx

nxnnx

−=−=

δδδδ

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Continuous-Time Unit Step Function

⎩⎨⎧

><

=01

00)(

t

ttu

Discontinuous at t = 0

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Unit Impulse Function

dt

tdut

)()( =δ dt

tdut

)()( ∆

∆ =δ

)(lim)(0

tt ∆→∆

= δδ

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∫ ∞−=

tdtu ττδ )()(

Relation between unit step and impulse functions

46

∫∞

−=0

)()( σστδ dtu

Relation between unit step and impulse functions

∞−

−−=

=

0)()(

)()(

σσδ

ττδ

dt

dtut

47

Scaled Unit Impulse Function

∫ ∞−=

tdktku ττδ )()(

48

Some Properties of Impulse Functions

)()()()(

)()0()()(

000 tttxtttx

txttx

−=−=

δδδδ

)()0()()( txttx ∆∆ ≈ δδ

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Continuous- and discrete-time systems

)()( tytx →

][][ nynx →

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Interconnection of Systems

Series (cascade)

parallel

Series-parallel

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Feedback Connections

53

Example of Feedback Systems

54

Systems without Memory

• Memoryless, depends on input at the same time

• Identity system

55

Systems with Memory

• Examples

• The system must remember or store something

56

Invertibility and Inverse Systems

57

Noninvertible Systems

• Examples

0][ =ny

)()( 2 txty =

|)(|)( txty =

58

Causality

• A system is causal if the output at any time depends on values of the input at present and past times.

• Causal system is nonanticipative

• Examples of causal system

• Examples of noncausal system

]1[][][ +−= nxnxny )1()( += txty

59

System Stability

• Stable Systems– Small perturbation does not give large

divergence.

• Examples of unstable systems– Inverted pendulum– Population growth??– Bank account with interest??– Accumulator

• Examples of stable systems– Normal pendulum– Population growth with limited resources.– RC circuits– Practical integrator

60

Time Invariance Systems

• The characteristic of the system are fixed over time.

• Mathematically,

• Example: Time-invariance

Time-variance

)()( tytx →

][][ nynx →

)()( 00 ttyttx −→−

][][ 00 nnynnx −→−

)](sin[)( txty =

][][ nnxny =

61

62

System Linearity

• Linear system

)()( 11 tytx → )()( 22 tytx →

)()()()( 2121 tytytxtx +→+

)()( 11 taytax →

if

then

Combined conditions:

)()()()( 2121 tbytaytbxtax +→+][][][][ 2121 tbytaytbxnax +→+

Superposition properties

Called additive and homogeneity properties

63

Examples

• Linear systems

• Nonlinear systems

)()( ttxty =

3][2][ += nxny

)()( 2 txty =

{ }][Re][ nxny =

64

Incrementally Linear System

Homework #1, due: Oct. 12O: 1.21, 22, 37,54B: 1.3(a),(b), 1.7(a),(b),(c)

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