signal & systems -...
Post on 12-Oct-2020
4 Views
Preview:
TRANSCRIPT
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
8
Other Signal Types
9
Continuous Time vs. Discrete Time
10
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)
16
Transformations of Signal
(Reflection)
17
Transformations of Signal (Scaling)
18
Transformations of Signal
(Examples)
19
Periodic Signals
20
Periodic Signals
21
Even and Odd Signals
22
Even and Odd Decomposition
23
Real Exponential Signals
x(t) = Ceat
24
Periodic Sinusoidal Signals (Imaginary Exponential)
if
25
Relationship betweenFrequency and period
26
27
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
30
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
31
32
33
General Complex Exponential Signalsx(t) = Ceat
34
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
36
Periodicity of Complex Exponentials (cont.)
• For a period of N
we need
or
• Fundamental period
37
38
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
40
Unit Step Function
⎩⎨⎧
≥<
=01
00][
n
nnu
]1[][][ −−= nununδ
41
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
43
Continuous-Time Unit Step Function
⎩⎨⎧
><
=01
00)(
t
ttu
Discontinuous at t = 0
44
Unit Impulse Function
dt
tdut
)()( =δ dt
tdut
)()( ∆
∆ =δ
)(lim)(0
tt ∆→∆
= δδ
45
∫ ∞−=
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
49
50
Continuous- and discrete-time systems
)()( tytx →
][][ nynx →
51
Interconnection of Systems
Series (cascade)
parallel
Series-parallel
52
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)
top related