signal & systems -...
TRANSCRIPT
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Signal & Systems
Hsin-chia Lu
https://ceiba.ntu.edu.tw/941s_and_s_vlsi
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Forward
Signal System Response
Input voltage, current
Circuit Output voltage,current
Depression of accelerator padel
AutomobileAutomobile
speed
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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, …
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• 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.
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What is a signal?
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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:
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Signal Energy & Power (cont.)
• For , the energy is
• Power is
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X(t)=t
X[n]=4
Finite duration signals
Examples
InfiniteInfiniteThird
FiniteInfiniteSecond
ZeroFiniteFirst
P ∞E∞Class of signals
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Transformations of Signal (Time shift)
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Transformations of Signal (Time shift)
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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
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Real Exponential Signals
x(t) = Ceat
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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
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Harmonically Related Complex Exponentials
• All complex exponentials with period of T0
or
• Define
• Sets of
with period of
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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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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
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Periodicity of Complex Exponentials (cont.)
• For a period of N
we need
or
• Fundamental period
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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
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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
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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
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∫∞
−=0
)()( σστδ dtu
Relation between unit step and impulse functions
∫
∫
∞
∞−
−−=
=
0)()(
)()(
σσδ
ττδ
dt
dtut
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Scaled Unit Impulse Function
∫ ∞−=
tdktku ττδ )()(
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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
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Example of Feedback Systems
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Systems without Memory
• Memoryless, depends on input at the same time
• Identity system
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Systems with Memory
• Examples
• The system must remember or store something
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Invertibility and Inverse Systems
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Noninvertible Systems
• Examples
0][ =ny
)()( 2 txty =
|)(|)( txty =
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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
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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
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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 =
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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
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Examples
• Linear systems
• Nonlinear systems
)()( ttxty =
3][2][ += nxny
)()( 2 txty =
{ }][Re][ nxny =
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Incrementally Linear System
Homework #1, due: Oct. 12O: 1.21, 22, 37,54B: 1.3(a),(b), 1.7(a),(b),(c)