eece 301 signals & systems
TRANSCRIPT
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EECE 301 Signals & Systems
Prof. Mark Fowler
Note Set #5• Basic Properties of Systems• Reading Assignment: Section 1.5 of Kamen and Heck
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Ch. 1 IntroC-T Signal Model
Functions on Real Line
D-T Signal ModelFunctions on Integers
System PropertiesLTI
CausalEtc
Ch. 2 Diff EqsC-T System Model
Differential EquationsD-T Signal Model
Difference Equations
Zero-State Response
Zero-Input ResponseCharacteristic Eq.
Ch. 2 Convolution
C-T System ModelConvolution Integral
D-T System ModelConvolution Sum
Ch. 3: CT Fourier Signal Models
Fourier SeriesPeriodic Signals
Fourier Transform (CTFT)Non-Periodic Signals
New System Model
New SignalModels
Ch. 5: CT Fourier System Models
Frequency ResponseBased on Fourier Transform
New System Model
Ch. 4: DT Fourier Signal Models
DTFT(for “Hand” Analysis)
DFT & FFT(for Computer Analysis)
New SignalModel
Powerful Analysis Tool
Ch. 6 & 8: Laplace Models for CT
Signals & Systems
Transfer Function
New System Model
Ch. 7: Z Trans.Models for DT
Signals & Systems
Transfer Function
New SystemModel
Ch. 5: DT Fourier System Models
Freq. Response for DTBased on DTFT
New System Model
Course Flow DiagramThe arrows here show conceptual flow between ideas. Note the parallel structure between
the pink blocks (C-T Freq. Analysis) and the blue blocks (D-T Freq. Analysis).
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1.5 Basic System PropertiesThere are some fundamental properties that many (but not all!) systems share regardless if they are C-T or D-T and regardless if they are electrical, mechanical, etc.
An understanding of these fundamental properties allows an engineer to develop tools that can be widely applied… rather than attacking each seemingly different problem anew!!
The three main fundamental properties we will study are:
1. Causality
2. Linearity
3. Time-Invariance
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Causality: A causal (or non-anticipatory) system’s output at a time t1 does not depend on values of the input x(t) for t > t1
The “future input” cannot impact the “now output”
Most systems in nature are causal
⇒ A Causal system (with zero initial conditions) cannot have a non-zero output until a non-zero input is applied.
But… we need to understand non-causal systems because theory shows that the “best” systems are non-causal! So we need to find causal systems that are as close to the best non-causal systems!!!
t
t
Input
Output
Causal System
t
t
Input
Output
Non-Causal System
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Linearity: A system is linear if superposition holds:
Linear Systemx1(t) y1(t)
Linear Systemx2(t) y2(t)
Linear Systemx(t) = a1 x1(t)+ a2 x2(t) y(t) = a1 y1(t)+ a2 y2(t)
Non-Linearx1(t) y1(t)
Non-Linearx2(t) y2(t)
Non-Linearx(t) = a1 x1(t)+ a2 x2(t) y(t) ≠ a1 y1(t)+ a2 y2(t)
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Systems with only R, L, and C are linear systems!Systems with electronics (diodes, transistors, op-amps, etc.)may be non-linear, but they could be linear… at least for inputs that do not exceed a certain range of inputs.
When superposition holds, it makes our life easier!We then can decompose complicated signals into a sum of simpler signals…and then find out how each of these simple signals goes through the system!!
This is exactly what the so-called Frequency Domain Methods of later chapters do!!!
Linear Systemx(t) y(t)If a system is linear: “a scaled
input gives a scaled output”
Same scaling factorLinear System
ky(t)kx(t)
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Time-InvariancePhysical View: The system itself does not change with time
Ex. A circuit with fixed R,L,C is time invariant.Actually, R,L,C values change slightly over time due to temperature & aging effects.
A circuit with, say, a variable R is time variant (assuming that someone or something is changing the R value)
systemx(t) y(t)
x(t)t
y(t)t
y(t-t0)x(t-t0)
t0
tx(t-t0)
t0
ty(t-t0)
Technical View: A system is time invariant (TI) if:
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Differential equations like this are LTI
- coefficients (a’s & b’s) are constants ⇒ TI
- No nonlinear terms ⇒ Linear
)()(...)()(...)()(0101
1
1 txbdt
tdxbdt
txdbtyadt
tydadt
tyda m
m
mn
n
nn
n
n +++=+++ −
−
−
.,)()(,)(,)()(),( etcdt
tyddt
tydtydtp
txddt
txdtx p
p
k
kn
p
k
kn
⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡
Examples Of Nonlinear terms
Systems described by Linear, Constant-Coefficient Differential Equations are Continuous-Time, Linear Time-Invariant (LTI) Systems
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][...]1[][][...]1[][ 1010 MnxbnxbnxbNnyanyanya MN −++−+=−++−+
Difference equations like this are LTI
- coefficients (a’s & b’s) are constants ⇒ TI
- No nonlinear terms ⇒ Linear
Systems described by Linear, Constant-Coefficient Difference Equations are Discrete-Time, Linear Time-Invariant (LTI) Systems
.,][][],[,][][],[ etcmnypnynymnxpnxnx nn −−−−
Examples Of Nonlinear terms