eece 301 signals & systems

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Page 1: EECE 301 Signals & Systems

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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