Signal and Systems Chapter 1: Signals and Systems Signals 1) - - PowerPoint PPT Presentation

Signal and Systems Chapter 1: Signals and Systems

1)

Signals

2)

Systems

3)

Some examples of systems

4)

System properties and examples

a)

Causality

b)

Linearity

c)

Time invariance

Reformatted version of open course notes from MIT opencourseware http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-003-signals-and-systems-spring-2010/lecture- notes/

Signals

 Signals are functions of independent variables that carry

• information. For example:

 Electrical signals

voltages and currents in a circuit

 Acoustic signals

audio or speech signals(analog or digital)

 Video signals

intensity variations in an image

 Biological signals

sequence of bases in a gene

Book Chapter#: Section# Department of Computer Engineering, Signal and Systems 2

The Independent Variables

 Can be continuous

 Trajectory of a space shuttle  Mass density in a cross-section of a brain

 Can be discrete

 DNA base sequence  Digital image pixels

 Can be 1-D, 2-D, ••• N-D

 For this course: Focus on a single (1-D) independent variable which

we call "time”

Continuous-Time (CT) signals: x(t), t continuous values Discrete-Time (DT) signals: x[n] , n

integer values only

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 3

CT Signals

 Most of the signals in the physical world are CT signals—

E.g. voltage & current, pressure, temperature, velocity, etc.

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 4

DT Signals

 x[n], n—integer, time varies discretely  Examples of DT signals in nature:

 DNA based sequence

 Population of the nth generation of certain species

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 5

Many human-made DT Signals

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 6

Ex.#1 Weekly Dow-Jones industrial average Ex.#2 digital image

Why DT? — Can be processed by modern digital computers and digital signal processors (DSPs).

SYSTEMS

 For the most part, our view of systems will be from an

input-output perspective:

 A system responds to applied input signals, and its

response is described in terms of one or more output signals

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 7

EXAMPLES OF SYSTEMS

 An RLC circuit

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 8

 Dynamics of an aircraft or space vehicle  An algorithm for analyzing financial and economic factors to

predict bond prices

 An algorithm for post-flight analysis of a space launch  An edge detection algorithm for medical images

SYSTEM INTERCONNECTIOINS

 An important concept is that of interconnecting systems

 To build more complex systems by interconnecting simpler subsystems  To modify response of a system

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 9

Cascade Parallel Feedback

SYSTEM EXAMPLES

 Example. #1 RLC circuit

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 10

) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (

2 2

t x t y dt t dy RC dt t y d LC dt t dy C t i t x t y dt t di L t Ri        



• Example. #2 Mechanical system

 Force Balance:  Observation: Very different physical systems may be modeled

mathematically in very similar ways. ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (

2 2 2 2

t x t Ky dt t dy D dt t y d M dt t dy D t Ky t x dt t y d M      

11



• Example. #3 Thermal system

 Cooling Fin in Steady State

 t = distance along rod  y(t) = Fin temperature as function of position  x(t) = Surrounding temperature along the fin

12



• Example. #3 Thermal system (Continued)

 Observations

 Independent variable can be something other than time,

such as space.

 Such systems may, more naturally, have boundary conditions,

rather than “initial” conditions.

13

) ( ) ( )] ( ) ( [ ) (

1 2 2

    T dt dy y T y t x t y K dt t y d



• Example. #4 Financial system

Fluctuations in the price of zero-coupon bonds

t = 0 Time of purchase at price y0 t = T Time of maturity at value yt y(t) = Values of bond at time t x(t)= Influence of external factors on fluctuations in bond price

Observation: Even if the independent variable is time,

there are interesting and important systems which have boundary conditions.

14

. ) ( , ) ( ), ( ),..., ( ), ( , ) ( ), ( ) (

2 1 2 2

yT T y y y t t x t x t x dt t dy t y f dt t y d

N

        



• Example. #5

 A Rudimentary “Edge” Detector

y[n]=x[n+1]-2x[n]+x[n-1] ={x[n+1]-x[n]}-{x[n]-x[n-1]} = “Second difference”

 This system detects changes in signal slope

a)

x[n]=n → y[n]=0

b)

x[n]=nu[n] → y[n]

15

Observations

1) A very rich class of systems (but by no means all systems

• f interest to us) are described by differential and

difference equations.

2) Such an equation, by itself, does not completely describe

the input-output behavior of a system: we need auxiliary conditions (initial conditions, boundary conditions).

3) In some cases the system of interest has time as the

natural independent variable and is causal. However, that is not always the case.

4) Very different physical systems may have very similar

mathematical descriptions.

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 16

SYSTEM PROPERTIES

(Causality, Linearity, Time-invariance, etc.)

 WHY?

 Important practical/physical implications  They provide us with insight and structure that we can

exploit both to analyze and understand systems more deeply.

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 17

CAUSALITY

 A system is causal if the output does not anticipate future

values of the input, i.e., if the output at any time depends

• nly on values of the input up to that time.

 All real-time physical systems are causal, because time only

moves forward. Effect occurs after cause. (Imagine if you

• wn a noncausal system whose output depends on

tomorrow’s stock price.)

 Causality does not apply to spatially varying signals. (We can

move both left and right, up and down.)

 Causality does not apply to systems processing recorded

signals, e.g. taped sports games vs. live broadcast.

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 18

CAUSALITY (continued)

 Mathematically (in CT):

 A system x(t) →y(t) is causal iff

when x1(t) →y1(t) x2(t) →y2(t) and x1(t) = x2(t) for all t≤ to Then y1(t) = y2(t) for all t≤ to

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 19

CAUSAL OR NONCAUSAL



ሻ 𝑧(𝑢ሻ = 𝑦2(𝑢 − 1 𝐹. 𝑕. ሻ 𝑧(5 depends on x(4) … causal



ሻ 𝑧(𝑢ሻ = 𝑦(𝑢 + 1 𝐹. 𝑕. 𝑧(5ሻ = 𝑦(6ሻ,y depends on future noncausal



ሿ 𝑧[𝑜ሿ = 𝑦[−𝑜 𝐹. 𝑕. ሿ 𝑧[5ሿ = 𝑦[−5 ok, but



ሿ 𝑧[−5ሿ = 𝑦[5 y depends on future noncausal

 𝑧[𝑜ሿ =

1 2 𝑜+1

𝑦3[𝑜 − 1] depends on x(4) … causal

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 20

TIME-INVARIANCE (TI)

 Informally, a system is time-invariant (TI) if its behavior

does not depend on what time it is.

 Mathematically (in DT): A system x[n] → y[n] is TI iff for

any input x[n] and any time shift n0, If x[n] →y[n] then x[n -n0] →y[n -n0]

 Similarly for a CT time-invariant system,

If x(t) →y(t) then x(t –t0) →y(t –t0) .

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 21

TIME-INVARIANT OR TIME- VARYING?



ሻ 𝑧(𝑢ሻ = 𝑦2(𝑢 + 1 is TI

 𝑧[𝑜ሿ =

1 2 𝑜+1

𝑦3[𝑜 − 1] is TV (NOT time-invariant)

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 22

NOW WE CAN DEDUCE SOMETHING!

 Fact: If the input to a TI System is periodic, then the

• utput is periodic with the same period.

“Proof”: Suppose x(t + T) = x(t) and x(t) → y(t) Then by TI x(t + T) →y(t + T). ↑ ↑

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 23

These are the same input! So these must be the same output, i.e., y(t) = y(t + T).

LINEAR AND NONLINEAR SYSTEMS

Many systems are nonlinear. For example: many circuit elements (e.g., diodes), dynamics of aircraft, econometric models,… However, in this course we focus exclusively on linear systems. Why?



Linear models represent accurate representations of behavior of many systems (e.g., linear resistors, capacitors, other examples given previously,…)



Can often linearize models to examine “small signal” perturbations around “operating points”



Linear systems are analytically tractable, providing basis for important tools and considerable insight

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 24

LINEARITY

 A (CT) system is linear iff it has the superposition

property: If x1(t) →y1(t) and x2(t) →y2(t) then ax1(t) + bx2(t) → ay1(t) + by2(t) y[n] = x2[n] Nonlinear, TI, Causal y(t) = x(2t) Linear, not TI, Noncausal Can you find systems with other combinations ?

• e.g. Linear, TI, Noncausal

Linear, not TI, Causal

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 25

EXAMPLES OF LINEAR SYSTEMS

• Ex. 1:

 𝑧(𝑢ሻ = 𝑦∗(𝑢ሻ

 Additive, but not scalablen(for x1(t)=jx(t) for example,

y1(tሻ≠jy(t) ), so not linear

• Ex. 2:

 𝑧(𝑢ሻ =

𝑦2(𝑢ሻ 𝑦(𝑢−1ሻ

 Scalable, but not additive, so not linear

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 26

PROPERTIES OF LINEAR SYSTEMS

 Superposition

ሿ 𝑦𝑙[𝑜ሿ → 𝑧𝑙[𝑜 ෌𝑙 𝑏𝑙𝑦𝑙[𝑜ሿ → ෌𝑙 ሿ 𝑏𝑙𝑧𝑙[𝑜

 For linear systems, zero input → zero output

"Proof" 0=0⋅x[n]→0⋅y[n]=0

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 27

If Then

Properties of Linear Systems (Continued)

 A linear system is causal if and only if it satisfies the

condition of initial rest: 𝑦(𝑢ሻ = 0 𝑔𝑝𝑠 𝑢 ≤ 𝑢0 → 𝑧(𝑢ሻ = 0 𝑔𝑝𝑠 𝑢 ≤ 𝑢0 (initial rest condition) “Proof” a) Suppose system is causal. Show that initial rest condition holds. b) Suppose the initial rest condition holds. Show that the system is causal.

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 28

Causality for Linear Systems

 Proof: a)

Let x(t) → y(t) be a linear system. Let x(t) = 0 for all 𝑢 ≤ 𝑢0 and let x’(t) = 0 be a zero input. Since the system is linear, y’(t) = 0 is a zero output. Note that x(t) = x’(t) for all 𝑢 ≤ 𝑢0. If the system is also causal, then y(t) = y’(t) = 0 for all 𝑢 ≤ 𝑢0. That is, the initial rest condition must be satisfied.

b)

Conversely, suppose that the initial rest condition is satisfied. Let x(t) = x’(t) for all 𝑢 ≤ 𝑢0. Then, x(t) − x’(t) = 0 for all 𝑢 ≤ 𝑢0. By linearity, x(t) − x’(t) → y(t) − y’(t). Furthermore, by the initial rest condition, the output y(t) − y’(t) = 0 for all 𝑢 ≤ 𝑢0. Thus, we have y(t) = y’(t) for all 𝑢 ≤ 𝑢0, and we conclude that the system must be causal.

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 29

LINEAR TIME-INVARIANT (LTI) SYSTEMS

 Focus of most of this course

• Practical importance
• The powerful analysis tools associated with LTI

systems

 A basic fact: If we know the response of an LTI system to

some inputs, we actually know the response to many inputs

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 30

Example: DT LTI System

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 31