signal and systems prof. h. sameti chapter #1: 1) signals 2) systems 3) some examples of systems 4)...
TRANSCRIPT
Signal and SystemsProf. H. Sameti
Chapter #1:1) Signals 2) Systems 3) Some examples of systems4) System properties and examples
a) Causalityb) Linearityc) 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/
Book Chapter#: Section#
2
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
Department of Computer Engineering, Signal and Systems
Book Chapter#: Section#
3
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
Computer Engineering Department, Signal and Systems
Book Chapter#: Section#
4
CT Signals Most of the signals in the physical world are CT signals—
E.g. voltage & current, pressure, temperature, velocity, etc.
Computer Engineering Department, Signal and Systems
Book Chapter#: Section#
5
DT Signals x[n], n—integer, time varies discretely
Examples of DT signals in nature: DNA base sequence Population of the nth generation of certain species
Computer Engineering Department, Signal and Systems
Book Chapter#: Section#
6
Many human-made DT Signals
Computer Engineering Department, Signal and Systems
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).
Book Chapter#: Section#
7
SYSTEMSFor 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
Computer Engineering Department, Signal and Systems
Book Chapter#: Section#
8
EXAMPLES OF SYSTEMS An RLC circuit
Computer Engineering Department, Signal and Systems
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
Book Chapter#: Section#
9
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
Computer Engineering Department, Signal and Systems
Cascade
Parallel
Feedback
Book Chapter#: Section#
10
SYSTEM EXAMPLES
Example. #1 RLC circuit
Computer Engineering Department, Signal and Systems
)()()()(
)()(
)()()(
)(
2
2
txtydt
tdyRC
dt
tydLC
dt
tdycti
txtydt
tdiLtRi
Example. #2 Mechanical system
Force Balance:
Observation: Very different physical systems may be modeled mathematically in very similar ways.
)()()()(
)()()(
)(
2
2
2
2
txtKydt
tdyD
dt
tydM
dt
tdyDtKytx
dt
tydM
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
0)(
)(
)]()([)(
1
00
2
2
Tdt
dy
yTy
txtyKdt
tyd
Example. #4 Financial system
Fluctuations in the price of zero-coupon bondst = 0 Time of purchase at price y0
t = T Time of maturity at value yt
y(t) = Values of bond at time tx(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
.)(,)0(
),(),...,(),(,)(
),()(
0
212
2
yTTyyy
ttxtxtxdt
tdytyf
dt
tydN
Example. #5
A Rudimentary “Edge” Detectory[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 slopea) x[n]=n → y[n]=0
b) x[n]=nu[n] → y[n]
15
Book Chapter#: Section#
16
Observations1) A very rich class of systems (but by no means all systems
of 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.
Computer Engineering Department, Signal and Systems
Book Chapter#: Section#
17
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.
Computer Engineering Department, Signal and Systems
Book Chapter#: Section#
18
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 only 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 own 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.
Computer Engineering Department, Signal and Systems
Book Chapter#: Section#
19
CAUSALITY (continued)
Mathematically (in CT):
A system x(t) →y(t) is causal if
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
Computer Engineering Department, Signal and Systems
Book Chapter#: Section#
20
CAUSAL OR NONCAUSAL depends on x(4) … causal y depends on future noncausal ok, but y depends on future noncausal ] depends on x(4) … causal
Computer Engineering Department, Signal and Systems
Book Chapter#: Section#
21
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 if 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 -to) →y(t -to) .
Computer Engineering Department, Signal and Systems
Book Chapter#: Section#
22
TIME-INVARIANT OR TIME-VARYING?
is TI
] is TV (NOT time-invariant)
Computer Engineering Department, Signal and Systems
Book Chapter#: Section#
23
NOW WE CAN DEDUCE SOMETHING!
Fact: If the input to a TI System is periodic, then the output 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). ↑ ↑
Computer Engineering Department, Signal and Systems
These are the same input!
So these must be the same output, i.e., y(t) = y(t + T).
Book Chapter#: Section#
24
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
Computer Engineering Department, Signal and Systems
Book Chapter#: Section#
25
LINEARITY A (CT) system is linear if 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
Computer Engineering Department, Signal and Systems
Book Chapter#: Section#
26
EXAMPLES OF LINEAR SYSTEMS
Ex. 1: Additive, but not scalable, so not linear
Ex. 2: Scalable, but not additive, so not linear
Computer Engineering Department, Signal and Systems
Book Chapter#: Section#
27
PROPERTIES OF LINEAR SYSTEMS Superposition
For linear systems, zero input → zero output "Proof" 0=0⋅x[n]→0⋅y[n]=0
Computer Engineering Department, Signal and Systems
If
Then
Book Chapter#: Section#
28
Properties of Linear Systems (Continued)
A linear system is causal if and only if it satisfies the condition of initial rest:
“Proof”a) Suppose system is causal. Show that (*) holds.
b) Suppose (*) holds. Show that the system is causal.
Computer Engineering Department, Signal and Systems
Book Chapter#: Section#
29
LINEAR TIME-INVARIANT (LTI) SYSTEMS
Focus of most of this course - Practical importance (Eg. #1-3 earlier this lecture are all LTI systems.) - 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
Computer Engineering Department, Signal and Systems
Book Chapter#: Section#
30
Example: DT LTI System
Computer Engineering Department, Signal and Systems