MIT Notes on SS

7/28/2019 MIT Notes on SS

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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 (e.g. a

CAT scan)

Biological signals --- sequence of bases in a gene .

.

.


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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


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4

CT Signals

Most of the signals in the physical world are CTsignalsE.g. voltage & current, pressure,

temperature, velocity, etc.


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5

DT Signals

Examples of DT signals in nature:

DNA base sequence

Population of the nth generation of certain

species

x[n], n integer, time varies discretely


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6

Many human-made DT Signals

Ex.#1 Weekly Dow-Jones

industrial average

Why DT? Can be processed by modern digital computers

and digital signal processors (DSPs).

Ex.#2 digital image

Courtesy of Jason Oppenheim.

Used with permission.


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7

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

x(t) y(t)CT System

DT Systemx[n] y[n]


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8

An RLC circuit

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

EXAMPLES OF SYSTEMS


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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

Signal flow (Block) diagram

Cascade

Feedback

Parallel +

+


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Signals and Systems

Fall 2003Lecture #2

9 September 2003

1) Some examples of systems

2) System properties and

examples

a) Causality

b) Linearity

c) Time invariance


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SYSTEM EXAMPLES

x(t) y(t)CT System DT Systemx[n] y[n]

Ex. #1 RLC circuit


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Force Balance:

Observation: Very different physical systems may be modeled

mathematically in very similar ways.

Ex. #2 Mechanical system


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Ex. #3 Thermal system

Cooling Fin in Steady State


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Ex. #3 (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.


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Ex. #4 Financial system

Observation: Even if the independent variable is time, there

are interesting and important systems which have boundary

conditions.

Fluctuations in the price of zero-coupon bonds

t = 0 Time of purchase at pricey0

t = T Time of maturity at valueyTy(t) = Values of bond at time t

x(t) = Influence of external factors on fluctuations in bond price


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A rudimentary edge detector

This system detects changes in signal slope

Ex. #5

0 1 2 3


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Observations

1) 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.


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SYSTEM PROPERTIES

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

Important practical/physical implications

They provide us with insight and structure that we

can exploit both to analyze and understand systemsmore deeply.

WHY ?


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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

tomorrows stock price.)

Causality does not apply to spatially varying signals. (Wecan move both left and right, up and down.)

Causality does not apply to systems processing recordedsignals, e.g. taped sports games vs. live broadcast.


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Mathematically (in CT): A systemx(t) y(t) is causal if

CAUSALITY (continued)

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


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CAUSAL OR NONCAUSAL


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TIME-INVARIANCE (TI)

Mathematically (in DT): A systemx[n] y[n] is TI if for

any inputx[n] and any time shift n0,

Informally, a system is time-invariant (TI) if its behavior does not

depend on what time it is.

Similarly for a CT time-invariant system,

If x[n] y[n]

then x[n - n0] y[n - n0] .

If x(t) y(t)

then x(t - to)

y(t - to) .


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TIME-INVARIANT OR TIME-VARYING ?

TI

Time-varying (NOT time-invariant)


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NOW WE CAN DEDUCE SOMETHING!

These are the

same input!

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).

So these must be

the same output,

i.e.,y(t) =y(t+ T).


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LINEAR AND NONLINEAR SYSTEMS

Many systems are nonlinear. For example: many circuit

elements (e.g., diodes), dynamics of aircraft, econometric

models,

However, in 6.003 we focus exclusively on linear systems.

Why?

Linear models represent accurate representations ofbehavior of many systems (e.g., linear resistors,

capacitors, other examples given previously,)

Can often linearize models to examine small signalperturbations around operating points

Linear systems are analytically tractable, providing basis

for important tools and considerable insight


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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)

LINEARITY

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


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PROPERTIES OF LINEAR SYSTEMS

Superposition

If

Then

For linear systems, zero input zero output

"Proof" 0 = 0 x[n] 0 y[n]= 0


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Properties of Linear Systems (Continued)

a) Suppose system is causal. Show that (*) holds.

b) Suppose (*) holds. Show that the system is causal.

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

condition of initial rest:

Proof


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LINEAR TIME-INVARIANT (LTI) SYSTEMS

Focus of most of this course

- Practical importance (Eg. #1-3 earlier this lectureare all LTI systems.)

- The powerful analysis tools associatedwith LTI systems

A basic fact: If we know the response of an LTIsystem to some inputs, we actually know the response

to many inputs


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Example: DT LTI System


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Signals and SystemsFall 2003

Lecture #3

11 September 2003

1) Representation of DT signals in terms of shifted unit samples

2) Convolution sum representation of DT LTI systems

3) Examples4) The unit sample response and properties

of DT LTI systems


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Exploiting Superposition and Time-Invariance

Question: Are there sets of basic signals so that:

a) We can represent rich classes of signals as linear combinations of

these building block signals.

b) The response of LTI Systems to these basic signals are both simple

andinsightful.

Fact: For LTI Systems (CT or DT) there are two natural choices for

these building blocks

Focus for now: DT Shifted unit samples

CT Shifted unit impulses


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Representation of DT Signals Using Unit Samples


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That is ...

Coefficients Basic Signals

The Sifting Property of the Unit Sample


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DT Systemx[n] y[n]

Suppose the system is linear, and define hk[n] as the

response to [n - k]:

From superposition:


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DT Systemx[n] y[n]

Now suppose the system is LTI, and define the unit

sample response h[n]:

From LTI:

From TI:


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Convolution Sum Representation of

Response of LTI Systems

Interpretation

n n

n n


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Visualizing the calculation of

y[0] = prod ofoverlap for

n = 0

y[1] = prod ofoverlap for

n = 1

Choose value ofn and consider it fixed

View as functions ofk with n fixed


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Calculating Successive Values: Shift, Multiply, Sum

-11 1 = 1

(-1) 2 + 0 (-1) + 1 (-1) = -3

(-1) (-1) + 0 (-1) = 1

(-1) (-1) = 1

4

0 1 + 1 2 = 2

(-1) 1 + 0 2 + 1 (-1) = -2


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Properties of Convolution and DT LTI Systems

1) A DT LTI System is completely characterizedby its unit sample

response


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Unit Sample response


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The Commutative Property

Ex: Step response s[n] of an LTI system

input Unit Sample response

of accumulator

step

input


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The Distributive Property

Interpretation

The Associative Property


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The Associative Property

Implication (Very special to LTI Systems)


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Properties of LTI Systems

1) Causality

2) Stability


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Lecture #4

16 September 2003

1. Representation of CT Signals in terms of shifted unit impulses

2. Convolution integral representation of CT LTI systems

3. Properties and Examples

4. The unit impulse as an idealized pulse that is

short enough: The operational definition of(t)


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Representation of CT Signals

Approximate any input x(t) as a sum of shifted, scaled

pulses


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has unit area

The Sifting Property of the Unit Impulse

Response of a CT LTI System


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Response of a CT LTI System

LTI

Operation of CT Convolution


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Example: CT convolution


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-1

-1 0

0 1

1 2

2

PROPERTIES AND EXAMPLES


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PROPERTIES AND EXAMPLES

1) Commutativity:

2)

4) Step response:

3) An integrator:

S


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DISTRIBUTIVITY

ASSOCIATIVITY


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ASSOCIATIVITY


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The impulse as an idealized short pulse


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Consider response from initial rest to pulses of different shapes and

durations, but with unit area. As the duration decreases, the responses

become similar for different pulse shapes.

p p

The Operational Definition of the Unit Impulse (t)


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The Operational Definition of the Unit Impulse (t)

(t) idealization of a unit-area pulse that is so short that, for

any physical systems of interest to us, the system responds

only to the area of the pulse and is insensitive to its duration

Operationally: The unit impulse is the signal which when

applied to any LTI system results in an output equal to theimpulse response of the system. That is,

(t) is defined by what it does under convolution.

The Unit Doublet Differentiator


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The Unit Doublet Differentiator

Impulse response = unit doublet

The operational definition of the unit doublet:

Triplets and beyond!


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Triplets and beyond!

n is number of

differentiations

Integrators


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-1 derivatives" = integral I.R. = unit step

Integrators (continued)


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g ( )

Notation


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Define

Then

E.g.

Sometimes Useful Tricks


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Differentiate first, then convolve, then integrate

Example


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1 21 2

Example (continued)


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Lecture #5

18 September 2003

1. Complex Exponentials as Eigenfunctions of LTI Systems

2. Fourier Series representation of CT periodic signals

3. How do we calculate the Fourier coefficients?

4. Convergence and Gibbs Phenomenon


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Signals & Systems, 2nd ed. Upper Saddle River, N.J.: Prentice Hall, 1997, p. 179.

Portrait of Jean Baptiste Joseph Fourier

Image removed due to copyright considerations.

Desirable Characteristics of a Set of Basic Signals


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a. We can represent large and useful classes of signalsusing these building blocks

b. The response of LTI systems to these basic signals is

particularly simple, useful, and insightful

Previous focus: Unit samples and impulses

Focus now: Eigenfunctions of all LTI systems

The eigenfunctions k(t) and their properties(Focus on CT systems now but results apply to DT systems as well )


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(Focus on CT systems now, but results apply to DT systems as well.)

eigenvalue eigenfunction

Eigenfunction in same function out with a gain

From the superposition property of LTI systems:

Now the task of finding response of LTI systems is to determine k.

Complex Exponentials as the Eigenfunctions of any LTI Systems


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DT:

What kinds of signals can we represent as

sums of complex exponentials?


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sums of complex exponentials?

For Now: Focus on restricted sets of complex exponentials

CT & DT Fourier Series and Transforms

CT:

DT:

Magnitude 1

Periodic Signals

Fourier Series Representation of CT Periodic Signals


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o =2

T

- smallest such Tis thefundamental period

- is thefundamental frequency

- periodic with period T

- {ak} are theFourier (series) coefficients

- k= 0 DC

- k= 1 first harmonic

- k= 2 second harmonic

Question #1: How do we find the Fourier coefficients?


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First, for simple periodic signals consisting of a few sinusoidal terms

0 no dc component

0

0

Euler's relation

(memorize!)

For realperiodic signals, there are two other commonly used

forms for CT Fourier series:


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Because of the eigenfunction property of ejt, we will usually

use the complex exponential form in 6.003.

- A consequence of this is that we need to include terms for

bothpositive and negative frequencies:

Now, the complete answer to Question #1


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Ex: Periodic Square Wave


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DC component

is just the

average

Convergence of CT Fourier Series


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How can the Fourier series for the square wave possibly makesense?

The key is: What do we meanby

One useful notion for engineers: there is no energy in the

difference

(just needx(t) to have finite energy per period)

Under a different, but reasonable set of conditions

(the Dirichlet conditions)


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Condition 1. x(t) is absolutely integrable over one period, i. e.

Condition 3. In a finite time interval,x(t) has only afinite

number of discontinuities.

Ex. An example that violates

Condition 3.

And

Condition 2. In a finite time interval,x(t) has afinite number

of maxima and minima.

Ex. An example that violates

Condition 2.

And

Dirichlet conditions are met for the signals we will

encounter in the real world. Then


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- The Fourier series =x(t) at points wherex(t) is continuous

- The Fourier series = midpoint at points of discontinuity

- AsN ,xN(t) exhibits Gibbs phenomenon atpoints of discontinuity

Demo: Fourier Series for CT square wave (Gibbs phenomenon).

Still, convergence has some interesting characteristics:


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Lecture #6

23 September 2003

1. CT Fourier series reprise, properties, and examples

2. DT Fourier series

3. DT Fourier series examples and

differences with CTFS

CT Fourier Series Pairs


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Skip it in future

for shorthand

Another (important!) example: Periodic Impulse Train


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All components have:

(1) the same amplitude,

&

(2) the same phase.

(A few of the) Properties of CT Fourier Series


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Linearity

Introduces a linear phase shift to

Conjugate Symmetry

Time shift

Example: Shift by half period


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Parsevals Relation


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Energy is the same whether measured in the time-domain or thefrequency-domain

Multiplication Property

Periodic Convolution

x(t),y(t) periodic with period T


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Periodic Convolution (continued)

P i di l ti I t t i d ( T/2 t T/2)


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Periodic convolution: Integrate overany one period (e.g. -T/2 to T/2)

Periodic Convolution (continued) Facts

1) z(t) is periodic with period T (why?)


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2) Doesnt matter what period over which we choose to integrate:

3)

Periodic

convolution

in time

Multiplication

in frequency!

Fourier Series Representation of DT Periodic Signals

x[n] - periodic with fundamental periodN, fundamental frequency


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Only ejn which are periodic with periodNwill appear in theFS

So we couldjust use

However, it is often useful to allow the choice ofNconsecutive

values ofkto be arbitrary.

There are onlyNdistinct signals of this form

DT Fourier Series Representation


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= Sum overany Nconsecutive values ofk

k=

This is afinite series

{ak} - Fourier (series) coefficients

Questions:

1) What DT periodic signals have such a representation?

2) How do we find ak?

Answer to Question #1:

Any DT periodic signal has a Fourier series representation


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A More Direct Way to Solve for ak

Finite geometric series


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So, from


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DT Fourier Series Pair


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Note: It is convenient to think ofakas being defined forallintegers k. So:

1) ak+N= ak Special property of DT Fourier Coefficients.

2) We only useNconsecutive values ofak in the synthesisequation. (Sincex[n] is periodic, it is specified byN

numbers, either in the time or frequency domain)

Example #1: Sum of a pair of sinusoids


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0

1/2

1/2

ej/4/2

e-j/4/2

0

0

a-1+16 = a-1 = 1/2

a2+416 = a2 = ej/4/2

Example #2: DT Square Wave


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Using n = m - N1

Example #2: DT Square wave (continued)


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Convergence Issues for DT Fourier Series:

Notan issue, since all series are finite sums.


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Properties of DT Fourier Series: Lots, just as with CT Fourier Series

Example:

Si l d S t


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Lecture #7

25 September 2003

1. Fourier Series and LTI Systems

2. Frequency Response and Filtering

3. Examples and Demos

The Eigenfunction Property of Complex Exponentials


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DT:

CT:

CT"System Function"

DT"System Function"

Fourier Series: Periodic Signals and LTI Systems


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The Frequency Response of an LTI System


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CT notation

Frequency Shaping and Filtering

By choice of H(j) (orH(ej

)) as a function of, we can shapethe frequency composition of the output


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the frequency composition of the output

- Preferential amplification- Selective filtering of some frequencies

Example #1: Audio System

AdjustableFilter

Equalizer Speaker

Bass, Mid-range, Treble controls

For audio signals, the amplitude is much more important than the phase.

Example #2: Frequency Selective Filters

L Fil

Filter out signals outside of the frequency range of interest


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Lowpass Filters:Only showamplitude here.

lowfrequency lowfrequency

Highpass Filters


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Remember:

high

frequency

highfrequency

Bandpass Filters


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Demo: Filtering effects on audio signals

IdealizedFilters

CT


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c cutoff

frequency

DT

Note: |H| = 1 andH= 0 for the ideal filters in the passbands,no need for the phase plot.

Highpass

CT


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DT

Bandpass

CT


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DT

lower cut-off upper cut-off

Example #3: DT Averager/Smoother


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LPF

FIR (Finite Impulse

Response) filters

Example #4: Nonrecursive DT (FIR) filters


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Rolls off at lower

as M+N+1

increases

Example #5: Simple DT Edge Detector

DT 2-point differentiator


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Passes high-frequency components

Demo: DT filters, LP, HP, and BP applied to DJ Industrial average


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Example #6: Edge enhancement using DT differentiator


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Example #7: A Filter Bank


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Demo: Apply different filters to two-dimensional image signals.

HPFace of a monkey.


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Note: To really understand these examples, we need to understandfrequency contents of aperiodic signals the Fourier Transform

LP

BP

BP

LP

HP

Image removed do to

copyright considerations

Signals and Systems


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g yFall 2003

Lecture #8

30 September 2003

1. Derivation of the CT Fourier Transform pair

2. Examples of Fourier Transforms

3. Fourier Transforms of Periodic Signals4. Properties of the CT Fourier Transform

Fouriers Derivation of the CT Fourier Transform

x(t) - an aperiodic signal

view it as the limit of a periodic signal as T


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- view it as the limit of a periodic signal as T

For a periodic signal, the harmonic components arespaced 0 = 2/T apart ...

As T , 0 0, and harmonic components are spaced

closer and closer in frequency

Fourier series Fourier integral

Motivating Example: Square wave

increases


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Discrete

frequency

pointsbecome

denser in

as T

increases

kept fixed

So, on with the derivation ...

For simplicity, assume

(t) h fi it d ti


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x(t) has a finite duration.

Derivation (continued)


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Derivation (continued)


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a) Finite energy

For what kinds of signals can we do this?

(1) It works also even ifx(t) is infinite duration, but satisfies:


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In this case, there iszero energy in the error

E.g. It allows us to considerFTforperiodic signals

c) By allowing impulses in x(t) or in X(j), we can represent

even more signals

b) Dirichlet conditions

Example #1

(a)


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(b)

Example #2: Exponential function


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Even symmetry Odd symmetry

Example #3: A square pulse in the time-domain


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Useful facts about CTFTs

Note the inverse relation between the two widths Uncertainty principle

Example #4: x(t) = eat2

A Gaussian, important in

probability, optics, etc.


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Also a Gaussian! Uncertainty Principle! Cannot makeboth tand arbitrarily small.

(Pulse width in t)(Pulse width in )

t~ (1/a1/2)(a1/2) = 1

CT Fourier Transforms of Periodic Signals


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periodic in twithfrequency o

All the energy is

concentrated in one

frequency o

Example #4:


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Line spectrum

Sampling functionExample #5:


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Same function in

the frequency-domain!

Note: (period in t) T

(period in ) 2/T

Inverse relationship again!

Properties of the CT Fourier Transform

1) Linearity


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2) Time Shifting

FTmagnitude unchanged

Linear change inFTphase

Properties (continued)

3) Conjugate Symmetry


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Even

Odd

Even

Odd

The Properties Keep on Coming ...

4) Time-Scaling


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a) x(t) real and even

b) x(t) real and odd

c)



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Fall 2003

Lecture #9

2 October 2003

1. The Convolution Property of the CTFT

2. Frequency Response and LTI Systems Revisited

3. Multiplication Property andParsevals Relation

4. The DT Fourier Transform

The CT Fourier Transform Pair


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Last lecture: some properties

Today: further exploration

(Synthesis Equation)

(Analysis Equation)

Convolution Property

A consequence of the eigenfunction property:


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Synthesis equation

fory(t)

The Frequency Response Revisited

impulse response


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The frequency response of a CT LTI system is simply the Fourier

transform of its impulse response

Example #1:

frequency response

Example #2: A differentiator

Differentiation property:


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1) Amplifies high frequencies (enhances sharp edges)

Larger at high o phase shift

Example #3: Impulse Response of an Ideal Lowpass Filter


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2) What is h(0)?

No.

Questions:

1) Is this a causal system?

3) What is the steady-state value of

the step response, i.e.s()?

Example #4: Cascading filtering operations


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H(j)

Example #5:


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Gaussian Gaussian = Gaussian Gaussian Gaussian = Gaussian

Example #6:

Example #2 from last lecture


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Example #7:


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Example #8: LTI Systems Described by LCCDEs

(Linear-constant-coefficient differential equations)

Using the Differentiation Property


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Using the Differentiation Property

1) Rational, can use

PFE to get h(t)

2) If X(j) is rationale.g.

then Y(j) is also rational

Parsevals Relation


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FTis highly symmetric,

We already know that:

Then it isnt a

surprise that:

A consequence ofDuality

Convolution in

Multiplication Property

Examples of the Multiplication Property


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For any s(t) ...

Example (continued)


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The Discrete-Time Fourier Transform


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DTFT Derivation (Home Stretch)


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DT Fourier Transform Pair

Analysis Equation


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Analysis Equation

FT

Synthesis Equation Inverse FT

Convergence Issues

Synthesis Equation: None, since integrating over a finite interval

Analysis Equation: Need conditions analogous to CTFT, e.g.


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Absolutely summable

Finite energy

ExamplesParallel with the CT examples in Lecture #8


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More Examples

Infinite sum formula


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Still More

4) DT Rectangular pulse (Drawn forN1 = 2)


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5)


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DTFT of Periodic Signals

DTFSsynthesis eq.


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Linearity

of DTFT

Example #1: DT sine function


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Example #2: DT periodic impulse train


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Also periodic impulse train in the frequency domain!

Properties of the DT Fourier Transform


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Different from CTFT

More Properties

Important implications in DT because of periodicity


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Example

Still More Properties


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Yet Still More Properties

7) Time Expansion

Recall CT property:

Time scale in CT is

infinitely fine

But in DT: x[n/2] makes no sense

x[2n] misses odd values ofx[n]


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Insert two zeros

in this example

(k=3)

But we can slow a DT signal down by inserting zeros:k an integer 1

x(k)[n] insert (k- 1) zeros between successive values

Time Expansion (continued)

Stretched by a factor

ofkin time domain


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-compressed by a factor

ofkin frequency domain

Is There No End to These Properties?

8) Differentiation in Frequency


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Total energy in

time domain

Total energy in

frequency domain

9) Parsevals Relation

Differentiation

in frequency

Multiplication

by n

The Convolution Property


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Example #1:

Example #2: Ideal Lowpass Filter


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Example #3:


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L #11


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Lecture #11

9 October 2003

1. DTFT Properties and Examples

2. Duality in FS & FT

3. Magnitude/Phase of Transforms

and Frequency Responses

Convolution Property Example


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DT LTI System Described by LCCDEs


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Rational function ofe-j,

use PFE to get h[n]

Example: First-order recursive system

with the condition of initial rest causal


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DTFT Multiplication Property


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Calculating Periodic Convolutions


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Example:


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Duality in Fourier AnalysisFourier Transform is highly symmetric

CTFT: Both time and frequency are continuous and in general aperiodic

Same except for

these differences


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Suppose f() and g() are two functions related by

Then

Example of CTFT dualitySquare pulse in either time or frequency domain


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DTFS

Duality in DTFS


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Duality in DTFS

Then

Duality between CTFS and DTFT

CTFS


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DTFT

CTFS-DTFT Duality


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Magnitude and Phase of FT, and Parseval Relation

CT:

Parseval Relation:


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Energy density in

DT:

Parseval Relation:


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Log-Magnitude and Phase


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Easy to add

Plotting Log-Magnitude and Phase

Plot for 0, often with alogarithmic scale for

frequency in CT

b) In DT, need only plot for 0 (with linearscale)

a) For real-valued signals and systems

c) For historical reasons log-magnitude is usually plotted in units


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So 20 dB or 2 bels:

= 10 amplitude gain

= 100 power gain

c) For historical reasons, log magnitude is usually plotted in units

ofdecibels (dB):

power magnitude


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A typical plot of the magnitude and phase of a second-

order DT frequency response

20log|H(ej)| and H(ej) vs.


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For real signals,

0 to is enough


Lecture #12


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1. Linear and Nonlinear Phase

2. Ideal and Nonideal Frequency-Selective

Filters

3. CT & DT Rational Frequency Responses

4. DT First- and Second-Order Systems

16 October 2003

Linear Phase

Result: Linear phase simply a rigid shift in time, no distortionNonlinear phase distortion as well as shift

CT


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Nonlinear phase distortion as well as shift

Question:

DT


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Demo: Impulse response and output of an all-pass

system with nonlinear phase


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How do we think about signal delay when the phase is nonlinear?

Group Delay


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Ideal Lowpass Filter

CT


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Noncausal h(t


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Often have specifications in time and frequency domain Trade-offs

Step responseFreq. Response

CT Rational Frequency Responses

CT: If the system is described by LCCDEs, then


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Prototypical

Systems First-order system, has only oneenergy storing element, e.g. L or C

Second-order system, has two

energy storing elements, e.g. L and C

DT Rational Frequency Responses

If the system is described by LCCDEs (Linear-Constant-Coefficient

Difference Equations), then


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Demo: Unit-sample, unit-step, and frequency response

of DT first-order systems


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DT Second-Order System


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oscillations

decaying

Demo: Unit-sample, unit-step, and frequency response of

DT second-order systems


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Lecture #13


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1. The Concept and Representation of PeriodicSampling of a CT Signal

2. Analysis of Sampling in the Frequency Domain

3. The Sampling Theorem the Nyquist Rate

4. In the Time Domain: Interpolation

5. Undersampling and Aliasing

21 October 2003

We live in a continuous-time world: most of the signals we

encounter are CT signals, e.g.x(t). How do we convert them into DTsignalsx[n]?

SAMPLING

Sampling, taking snap shots ofx(t) every Tseconds.

T sampling periodx[n] x(nT), n = ..., -1, 0, 1, 2, ... regularly spaced samples


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How do we perform sampling?

Applications and Examples

Digital Processing of Signals

Strobe

Images in Newspapers

Sampling Oscilloscope

Why/When Would a Set of Samples Be Adequate?

Observation:Lots of signals have the same samples


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By sampling we throw out lots of information all values ofx(t) between sampling points are lost.

Key Question for Sampling:

Under what conditions can we reconstruct the original CT signalx(t) from its samples?


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Analysis of Sampling in the Frequency Domain

I t t t


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Important to

note: s1/T

Illustration of sampling in the frequency-domain for a

band-limited (X(j)=0 for | |> M) signal


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No overlap between shifted spectra

Reconstruction ofx(t) from sampled signals


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If there is no overlap between

shifted spectra, a LPF can

reproducex(t) fromxp(t)


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Observations on Sampling

(1) In practice, we obviously

dont sample with impulsesor implement ideal lowpass

filters.

One practical example:

The Zero-Order Hold


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Observations (Continued)

(2) Sampling is fundamentally a time-varyingoperation, since we

multiplyx(t) with a time-varying functionp(t). However,


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is the identity system (which is TI) for bandlimitedx(t) satisfying

the sampling theorem (s > 2M).

(3) What ifs 2M? Something different: more later.

Time-Domain Interpretation of Reconstruction ofSampled Signals Band-Limited Interpolation


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The lowpass filter interpolates the samples assuming x(t) contains

no energy at frequencies c

T

h(t)

Graphic Illustration of Time-Domain Interpolation

Original

CT signal

After sampling


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After passing the LPF

Interpolation Methods

Bandlimited Interpolation

Zero-Order Hold

First-Order Hold Linear interpolation


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Undersampling and Aliasing

When s 2 M Undersampling


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Undersampling and Aliasing (continued)


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Higher frequencies ofx(t) are folded back and take on thealiases of lower frequencies

Note that at the sample times,xr(nT) =x(nT)

Xr(j

)

X(j

)Distortion because

ofaliasing

A Simple Example

Picture would be

Modified


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Demo: Sampling and reconstruction of cosot

Modified


Lecture #1423 October 2003


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1. Review/Examples of Sampling/Aliasing

2. DT Processing of CT Signals

Sampling Review


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Demo: Effect of aliasing on music.

Strobe Demo


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> 0, strobed image moves forward, but at a slower pace

= 0, strobed image still

< 0, strobed image moves backward.

Applications of the strobe effect (aliasingcan be useful sometimes):

E.g., Sampling oscilloscope

DT Processing ofBand-LimitedCT Signals

Why do this? Inexpensive, versatile, and higher noise margin.


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How do we analyze this system?

We will need to do it in the frequency domain in both CT andDT In order to avoid confusion about notations, specify

CT frequency variable

DT frequency variable ( = )

Step 1: Find the relation betweenxc(t) andxd[n], orXc(j) andXd(ej)

Time-Domain Interpretation of C/D Conversion

Note: Not full

analog/digital

(A/D) conversion

not quantizingthe x[n] values


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Frequency-Domain Interpretation of C/D Conversion


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Note: s 2

CT DT

Illustration of C/D Conversion in the Frequency-Domain


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)(eX jd)(eX jd

1T = 2T =

D/C Conversion yd[n] yc(t)Reverse of the process of C/D conversion


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Now the whole picture

Overall system is time varying if sampling theorem is not satisfied


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Overall system is time-varying if sampling theorem is notsatisfied

It is LTI if the sampling theorem is satisfied, i.e. for bandlimitedinputsxc(t), with

When the inputxc(t) is band-limited (X(j) = 0 at || > ) and the

sampling theorem is satisfied (s > 2M), then

M M

Synchronous Demodulation of Sinusoidal AM

Suppose

= 0 for now, Local oscillator is in

phase with the carrier.


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Synchronous Demodulation in the Time Domain

Now suppose there is a phase difference, i.e. 0, then


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Two special cases:

1) = /2, the local oscillator is 90o out of phase with the carrier,

r(t) = 0, signal unrecoverable.2) = (t) slowly varying with time, r(t) cos[(t)] x(t),

time-varying gain.

Synchronous Demodulation (with phase error) in theFrequency Domain

Demodulating signal

has phase difference w.r.t.

the modulating signal


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Again, the low-frequency signal ( < M) = 0 when = /2.

Alternative: Asynchronous Demodulation

Assume c >> M, so signal envelope looks likex(t)

Add same carrier with amplitude A to signal


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A = 0 DSB/SC (Double Side Band, Suppressed Carrier)

A > 0 DSB/WC (Double Side Band, With Carrier)

Time Domain

Frequency Domain

Asynchronous Demodulation (continued)Envelope Detector

In order for it to function properly, the envelope function must be positivefor all time, i.e. A +x(t) > 0 for all t.

Demo: Envelope detection for asynchronous demodulation.


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Disadvantages of asynchronous demodulation: Requires extra transmitting power [Acosct]

2 to make sure

A +x(t) > 0 Maximum power efficiency = 1/3 (P8.27)

Advantages of asynchronous demodulation:

Simpler in design and implementation.

Double-Sideband (DSB) and Single-Sideband (SSB) AM

Sincex(t) andy(t) are

real, from conjugatesymmetry bothLSB

and USB signals carry

exactly the same

information.

DSB, occupies

2Mbandwidth

in

> 0.

Each sidebandUSB


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Each sideband

approach only

occupies Mbandwidth in

> 0.LSB

Single Sideband Modulation


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Can also get SSB/SC

or SSB/WC

Frequency-Division Multiplexing (FDM)(Examples: Radio-station signals and analog cell phones)

All the channels

can share the same

medium.


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air

FDM in the Frequency-Domain

Baseband

signals

Channel a

Channel b


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Channel c

Multiplexed

signals


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The Superheterodyne Receiver

AM,c

2

= 535 1605 kHz RF

FCC:IF

2= 455 kHz IF


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Operation principle: Down convert from c to IF, and use a coarse tunable BPF for the front end.

Use a sharp-cutofffixedBPF at IF to get rid of other signals.


Lecture #16

30 October 2003

1. AM with an Arbitrary Periodic Carrier


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2. Pulse Train Carrier and Time-Division Multiplexing

3. Sinusoidal Frequency Modulation

4. DT Sinusoidal AM

5. DT Sampling, Decimation,and Interpolation


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Modulating a (Periodic) Rectangular Pulse Train


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Modulating a Rectangular Pulse Train Carrier, contd


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for rectangular pulse

Observations1) We get a similar picture with any c(t) that is periodic with period T

x(t) can be recovered by passingy(t) through a LPF

2) As long as c = 2/T> 2M, there is no overlap in the shifted and

scaled replicas ofX(j). Consequently, assuming ao 0:


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4) Really only needsamples {x(nT)} when c > 2 M Pulse Amplitude Modulation

3) Pulse Train Modulation is the basis for Time-Division Multiplexing

Assign time slots instead offrequency slots to different channels,

e.g. AT&T wireless phones

Sinusoidal Frequency Modulation (FM)

FM

x(t) is signal

to be

transmitted


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FM

Sinusoidal FM (continued)

Transmitted power does not depend onx(t): average power = A2/2

Bandwidth of y(t) can depend on amplitude ofx(t)

Demodulationa) Direct tracking of the phase (t) (by usingphase-locked loop)

b) Use of an LTI system that acts like a differentiator


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H(j) Tunable band-limited differentiator, over the bandwidth ofy(t)

looks like AM

envelope detection

DT Sinusoidal AM

Multiplication Periodic convolution

Example #1:


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Example #2: Sinusoidal AM


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No overlap of

shifted spectra

Example #2 (continued): Demodulation

Possible as long as there is

no overlap of shifted replicas

ofX(ej):


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Misleading drawing shown for a

very special case ofc = /2

Example #3: An arbitrary periodic DT carrier


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Example #3 (continued):

2a3 = 2a0


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No overlap when: c > 2M (Nyquist rate)M < /N

DT Sampling

Motivation: Reducing the number of data points to be stored or

transmitted, e.g. in CD music recording.


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DT Sampling (continued)


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Decimation Downsampling

xp[n] has (n - 1) zero values between nonzero values:

Why keep them around?

Useful to think of this as sampling followed by discarding the zero values


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compressed in

time byN

Illustration of Decimation in the Time-Domain (forN= 3)


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Decimation in the Frequency Domain


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Squeeze in time

Expand in frequency

Illustration of Decimation in the Frequency Domain

After sampling


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After discarding zeros

The Reverse Operation: Upsampling (e.g. CD playback)

Nx[n]

s s


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Lecture #18

6 November 2003

Inverse Laplace Transforms Laplace Transform Properties


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Laplace Transform Properties

The System Function of an LTI System

Geometric Evaluation of Laplace Transforms

and Frequency Responses

Inverse Laplace Transform

Fix ROC and apply the inverse Fourier transform


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But s = + j( fixed) ds = jd

Inverse Laplace Transforms Via Partial FractionExpansion and Properties

Example:

Three possible ROCs corresponding to three differentsignals


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Recall

ROC I: Left-sided signal.

ROC III: Right sided signal

ROC II: Two-sided signal, has Fourier Transform.


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ROC III: Right-sided signal.

Properties of Laplace Transforms

For example:

Linearity

ROC at least the intersection of ROCs ofX1(s) and X2(s)

ROC b bi (d t l ll ti )

Many parallel properties of the CTFT, but for Laplace transforms

we need to determine implications for the ROC


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ROC can be bigger (due to pole-zero cancellation)

ROC entire s-plane

Time Shift


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Time-Domain Differentiation

ROC could be bigger than the ROC ofX(s), if there is pole-zero

cancellation. E.g.,

s-Domain Differentiation


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s o a e e t at o

Convolution Property

ForThen

ROC of Y(s) = H(s)X(s): at least the overlap of the ROCs ofH(s) & X(s)

ROC could be empty if there is no overlap between the two ROCs

E.g.

ROC could be larger than the overlap of the two. E.g.

)t(ue)t(h),t(ue)t(x tt == and


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g p g

The System Function of an LTI System

The system function characterizes the system

System properties correspond to properties ofH(s) and its ROC

A first example:


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Geometric Evaluation of Rational Laplace Transforms

Example #1: A first-order zero

Graphic evaluation

of

Can reason about- vector length

- angle w/ real axis


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Example #2: A first-order pole

Example #3: A higher-order rational Laplace transform

Still reason with vector, but

remember to "invert" for poles


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First-Order System

Graphical evaluation ofH(j):


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Bode Plot of the First-Order System

-20 dB/decade


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changes by -/2

Second-Order System

0 <


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>1 2 poles on negative real axis

Overdamped

Demo Pole-zero diagrams, frequency response, and stepresponse of first-order and second-order CT causal systems


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Bode Plot of a Second-Order System

-40 dB/decade

Top is flat when

= 1/2 = 0.707 a LPF for

< n

changes by -


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Unit-Impulse and Unit-Step Response of a Second-

Order System

No oscillations when

1 Critically (=) andover (>) damped.


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First-Order All-Pass System

1. Two vectors have

the same lengths

2.

aa


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Lecture #19

18 November 2003

1. CT System Function Properties2. System Function Algebra and


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Block Diagrams

3. Unilateral Laplace Transform and

Applications

CT System Function Properties

2) Causality h(t) right-sided signal ROC ofH(s) is a right-half plane

Question:

If the ROC ofH(s) is a right-half plane, is the system causal?

|h(t) | dt<

1) System is stable ROC ofH(s) includesjaxis

Ex.

H(s) = system function


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Ex.

Non-causal

Properties of CT Rational System Functions

a) However, if H(s) is rational, then

The system is causal The ROC ofH(s) is to theright of the rightmost pole

j axis is in ROC

b) IfH(s) is rational and is the system function of a causal

system, then

The system is stable


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j-axis is in ROC all poles are in LHP

The system is stable

Checking if All Poles Are In the Left-Half Plane

Method #1: Calculate all the roots and see!

Method #2: Routh-Hurwitz Without having to solve for roots.


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Initial- and Final-Value Theorems

Ifx(t) = 0 for t< 0 and there are no impulses or higher order

discontinuities at the origin, then

Initial value

Ifx(t) = 0 for t< 0 andx(t) has a finite limit as t , then

Final value


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Applications of the Initial- and Final-Value Theorem

Initial value:

Final value

For


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LTI Systems Described by LCCDEs

roots of numerator zerosroots of denominator poles


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ROC =? Depends on: 1) Locations of allpoles.2) Boundary conditions, i.e.

right-, left-, two-sided signals.

System Function AlgebraExample: A basic feedback system consisting ofcausalblocks


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ROC: Determined by the roots of1+H1(s)H2(s), instead ofH1(s)

More on this later

in feedback

Block Diagram for Causal LTI Systems

with Rational System Functions

Can be viewed

as cascade of

two systems.

Example:


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Example (continued)

Instead of1

s2 + 3s + 2

2s2 + 4s 6

H(s)

We can constructH(s) using:

x(t) y(t)


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Notation: 1/s an integrator

Note also that


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Lesson to be learned: There are many differentways to construct a

system that performs a certain function.

The Unilateral Laplace Transform

(The preferred tool to analyze causal CT systemsdescribed by LCCDEs with initial conditions)

Note:1) Ifx(t) = 0 fort< 0, then

2) Unilateral LT ofx(t) = Bilateral LT ofx(t)u(t-)

3) For example, ifh(t) is the impulse response of a causal LTI

system, then

4) Convolution property:Ifx1(t) =x2(t) = 0 fort< 0, then


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Same as Bilateral Laplace transform

) p p y 1( ) 2( ) ,

Differentiation Property for Unilateral Laplace Transform

Note:

Derivation:

Initial condition!


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Use of ULTs to Solve Differentiation Equations

with Initial Conditions

Example:

Take ULT:


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ZIR Response forzero inputx(t)=0

ZSR Response for zero state,==0, initially at rest

Example (continued)

Response for LTI system initially at rest ( = = 0)

Response to initial conditions alone ( = 0).

For example:


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Lecture #20

20 November 2003

1. Feedback Systems


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y

2. Applications of Feedback Systems

Why use Feedback?

Reducing Effects of Nonidealities

Reducing Sensitivity to Uncertainties and Variability Stabilizing Unstable Systems

Reducing Effects of Disturbances

T ki

A Typical Feedback System


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Tracking

Shaping System Response Characteristics (bandwidth/speed)

One Motivating Example


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Open-Loop System Closed-Loop Feedback System

Analysis of (Causal!) LTI Feedback Systems: Blacks Formula

CT System

Blacks formula (1920s)

Closed - loop system function =forward gain


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Closed loop system function1 - loop gain

Forward gain total gain along the forward path from the input to the output

Loop gain total gain around the closed loop

Applications of Blacks Formula

Example:

1)


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2)

The Use of Feedback to Compensate for Nonidealities

AssumeKP(j) is very large over the frequency range of interest.

In fact, assume

I d d t f P( )!!


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Independent of P(s)!!

Example of Reduced Sensitivity

10)0990)(1000(1

1000)(

0990)(1000)(

1

11

=

+

=

==

.jQ

.jG,jKP

1)The use of operational amplifiers

2)Decreasing amplifier gain sensitivity

Example:

(a) Suppose

(b) Suppose

(50% gain change)

0990)(500)( 22 .jG,jKP ==


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99)0990)(500(1

500)( 2 .

.jQ

+

= (1% gain change)

Fine, but why doesnt G(j) fluctuate ?

Note:

Needs a large loop gain to produce asteady (and linear) gain for the

whole system

For amplification, G(j) must attenuate, and it is much easier to

build attenuators (e.g. resistors) with desired characteristics

There is a price:


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whole system.

Consequence of the negative (degenerative) feedback.

Example: Operational Amplifiers

If the amplitude of the loop gain

|KG(s)| >> 1 usually the case, unless the battery is totally dead.

Then Steady State


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The closed-loop gain only depends on thepassive components

(R1 &R2), independent of the gain of the open-loop amplifierK.

The Same Idea Works for the Compensation for Nonlinearities

Example and Demo:

Amplifier with a Deadzone

The second system in the forward path has a nonlinear input-outputrelation (a deadzone for small input), which will cause distortion if it is

used as an amplifier. However, as long as the amplitude of the loop gain

is large enough the input output response 1/K


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is large enough, the input-output response 1/K2

Improving the Dynamics of Systems

Example: Operational Amplifier 741

The open-loop gain has a very large value at dc but very limited bandwidth

Not very useful on its own


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Stabilization of Unstable Systems

P(s) unstable

Design C(s), G(s) so that the closed-loop system


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is stable

poles ofQ(s) = roots of 1+C(s)P(s)G(s) in LHP

Example #1: First-order unstable systems


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Example #2: Second-order unstable systems

Unstable forallvalues ofK

Physically, need damping a term proportional tos d/dt


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Example #2 (continued):

Attempt #2: Try Proportional-Plus-Derivative (PD) Feedback

Stable as long asK2 > 0 (sufficient damping)


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andK1 > 4 (sufficient gain).

Example #2 (one more time):

Why didnt we stabilize by canceling the unstable poles?

There are at least two reasons why this is a really bad idea:

a) In real physical systems, we can neverknow the precisevalues of the poles, it could be 2.

b) Disturbance between the two systems will cause instability


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b) Disturbance between the two systems will cause instability.

Demo: Magnetic Levitation

io = current needed to balance the weight W at the rest heightyoForce balance

Linearize about equilibrium with specific values for parameters


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Second-order unstable system

Magnetic Levitation (Continued):


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Stable!

Signals and Systems

Fall 2003

Lecture #21

25 November 2003

1. Feedback

a) Root Locus

b) Tracking

c) Disturbance Rejection

d) The Inverted Pendulum


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)

2. Introduction to the Z-Transform

The Concept of a Root Locus

C(s), G(s) Designed with one or more free parameters

Question: How do the closed-loop poles move as we vary


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these parameters? Root locus of 1+ C(s)G(s)H(s)

The Classical Root Locus Problem

C(s) =K a simple linear amplifier

Closed-loop

poles are

the same.


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A Simple Example

Becomes more stable Becomes less stable

Sketch where

pole moves

as |K| increases...

In either case, pole is atso = -2 -K


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What Happens More Generally ?

For simplicity, suppose there is no pole-zero cancellation in G(s)H(s)

Difficult to solve explicitly for solutions given anyspecific

value ofK, unless G(s)H(s) is second-order or lower.

That is

Closed-loop poles are the solutions of

Much easier to plot the root locus, the values ofs that are

solutions forsome value ofK, because:

1) It is easier to find the roots in the limiting cases for

K = 0


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K= 0, .

2) There are rules on how to connect between these

limiting points.

Rules for Plotting Root Locus

End points

AtK= 0, G(so)H(so) =

so arepoles of the open-loop system function G(s)H(s).

At |K| = , G(so)H(so) = 0

so arezeros of the open-loop system function G(s)H(s). Thus:

Rule #1:A root locus starts (atK= 0) from apole ofG(s)H(s) and ends (at

|K| = ) at azero ofG(s)H(s).

Question: What if the number ofpoles the number ofzeros?

Answer: Start or end at .


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Rule #2: Angle criterion of the root locus

Thus, s0 is a pole for somepositive value of K if:

In this case,s0

is a pole ifK = 1/|G(s0

)H(s0

)|.

Similarlys0

is a pole for some negative value of K if:


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In this case,s0 is a pole ifK = -1/|G(s0)H(s0)|.

Example of Root Locus.

One zero at -2,

two poles at 0, -1.


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In addition to stability, we may want good tracking behavior, i.e.

for at least some set of input signals.

Tracking

+= )(

)()(1

1)( sX

sHsCsE

)()()(1

1)(

jXjHjC

jE+

=


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We want to be large in frequency bands in which wewant good tracking

)()( jPjC

Tracking (continued)

Using the final-value theorem

Basic example: Tracking error for a step input


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Disturbance Rejection

There may be otherobjectives in feedback controls due to unavoidabledisturbances.

Clearly, sensitivities to the disturbancesD1(s) andD2(s) are much


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reduced when the amplitude of the loop gain

Internal Instabilities Due to Pole-Zero Cancellation

Hw(t)

)(33

1)()()(1

)()()(

2)(

)1(1)(

Stable

2 sXsssXsHsC

sHsCsY

sssH,

sssC

++=+=

+=+=

However

2)( ssC +


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)()33()()()(1)(

Unstable

2 sXssssXsHsCsW++=+=

Inverted Pendulum


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Unstable!

Feedback System to Stabilize the Pendulum

PI feedback stabilizes

Additional PD feedback around motor / amplifier

Subtle problem: internal instability in x(t)!

a


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centers the pendulum

Root Locus & the Inverted Pendulum Attempt #1: Negative feedback driving the motor

Remains unstable!

Root locus of M(s)G(s)


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after K. Lundberg

Root Locus & the Inverted Pendulum Attempt #2: Proportional/Integral Compensator

Stable for large enough K

Root locus of K(s)M(s)G(s)


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after K. Lundberg

Root Locus & the Inverted Pendulum BUT x(t) unstable:

System subject to drift...

Solution: add PD feedback

around motor and

compensator:


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after K. Lundberg

The z-Transform

The (Bilateral) z-Transform

Motivation: Analogous to Laplace Transform in CT

We now do not

restrict ourselves

just toz = ej


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The ROC and the Relation Between zT and DTFT

Unit circle (r= 1) in the ROC DTFTX(ej) exists

depends only on r= |z|, just like the ROC ins-plane

only depends onRe(s)

, r = |z|


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Example #1

That is, ROC |z| > |a|,

outside a circle

This form to find

pole and zero locations

This form

for PFE

and

inverse z-

transform

11

1

=az az

z

=


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Example #2:


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SameX(z) as in Ex #1, but different ROC.

Rational z-Transforms

x[n] = linear combination of exponentials forn > 0 and forn < 0

characterized (except for a gain) by its poles and zeros

Polynomials inz


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Signals and Systems

Fall 2003

Lecture #22

2 December 2003

1. Properties of the ROC of the z-Transform

2. Inverse z-Transform

3. Examples

4. Properties of the z-Transform

5. System Functions of DT LTI Systems

a Causality


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a. Causalityb. Stability

The z-Transform

Last time:

Unit circle (r= 1) in the ROCDTFTX(ej) exists

Rational transforms correspond to signals that are linear

combinations of DT exponentials

-depends only on r= |z|, just like the ROC ins-plane

only depends onRe(s)


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Some Intuition on the Relation between zT and LT

Can think of z-transform as DT

version of Laplace transform with

The (Bilateral) z-Transform


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version of Laplace transform with

More intuition on zT-LT, s-plane - z-plane relationship

LHP ins-plane,Re(s) < 0 |z| = | esT| < 1, inside the |z| = 1 circle.

Special case,Re(s) = - |z| = 0.

RHP ins-plane,Re(s) > 0 |z| = | esT| > 1, outside the |z| = 1 circle.

Special case,Re(s) = + |z| = .

A vertical line in s-plane Re(s) = constant | esT| = constant a


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A vertical line ins-plane,Re(s) constant | e | constant, acircle inz-plane.

Properties of the ROCs ofz-Transforms

(1) The ROC ofX(z) consists of a ring in thez-plane centered about

the origin (equivalent to a vertical strip in thes-plane)

(2) The ROC does not contain any poles (same as inLT).


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More ROC Properties

(3) If x[n] is of finite duration, then the ROC is the entirez-plane,

except possibly atz= 0 and/orz= .

Why?

CT counterpartExamples:


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ROC Properties Continued

(4) If x[n] is a right-sided sequence, and if |z| = ro is in the ROC, then

all finite values ofzfor which |z| > ro are also in the ROC.


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Side by Side

(6) Ifx[n] is two-sided, and if |z| = ro is in the ROC, then the ROCconsists of a ring in thez-plane including the circle |z| = ro.

What types of signals do the following ROC correspond to?

right-sided left-sided two-sided

(5) Ifx[n] is a left-sided sequence, and if |z| = ro is in the ROC,

then all finite values ofzfor which 0 < |z| < ro are also in the ROC.


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right sided left sided two sided

Example #1


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Example #1 continued

Clearly, ROC does notexist ifb > 1 No z-transform forb|n|.


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y,

Inverse z-Transforms

for fixed r:


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Example #2

2) When doing inversez-transform

using PFE, expressX(z) as ai 1

Partial Fraction Expansion Algebra: A = 1,B = 2

Note, particular toz-transforms:

1) When finding poles and zeros,

expressX(z) as a function ofz.


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function ofz-1.

ROC I:

ROC III:

ROC II:


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Inversion by Identifying Coefficients

in the Power Series

A finite-duration DT sequence

Example #3:

3

-1

2

0 for all other ns


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A finite duration DT sequence

Example #4:

(a)

(b)


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Properties of z-Transforms

(1) Time Shifting

The rationality ofX(z) unchanged, differentfrom LT. ROC unchanged

except for the possible addition or deletion of the origin or infinityno> 0 ROCz 0 (maybe)

no< 0 ROCz (maybe)

(2) z-Domain Differentiation same ROC

Derivation:


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Convolution Property and System Functions

Y(z) =H(z)X(z) , ROC at least the intersection ofthe ROCs ofH(z) andX(z),

can be bigger if there is pole/zero

cancellation. e.g.

H(z) + ROC tells us everything about system


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H(z) + ROC tells us everything about system

CAUSALITY

(1) h[n] right-sided ROC is the exterior of a circlepossibly

includingz= :

A DT LTI system with system functionH(z) is causal the ROC of

H(z) is the exterior of a circle includingz=


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Causality for Systems with Rational System Functions

A DT LTI system with rational system functionH(z) is causal

(a) the ROC is the exterior of a circle outside the outermost pole;

and (b) if we writeH(z) as a ratio of polynomials

then


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Stability

A causal LTI system with rational system function is stable all

poles are inside the unit circle, i.e. have magnitudes < 1

LTI System Stable ROC ofH(z) includes

the unit circle |z| = 1

Frequency ResponseH(ej) (DTFT ofh[n]) exists.


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Signals and Systems

Fall 2003Lecture #234 December 2003

1. Geometric Evaluation of z-Transforms and DT FrequencyResponses

2. First- and Second-Order Systems

3. System Function Algebra and Block Diagrams

4. Unilateral z-Transforms


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Geometric Evaluation of a Rational z-Transform

Example #1:

Example #3:

Example #2:

All same asin s-plane


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Geometric Evaluation of DT Frequency Responses

First-Order System one realpole


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Second-Order System

Two poles that are a complex conjugate pair (z1= rej

=z2*

)

Clearly, |H| peaks near =


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Demo: DT pole-zero diagrams, frequency response, vector

diagrams, and impulse- & step-responses


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DT LTI Systems Described by LCCDEs

ROC: Depends on Boundary Conditions, left-, right-, or two-sided.

Rational

Use the time-shift property

For Causal Systems ROC is outside the outermost pole


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Feedback System(causal systems)

System Function Algebra and Block Diagrams

Example #1:

negative feedbackconfiguration

z-1 D

Delay


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Example #2:

Cascade oftwo systems


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Unilateral z-Transform

Note:

(1) Ifx[n] = 0 forn < 0, then

(2) UZT ofx[n] = BZT ofx[n]u[n] ROC always outside a circleandincludes z =

(3) For causal LTI systems,


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But there are important differences. For example, time-shift

Properties of Unilateral z-Transform

Many properties are analogous to properties of the BZT e.g.

Convolution property (forx1[n


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Use of UZTs in Solving Difference Equations

with Initial Conditions

ZIR Output purely due to the initial conditions,

ZSR Output purely due to the input.

UZT of Difference Equation


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= 0 System is initially at rest:

ZSR

Example (continued)

= 0 Get response to initial conditions

ZIR


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Documents

MIT Notes on SS