signals and system 6.003fall 2003

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1) Administrative details

2) Signals3) Systems

4) For examples ...“Figures and images used in these lecture notes by permission,

copyright 1997 by Alan V. Oppenheim and Alan S. Willsky”

Signals and SystemsFall 2003

Lecture #1

Prof. Alan S. Willsky

4 September 2003



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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



4

CT Signals

• Most of the signals in the physical world are CTsignals—E.g. voltage & current, pressure,

temperature, velocity, etc.



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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.



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 System x[n] y[n]



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• 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



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 +

+



SYSTEM EXAMPLES

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

Ex. #1 RLC circuit



Force Balance:

Observation: Very different physical systems may be modeled

mathematically in very similar ways.

Ex. #2 Mechanical system



Ex. #3 Thermal system

Cooling Fin in Steady State



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.



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 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



• A rudimentary “edge” detector

• This system detects changes in signal slope

Ex. #5

0 1 2 3



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.



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 ?



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. (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.



• Mathematically (in CT): A system x(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 ≤ t o

Then y1(t ) = y2(t ) for all t ≤ t o



CAUSAL OR NONCAUSAL



TIME-INVARIANCE (TI)

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

any input x[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 - t o)→

y(t - t o) .



TIME-INVARIANT OR TIME-VARYING ?

TI

Time-varying (NOT time-invariant)



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 ).



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 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



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



PROPERTIES OF LINEAR SYSTEMS

• Superposition

If

Then

• For linear systems, zero input → zero output

"Proof" 0 = 0 ⋅ x[n]→ 0 ⋅ y[n]= 0



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



Example: DT LTI System




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



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

and insightful.

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



Representation of DT Signals Using Unit Samples



That is ...

Coefficients Basic Signals

The Sifting Property of the Unit Sample



DT System x[n] y[n]

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

response to δ [n - k ]:

From superposition:



DT System x[n] y[n]

• Now suppose the system is LTI, and define the unit

sample response h[n]:

From LTI:

From TI:



Convolution Sum Representation of

Response of LTI Systems

Interpretation

n n

n n



Visualizing the calculation of

y[0] = ∑ prod of

overlap for

n = 0

y[1] = ∑ prod of

overlap for

n = 1

Choose value of n and consider it fixed

View as functions of k with n fixed



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



Properties of Convolution and DT LTI Systems

1) A DT LTI System is completely characterized by its unit sample

response



Unit Sample response



The Commutative Property

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

“input” Unit Sample response

of accumulator

step

input



The Distributive Property

Interpretation

The Associative Property



The Associative Property

Implication (Very special to LTI Systems)



Properties of LTI Systems

1) Causality ⇔

2) Stability ⇔




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 δ (t)



Representation of CT Signals

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

pulses



has unit area

The Sifting Property of the Unit Impulse

Response of a CT LTI System



Response of a CT LTI System

LTI⇒

Operation of CT Convolution



Example: CT convolution



-1

-1 0

0 1

1 2

2

PROPERTIES AND EXAMPLES



PROPERTIES AND EXAMPLES

1) Commutativity:

2)

4) Step response:

3) An integrator:

S



DISTRIBUTIVITY

ASSOCIATIVITY



ASSOCIATIVITY



The impulse as an idealized “short” pulse



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)



The Operational Definition of the Unit Impulse δ(t )

δ(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,

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

The Unit Doublet — Differentiator



The Unit Doublet Differentiator

Impulse response = unit doublet

The operational definition of the unit doublet:

Triplets and beyond!



Triplets and beyond!

n is number of

differentiations

Integrators



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

Integrators (continued)



g ( )

Notation



Define

Then

E.g.

Sometimes Useful Tricks



Differentiate first, then convolve, then integrate

Example



1 21 2

Example (continued)




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



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.



The eigenfunctions φk ( t ) and their properties

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



(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 λk .

Complex Exponentials as the Eigenfunctions of any LTI Systems



DT:

What kinds of signals can we represent as

“sums” of complex exponentials?



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



ω o =2π

T

- smallest such T is the fundamental period

- is the fundamental frequency

- periodic with period T

- ak are the Fourier (series) coefficients

- k = 0 DC

- k = ±1 first harmonic

- k = ±2 second harmonic

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



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

0 – no dc component

0

0

Euler's relation

(memorize!)

• For real periodic signals, there are two other commonly used

forms for CT Fourier series:



• Because of the eigenfunction property of e jω t , we will usually

use the complex exponential form in 6.003.

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

both positive and negative frequencies:

Now, the complete answer to Question #1



Convergence of CT Fourier Series



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

• The key is: What do we mean by

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

difference

(just need x(t ) to have finite energy per period)

Under a different, but reasonable set of conditions

(the Dirichlet conditions)



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

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

number of discontinuities.

Ex. An example that violates

Condition 3.

And

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



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

- The Fourier series = “midpoint” at points of discontinuity

- As N → ∞, x N (t ) exhibits Gibbs’ phenomenon at points of discontinuity

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

• Still, convergence has some interesting characteristics:




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



Skip it in future

for shorthand

Another (important!) example: Periodic Impulse Train



— All components have:

(1) the same amplitude,

&

(2) the same phase.

(A few of the) Properties of CT Fourier Series



• Linearity

Introduces a linear phase shift ∝ t o

• Conjugate Symmetry

• Time shift

Example: Shift by half period



• Parseval’s Relation



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



Periodic Convolution (continued)

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



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

Periodic Convolution (continued) Facts

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



2) Doesn’t 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 period N , fundamental frequency



• Only e jω n which are periodic with period N will appear in the FS

• So we could just use

• However, it is often useful to allow the choice of N consecutive

values of k to be arbitrary.

⇓

• There are only N distinct signals of this form

DT Fourier Series Representation



= Sum over any N consecutive values of k

k =< N >

∑

— This is a finite 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



A More Direct Way to Solve for ak

Finite geometric series



So, from



Example #1: Sum of a pair of sinusoids



0

1/2

1/2

e jπ /4/2

e-jπ /4/2

0

0

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

a2+4×16= a2 = e jπ /4/2

Example #2: DT Square Wave



Using n = m - N 1

Example #2: DT Square wave (continued)



Convergence Issues for DT Fourier Series:

Not an issue, since all series are finite sums.



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

Example:

Si l d S t




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



DT:

CT:

CT"System Function"

DT"System Function"

Fourier Series: Periodic Signals and LTI Systems



The Frequency Response of an LTI System



CT notation

Frequency Shaping and Filtering

• By choice of H(jω ) (or H(e jω

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



the frequency composition of the output

- Preferential amplification

- Selective filtering of some frequencies

Example #1: Audio System

Adjustable

FilterEqualizer 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



Lowpass Filters:

Only show

amplitude here.

lowfrequency

lowfrequency

Highpass Filters



Remember:

high

frequency

high

frequency

Bandpass Filters



Demo: Filtering effects on audio signals

Idealized Filters

CT



ωc — cutoff

frequency

DT

Note: | H | = 1 and ∠ H = 0 for the ideal filters in the passbands,

no need for the phase plot.

Highpass

CT



DT

Bandpass

CT



DT

lower cut-off upper cut-off

Example #3: DT Averager/Smoother



LPF

FIR (Finite Impulse

Response) filters

Example #4: Nonrecursive DT ( FIR) filters



Rolls off at lower

ω as M+N+1

increases

Example #5: Simple DT “Edge” Detector

— DT 2-point “differentiator”



Passes high-frequency components

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



Example #6: Edge enhancement using DT differentiator







Example #7: A Filter Bank



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

HPFace of a monkey.



Note: To really understand these examples, we need to understand

frequency contents of aperiodic signals⇒ the Fourier Transform

LP

BP

BP

LP

HP

Image removed do to

copyright considerations

Signals and Systems



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

Fourier’s Derivation of the CT Fourier Transform

• x(t ) - an aperiodic signal

view it as the limit of a periodic signal as T → ∞



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

• For a periodic signal, the harmonic components arespaced ω0 = 2π/T apart ...

• As T → ∞, ω 0 → 0, and harmonic components are spaced

closer and closer in frequency

⇓

Fourier series Fourier integral ⎯ → ⎯

Motivating Example: Square wave

increases



Discrete

frequency

points become

denser in

ω as T

increases

kept fixed

So, on with the derivation ...

For simplicity, assume

(t) h fi it d ti



x(t ) has a finite duration.

Derivation (continued)



Derivation (continued)



a) Finite energy

For what kinds of signals can we do this?

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



In this case, there is zero energy in the error

E.g. It allows us to consider FT for periodic signals

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

even more signals

b) Dirichlet conditions



Example #2: Exponential function



Even symmetry Odd symmetry

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



Useful facts about CTFT’s

Note the inverse relation between the two widths ⇒ Uncertainty principle

Example #4: x(t ) = e−at 2 — A Gaussian, important in

probability, optics, etc.



Also a Gaussian! Uncertainty Principle! Cannot make

both ∆t and ∆ω arbitrarily small.

(Pulse width in t )•(Pulse width in ω)

⇒ ∆t• ∆ω ~ (1/a1/2)•(a1/2) = 1

CT Fourier Transforms of Periodic Signals



— periodic in t withfrequency ωo

— All the energy is

concentrated in one

frequency — ωo

Example #4:



“Line spectrum”

— Sampling functionExample #5:



Same function in

the frequency-domain!

Note: (period int ) T

⇔ (period in ω) 2π/T

Inverse relationship again!

Properties of the CT Fourier Transform

1) Linearity



2) Time Shifting

FT magnitude unchanged

Linear change in FT phase

Properties (continued)

3) Conjugate Symmetry



Even

Odd

Even

Odd

The Properties Keep on Coming ...

4) Time-Scaling



a) x(t ) real and even

b) x(t ) real and odd

c)



The CT Fourier Transform Pair



Last lecture: some properties

Today: further exploration

(Synthesis Equation)

(Analysis Equation)



The Frequency Response Revisited

impulse response



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:



1) Amplifies high frequencies (enhances sharp edges)

Larger at high ωo phase shift

Example #3: Impulse Response of an Ideal Lowpass Filter



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



H(jω)

Example #5:



Gaussian × Gaussian = Gaussian⇒ Gaussian ∗ Gaussian = Gaussian

Example #6:

Example #2 from last lecture



Example #7:



Example #8: LTI Systems Described by LCCDE’s

(Linear-constant-coefficient differential equations)

Using the Differentiation Property



Using the Differentiation Property

1) Rational, can use

PFE to get h(t )

2) If X ( jω ) is rationale.g.

then Y ( jω ) is also rational



Examples of the Multiplication Property



⇓For any s(t) ...

Example (continued)



The Discrete-Time Fourier Transform



DTFT Derivation (Continued)

DTFS synthesis eq.

DTFS analysis eq.



Define

DTFT Derivation (Home Stretch)



Lecture #10

7 October 2003

1. Examples of the DT Fourier Transform

2. Properties of the DT Fourier Transform

3. The Convolution Property and its

Implications and Uses

DT Fourier Transform Pair

Analysis Equation



– 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.



— Absolutely summable

— Finite energy

ExamplesParallel with the CT examples in Lecture #8



More Examples

Infinite sum formula



Still More

4) DT Rectangular pulse (Drawn for N 1 = 2)



5)



DTFTs of Sums of Complex Exponentials

Recall CT result:

What about DT:

a) We expect an impulse (of area 2π) at ω = ω



Note: The integration in the synthesis equation is over 2π period,

only need X (e jω ) in one 2π period. Thus,

a) We expect an impulse (of area 2π) at ω = ω

o b) But X (e jω ) must be periodic with period 2π

In fact

DTFT of Periodic Signals

DTFSsynthesis eq.



Linearity

of DTFT

Example #1: DT sine function



Example #2: DT periodic impulse train



— Also periodic impulse train – in the frequency domain!

Properties of the DT Fourier Transform



— Different from CTFT

More Properties

— Important implications in DT because of periodicity



Example

Still More Properties



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 of x[n]



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

of k in time domain



-compressed by a factor

of k in frequency domain

Is There No End to These Properties?

8) Differentiation in Frequency



Total energy in

time domain

Total energy in

frequency domain

9) Parseval’s Relation

Differentiation

in frequency

Multiplication

by n

The Convolution Property



Example #1:



Example #3:




L #11



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



DT LTI System Described by LCCDE’s



— Rational function of e-jω,

use PFE to get h[n]

Example: First-order recursive system

with the condition of initial rest⇔ causal



DTFT Multiplication Property



Calculating Periodic Convolutions



Example:



Duality in Fourier AnalysisFourier Transform is highly symmetric

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

Same except for

these differences



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

Then

Example of CTFT dualitySquare pulse in either time or frequency domain



DTFS

Duality in DTFS



Duality in DTFS

Then

Duality between CTFS and DTFT

CTFS



DTFT

CTFS-DTFT Duality



Magnitude and Phase of FT, and Parseval Relation

CT:

Parseval Relation:



Energy density in ω

DT:

Parseval Relation:

Effects of Phase

• Not on signal energy distribution as a function of frequency

• Can have dramatic effect on signal shape/character



g p

— Constructive/Destructive interference

• Is that important?

— Depends on the signal and the context

Demo: 1) Effect of phase on Fourier Series

2) Effect of phase on image processing



Log-Magnitude and Phase



Easy to add

Plotting Log-Magnitude and Phase

Plot for ω ≥ 0, often with alogarithmic scale for

frequency in CT

b) In DT, need only plot for 0≤ ω ≤ π (with linear scale)

a) For real-valued signals and systems

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



So… 20 dB or 2 bels:

= 10 amplitude gain

= 100 power gain

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

of decibels (dB):

power magnitude

A Typical Bode plot for a second-order CT system20 log| H ( jω )| and ∠ H ( jω ) vs. log ω

40 dB/decade



Changes by -π

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

order DT frequency response

20log| H (e jω )| and ∠ H (e jω ) vs. ω



For real signals,

0 to π is enough


Lecture #12



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



Nonlinear phase ⇔ distortion as well as shift

Question:

DT

All-Pass Systems

CT



DT

Demo: Impulse response and output of an all-pass

system with nonlinear phase



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

Group Delay

φ



Ideal Lowpass Filter

CT

← → ⎯



• Noncausal h(t <0) ≠ 0

• Oscillatory Response — e.g. step response Overshoot by 9%,

Gibbs phenomenon

Nonideal Lowpass Filter

• Sometimes we don’t want a sharp cutoff, e.g.



• 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



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 LCCDE’s (Linear-Constant-Coefficient

Difference Equations), then



DT First-Order Systems



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

of DT first-order systems



DT Second-Order System



oscillations

decaying

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

DT second-order systems




Lecture #13



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 DTsignals x[n]?

SAMPLING

— Sampling, taking snap shots of x(t ) every T seconds.

T – sampling period x[n] ≡ x(nT ), n = ..., -1, 0, 1, 2, ... — regularly spaced samples



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



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

• Key Question for Sampling:

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

Impulse Sampling — Multiplying x(t ) by the sampling function



Analysis of Sampling in the Frequency Domain

I t t t



Important to

note: ωs∝1/T

Illustration of sampling in the frequency-domain for a

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



No overlap between shifted spectra

Reconstruction of x (t ) from sampled signals



If there is no overlap between

shifted spectra, a LPF can

reproduce x(t ) from x p(t )

The Sampling Theorem

Suppose x(t ) is bandlimited, so that

Then x(t ) is uniquely determined by its samples x(nT ) if



Observations on Sampling

(1) In practice, we obviously

don’t sample with impulsesor implement ideal lowpass

filters.

— One practical example:

The Zero-Order Hold



Time-Domain Interpretation of Reconstruction of Sampled Signals — Band-Limited Interpolation



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

no energy at frequencies ≥ ωc

T

h(t)

Graphic Illustration of Time-Domain Interpolation

Original

CT signal

After sampling



After passing the LPF

Interpolation Methods

• Bandlimited Interpolation

• Zero-Order Hold

• First-Order Hold — Linear interpolation



Undersampling and Aliasing

When ωs ≤ 2 ωM⇒ Undersampling



Undersampling and Aliasing (continued)



— Higher frequencies of x(t ) are “folded back” and take on the“aliases” of lower frequencies

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

X r ( jω

)≠

X ( jω

)Distortion because

of aliasing

A Simple Example

Picture would be

Modified…



Demo: Sampling and reconstruction of cosω ot

Modified…


Lecture #1423 October 2003



1. Review/Examples of Sampling/Aliasing

2. DT Processing of CT Signals

Sampling Review



Demo: Effect of aliasing on music.



DT Processing of Band-Limited CT Signals

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



How do we analyze this system?

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

ω — CT frequency variable

Ω — DT frequency variable (Ω = ωΤ)

Step 1: Find the relation between xc(t ) and xd[n], or X c( jω ) and X d(e jΩ)

Time-Domain Interpretation of C/D Conversion

Note: Not full

analog/digital

(A/D) conversion

– not quantizingthe x[n] values



Frequency-Domain Interpretation of C/D Conversion



Note: ωs ⇔ 2π

CT DT

Illustration of C/D Conversion in the Frequency-Domain



)(eX jΩ

d)(eX jΩ

d

1ωTΩ = 2ωTΩ =

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



Now the whole picture

• Overall system is time varying if sampling theorem is not satisfied



• Overall system is time-varying if sampling theorem is not satisfied

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

• When the input xc(t ) is band-limited ( X ( jω ) = 0 at |ω | > ω Μ) and the

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

ω M <ω s

2

DT omege needs to changed

Frequency-Domain Illustration of DT Processing of CT Signals

Sampling

DT filter

DT freq→ CT freq



Interpolate

(LPF)

⇓

equivalent

CT filter

CT freq→ DT freq

Assuming No Aliasing



In practice, first specify the desired H c( jω ), then design H d(e jΩ).

Example: Digital DifferentiatorApplications: Edge Enhancement







Construction of Digital Differentiator

Bandlimited Differentiator



Band-Limited Digital Differentiator (continued)



CT DT


Lecture #15

28 October 2003



1. Complex Exponential Amplitude Modulation2. Sinusoidal AM

3. Demodulation of Sinusoidal AM

4. Single-Sideband (SSB) AM

5. Frequency-Division Multiplexing

6. Superheterodyne Receivers

The Concept of Modulation

Why?

• More efficient to transmit E&M signals at higher frequencies

• Transmitting multiple signals through the same medium usingdifferent carriers

• Transmitting through “channels” with limited passbands

Others

Transmitted Signalx(t)

Carrier Signal



• Others...

• Many methods

• Focus here for the most part on Amplitude M odulation (AM)

How?

Amplitude M odulation (AM) of a

Complex Exponential Carrier



Demodulation of Complex Exponential AM

Corresponds to two separate modulation channels (quadratures)

with carriers 90o out of phase



Sinusoidal AM



Drawn assumingωc > ωM

Synchronous Demodulation of Sinusoidal AM

Supposeθ

= 0 for now,⇒ Local oscillator is in

phase with the carrier.



Synchronous Demodulation in the Time Domain

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



Two special cases:

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

⇒ r (t ) = 0, signal unrecoverable.2) θ = θ(t ) — slowly varying with time, ⇒ r (t ) ≅ cos[θ(t )] • x(t ),

⇒ time-varying “gain”.

Synchronous Demodulation (with phase error) in theFrequency Domain

Demodulating signal –

has phase difference θ w.r.t.

the modulating signal



Again, the low-frequency signal (ω < ωM) = 0 when θ = π/2.

Alternative: Asynchronous Demodulation

• Assume ωc >> ωM, so signal envelope looks like x(t )

• Add same carrier with amplitude A to signal



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.



Disadvantages of asynchronous demodulation: — Requires extra transmitting power [ Acosωct ]

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

Since x(t ) and y(t ) are

real , from conjugatesymmetry both LSB

and USB signals carry

exactly the same

information.

DSB, occupies

2ωM bandwidth

inω

> 0.

Each sidebandUSB



Each sideband

approach only

occupies ωM

bandwidth in

ω > 0.LSB



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

All the channels

can share the same

medium.



air

FDM in the Frequency-Domain

“Baseband”

signals

Channel a

Channel b



Channel c

Multiplexed

signals

Demultiplexing and Demodulation

ωa needs to be tunable



• Channels must not overlap ⇒ Bandwidth Allocation

• It is difficult (and expensive) to design a highly selective

bandpass filter with a tunable center frequency

• Solution – Superheterodyne Receivers

The Superheterodyne Receiver

AM,ω c

2π

= 535 −1605 kHz — RF

FCC:ω IF

2π= 455 kHz — IF



Operation principle: — Down convert from ωc to ωIF, and use a coarse tunable BPF for the front end.

— Use a sharp-cutoff fixed BPF at ωIF to get rid of other signals.


Lecture #16

30 October 2003

1. AM with an Arbitrary Periodic Carrier



2. Pulse Train Carrier and Time-Division Multiplexing

3. Sinusoidal Frequency Modulation

4. DT Sinusoidal AM

5. DT Sampling, Decimation,and Interpolation

AM with an Arbitrary Periodic Carrier



Modulating a Rectangular Pulse Train Carrier, cont’d



for rectangular pulse



Sinusoidal F requency M odulation (FM)

FM

x(t) is signal

to be

transmitted



FM

Sinusoidal FM (continued)

• Transmitted power does not depend on x(t ): average power = A2/2

• Bandwidth of y(t ) can depend on amplitude of x(t )

• Demodulationa) Direct tracking of the phase θ (t ) (by using phase-locked loop)

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



H ( jω ) — Tunable band-limited differentiator, over the bandwidth of y(t )

… looks like AM

envelope detection

DT Sinusoidal AM

Multiplication ↔ Periodic convolution

Example #1:



Example #2: Sinusoidal AM



No overlap of

shifted spectra

Example #2 (continued): Demodulation

Possible as long as there is

no overlap of shifted replicas

of X (e jω ):



Misleading drawing – shown for a

very special case of ωc = π/2

Example #3: An arbitrary periodic DT carrier



Example #3 (continued):

2πa3 = 2πa0



No overlap when: ωc > 2ωM (Nyquist rate) ⇒ ωM < π/N

DT Sampling

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

transmitted, e.g. in CD music recording.



DT Sampling Theorem

We can reconstruct x[n]

if ωs = 2π/ N > 2ωM

Drawn assuming

ωs > 2ωM

Nyquist rate is met

⇒ ωM < π/N



Drawn assuming

ωs < 2ωM

Aliasing!

/ N

Decimation — Downsampling

x p[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



compressed in

time by N

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



Decimation in the Frequency Domain



Squeeze in time

Expand in frequency

Illustration of Decimation in the Frequency Domain

After sampling



After discarding zeros




Lecture #17

4 November 2003

1. Motivation and Definition of the (Bilateral)

Laplace Transform



p

2. Examples of Laplace Transforms and Their

Regions of Convergence (ROCs)

3. Properties of ROCs

• CT Fourier transform enables us to do a lot of things, e.g.

— Analyze frequency response of LTI systems

— Sampling — Modulation

Motivation for the Laplace Transform

• In particular, Fourier transform cannot handle large (and important)classes of signals and unstable systems i e when

• Why do we need yet another transform?

• One view of Laplace Transform is as an extension of the Fourier

transform to allow analysis of broader class of signals and systems



classes of signals and unstable systems, i.e. when

Motivation for the Laplace Transform (continued)

• How do we analyze such signals/systems?

Recall from Lecture #5, eigenfunction property of LTI systems:

• In many applications, we do need to deal with unstable systems, e.g.

— Stabilizing an inverted pendulum

— Stabilizing an airplane or space shuttle

— Instability is desired in some applications, e.g. oscillators and

lasers



— e st is an eigenfunction of any LTI system

— s = σ + jω can be complex in general

(2) A critical issue in dealing with Laplace transform is convergence:

— X ( s) generally exists only for some values of s,

located in what is called the region of convergence (ROC)

The (Bilateral) Laplace Transform

Basic ideas:

(1)

absolute integrability needed



(3) If s = jω is in the ROC (i.e. σ = 0), then absoluteintegrability

condition

Example #1:

Unstable:

• no Fourier Transform• but Laplace Transform

exists



This converges only if Re( s+a) > 0, i.e. Re( s) > - Re(a)

Example #2:

This converges only if Re( s+a) < 0, i.e. Re( s) < - Re(a)



Key Point (and key difference from FT ): Need both X ( s) and ROC touniquely determine x(t ). No such an issue for FT .

Graphical Visualization of the ROC

Example #2Example #1



Rational Transforms

• Many (but by no means all) Laplace transforms of interest to us

are rational functions of s (e.g., Examples #1 and #2; in general,

impulse responses of LTI systems described by LCCDEs), where

• Roots of N(s) = zeros of X ( s)

• Roots of D(s) = poles of X ( s)



• Any x(t ) consisting of a linear combination of complex

exponentials for t > 0 and for t < 0 (e.g., as in Example #1 and #2)

has a rational Laplace transform.

Example #3

Notation:

× — pole

° — zero

BOTH required→

ROC intersection



Q: Does x(t ) have FT ?



Properties of the ROC

1) The ROC consists of a collection of lines parallel to the jω -axis in

the s-plane (i.e. the ROC only depends on σ).

Why?

• The ROC can take on only a small number of different forms

2) If X ( s) is rational, then the ROC does not contain any poles.Why?



y

Poles are places where D( s) = 0

⇒ X ( s) =

N ( s)

D( s) = ∞ Not convergent.

More Properties

3) If x(t ) is of finite duration and is absolutely integrable, then the ROC

is the entire s-plane.



ROC Properties that Depend on Which Side You Are On - I

4) If x(t ) is right-sided (i.e. if it is zero before some time), and if

Re( s) = σo is in the ROC, then all values of s for which

Re( s) > σo are also in the ROC.



ROC is a right half plane (RHP)

ROC Properties that Depend on Which Side You Are On - II

5) If x(t ) is left-sided (i.e. if it is zero after some time), and if

Re( s) = σo is in the ROC, then all values of s for which

Re( s) < σo are also in the ROC.



ROC is a left half plane (LHP)

Still More ROC Properties6) If x(t ) is two-sided and if the line Re( s) = σo is in the ROC,

then the ROC consists of a strip in the s-plane that includes

the line Re( s) = σo

.

ROC is

RHP

Strip =

RHP ∩ LHP



ROC is

LHP

Example:

Intuition?

• Okay: multiply by

constant (e0t) and

will be integrable

• Looks bad: no eσ t

will dampen both

id



sides

Example (continued):



What if b < 0? ⇒ No overlap ⇒ No Laplace Transform

Properties, Properties

7) If X ( s) is rational, then its ROC is bounded by poles or extends to

infinity. In addition, no poles of X ( s) are contained in the ROC.

8) Suppose X ( s) is rational, then

(a) If x(t ) is right-sided, the ROC is to the right of the rightmost pole.

(b) If x(t ) is left-sided, the ROC is to the left of the leftmost pole.



(a) (b) (c)

9) If ROC of X ( s) includes the jω -axis, then FT of x(t ) exists.

Example:

9) If ROC of X ( s) includes the jω -axis, then FT of x(t ) exists.

Three possible ROCs

Fourier

Transform



x(t ) is right-sided ROC:

x(t ) is left-sided ROC: x(t ) extends for all time

exists?

No

NoROC: Yes

III

III


Lecture #18

6 November 2003

• Inverse Laplace Transforms

• Laplace Transform Properties

Th S t F ti f LTI S t



• The System Function of an LTI System

• Geometric Evaluation of Laplace Transforms

and Frequency Responses

Inverse Laplace Transform

But s = σ + jω (σ fixed) ⇒ ds = jd ω

Fix σ ∈ ROC and apply the inverse Fourier transform



Inverse Laplace Transforms Via Partial Fraction

Expansion and Properties

Example:

Three possible ROC’s — corresponding to three different signals



Recall

ROC I: — Left-sided signal.

ROC III:— Right-sided signal.

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



Properties of Laplace Transforms

• For example:

Linearity

ROC at least the intersection of ROCs of X 1( s) and X 2( s)

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

• Many parallel properties of the CTFT, but for Laplace transforms

we need to determine implications for the ROC



⇒ ROC entire s-plane

Time Shift



Time-Domain Differentiation

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

cancellation. E.g.,

s-Domain Differentiation



Convolution Property

For Then

• ROC of Y ( s) = H ( s) X ( s): at least the overlap of the ROCs of H ( 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 t t −−== − and



The System Function of an LTI System

The system function characterizes the system

⇓System properties correspond to properties of H ( s) and its ROC

A first example:



Geometric Evaluation of Rational Laplace Transforms

Example #1: A first-order zero

Graphic evaluation

of

Can reason about- vector length

- angle w/ real axis



Example #2: A first-order pole

Example #3: A higher-order rational Laplace transform

Still reason with vector, but

remember to "invert" for poles



First-Order System

Graphical evaluation of H ( jω ):



Bode Plot of the First-Order System

-20 dB/decade

changes by -π/2



Second-Order System

0 < ζ <1 ⇒ complex poles

ζ =1 ⇒

— Under damped

double pole at s = −ω n — Critically damped



ζ >1 ⇒ 2 poles on negative real axis

— Over damped

Demo Pole-zero diagrams, frequency response, and step

response of first-order and second-order CT causal systems



Bode Plot of a Second-Order System

-40 dB/decade

Top is flat when

ζ= 1/√2 = 0.707⇒ a LPF for

ω < ωn

changes by -π



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

Order System

No oscillations when

ζ ≥ 1⇒ Critically (=) and

over (>) damped.



First-Order All-Pass System

1. Two vectors havethe same lengths

2.

a

a




Lecture #19

18 November 2003

1. CT System Function Properties2. System Function Algebra and

Block Diagrams



3. Unilateral Laplace Transform and

Applications

CT System Function Properties

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

Question:

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

|h(t ) | dt < ∞−∞

∞

∫ 1) System is stable ⇔ ⇔ ROC of H ( s) includes jω axis

Ex.

H(s) = “system function”



Non-causal

Properties of CT Rational System Functions

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

The system is causal ⇔ The ROC of H ( s) is to the

right of the rightmost pole

jω-axis is in ROC

⇔ all poles are in LHP

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

system, then

The system is stable ⇔



⇔ all poles are in LHP

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.



Initial- and Final-Value Theorems

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

discontinuities at the origin, then

Initial value

If x(t ) = 0 for t < 0 and x(t ) has a finite limit as t → ∞, then

Final value



Applications of the Initial- and Final-Value Theorem

• Initial value:

• Final value

For



LTI Systems Described by LCCDEs

roots of numerator ⇒ zerosroots of denominator ⇒ poles



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

right-, left-, two-sided signals.

System Function Algebra

Example: A basic feedback system consisting of causal blocks

More on this later



ROC: Determined by the roots of 1+ H 1( s) H 2( s), instead of H 1( 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:



Example (continued)

Instead of 1

s2 + 3 s + 2

2 s2 + 4 s − 6

H ( s)

We can construct H ( s) using:

x(t)



Notation: 1/ s — an integrator

Note also that



Lesson to be learned: There are many different ways to construct asystem that performs a certain function.

The Unilateral Laplace Transform

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

Note:1) If x(t ) = 0 for t < 0, then

2) Unilateral LT of x(t ) = Bilateral LT of x(t )u(t-)

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

system, then

4) Convolution property:If x1(t ) = x2(t ) = 0 for t < 0, then



Same as Bilateral Laplace transform

Differentiation Property for Unilateral Laplace Transform

Note:

Derivation:

Initial condition!



Use of ULTs to Solve Differentiation Equations

with Initial Conditions

Example:

Take ULT:



ZIR — Response for zero input x(t )=0

ZSR — Response for zero state, β=γ= 0, initially at rest



Signals and Systems

Fall 2003

Lecture #20

20 November 2003

1. Feedback Systems

2. Applications of Feedback Systems



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

• Tracking

Sh i S t R Ch t i ti (b d idth/ d)

A Typical Feedback System



• Shaping System Response Characteristics (bandwidth/speed)

One Motivating Example



Open-Loop System Closed-Loop Feedback System

Analysis of (Causal!) LTI Feedback Systems: Black’s Formula

CT System

Black’s formula (1920’s)

Closed - loop system function =

forward gain

1 - loop gain



Forward gain — total gain along the forward path from the input to the outputLoop gain — total gain around the closed loop

Applications of Black’s Formula

Example:

1)

2)



)

The Use of Feedback to Compensate for Nonidealities

Assume KP ( jω ) is very large over the frequency range of interest.

In fact, assume

— Independent of P(s)!!



Example of Reduced Sensitivity

10)0990)(1000(1

1000)(

0990)(1000)(

1

11

=

+

=

==

. jQ

. jG , j KP

ω

ω ω

1)The use of operational amplifiers

2)Decreasing amplifier gain sensitivity

Example:

(a) Suppose

(b) Suppose

(50% gain change)

0990)(500)( 22 . jG , j KP == ω ω



99)0990)(500(1

500)( 2 ..

jQ ≅

+

=ω (1% gain change)

Fine, but why doesn’t G ( j ω) fluctuate ?

Note:

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

whole system.

⇒

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

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

There is a price:



⇒

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.

The closed-loop gain only depends on the passive components

Then Steady State



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



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



Stabilization of Unstable Systems

• P ( s) — unstable

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

is stable



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

Example #1: First-order unstable systems



Example #2: Second-order unstable systems

— Unstable for all values of K

— Physically, need damping — a term proportional to s⇔ d /dt



Example #2 (continued):

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

— Stable as long as K 2 > 0 (sufficient damping)

and K 1 > 4 (sufficient gain).



Example #2 (one more time):

Why didn’t 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 never know the precisevalues of the poles, it could be 2±∆.

b) Disturbance between the two systems will cause instability.



Demo: Magnetic Levitation

io

= current needed to balance the weight W at the rest height yoForce balance

Linearize about equilibrium with specific values for parameters



— Second-order unstable system



Signals and Systems

Fall 2003Lecture #21

25 November 2003

1. Feedback

a) Root Locus

b) Tracking

c) Disturbance Rejection

d) The Inverted Pendulum

2 I t d ti t th Z T f



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

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.



A Simple Example

Becomes more stable Becomes less stable

Sketch where

pole moves

as |K| increases...

In either case, pole is at so = -2 - K



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 any specific

value of K , 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 of s that aresolutions for some value of K , because:

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

K = 0, ±∞.

2) There are rules on how to connect between these



2) There are rules on how to connect between theselimiting points.

Rules for Plotting Root Locus

• End points

— At K = 0, G( so) H ( so) = ∞

⇒ so are poles of the open-loop system function G( s) H ( s).

— At | K| = ∞, G( so) H ( so) = 0

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

Rule #1:A root locus starts (at K = 0) from a pole of G( s) H ( s) and ends (at

| K| = ∞) at a zero of G( s) H ( s).

Question: What if the number of poles ≠ the number of zeros?

Answer: Start or end at ±∞.



Rule #2: Angle criterion of the root locus

• Thus, s0 is a pole for some positive value of K if:

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

• Similarly s0 is a pole for some negative value of K if:

In this case s0 is a pole if K = -1/|G(s0) H(s0)|



In this case, s is a pole if K -1/|G( s ) H ( s )|.

Example of Root Locus.

One zero at -2,

two poles at 0, -1.



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

for at least some set of input signals.

Tracking

⇓

+= )(

)()(1

1)( s X

s H sC s E

)()()(1

1)( ω

ω ω

ω j X j H jC

j E +

=

We want to be large in frequency bands in which we)()( ω ω jPjC



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

)()( j P jC

Tracking (continued)

Using the final-value theorem

Basic example: Tracking error for a step input



Disturbance Rejection

There may be other objectives in feedback controls due to unavoidabledisturbances.

Clearly, sensitivities to the disturbances D1( s) and D2( s) are much

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 s X s s s X s H sC

s H sC

sY

s

s s H , s s sC

++=+=

+=+=

However

)()33(

2)(

)()(1

)()(

2 s X

s s s

s s X

s H sC

sC sW

++

+=

+=



)()()(Unstable

+++

Inverted Pendulum

— Unstable!



Feedback System to Stabilize the Pendulum

• PI feedback stabilizes θ

– Additional PD feedback around motor / amplifier

centers the pendulum

• Subtle problem: internal instability in x(t)!

a



p

Root Locus & the Inverted Pendulum

• Attempt #1: Negative feedback driving the motor

– Remains unstable!

• Root locus of M(s)G(s)



after K. Lundberg


• Attempt #2: Proportional/Integral Compensator

– Stable for large enough K

• Root locus of K(s)M(s)G(s)



after K. Lundberg


• BUT – x(t) unstable:

System subject to drift...

• Solution: add PD feedback

around motor and

compensator:



after K. Lundberg

The z-Transform

The (Bilateral) z-Transform

Motivation: Analogous to Laplace Transform in CT

We now do not

restrict ourselves

just to z = e jω



The ROC and the Relation Between z T and DTFT

• Unit circle (r = 1) in the ROC ⇒ DTFT X (e jω ) exists

•

— depends only on r = | z |, just like the ROC in s-plane

only depends on Re( s)

, r = |z|



Example #1

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

outside a circle

This form to find

pole and zero locations

This formfor PFE

and

inverse z-

transform

11

1−−

=az a z

z

−=



Example #2:

Same X(z) as in Ex #1 but different ROC



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

Rational z-Transforms

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

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

Polynomials in z



Signals and Systems

Fall 2003Lecture #22

2 December 2003

1. Properties of the ROC of the z-Transform

2. Inverse z-Transform

3. Examples4. Properties of the z-Transform

5. System Functions of DT LTI Systems

a. Causality

b. Stability



y

The z-Transform

• Last time:

•Unit circle (r = 1) in the ROC⇒DTFT X (e jω ) exists

•Rational transforms correspond to signals that are linear

combinations of DT exponentials

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

only depends on Re( s)



Some Intuition on the Relation between z T and LT

Can think of z-transform as DT

version of Laplace transform with

The (Bilateral) z-Transform



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

• LHP in s-plane, Re( s) < 0 ⇒ | z | = | e sT | < 1, inside the | z | = 1 circle.

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

• RHP in s-plane, Re( s) > 0 ⇒ | z | = | e sT | > 1, outside the | z | = 1 circle.

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

• A vertical line in s-plane, Re( s) = constant ⇔ | e sT

| = constant, acircle in z -plane.



Properties of the ROCs of z -Transforms

(1) The ROC of X ( z ) consists of a ring in the z -plane centered about

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

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



More ROC Properties

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

except possibly at z = 0 and/or z = ∞.

Why?

CT counterpartExamples:



ROC Properties Continued

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

all finite values of z for which | z | > r o are also in the ROC.



Side by Side

(6) If x[n] is two-sided, and if | z | = r o is in the ROC, then the ROC

consists of a ring in the z -plane including the circle | z | = r o.

What types of signals do the following ROC correspond to?

right-sided left-sided two-sided

(5) If x[n] is a left-sided sequence, and if | z | = r o is in the ROC,

then all finite values of z for which 0 < | z | < r o are also in the ROC.



Example #1



Example #1 continued

Clearly, ROC does not exist if b > 1⇒ No z -transform for b|n|

.



Inverse z-Transforms

for fixed r:



Example #2

2) When doing inverse z -transform

using PFE, express X ( z ) as afunction of z -1.

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

Note, particular to z -transforms:

1) When finding poles and zeros,

express X ( z ) as a function of z .



ROC I:

ROC III:

ROC II:



Inversion by Identifying Coefficients

in the Power Series

— A finite-duration DT sequence

Example #3:

3

-1

2

0 for all other n’s



Example #4:

(a)

(b)



Properties of z-Transforms

(1) Time Shifting

The rationality of X ( z ) unchanged, different from LT. ROC unchanged

except for the possible addition or deletion of the origin or infinityno> 0⇒ ROC z ≠ 0 (maybe)

no< 0⇒ ROC z ≠ ∞ (maybe)

(2) z -Domain Differentiation same ROC

Derivation:



Convolution Property and System Functions

Y ( z ) = H ( z ) X ( z ) , ROC at least the intersection of

the ROCs of H ( z ) and X ( z ),

can be bigger if there is pole/zero

cancellation. e.g.

H(z) + ROC tells us everything about system



CAUSALITY

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

including z = ∞:

A DT LTI system with system function H ( z ) is causal⇔ the ROC of

H ( z ) is the exterior of a circle including z = ∞

⇓



Causality for Systems with Rational System Functions

A DT LTI system with rational system function H ( z ) is causal

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

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

then



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 of H ( z ) includes

the unit circle | z | = 1

⇒ Frequency Response H (e jω

) (DTFT of h[n]) exists.



Signals and Systems

Fall 2003Lecture #234 December 2003

1. Geometric Evaluation of z-Transforms and DT Frequency

Responses

2. First- and Second-Order Systems

3. System Function Algebra and Block Diagrams

4. Unilateral z-Transforms



Geometric Evaluation of a Rational z-Transform

Example #1:

Example #3:

Example #2:

All same as

in s-plane



Geometric Evaluation of DT Frequency Responses

First-Order System— one real pole



Second-Order System

Two poles that are a complex conjugate pair ( z1

= re jθ = z2

*)

Clearly, | H | peaks near ω = ±θ



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

diagrams, and impulse- & step-responses



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



Feedback System

(causal systems)

System Function Algebra and Block Diagrams

Example #1:

negative feedback

configuration

z-1 D⇔

Delay



Example #2:

— Cascade of

two systems



Unilateral z-Transform

Note:

(1) If x[n] = 0 for n < 0, then

(2) UZT of x[n] = BZT of x[n]u[n] ⇒ ROC always outside a circle

and includes z = ∞

(3) For causal LTI systems,



• 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 (for x1[n<0] = x2[n<0] = 0)

Derivation:Initial condition



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



β = 0 ⇒ System is initially at rest:

ZSR

Example (continued)

α = 0 ⇒ Get response to initial conditions

ZIR

signals and system 6.003fall 2003

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