Signals and Systems Spring Lecture #1 Jacob White (Slides thanks to A

Signals and Systems Spring Lecture #1 Jacob White (Slides thanks to A. Willsky, T. Weiss, Q. Hu, and D. Boning) “Content, Figures and images used in these lecture notes by permission, copyright 2005 by Alan V. Oppenheim and Alan S. Willsky”

Signals and Systems

Different Types of Signals

Signal Classification
Type of Independent Variable

Cervical MRI

Independent Variable Dimensionality

Continuous Time (CT) and Discrete-Time (DT) Signals

Mandril Example Blurred Image

Unblurred Image – No Noise
Mandril Example Unblurred Image – No Noise

Unblurred Image – 0.1% Noise
Mandril Example Unblurred Image – 0.1% Noise

Real and Complex Signals

Periodic and A-periodic Signals

Right- and Left-Sided Signals

Bounded and Unbounded Signals

Even and Odd Signals

Building Block Signals
Eternal Complex Exponentials

Why are eternal complex exponentials so important

Cervical Spine MRI

Unit Impulse Function

Narrow Pulse Approximation

Intuiting Impulse Definition

Uses of the Unit Impulse

Robot Arm System

Unit Step Function

Successive Integrations of the Unit Impulse Function

Building Block Signals can be used to create a rich variety of Signals

Conclusions

Signals and Systems Spring Lecture #2 Jacob White (Slides thanks to A. Willsky, T. Weiss, Q. Hu, and D. Boning) “Figures and images used in these lecture notes by permission, copyright 2005 by Alan V. Oppenheim and Alan S. Willsky”

Outline - Systems How do we construct complex systems
Using Hierarchy Composing simpler elements System Representations Physical, differential/difference Equations, etc. System Properties Causality, Linearity and Time-Invariance

Hierarchical Design Robot Car

Robot Car Block Diagram
Top Level of Abstraction

Wheel Position Controller Block Diagram
2nd Level of the Hierarchy

Motor Dynamics Differential Equations
3nd Level of the Hierarchy

Observations If we “flatten” the hierarchy, the system becomes very complex Human designed systems are often created hierarchically. Block input/output relations provide communication mechanisms for team projects

Mechanics - Sum Element Forces
Compositional Design Mechanics - Sum Element Forces

Circuit - Sum Element Currents

System Representation
Differential Equation representation Mechanical and Electrical Systems Dynamically Analogous Can reason about the system using either physical representation.

Integrator-Adder-Gain Block Diagram

Four Representations for the same dynamic behavior
Pick the representation that makes it easiest to solve the problem

Discrete-Time Example - Blurred Mandril
Blurrer (system model) Blurred Image Original Image Deblurrer System Deblurred Image

Difference Equation Representation
Difference Equation Representation of the model of a Blurring System Deblurring System 48 Note Typo on handouts How do we get ?

Observations CT System representations include circuit and mechanical analogies, differential equations, and Integrator-Adder-Gain block diagram. Discrete-Time Systems can be represented by difference equations. The Difference Equation representation does not help us design the mandril deblurring New representations and tools for manipulating are needed!

System Properties Important practical/physical implications
Help us select appropriate representations They provide us with insight and structure that we can exploit both to analyze and understand systems more deeply.

Causal and Non-causal Systems

Observations on Causality
A system is causal if the output does not anticipate future values of the input, i.e., if the output at any time depends only on values of the input up to that time. All real-time physical systems are causal, because time only moves forward. Effect occurs after cause. (Imagine if you own a noncausal system whose output depends on tomorrow’s stock price.) Causality does not apply to spatially varying signals. (We can move both left and right, up and down.) Causality does not apply to systems processing recorded signals, e.g. taped sports games vs. live broadcast.

Linearity

Key Property of Linear Systems
Superposition If Then

Linearity and Causality
A linear system is causal if and only if it satisfies the conditions of initial rest: “Proof” Suppose system is causal. Show that (*) holds. Suppose (*) holds. Show that the system is causal.

Time-Invariance Mathematically (in DT): A system x[n]  y[n] is TI if for any input x[n] and any time shift n0, If x[n]  y[n] then x[n - n0]  y[n - n0] . Similarly for CT time-invariant system, If x(t)  y(t) then x(t - to)  y(t - to) .

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

Example - Multiplier

Multiplier Linearity

Multiplier – Time Varying

Example – Constant Addition

Linear Time-Invariant (LTI) Systems
Focus of most of this course - Practical importance (Eg. #1-3 earlier this lecture are all LTI systems.) - The powerful analysis tools associated with LTI systems A basic fact: If we know the response of an LTI system to some inputs, we actually know the response to many inputs

Example – DT LTI System

Conclusions

Amazing Property of LTI Systems

Outline Superposition Sum for DT Systems
Representing Inputs as sums of unit samples Using the Unit Sample Response Superposition Integral for CT System Use limit of tall narrow pulse Unit Sample/Impulse Response and Systems Causality, Memory, Stability

Representing DT Signals with Sums of Unit Samples

Written Analytically Note the Sifting Property of the Unit Sample
Coefficients Basic Signals Note the Sifting Property of the Unit Sample

The Superposition Sum for DT Systems
Graphic View of Superposition Sum

Derivation of Superposition Sum

Convolution Sum

Convolution Notation Notation is confusing, should not have [n]
takes two sequences and produces a third sequence makes more sense Learn to live with it.

Convolution Computation Mechanics

DT Convolution Properties
Commutative Property

Associative Property

Distributive Property
+

Delay Accumulation

Superposition Integral for CT Systems
Graphic View of Staircase Approximation

Tall Narrow Pulse

Derivation of Staircase Approximation of Superposition Integral

The Superposition Integral

Sifting Property of Unit Impulse

CT Convolution Mechanics

CT Convolution Properties

Computing Unit Sample/Impulse Responses
Circuit Example

Narrow pulse approach

Narrow pulse response

Narrow pulse response cont’d

Convergence of Narrow pulse response

Alternative Approach – Use Differentiation

Alternative Approach – Use Differentiation cont’d

How to measure Impulse Responses

Unit Sample/Impulse Responses of Different Classes of Systems

Bounded-Input Bounded-Output Stability

Conclusions

Outline Unfiltering (Deconvolving)
Candidate Unit Sample Responses Causality and BIBO stability Additional Conditions for Difference Equations Compute Impulse responses of a CT System Use limit of tall narrow pulse Differentiating the step response

“Unfiltering an Audio Signal”
Problem Statement “Filtered” Audio Original Audio

DT LTI System Specification?
Unit Sample Response! 1.0 n -0.9 We know how the Audio Signal was filtered

Block Diagram DT LTI Filtered Audio Original Audio “Unfiltered” Audio

Associative Property

Use Associative Property
Make an Identity System

A Candidate Unfilterer
1.0 -0.9 n ......

Unit Sample/Impulse Responses of Different Classes of Systems

Bounded-Input Bounded-Output Stability

Candidate Unfilterer is Causal and definitely not BIBO stable
......

Give Up Causality to Gain Stabilty
1.0 -0.9 ...... n

Convolving is expensive
...... m One output point 2N terms in summation

Use A Difference Equation
Cheap to generate y[n] Two possible unit sample responses

A Difference Equation Does Not Uniquely Determine a system!
Need Additional Conditions: Causal or Anti-causal BIBO Stability Some values of the output OR OR

Computing Impulse Responses
Circuit Example

Narrow pulse approach

Narrow pulse response

Narrow pulse response cont’d

Convergence of Narrow pulse response

Alternative Approach – Use Differentiation

Alternative Approach – Use Differentiation cont’d

How to measure Impulse Responses

Conclusions Unfiltering (Deconvolving)
There can be multiple deconvolving systems Causality and BIBO stability are issues Difference Equations need similar constraints Computing Impulse responses of a CT System Use limit of tall narrow pulse Differentiating the step response

Outline Complex Exponentials as Eigenfunctions of LTI Systems
Fourier Series representation of CT periodic signals How do we calculate the Fourier coefficients? Convergence and Gibbs’ Phenomenon

The eigenfunctions k(t) and their properties
(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. The solution is simple, general, and insightful.

What are the eigenfunctions of LTI systems?
Two Key Questions What are the eigenfunctions of LTI systems? What kinds of signals can be expressed as superpositions of these eigenfunctions?

A specific LTI system can have more than one type of eigenfunction
Ex. #1: Identity system Any function is an eigenfunction for this LTI system. Any periodic function x(t) = x(t+T) is an eigenfunction. Ex. #2: A delay Ex. #3: h(t) even (for this system)

Complex Exponentials are the only Eigenfunctions of any LTI Systems
that work for any and all Complex Exponentials are the only Eigenfunctions of any LTI Systems eigenvalue eigenfunction eigenvalue eigenfunction

CT & DT Fourier Series and Transforms
What kinds of signals can we represent as “sums” of complex exponentials? For Now: Focus on restricted sets of complex exponentials CT: DT: Magnitude 1 131 ß CT & DT Fourier Series and Transforms Periodic Signals

Joseph Fourier ( )

Fourier Series Representation of CT Periodic Signals
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 Euler's relation (memorize!) 0 – no dc component

(periodic) For real signals, there are two other commonly used forms for CT Fourier series: Because of the eigenfunction property of ejt, we will usually use the complex exponential form in - A consequence of this is that we need to include terms for both positive and negative frequencies:

Now, the complete answer to Question #1

Ex: Periodic Square Wave
DC component is just the average

Convergence of CT Fourier Series
How can the Fourier series for the square wave possibly make sense? 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 2. In a finite time interval, x(t) has a finite number of maxima and minima. Ex. An example that violates Condition 2. And Condition 3. In a finite time interval, x(t) has only a finite number of discontinuities. Ex. An example that violates Condition 3. And

Dirichlet conditions are met for most of 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 Still, convergence has some interesting characteristics: - As N ∞, xN(t) exhibits Gibbs’ phenomenon at points of discontinuity Demo: Fourier Series for CT square wave (Gibbs phenomenon).

Conclusions Exponentials are Eigenfunctions of CT Systems
zn are Eigenfunctions of DT Systems Periodic CT Functions Can Be Represented with a Fourier Series Synthesis and analysis formulas using orthogonality Series convergence for square wave

Representation with Fourier Series
CT Fourier Series Energy View and Parsevals Convergence Rate Periodic Impulse Train Differentiation, Linearity and Time Shift DT Fourier Series Finite Frequency Set System of Equations

Fourier Series for Periodic CT Functions

Two Uses for Fourier Series
Use Eigenfunction property to simplify LTI system analysis Compress Data using truncated Fourier Series

Fourier Representation Issue
Not Infinite Enough Uncountably Infinite number of “points” Countably Infinite number of Fourier Coefficients

Energy View Signals are equal “Energy-wise”
Fourier Series Preserves Energy (Parsevals) 153

Truncated Fourier Series for Square Wave

Square Wave From Series Matches in Energy
x(t) Equal energy-wise Not equal pointwise

Fourier Convergence Rate: Useful Result
Impulse Train or Sampling Function

Truncated Fourier Series for Impulse Train

Truncated Impulse Train Series Cont’d

Fourier Convergence Rate: Useful Properties
Linearity Time Differentiation and Integration Time Shift

Square Wave Convergence Rate
Derivative of a square wave is an impulse train

Triangle Wave Convergence Rate
Second derivative of a triangle wave is an impulse train

Fourier Series Representation of DT Periodic Signals
x[n] - periodic with fundamental period N Only ejn which is periodic with period N will appear in the FS There are only N distinct signals of this form So we could just use However, it is often useful to allow the choice of N consecutive values of k to be arbitrary (E.g. choose {-(N-1)/2, (N-1)/2} if N is odd and x[n] has definite parity).

DT Fourier Series Representation
{ak} Fourier (series) coefficients Questions: What DT periodic signals have such a representation? How do we find ak?

Computing Fourier Series Coefficients
Any DT periodic signal has a Fourier series representation

Using Orthogonality to Solve for ak
Finite geometric series

Computing Coefficients Cont’d

DT Fourier Series Pair Note: It is convenient to think ak as being defined for all integers k. So: ak+N = ak — Unique for DT. Since x[n] = x[n+N], there are only N pieces of information, whether in time-domain or in frequency-domain. We only use N consecutive values of ak in the synthesis equation

Example: DT Rectangular Wave

Conclusions CT Fourier Series DT Fourier Series
Fourier Series Converge energy-wise not ptwise Coefficient Decay Periodic Impulse Train Coeffs do not decay Decay rate related to differentiability DT Fourier Series Finite Frequency Set System of Equations

Frequency Response Filters DT Filters Application Examples
Types of filters High Pass, Low Pass DT Filters Finite Unit Sample Response - FIR Infinite Unit Sample Response - IIR

The Eigenfunction Property of Complex Exponentials
DT:

Fourier Series and LTI Systems

The Frequency Response of an LTI System

Finite Set of Discrete-Time Frequencies

Frequency Shaping and Filtering
By choice of H(jw) (or H(ej)) as a function of , we can shape the frequency composition of the output Preferential amplification Selective filtering of some frequencies Audio System Adjustable Filter Equalizer Speaker Bass, Mid-range, Treble controls For audio signals, the amplitude is much more important than the phase.

Signal Processing with Filters

Touch Tone Phone

Low and High Pass Circuit Filters
Y X Y X

Frequency Selective Filters
— Filter out signals outside of the frequency range of interest Lowpass Filters: only look at amplitude here.

Highpass Filters Remember:

Bandpass Filters BPF if LP HP cl ch

Idealized Filters CT DT
c — 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 lower cut-off upper cut-off DT

DT Averager/Smoother LPF

Nonrecursive DT (FIR) filters
Rolls off at lower as M+N+1 increases

Simple DT “Edge” Detector — DT 2-points “differentiator”
Passes high-frequency components

Edge enhancement using DT differentiator

Notch Filter - IIR

Conclusions Filters DT Filters Signal Seperation Types of filters
High Pass, Low Pass DT Filters FIR Edge Detector IIR Notch

System Frequency Response and Unit Sample Response
Fourier Transform System Frequency Response and Unit Sample Response Derivation of CT Fourier Transform pair Examples of Fourier Transforms Fourier Transforms of Periodic Signals Properties of the CT Fourier Transform

The Frequency Response of an LTI System

First Order CT Low Pass Filter
Direct Solution of Differential Equation

Using Impulse Response
Note map from unit sample response to frequency response

Fourier’s Derivation of the CT Fourier Transform
x(t) - an aperiodic signal view it as the limit of a periodic signal as T ! 1 For a periodic sign, the harmonic components are spaced w0 = 2p/T apart ... as T  and o  0, then w = kw0 becomes continuous  Fourier series  Fourier integral

Discrete frequency points become
Square Wave Example Discrete frequency points become denser in  as T increases

“Periodify” a non-periodic signal
For simplicity, assume x(t) has a finite duration.

Fourier Series For Periodified x(t)

Limit of Large Period

What Signals have Fourier Transforms?
(1) x(t) can be of infinite duration, but must satisfy: a) Finite energy In this case, there is zero energy in the error b) Dirichlet conditions (including ) c) By allowing impulses in x(t) or in X(j), we can represent even more signals

Fourier Transform Examples
Impulses (a) (b)

Fourier Transform of Right-Sided Exponential
Even symmetry Odd symmetry

Fourier Transform of square pulse
Note the inverse relation between the two widths  Uncertainty principle Useful facts about CTFT’s

Fourier Transform of a Gaussian
(Pulse width in t)•(Pulse width in ) ∆t•∆ ~ (1/a1/2)•(a1/2) = 1

CT Fourier Transforms of Periodic Signals

Fourier Transform of Cosine

Impulse Train (Sampling Function)
Note: (period in t) T (period in ) 2/T

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 Or Odd When x(t) is real (all the physically measurable signals are real), the negative frequency components do not carry any additional information beyond the positive frequency components:  ≥ 0 will be sufficient.

More Properties 4) Time-Scaling x(t) real and even x(t) real and odd

Conclusions System Frequency Response and Unit Sample Response
Derivation of CT Fourier Transform pair CT Fourier Transforms of pulses, exponentials FT of Periodic Signals  Impulses Time shift, Scaling, Linearity

Fourier Transform Fourier Transforms of Periodic Functions
The convolution Property of the CTFT Frequency Response and LTI Systems Revisited Multiplication Property and Parseval’s Relation

Fourier Transform Duality
Fourier Transform Synthesis and Analysis formulas

Narrow in Time – Wide in Frequency

CT Fourier Transforms of Periodic Signals

Fourier Transform of Cosine

Impulse Train (Sampling Function)

Convolution Property A consequence of the eigenfunction property:

Reminder

Computing General Responses

Periodic Response Using Fourier Transforms
The frequency response of a CT LTI system is simply the Fourier transform of its impulse response

Ideal Low Pass Filter Example

Cascading Nonideal Filters

Cascading Ideal Filters
Cascading Gaussians GaussianGaussian = Gaussian ) GaussianGaussian = Gaussian

Differentiation Property
1) Amplifies high frequencies (enhances sharp edges)

Rational, can use PFE to get h(t) If X(j) is rational e.g.
LTI Systems of LCCDE’s (Linear-constant-coefficient differential equations) Using the Differentiation Property Rational, can use PFE to get h(t) If X(j) is rational e.g. then Y(j) is also rational

Multiplication Property
Parseval’s Relation Multiplication Property Since FT is highly symmetric, thus if then the other way around is also true — Definition of convolution in  — A consequence of Duality

Examples of the Multiplication Property
+

Example (continued)

Conclusions Fourier Transforms of Periodic Functions are series of impulses Convolution in Time, Multiplication in Fourier and vice-versa. Fourier picture makes filters easy to manipulate Multiplication Property and Parseval’s Relation

Discrete-Time Fourier Transform
The DT Fourier Transform Examples of the DT Fourier Transform Properties of the DT Fourier Transform The Convolution Property and its Implications and Uses

The Discrete-Time Fourier Transform

DTFT Derivation (Continued)
DTFS synthesis eq. DTFS analysis eq. Define

DTFT Derivation (Home Stretch)

DT Fourier Transform Equations
– Analysis Equation – DTFT – Synthesis Equation – Inverse DTFT

Transform Requirements
Synthesis Equation: Finite Integration Interval, X must be finite Analysis Equation: Need conditions analogous to CTFT, e.g. — Finite energy — Absolutely summable

Examples Unit Samples

Decaying Exponential Infinite sum formula

Rectangular Pulse FIR LPF

Ideal DT LPF

DTFTs of Periodic Functions
Complex Exponentials Recall CT result: What about DT: We expect an impulse (of area 2) at o But X(ej) must be periodic with period 2 In fact Note: The integration in the synthesis equation is over 2π period, only need X(ej) in one 2π period. Thus,

DTFT of General Periodic Functions Using FS
by superposition

DTFT of Sine Function

DTFT of DT Unit Sample Train
— Also periodic impulse train – in the frequency domain!

Properties of the DT Fourier Transform
Periodicity and Linearity — Different from CTFT

Time and Frequency Shifting
Example — Important implications in DT because of periodicity

Time Reversal and Conjugate Symmetry

Time Expansion 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] 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 Insert two zeros in this example

Time Expansion (continued)
— Stretched by a factor of k in time domain — compressed by a factor of k in frequency domain

Differentiation and Parsevals
8) Differentiation in Frequency Differentiation in frequency Multiplication by n Total energy in time domain Total energy in frequency domain 9) Parseval’s Relation

The Convolution Property

Ideal Low Pass Filter (Reminder)

Composing Filters Using DTFT

Conclusions DTFT maps from discrete-time function to continuous frequency function. DTFT is 2pi periodic. Many similar properties to CTFT (linearity, time-frequency shifting) with small differences Convolution in Time, Multiplication in Fourier and vice-versa. Fourier picture makes filters easy to manipulate

Fourier Transform Review
The DT and CT Fourier Transforms Examples of DT and CT Transforms DT and CT Properties Next time – Sampling (Chapter 7)

DT and CT Fourier Transform Equations
DT Transform CT Transform Discrete  Continuous Transform 2pi periodic Continuous Continuous Strong Duality

Square pulse in time and LPF in frequency domain
Examples Square pulse in time and LPF in frequency domain

DTFT of Rectangular Pulse
FIR LPF

Ideal DT LPF

CT Fourier Transform of Cosine

DTFT of Sine Function

CT FT Impulse Train (Sampling Function)

DTFT of DT Unit Sample Train
— Also periodic impulse train – in the frequency domain!

FT Convolution Property
FT Properties FT Convolution Property CT DT

CT Convolution Example

DT Convolution Example

Differentiation/Shift Property
CT DT

Using the Differentiation Property
CT LTI Systems of LCCDE’s (Linear-constant-coefficient differential equations) Using the Differentiation Property Rational, can use PFE to get h(t) If X(j) is rational e.g. then Y(j) is also rational

DT LTI System Described by LCCDE’s
Rational function of e-j, use PFE to get h[n]

CT Multiplication Property
The CTFT has the duality property, thus if then the other way around is also true

Examples of the CTFT Multiplication Property
+

CTFT Multiplication Property Example (continued)

DTFT Multiplication Property

Example:

CT and DT Parsevals CT: DT: Total energy in time domain
Total energy in frequency domain

DT Time and Frequency Shifting
Example — Important implications in DT because of periodicity

DT Time Expansion 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] 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 Insert two zeros in this example

DT Time Expansion (continued)
— Stretched by a factor of k in time domain — compressed by a factor of k in frequency domain

Conclusions DTFT maps from discrete-time function to continuous frequency function CTFT maps from a continuous function to a continuous function. CTFT has duality. DTFT is 2pi periodic. DTFT and CTFT have many properties in common. Next time – Sampling (Chapter 7).

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