1 Course Introduction

The concepts and techniques in this course are probably the most useful in engineering. A signal is a function of one or more independent variables conveying information about a physical (or virtual) phenomena. A system may respond to signals to produce other signals, or produce signals directly.

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This course is about the mathematical models and related techniques for the design and understanding of systems as signal transformations. We focus on a broadly useful class of systems, known as linear, time-invariant systems. You will learn about:

the representation and analysis of signals as information carrying channels
and how to analyze and implement linear, time-invariant systems to transform those signals.

1.1 Example Signals and Systems

Electrical Circuits. This is a Sallen-Key filter, a second-order system commonly use to select frequencies from a signal:

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There are two signals we can easily identify, the input signal as the voltage applied across \(x(t)\), and the output voltage measured across \(y(t)\). We build on your circuits course by viewing this circuit as an implementation of a more abstract linear system. We see how it can be viewed as a frequency selective filter. We will see how to answer questions such as: how do we choose the values of the resistors and capacitors to select the frequencies we are interested in? and how do we determine what those frequencies are?

Robotic Joint. This is a Linear, Time-Invariant model of a DC motor, a mixture of electrical and mechanical components.

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How do we convert the motor into a servo for use in a robotic joint? What are its characteristics (e.g. how fast can it move)?

Audio Processing. Suppose you record an interview for a podcast, but during an important part of the discussion, the HVAC turns on and there is an annoying noise in the background.

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How could you remove the noise minimizing distortion to the rest of the audio?

Communications. Consider a wireless sensor, that needs to transmit to a base station, e.g. a wireless mic system.

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How should the signal be processed so it can be transmitted? How should the received signal be processed?

1.2 Types of Problems

Applications of this material occur in all areas of science and engineering. When we have a measured output but are unsure what combination of inputs and system components could have produced it, we have a modeling problem.

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Models are the bedrock of the scientific method and are required to apply the concepts of this course to engineering problems.

When we know the input and the system description and desire to know the output we have an analysis problem.

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Analysis problems are the kind you have encountered most often already. For example, given an electrical circuit and an applied voltage or current, what are the voltages and currents across and through the various components.

When we know either the input and desired output and seek the system to perform this transformation,

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or we know the system description and output and desire the input that would generate the output,

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we have a design problem.

This course focuses on modeling and analysis with applications to electrical circuits and devices for measurement and control of the physical world and is broadly applicable to all ECE majors. Some Examples:

Controls, Robotics, & Autonomy: LTI systems theory forms the basis of perception and control of machines.
Communications & Networking: LTI systems theory forms the basis of transmission and reception of signals, e.g. AM and FM radio.
Machine Learning: LTI systems are often used to pre-process samples or to create basis functions to improve learning.
Energy & Power Electronic Systems: linear circuits are often modeled as LTI systems.

Subsequent courses, e.g. ECE 3704, focus more on analysis and design.

1.3 Learning Objectives

The learning objectives (LOs) for the course are:

Describe a given system using a block-level description and identify the input/output signals.
Mathematically model continuous and discrete linear, time-invariant systems using differential and difference equations respectively.
Analyze the use of filters and their interpretation in the time and frequency domains and implement standard filters in hardware and/or software.
Apply computations of the four fundamental Fourier transforms to the analysis and design of linear systems.
Communicate solutions to problems and document projects within the domain of signals and systems through formal written documents.

These are broken down further into the following topic learning objectives (TLOs). The TLOs generally map onto one class meeting but are used extensively in later TLOs.

Course introduction (OW Forward and §1.0)
1. Signals as models
2. Systems as transformation of signals
3. Prerequisites
Continuous-time (CT) signals (OW §1.1 through 1.4 and 2.5): A continuous-time (CT) signal is a function of one or more independent variables conveying information about a physical phenomena. This lecture gives an introduction to continuous-time signals as functions. You learn how to characterize such signals in a number of ways and are introduced to two very important signals: the unit impulse and the complex exponential.
1. Continuous-time signals as functions \(\mathbb{R}\mapsto\mathbb{C}\)
2. Transformations of time
3. Characterizing signals
  1. periodic/aperiodic
  2. even/odd
  3. energy or/nor power
4. Impulse function
5. Step function
6. Complex exponential
Discrete-time (DT) signals (OW §1.1 through 1.4)
1. Discrete-time signals as functions \(\mathbb{Z}\mapsto\mathbb{C}\)
2. Transformations of time index
3. Characterizing signals
  1. periodic/aperiodic
  2. even/odd
  3. energy or/nor power
4. Impulse function
5. Step function
6. Complex exponential
CT systems as linear constant coefficient differential equations (OW §2.4.1)
1. LCCDE and their solution (1st and 2nd order)
2. impulse response from LCCDE
DT systems as linear constant coefficient difference equations (OW §2.4.2)
1. LCCDE and their solution (1st and 2nd order)
2. impulse response from LCCDE
Linear time invariant CT systems (OW §1.5, 1.6, 2.3)
1. Memory
2. Invertability
3. Causality
4. Stability
5. Time-invariance
6. Linearity
7. Define LTI system
Linear time invariant DT systems (OW §1.5, 1.6, 2.3)
1. Memory
2. Invertability
3. Causality
4. Stability
5. Time-invariance
6. Linearity
7. Define LTI system
CT convolution (OW §2.2)
1. Review CT LTI systems and superposition property
2. CT Convolution Integral
3. Properties of convolution
  1. communative
  2. distributive
  3. associative
4. Determining system response using convolution with impulse response
DT convolution (OW §2.1)
1. Review DT LTI systems and superposition property
2. DT Convolution Sum
3. Properties of convolution
  1. communative
  2. distributive
  3. associative
4. Determining system response using convolution with impulse response
CT block diagrams (OW §1.5.2 and 2.4.3)
1. blocks represented by impulse response
2. series and parallel connections, reductions
3. scale, sum, and integrator blocks
4. equivalence of LCCDE’s and block diagrams
5. first-order differential equation as feedback motif
6. second-order differential equation as a feedback motif
7. implementing a LCCDE using adders, multipliers, and integrators
DT block diagrams (OW §1.5.2 and 2.4.3)
1. blocks represented by impulse response
2. series and parallel connections, reductions
3. scale, sum, and unit delay blocks
4. equivalence of LCCDE’s and block diagrams
5. first-order difference equation as feedback motif
6. second-order difference equation as a feedback motif
7. implementing a LCCDE using adders, multipliers, and delays
Eigenfunctions of CT systems (OW §3.2 and 3.8)
1. Eigenfunction \(e^{st}\)
2. Transfer Function \(H(s)\)
3. Stability and Frequency Response (FR) \(H(j\omega)\)
4. How this is useful - decomposition of input signal into complex exp
5. What signals can be decomposed this way, foreshadow Fourier Analysis
Eigenfunctions of DT systems (OW §3.2 and 3.8)
1. Eigenfunction \(z^{n}\)
2. Transfer Function \(H(z)\)
3. Stability and Frequency Response (FR) \(H\left(e^{j\omega}\right)\)
4. How this is useful - decomposition of input signal into complex exp
5. What signals can be decomposed this way, foreshadow Fourier Analysis
CT Fourier Series representation of signals (OW §3.3 through 3.5)
1. review CT periodic functions
2. harmonic sums
3. derive synthesis equation
4. derive analysis equation
5. spectrum plots
6. define mean-square convergence
7. truncated CT FS
8. stable LTI system response using CTFS
9. example of the impulse train (for sampling theory later)
10. formal Dirichlet conditions
11. properties of CT FS
DT Fourier Series representation of signals (OW §3.6 and 3.7)
1. review DT periodic functions
2. harmonic sums
3. derive synthesis equation
4. derive analysis equation
5. spectrum plots
6. stable LTI system response using DTFS
7. properties of DT FS
CT Fourier Transform (OW §4.0 through 4.7)
1. derive the CTFT pair from the CTFS
2. Dirichlet existence conditions
3. CTFT of the CTFS
4. Properties of the CT Fourier Transform
  1. linearity
  2. time shift
  3. conjugacy
  4. integration and differentiation: application to LCCDE \(\mapsto\) CTFR
  5. time scaling
  6. duality
  7. convolution: stable LTI system response using CTFT
  8. multiplication/modulation
  9. application of the properties in combination
DT Fourier Transform (OW §5.0 though 5.8)
1. derive the DTFT from DTFS
2. DTFT of DTFS
3. Properties of the DT Fourier Transform
  1. periodicity
  2. linearity
  3. index-shift: application to LCCDE \(\mapsto\) DTFR
  4. frequency shift
  5. conjugation
  6. finite difference and accumulation
  7. interpolation /index expansion
  8. frequency differentiation
  9. Parseval’s
  10. convolution: stable LTI system response using DTFT
  11. multiplication/modulation
  12. application of the properties in combination
CT Frequency Response (OW §6.1, 6.2, 6.5)
1. review CTFR as CTFT of impulse response
2. review CTFR to/from LCCDE
3. review CTFR to/from block diagram
4. magnitude-phase representation of the frequency response
5. frequency response acting on sinusoids
6. Bode plots
  1. why plot it this way: dB units and log time axis
  2. how to read them (not construct them manually)
  3. Bode plots in software, e.g. Matlab/Python/Julia
7. CTFR of first and second order systems
DT Frequency Response (OW §6.1, 6.2, 6.6)
1. review DTFR as DTFT of impulse response
2. review DTFR to/from LCCDE
3. review DTFR to/from block diagram
4. magnitude-phase representation of the frequency response
5. frequency response acting on sinusoids
6. DTFR plots
  1. periodicity
  2. dB units
  3. DTFR plots in software, e.g. Matlab/Python/Julia
7. DTFR of first and second order systems
Frequency Selective Filters in CT (OW §3.9, 3.10, 6.3, 6.4)
1. ideal low-pass
2. ideal high-pass
3. ideal bandpass
4. ideal notch/bandstop
5. practical filters
6. transformations
7. first and second order systems as building blocks
  1. review LCCDE representation
  2. review block diagram representation
  3. review CTFR representation
  4. CT 1st order RC+buffer
  5. CT Sallen-key
Frequency Selective Filters in DT (OW §3.11, 6.3, 6.4)
1. ideal low-pass
2. ideal high-pass
3. ideal bandpass
4. ideal notch/bandstop
5. practical filters
6. transformations
7. first and second order systems as building blocks
  1. review LCCDE representation
  2. review block diagram representation
  3. review DTFR representation
  4. DT 1st order implementation in code
  5. DT 2nd order implementation in code
The Discrete Fourier Transform
1. time window the DTFT to get the DFT
2. interpreting the index axis as DT and CT frequency
3. zero-padding
4. offline or batched filtering using the DFT
5. briefly mention fast algorithms to compute the DFT = FFT
Sampling (OW §7.1, 7.3, 7.4)
1. sampling using the impulse train
2. derive the Nyquist rate
3. effects of aliasing
4. practical ADC (sample and hold, SAR, bit-width)
5. designing anti-aliasing filters
Reconstruction (OW §7.2)
1. reconstruction as removal of images
2. reconstruction as interpolation
3. practical DAC: R-2R ladder
4. designing reconstruction filters

1.4 Graphical Outline

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