ECE 280/Concept List/F26

Lecture 1 - 8/28 - Class introduction; basic signal classifications

Class logistics and various resources on Canvas
Systems will often be represented with block diagrams. System operations for linear, time-invariant (more on that later) systems may be characterized in the frequency domain using transfer functions.
Review of ACSS, phasors, transfer functions
Signal classifications
- Dimensionality ( $$ x(t) $$ , $$ g(x, y) $$ , etc)
- Continuous versus discrete
- Analog versus digital and/or quantized
- Periodic versus aperiodic
- Generally $$ f(t)=f(t+nT) $$ for all integers n (i.e. $f(t)=f(t+nT), n\in \mathbb{Z}$ ). The period $$ T $$ (sometimes called the fundamental period $$ T_0 $$ ) is the smallest value for which this relation is true
- A periodic signal can be defined as an infinite sum of shifted versions of one period of the signal: $f(t)=\sum_{k=-\infty}^{\infty}\hat{f}(t-kT)$ where $\hat{f}(t)$ is only possibly nonzero within one particular period of the signal and 0 outside of that period.

Lecture 2 - 8/26 - Power and energy signals, Even and odd signals and components, unit steps and ramps

Energy, power, or neither
- Energy signals have a finite amount of energy: $ E_{\infty}=\int_{-\infty}^{\infty}|x(\tau)|^2\,d\tau<\infty $
  - Examples: Bounded finite duration signals; exponential decay
  - $E_{\infty}$ of $Ae^{-at}u(t)$ is $\frac{A^2}{2a}$
- Power signals have an infinite amount of energy but a finite average power over all time: $ P_{\infty}=\lim_{T\rightarrow\infty}\frac{1}{T}\int_{-T/2}^{T/2}|x(\tau)|^2\,d\tau=\lim_{T\rightarrow\infty}\frac{1}{2T}\int_{-T}^{T}|x(\tau)|^2\,d\tau<\infty $ and $ E_{\infty}=\infty $
  - Examples: Bounded infinite duration signals, including periodic signals
  - For periodic signals, only need one period (that is, remove the limit and use whatever period definition you want): $P_{\infty}=\frac{1}{T}\int_{T}|x(\tau)|^2\,d\tau$
  - $P_{\infty}$ of $A\,\cos(\omega t+\phi)$ is $\frac{A^2}{2}$ .
- If both the energy and the overall average power are infinite, the signal is neither an energy signal nor a power signal.
  - Examples: Certain unbounded signals such as $$ x(t)=e^t $$
Useful math shortcut
- For a trapezoidal pulse $x(t)=\begin{cases}mt+b, &0<t\leq\Delta t\\0,&\mathrm{otherwise}\end{cases}$ where $x(0)=b=H_1,~x(\Delta t)=b+m\,\Delta t=H_2$ the energy is: $E=\frac{(b+m\,\Delta t)^3-b^3}{3m}=\frac{H_1^2+H_1H_2+H_2^2}{3}\Delta t$
- For a rectangular pulse where $$ H_1=H_2=A $$ , this yields: $E=A^2\,\Delta t$
- For a triangle pulse where $$ H_1=0 $$ and $$ H_2=A $$ , this yields: $E=\frac{1}{3}A^2\,\Delta t$
Singularity functions - see Singularity_Functions and specifically Singularity_Functions#Accumulated_Differences
Even and Odd
- Purely even signals: $$ x(t)=x(-t) $$ (even powered polynomials, cos, $$ |t| $$ )
- Purely odd: $$ x(t)=x(-t) $$ (odd-powered polynomials, sin)
- Even component: $\mathcal{Ev}\{x(t)\}=x_e(t)=\frac{x(t)+x(-t)}{2}$
- Odd component: $\mathcal{Od}\{x(t)\}=x_o(t)=\frac{x(t)-x(-t)}{2}$
- $$ x_e(t)+x_o(t)=x(t) $$
- The even and odd components of $x(t)=e^{at}$ end up being $\cosh(at)$ and $\sinh(at)$
- The even and odd components of $x(t)=e^{j\omega t}$ end up being $\cos(\omega t)$ and $j\,\sin(\omega t)$
Steps and ramps
- Unit step: $ {\displaystyle u(t)=\begin{cases}1, t>0\\0, t<0\end{cases}} $
  - Basic integrals with steps
- Unit ramp: $r(t)=\int_{-\infty}^{t}u(\tau)\,d\tau=\begin{cases}t, t>0\\0, t<0\end{cases}$

Lecture 3 - 8/31 - More with steps and ramps; more with periodic signals

Integrals with unit steps - figure out when integrand might be non-zero and work from there
- See Singularity_Functions and especially Singularity_Functions#General_Simplification_of_Integrals and Singularity_Functions#Convolution_Integral_Simplification_with_Step_Function_Product_as_Part_of_Integrand
Properties of sums and products of even and/or odd signals
Properties of combinations of periodic and/or aperiodic signals
- The sum or difference of two periodic signals will be periodic if their periods are commensurable (i.e. if their periods form a rational fraction) or if any aperiodic components are removed through addition or subtraction.
- The period of a sum of periodic signals will be at most the least common multiple of the component signal periods; the actual period could be less than this period depending on interference
- The product of two signals with periodic components will have elements at frequencies equal to the sums and differences of the frequencies in the first signal and the second signal. If the periods represented by those components are commensurable, the signal will be periodic, and again the upper bound on the period will be the least common multiple of the component periods.
- Best bet is to combine the signals, determine the angular frequencies of each component, and determine if all pairs of frequencies are commensurable; if they are, find the largest number that can be multiplied by integers to get all the component frequencies - that number is the fundamental frequency $\omega_0$ .
Signal transformations
- $z(t)=K\,x(\pm a(t-t_0))+c$ with
- $$ K $$ : vertical scaling factor
- $\pm a$ : time scaling (with reversal if negative); $$ |a|>1 $$ speeds things up / compresses the signal while $$ |a|<1 $$ slows things down / expands the signal
- $$ t_0 $$ : time shift
- $$ c $$ : vertical shift
- Get into the form above first; for example, rewrite $3\,x\left(\frac{t}{2}+4\right)$ as $3\,x\left(\frac{1}{2}(t+8)\right)$ first

Lecture 4 - 9/4 - P/E for transformation, Impulses and integration with impulses, Integration with impulses and steps, Introduction to discrete signals

Power and Energy transformations
- If $$ x(t) $$ is an energy signal with total energy $$ E_x $$ , and $$ y(t)=Kx(a(t-t_0)) $$ , then $E_y=\frac{K^2}{|a|}E_x$
- If $$ x(t) $$ is an power signal with average power $$ P_x $$ , and $$ y(t)=Kx(a(t-t_0)) $$ , then $$ P_y=K^2 P_x $$
Definition of the impulse function: Area of 1 at time 0; 0 elsewhere
- Sifting property - figure out when $\delta$ fires off, see if that argument happens or if there are restrictions based on integral limits
Discrete signals $ x[n] $ versus continuous $ x(t) $
- Discrete: steps and impulses
  - Unit step: $u[n]=\begin{cases}1, n\ge 0\\0, n<0\end{cases}$
  - Unit impulse: $\delta[n]=u[n]-u[n-1]=\begin{cases}0, n\neq 0\\1, n=0\end{cases}$
- Periodic discrete signals must have integer periods; if the continuous version is periodic with some rational period $$ T=a/b $$ with $$ a $$ and $$ b $$ both being integers, the period $$ N $$ of the discrete version is $$ a $$ .
- Some weirdness can occur (for instance, $\sin[2\pi n]$ is 0 everywhere
- Signals can radically different periods when discretely sampled
- $ \cos[1.9\pi n] $ has a period of $ N=20 $ since $ T=20/19 $ is rational with a numerator of 20; this means its fundamental frequency is $ \pi/10 $ not $ 19\pi/10 $
  - This frequency shift from continuous to discrete (or sampled) is called aliasing - more on that later

Lecture 5 - 9/7 - System Configurations / Math Review

This is to be done outside of class (and was actually presented after lecture 6)

Systems can be connected together in cascade, parallel, and feedback configurations
- For LTI systems, systems in cascade will have their transfer functions multiplied by each other
- For LTI systems, systems in parallel will have their transfer functions added together
- For LTI systems, systems with negative feedback are a little more complicated
Review of exponential integral

Lecture 6 - 9/9 - System Properties

System properties - see System_Properties for some ways to check some of the properties
- Linearity (linear versus nonlinear)
  - Common nonlinearities include additive constants, non-unity powers of signals
- Time invariance (time invariant versus time-varying)
  - Common time-varying elements include $$ t $$ outside of arguments of signals, time reversals, or time scales other than 1
- Stability (stable versus unstable)
  - Common instabilities involve inverses, some integrals (growing domain), some trig functions, and derivatives if you are including discontinuities
- Memoryless (memoryless versus having memory)
  - Memoryless signals can *only* depend on "right now"; some debate about derivatives. Depending on the future also means memory.
- Causality (causal versus non-causal)
  - Systems whose responses depend only on current and previous values of the independent variable are causal; if they depend at all on future values, they are non-causal.
  - Real-time systems with time $$ t $$ as the independent variable are causal; systems with location as the independent value may be non-causal
- Invertibility (invertible versus non-invertible)
  - Invertible systems are systems where you can uniquely determine what the input was based on what the output is.
  - Determining invertibility is...complicated...
  - So much so that we just skipped this in class for now...
Summations - see Summation Formulas

Lecture 7 - 9/14 - Common Homogeneous Responses; Integrals and Sums of Homogeneous Responses; Discrete LTI Systems and Convolution

Homogeneous solutions and integrals / summations
- Continuous:
  - $y_h(t)=Ke^{st}u(t)$ sometimes written as $Ke^{-t/\tau}u(t)$ or $Ke^{-at}u(t)$
  - $\int_{-\infty}^tKe^{-at}\,dt=K\left(\frac{1-e^{-at}}{a}\right)u(t)$
- Discrete:
  - $y_h[n]=K\gamma^nu[n]$
  - $\sum_{k=-\infty}^{n}K\gamma^nu[k]=K\left(\frac{1-\gamma^{n+1}}{1-\gamma}\right)u[n]$
Introduction to LTI system analysis:
- Define the discrete step and impulse functions
- Define the impulse response $$ h[n] $$ as the response to an impulse $\delta[n]$ ; that is, $\delta[n]\,\longrightarrow\,h[n]$
- Define the step response $$ s_r[n] $$ as the response to a step $$ u[n] $$ ; that is, $u[n]\,\longrightarrow\,s_r[n]$
- From linearity and time-invariance: note that just as $\delta[n]=u[n]-u[n-1]$ , $$ h[n]=s_r[n]-s_r[n-1] $$ and just as $u[n]=\sum_{k=-\infty}^{n}\delta[n]$ , $s_r[n]=\sum_{k=-\infty}^{n}h[n]$
- Punchline: If a linear, time invariant system has an impulse response $ h[n] $, the response $ y[n] $ to any $ x[n] $ can be found by convolving $ x[n] $ with $ h[n] $ (or $ h[n] $ with $ x[n] $); that is:$ {\displaystyle \begin{aligned} y[n]&=x[n]*h[n]=\sum_{k=-\infty}^{\infty}x[k]h[n-k]\\ y[n]&=h[n]*x[n]=\sum_{k=-\infty}^{\infty}h[k]x[n-k]\\ \end{aligned}} $
  - Rule of thumb: pick the more "complicated" signal to get the $$ [k] $$ argument and the "simpler" signal to get the $$ [n-k] $$ argument.
Cave example of $y[n]=\frac{1}{2}y[n-1]+x[k]$
MATLAB demonstration

Lecture 8 - 9/16 - Discrete LTI Systems and Convolution

Recap of discrete convolution
Relationship between impulse and step similar to relationship between impulse resonse and step response: $\begin{align} u[n]&=\sum_{k=-\infty}^{n}\delta[k] & \delta[n]&=u[n]-u[n-1]\\ s_r[n]&=\sum_{k=-\infty}^{n}h[k] & h[n]&=s_r[n]-s_r[n-1] \end{align}$
Convolution of two geometric series: $\begin{aligned} \left(\alpha^{n}\,u[n]\right)*\left(\beta^{n}\,u[n]\right)&=\begin{cases} \alpha=\beta, & (n+1)\,\alpha^n\,u[n]\\ \alpha\neq\beta, & \left(\frac{\alpha^{n+1}-\beta^{n+1}}{\alpha-\beta}\right)\,u[n] \end{cases} \end{aligned}$
System properties based on $ h[n] $:
- LTI systems have impulse and step responses; if you are given an impulse or step response for a system, it is most likely an LTI system (or else the impulse response is not as useful...)
- Memoryless: $h[n]=K\delta[n]$ , $$ s_r[n]=Ku[n] $$
- Causal: $$ h[n]=0 $$ and $$ s_r[n]=0 $$ for all $$ n<0 $$
- Stable: $\sum_{n=-\infty}^{\infty}|h[n]|<\infty$ ; this one is a little tricky to get directly from step response so calculate $$ h[n]=s_r[n]-s_r[n-1] $$ first and then analyze $$ h[n] $$
Quick convolutions / shifting property $\begin{align} \delta[n]*\delta[n]&=\delta[n]\\ \delta[n]*\delta[n-n_0]&=\delta[n-n_0]\\ \delta[n]*f[n]&=f[n]\\ \delta[n]*f[n-n_0]=\delta[n-n_0]*f[n]&=f[n-n_0]\end{align}$
- When dealing with shifted signals, unshift them by pulling out a $\delta[n-n_0]$ , work with the unshifted signals, then put the shifts back in.

Lecture 9 - 9/21 - Introduction to Continuous Convolution

Example of finding discrete impulse response step-by-step
Introduction to LTI system analysis:
- Define the step and impulse functions as given above
- Define the impulse response $$ h(t) $$ as the response to an impulse $\delta(t)$ ; that is, $\delta(t)\,\longrightarrow\,h(t)$
- This will be mathematically very useful and physically impossible to measure, though we may be able to measure it approximately using a high-amplitude, short duration rectangular or other pulse with an area of 1.
- Define the step response $y_{\mbox{step}}(t)$ as the response to an impulse $$ u(t) $$ ; that is, $u(t)\,\longrightarrow\,y_{\mbox{step}}(t)$
- This will be more likely to be physically obtainable but mathematically not quite as useful. Forutunately...
- The step and impulse responses are related in the same ways as the step and impulse:
  $\begin{aligned} \delta(t)&=\frac{d}{dt}u(t) & u(t)&=\int_{-\infty}^t\delta(\tau)\,d\tau\\ h(t)&=\frac{d}{dt}y_{\mbox{step}}(t) & y_{\mbox{step}}(t)&=\int_{-\infty}^th(\tau)\,d\tau \end{aligned}$
- Given those definitions, and assuming a linear-time invariant system:
  $\begin{aligned} \mbox{Definition}&~ & \delta(t)\,&\longrightarrow\,h(t)\\ \mbox{Time Invariant}&~ & \delta(t-\tau)\,&\longrightarrow\,h(t-\tau)\\ \mbox{Linearity (Homogeneity)}&~ & x(\tau)\,\delta(t-\tau)\,&\longrightarrow\,x(\tau)\,h(t-\tau)\\ \mbox{Linearity (Superposition)}&~ & \int_{-\infty}^{\infty}x(\tau)\,\delta(t-\tau)\,d\tau\,&\longrightarrow\,\int_{-\infty}^{\infty}x(\tau)\,h(t-\tau)\,d\tau\\ \mbox{Sifting}&~ & \int_{-\infty}^{\infty}x(\tau)\,\delta(t-\tau)\,d\tau=x(t)\,&\longrightarrow\,y(t)=\int_{-\infty}^{\infty}x(\tau)\,h(t-\tau)\,d\tau\\ \end{aligned}$
- Punchline: For an LTI system with impulse response $$ h(t) $$ and input signal $$ x(t) $$ the output signal is given by the convolution integral:
  $\begin{aligned} y(t)=x(t)*h(t)=\int_{-\infty}^{\infty}x(\tau)\,h(t-\tau)\,d\tau \end{aligned}$

and through a transformation of variables can also be given by:

\begin{aligned} y(t)=h(t)*x(t)=\int_{-\infty}^{\infty}x(t-\tau)\,h(\tau)\,d\tau \end{aligned}

Example using shortcuts
System properties based on $ h(t) $:
- LTI systems have impulse responses; if you are given an impulse response for a system, it is most likely an LTI system (or else the impulse response is not as useful...)
- Memoryless: $h(t)=K\delta(t)$ , $s_r(t)=K\,u(t)$
- Causal: $$ h(t)=0 $$ , $$ s_r(t)=0 $$ for all $$ t<0 $$
- Stable: $\int_{-\infty}^{\infty}|h(t)|\,dt<\infty$
Basic convolution properties - see Convolution Shortcuts
Step and impulse response of capacitor voltage for an RC circuit

Lecture 10 - 9/23 - More Continuous Convolution; Block Diagrams

Step and impulse response of resistor voltage for an RC circuit
- Using Kirchhoff's Voltage Law
Quick review of frequency analysis using impedance and division to get a transfer function
- Reminder of translating between time and frequency domain with $\frac{d}{dt}\leftrightarrows j\omega$
Block Diagrams
- Basic mathematical relationships in time and frequency
- Parallel, Series, and Cascade Connections
- Interpreting block diagrams with integral and delay blocks

Lecture 11 - 9/28 - Linear Constant Coefficient Differential Equations

Step and impulse response for general first-order differential equations
- $\begin{align*} s_r(t)&=\left(1-e^{-t/\tau}\right)\,u(t) & h(t)&=\frac{e^{-t/\tau}}{\tau}\,u(t)\end{align*}$
Working with derivatives on input side
- After normalizing, rewrite equation to use $$ z(t) $$ instead of $$ y(t) $$ everywhere on the left and just have $$ x(t) $$ on the right
- Solve for step or impulse response of $$ z(t) $$ to $$ x(t) $$
- Whatever was actually happening to $$ x(t) $$ on the right hand side, do that to the response of $$ z(t) $$ to get the actual response of $$ y(t) $$
Characteristic polynomial for general second-order differential equation:
- $\begin{align*} \frac{d^2y(t)}{dt^2}+2\zeta \omega_n\frac{dy(t)}{dt}+\omega_n^2y(t)&=x(t)\\ s^2+2\zeta\omega_n s+\omega_n^2&=0\\ s&=-\zeta\omega_n\pm\omega_n\sqrt{\zeta^2-1} \end{align*}$ where $\zeta$ is the damping ratio and $\omega_n$ is the natural frequency.
- Depending on value of $ \zeta $, system can be:
  - Undamped: $\zeta=0$ , $$ s $$ =purely imaginary complex conjugates $\pm j\omega_n$ , homogeneous response is sinusoidal
  - Unerdamped: $0<\zeta<1$ , $$ s $$ =complex conjugates with real parts $-\zeta\omega_n\pm j\omega_n\sqrt{1-\zeta^2}$ , homogeneous response is exponential sinusoid with exponential of $-\zeta\omega_n$ and frequency of oscillation $\omega_d=\omega_n\sqrt{1-\zeta^2}$
  - Critically damped: $\zeta=1$ , $$ s $$ =repeated real roots at $-\zeta\omega_n$ , homogeneous response is polynomial exponential
  - Overdamped: $\zeta>1$ , $$ s $$ =two different purely real roots, homogeneous response is exponential
  - If $\zeta$ and $\omega_n$ are both positive, real part is negative meaning exponential decay
SeeLinear Constant Coefficient Differential Equations

Lecture 12 - 9/30 - Linear Constant Coefficient Differential Equations

Advance equation to get all non-negative shifts in $$ y[n] $$
Get characteristic polynomial
For n-th differences, use n time values to get coefficients
See Linear Constant Coefficient Discrete Difference Equations