ECE 280/Concept List/F25

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

• 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
  • 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.
• Energy, power, or neither
  • Energy signals have a finite amount of energy: $$E_{\infty}=\int_{-\infty}^{\infty}|x(\tau)|^2\,d\tau<\infty$$
  • 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 $$E_{\infty}=\infty$$
    • For periodic signals, only need one period (that is, remove the limit and use whatever period definition you want): $$P_{\infty}=\frac{1}{T}\int_{T}|x(\tau)|^2\,d\tau$$
  • If both the energy and the overall average power are infinite, the signal is neither an energy signal nor a power signal.
• Singularity functions - see Singularity_Functions and specifically Singularity_Functions#Accumulated_Differences
  • Unit step: $$u(t)=\begin{cases}1, t>0\\0, t<0\end{cases}$$
  • Unit ramp: $$r(t)=\int_{-\infty}^{t}u(\tau)\,d\tau=\begin{cases}t, t>0\\0, t<0\end{cases}$$

Lecture 2 - 8/30 Periodicity, even and odd, basic transformations, 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:$$E=\frac{1}{3}A^2\,\Delta t$$

Lecture 3 - 9/2 - More on periodicity, even and odd; basic transformations, steps and ramps

• More on Evan 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 $$j\,\sin(\omega t)$$
• More on periodic 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
  • $$z(t)=K\,x(\pm a(t-t_0))$$ 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
  • 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
• 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

Lecture 4 - 9/4 - Integration with impulses and steps, Introduction to discrete 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
• 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 - 9/9 - 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, integrals, 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
  • 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 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...

Lecture 6 - 9/11 - System Configurations / Math Review

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

Lecture 7 - 9/16 - Discrete LTI Systems and Convolution

• 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:\(\begin{align*} y[n]&=x[n]*h[n]=\sum_{k=-\infty}^{\infty}x[k]h[n-k]\\ y[n]&=h[n]*x[n]=\sum_{k=-\infty}^{\infty}h[k]x[n-k]\\ \end{align*}\)
    • Rule of thumb: pick the more "complicated" signal to get the $$[k]$$ argument and the "simpler" signal to get the $$[n-k]$$ argument.
  • From class:\(\begin{align*} \alpha^{n}\,u[n]*\beta^{n}\,u[n]&=\begin{cases} \alpha=\beta, & (n+1)\,\alpha^n\,u[n]\\ \alpha\neq\beta, & \left(\frac{\alpha^{n+1}-\beta^{n+1}}{\alpha-\beta}\right)\,u[n] \end{cases} \end{align*}\)

Lecture 8 - 9/18 - More discrete Convolutions

• Dealing with shifts
• Periodic impulse train notation: $$\delta_T(t)$$ means an impulse train centered at 0 with period $$T$$
• Time scale with singularities:
• ERROR: $$r[n]$$ was defined incorrectly in lecture - it should be $$r[n]=n\,u[n]$$
• Graphical discrete convolution

Lecture 9 - 9/23 - Continuous LTI Systems and Convolution

• 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:
    $$\begin{align*} \delta(t)&=\frac{d}{dt}u(t) & u(t)&=\int_{-\infty}^t\delta(\tau)\,d\tau\\ h(t)&=\frac{d}{dt}y_{\mbox{step}}(t) & y_{\mbox{step}}(t)&=\int_{-\infty}^th(\tau)\,d\tau \end{align*}$$
  • Given those definitions, and assuming a linear-time invariant system:
    $$\begin{align*} \mbox{Definition}&~ & \delta(t)\,&\longrightarrow\,h(t)\\ \mbox{Time Invariant}&~ & \delta(t-\tau)\,&\longrightarrow\,h(t-\tau)\\ \mbox{Linearity (Homogeneity)}&~ & x(\tau)\,\delta(t-\tau)\,&\longrightarrow\,x(\tau)\,h(t-\tau)\\ \mbox{Linearity (Superposition)}&~ & \int_{-\infty}^{\infty}x(\tau)\,\delta(t-\tau)\,d\tau\,&\longrightarrow\,\int_{-\infty}^{\infty}x(\tau)\,h(t-\tau)\,d\tau\\ \mbox{Sifting}&~ & \int_{-\infty}^{\infty}x(\tau)\,\delta(t-\tau)\,d\tau=x(t)\,&\longrightarrow\,y(t)=\int_{-\infty}^{\infty}x(\tau)\,h(t-\tau)\,d\tau\\ \end{align*}$$
  • Punchline: For an LTI system with impulse response $$h(t)$$ and input signal $$x(t)$$ the output signal is given by the convolution integral:
    $$ \begin{align*} y(t)=x(t)*h(t)=\int_{-\infty}^{\infty}x(\tau)\,h(t-\tau)\,d\tau \end{align*}$$

and through a transformation of variables can also be given by:
$$ \begin{align*} y(t)=h(t)*x(t)=\int_{-\infty}^{\infty}x(t-\tau)\,h(\tau)\,d\tau \end{align*}$$

• 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: $$h(t)=K\delta(t)$$
  • Causal: $$h(t)=0$$ 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 - 9/25 - More Continuous Convolution

• Graphical convolution (see ECE_280/Examples/Convolution) ("flip 'n' shift")
• Step and impulse response of resistor voltage for an RC circuit
• Quick review of frequency analysis using impedance and division to get a transfer function
  • Definitions of phasor, impedance, resistance, reactance, admittance, conductance, susceptance
  • Reminder of translating between time and frequency domain with $$\frac{d}{dt}\leftrightarrows j\omega$$
  • Discussion about "illegal" circuit conditions (instant voltage change across capacitor or instant current change through inductor) and "weird" circuit conditions (voltage in parallel with an inductor or current source in series with a capacitor)
  • ECE 110 use $$e^{j\omega t}$$ as the model signal for frequency analysis; we will eventually use $$e^{st}$$ where $$s=\sigma+j\omega$$

Lecture 11 - 9/30 - Linear Constant Coefficient Differential Equations

• Stand by - lecture did not connect...

Lecture 12 - 10/2 - Linear Constant Coefficient Discrete Difference 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 more at Linear Constant Coefficient Discrete Difference Equations

Lecture 13 - 10/6 - Fourier Series

See Fourier Series

Lecture 14 - 10/9 - More Fourier Series

• FS for impulse train
• FS for rectangular pulse train
• sinc function

Lecture 15 - 10/16 - Fourier Series Properties

• Time shift property
• Linearity
• Derivative and integral properties
• Singularity shortcuts

Lecture 17 - 10/23 - Fourier Transform

• Synthesis and analysis equations
• FT of an impulse
• FT of a decaying exponential
• Derivative property
• Finding $$h(t)$$ based on a differential equation using FT
• Finding $$h(t)$$ for a circuit using FT
• "Cover-up" method for inverse FT
• Frequency derivative property and repeated roots

Lecture 18 - 10/28 - More Fourier Transform

• Dealing with repeated roots
• Dealing with complicated numerators
• Time shift property
• Frequency shift property
• FT of a unit step

Lecture 19 - 10/30 - Even More Fourier Transform

• Dealing with partial time shifts
• MOAT: \( e^{-at}\left(A\,\cos(\omega_xt)+B\,\sin(\omega_xt)\right)\,u(t) \overset{FT}{\longleftrightarrow} \frac{A(j\omega+a)+B(\omega_x)}{(j\omega+a)^2+\omega_x^2} \)
• Identifying MOATs using discriminant
• Completing the polynomial to get denominator in correct form
• Fourier Transform of a constant
• Fourier Transform of a periodic signal
  • Fourier Transform of cos and sin
• Convolution property: $$x(t)\ast h(t) \overset{FT}{\longleftrightarrow} X(j\omega)\times H(j\omega)$$
• Multiplication property: $$x(t)\times h(t) \overset{FT}{\longleftrightarrow} \frac{1}{2\pi}X(j\omega)\ast H(j\omega)$$

Lecture 20 - 10/4 - Sampling

• Impulse sampling
• Nyquist criterion for recoverability
  • Aliasing issue if sampling frequency is too low
• Low pass filtering to recover central sampling
• Zero-order hold reconstruction

Lecture 21 - 10/6 - AM Radio

• Amplitude modulation
  • Synchronous demodulation with multiplication by carrier frequency followed by LPF
  • May include BPF to isolate channel
  • Very sensitive to any difference in frequency or phase between modulation and demodulation signals
• Asynchronous Demodulation
  • May include BPF to isolate channel
  • Uses nonlinear circuit to do envelope detection
  • Requires adding DC component to signal to make original message purely positive - costs power

Lecture 22 - 11/11 - Parseval's Theorems / Intro to Laplace Transforms

• The average power in a periodic signal can be found using either the time domain or frequency domain using Parseval's Theorem:

\( \begin{align*} P_{avg}&=\frac{1}{T}\int_T|x(t)|^2\,dt=\sum_{k=-\infty}^{\infty}|\mathbb{X}[k]|^2 \end{align*} \)

• The total energy in an energy signal can be found using either the time domain or frequency domain using Parseval's Theorem:

\( \begin{align*} E_{\infty}&=\int_T|x(t)|^2\,dt=\frac{1}{2\pi}\int_{-\infty}^{\infty}|\mathbb{X}(j\omega)|^2\,d\omega \end{align*} \)

• Recap of LTI systems and definitions of impulse and step responses
• Recap of input/output relationship of a generic exponential and definition of Laplace Transform and Inverse Laplace Transform
• Laplace Transforms of impulses and steps, the latter leading to the necessity of a Region of Convergence for the integrals

Lecture 23 - 11/13 - More Laplace Transform

• System characteristics based on $$H(s)$$ and ROC
  • Stable if $$\sigma=0$$ is included in ROC
  • Causal if ROC is everything or the right-half-plane AND there are no time shifts to negative $$t$$ values
• Signal characteristics based on ROC:
  • The ROC is bounded by poles
    • Time shifts may create zeros that cancel poles and change ROC
  • If ROC is everything, signal is of limited duration
  • If ROC is left or right-sided, the signal is left or right-sided
  • If the ROC is bounded, the signal is two-sided
• Laplace Properties
  • Time shift and frequency shift
  • Time derivative and frequency derivative
• Inverse Laplace requires knowledge of ROC

Lecture 25 - 11/20 - More Laplace Transforms

• Laplace transform of semi-periodic signals
  • Find transform of one period
  • Divide by $$1-e^{-sT}$$
  • If right-sided, ROC is $$\sigma>0$$
• The Laplace Transform we have been looking at so far is the Bilateral Laplace Transform - it is useful when we know everything about the system for all time. Now we will cover the Unilateral Laplace Transform which needs to know a value at some time and then everything about the system after that.
• Unilateral Laplace Transforms
  • Properties different from BLT include time shift, time reversal, and derivative
  • Derivative property allows for solving differential equations with non-zero initial conditions
• Initial and Final Value Theorems
• Element models for use in switched circuits

Lecture 26 - 11/25 - Turbo Encabulators

• Video