ECE 280/Concept List/F21

This page will be a summary of the topics covered during lectures. It is not meant to be a replacement for taking good notes!

Lecture 1

• Class logistics and various resources on [sakai.duke.edu sakai]
• Signals: "information about how one parameter changes depending on another parameter" - zyBooks
• Systems: "processes that create output signals in response to input signals" paraphrased from zyBooks
• Signal classifications
  • Continuous versus discrete
• Analog versus digital and/or quantized
• Periodic
  • Generally $$x(t)=x(t+kT)$$ for all integers k (i.e. $$x(t)=x(t+kT), k\in \mathbb{Z}$$). 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: $$x(t)=\sum_{n=-\infty}^{\infty}g(t-nT)$$ where $$g(t)$$ is only possibly nonzero within one particular period of the signal and 0 outside of that period.
• Energy, power, or neither (more on this on Friday)

Lecture 2

• More on periodic signals
  • The sum / 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 period 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.
• Evan and Odd
  • Purely even signals: $$x(t)=x(-t)$$ (even powered polynomials, cos)
  • 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 $$\sin(\omega t)$$
• 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: $$r(t)=\int_{-\infty}^{t}u(\tau)\,d\tau=\begin{cases}t, t>0\\0, t<0\end{cases}$$
• 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 and $$|a|<1$$ slows down
  • $$t_0$$: 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
  • Do flip and scalings first, then shift the flipped and scaled versions.
• Energy and Power
  • 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
  • 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): $$P_{\infty}=\frac{1}{T}\int_{T}|x(\tau)|^2\,d\tau$$
  • 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 $$x(t)=e^t$$
  • Vertical scaling of $$K$$ changes the energy or power by a factor of $$K^2$$
  • Time scaling of $$a$$ changes the energy or power by a factor of $$\frac{1}{a}$$
  • Neither time shift nor reversal impact energy or power, so you can shift and flip signal components to more mathematically convenient locations.

Lecture 3

• Unit step and the $$u(t)\approx \frac{1}{2\epsilon}\left(r(t+\epsilon)-r(t-\epsilon)\right)$$ approximation
• Unit ramp
• Rectangular pulse function
• Definition of the impulse function: Area of 1 at time 0; 0 elsewhere
  • Limit as $$\epsilon$$ goes to 0 of the derivative of the unit step approximation
  • Sifting property - figure out when $$\delta$$ fires off, see if that argument happens or if there are restrictions based on integral limits
• 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
• Exponentials and sketches
  • Solution to $$\tau\frac{dv(t)}{dt}+v(t)=v_f$$ given $$v(t_0)=v_i$$ is $$v(t)=v_f+\left(v_i-v_f\right)\exp\left(-\frac{t-t_0}{\tau}\right)$$
  • Went over how to make an accurate sketch using approximate values and slopes