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)
  • Energy signals have a finite amount of energy: $$E_{\infty}=\int_{-\infty}^{\infty}|x(\tau)|\,d\tau<\infty$$
    • 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 $$x(t)=e^t$$