ECE 280/Concept List/S24

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

• Class logistics and various resources on Canvas
• Definition of signals and systems from OW
• 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.
• Signal classifications
  • Dimensionality ($$x(t)$$, $$g(x, y)$$, etc)
  • Continuous versus discrete
• Analog versus digital and/or quantized
  • Periodic versus aperiodic
    • 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: $$x(t)=\sum_{n=-\infty}^{\infty}g(t\pm nT)$$ where $$g(t)$$ is only possibly nonzero within one particular period of the signal and 0 outside of that period.
• Energy, power, or neither
  • 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 $$x(t)=e^t$$
• 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:$$E=\frac{1}{3}A^2\,\Delta t$$
• If a signal has components that have finite total energy and other components that have finite average power, the finite total energy parts do not contribute to the average power.