ECE 280/Concept List/F23

Lecture 1 - 8/28

• 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
  • 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:$$E=\frac{1}{3}A^2\,\Delta t$$

Lecture 2 - 9/1

• Conclusion of "homework" from previous class: $$x(t)=1/\sqrt{t}$$ for $$t>1$$ has infinite total energy but 0 average power. This is related to Gabriel's horn, which has a finite volume but an infinite surface area.
• 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 $$\omega_0$$.
• 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 $$j\,\sin(\omega t)$$
• 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 / 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