ECE 280/Fall 2023/ld

The following represents the information that would have been in Lecture 3 but isn't because Labor Day!

• Power and energy and transformations
  • In previous lectures, we looked at power signals and their overall average power $$P_{\infty}$$ and energy signals and their total energy $$E_{\infty}$$
  • We also looked at a transformation of variables $$z(t)=K\,x(\pm a(t-t_0))$$
  • If $$x(t)$$ is an energy signal with energy $$E_{\infty,x}$$ or a power signal with overall average power $$P_{\infty,x}$$, then the different transformations above have the following individual impacts:
    • A multiplicative factor of $$K$$ in the transformation means the energy or power of $$z(t)$$ will be the energy or power of $$x(t)$$ multiplied by $$K^2$$
    • A time scale of $$a$$ in the transformation means the energy of $$z(t)$$ will be the energy of $$x(t)$$ multiplied by $$\frac{1}{a}$$
    • Time reversal has no influence on energy or power
    • Time shifts have no influence on energy or power
• Also in a previous lecture, we looked at the unit step function $$u(t)$$ and the unit ramp function $$r(t)$$
  • We used scaled, shifted versions of steps and ramps to write equations for piecewise linear functions - see Singularity_Functions#Accumulated_Differences
• Read OW 1.3.1 to review complex exponential signals and sinusoids - pay close attention to Example 1.5!
• Skim OW 1.3.2 on discrete-time exponential signals and sinusoids
• Skip (or skim) 1.3.3 for now
• Skim 1.4.1 - note that the discrete-time unit impulse is defined at 1 at index 0 and 0 everywhere else. Also, the discrete-time unit step is defined as 1 for all indices 0 or greater and 0 elsewhere. Because these are discrete-time functions, there is no instantaneous transition like there is with continuous.
• Read 1.4.2 very carefully. The executive summary is that the unit impulse $$\delta(t-t_0)$$ has an area when $$t=t_0$$ and is 0 otherwise. The unit impulse is almost only ever used in contexts where it will eventually become an integrand, so the argument of the impulse can be used to figure out when the integrand will be non-zero.
  • Look through Singularity_Functions#General_Simplification_of_Integrals - equations 42, 43, 45, 48, 49, 51, 57, 61, and 63 are the most common ones we will see.
  • Look through Singularity_Functions#Opposite_Directions_in_Integrand - don't worry about the context of "convolution" yet - just look at how the two step functions interact with the limits on the integral and the outer step function that gets generated. Note that there are three changes when opposite-facing steps are in an integrand - the upper limit, the lower limit, and the step on the outside that will generally be a step function with an argument that is the difference between the upper and lower limits.