EE 313 Linear Systems and Signals - Lecture 2

Before Lecture

• "Roll the Bones", Shakey Graves, 2011.

Announcements

• UT fall job fairs, graduate studies, and resume tips

Lecture

• Lecture slides on Periodic Signals in PowerPoint format.
• A signal x(t) is periodic with period T seconds if x(t + T) = x(t) for all time t.
  • If we shift x(t) to the right by T seconds, then the signal looks identical to x(t)
  • The signal is also periodic for any integer multiple of T

Supplemental Material

• Fall 2025, Part 1. Notes in Word and PDF formats.
  • Review. The Sampling Theorem says choose the sampling rate f_s so that f_s > 2 f_max where so that frequencies up to and including the frequency of interest, f_max, are captured.
  • Slide 2-4. Phase shift. A phase shift of pi/4 looks the same as a phase shift of 2 pi + pi/4.
  • Slide 2-6. The sampling rate f_s is 1 / T_s.
• Fall 2025, Part 2. Notes in Word and PDF formats.
  • Review. Interpreting the Sampling Theorem. We can divide both sides of the inequality f_s > 2 f_max to obtain f_max < (1/2) f_s. That is, sampling at a sampling rate of f_s captures frequencies up to, but not including, (1/2) f_s.
  • Slide 2-8. Inverse Euler's formulas. Convert sinusoidal signals in expressions to complex sinusoids to aid in simplifying the expression.
    • e^{j theta} + e^{-j theta} = 2 cos(theta)
    • e^{j theta} - e^{-j theta} = 2 j sin(theta)
  • Slide 2-10. x(t) = cos(2 pi 440 t) which takes values on [-1, 1]. For x(t),
    A₀ = 0
    A₁ = 1, f₁ = 440, phi₁ = 0
    Input x(t) into a squaring block to produce output
    y(t) = x²(t) = cos²(2 pi 440 t) which takes values on [0, 1]. y(t) = (1/2) + (1/2) cos(2 pi (880) t) by using the trigonometric identity for cos²(theta) = (1/2) + (1/2) cos(2 theta). For y(t),
    A₀ = 1/2
    A₁ = 0, f₁ = 440, phi₁ = 0
    A₂ = 1/2, f₂ = 880, phi₂ = 0
  • Fall 2024
    • Slide 2-9. Even and odd symmetry
  • Fall 2023
    • Board. Fall 2023 Part 1 Review of continuous vs. discrete time and analog vs. digital amplitude
    • Board. Fall 2023 Part 2: dB scale, exponential response in RC circuit, Euler's formula and summing sinusoids to create a more complicated signal
  • Fall 2021: Periodicity review and Fourier analysis of cosine and cosine squared
  • Audio recording of the presentation on Sept. 5, 2017, of slides 1-13 to 2-4 in MP3 format
  • Audio recording of the presentation on Sept. 7, 2017, of slides 2-5 to 2-12 (last slide) in MP3 format
  Last updated %G%. Send comments to Prof. Evans at bevans@ece.utexas.edu