ECE 2020 Digital Design

Last time

• Last time
  • CMOS logic
  • Gates and mixed logic
• To be effectively prepared for today you should have:
  • Read your notes
• Exam on Wednesday October 15, 2025
  • Coverage comprehensive but focus on CMOS logic and gates
• Today
  • Number systems
• Be ready!
  • I expect you to take notes
  • Take your quizz at 10:15am

General feedback for Exam 1

• Statistics
  • Average of 84%, standard deviation of 13%
• Common issues
  • Errors in identifying essential and non-essential prime implicants
  • Errors in truth table construction
  • Not being clear in arguments
  • Poor presentation
• What I would like you to do
  • I saw no errors that I couldn't have been avoided but…
  • I need everyone to use my solutions as a template for how to present your reasoning
  • You need to find "safety nets" to check your results

General feedback for Lab 1

• Thanks for helping with the logistics: it takes time to check out 75 students

• My ask
  • Prepare the prelab diligently before the lab
  • Have your own kit and myDAQ (We'll work with you though)
  • Next time I'll have you work on a prelab/lab report to learn about
• Congrats!: that was your first lab with a breadboard
  • Learn to debug your board (broken pins?)
  • Don't forget to power the chips

Positional Number Systems

• Key objective: go beyond binary operations and represent others

• Positional number system
  • Number is represented by a string of digits where each digit position has an associated weight
  • Value of the number is the weighted sum of the digits \[ D = d_{p-1}d_{p-2}\cdots d_{1}d_0.d_{-1}\cdots d_{-n} = \sum_{i=-n}^{p-1}d_i r^i \]
    • \(r\) is the radix (basis) of the number system (typically \(r\) is an integer with \(r\geq 2\))
    • a digit in position \(i\) (\(d_i\)) has weight \(r^i\)
• Typical systems
  • \(r=2\): binary (\(0,1\))
  • \(r=8\) octal (\(0,1,2,3,4,5,6,7\))
  • \(r=16\) hexadecimal (\(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F\))
  • \(r=10\) decimal (\(0, 1, 2, 3, 4, 5, 6, 7, 8, 9\))

Conversion between number systems

• The formula \(D = \sum_{i=-n}^{p-1}d_i r^i\) converts any number system to decimal

• We can use binary representations as pivot point between binary, hexadecimal, octal

• We will mainly worry about decimal to binary and vice versa

Unsigned integers and addition

Signed integers and addition/subtraction

• Signed integer representation
  • signed magnitude
  • 1's complement
  • 2's complement
• We will only focus on 2's complement for addition and subtraction

Until next time

• To be effectively prepared for Wednesday October 15, 2025, you should:
  • Review your notes and homework solution for CMOS logic
  • Don't forget what we did before CMOS logic - that is still relevant