ECE2072 - Digital Systems Summary Notes

ECE2072 - Digital Systems Summary Notes Contents Combinational Logic ......................................................................................................... 2 De Morgan’s Law .................................................................................................................... 2 Logic Gates ............................................................................................................................. 2 Canonical Form....................................................................................................................... 2 Min and Maxterms ................................................................................................................. 3 Definitions .............................................................................................................................. 4 Logic Minimization ........................................................................................................... 4 Karnaugh Maps ...................................................................................................................... 4 Quine-McCluskey Method ..................................................................................................... 4 Logic Hazard ........................................................................................................................... 5 Logic Components ............................................................................................................ 6 Multiplexers ........................................................................................................................... 6 Programmable Logic Arrays ................................................................................................... 7 Tri-state Outputs .................................................................................................................... 7 Numerical Operations ............................................................................................................ 8 Sequential Circuits ............................................................................................................ 9 R-S Latches ........................................................................................................................... 10 Shift Register ........................................................................................................................ 12 Finite State Machine ............................................................................................................ 12 State Minimization ............................................................................................................... 14 Verilog ........................................................................................................................... 16 Sequential Logic Implementation.................................................................................... 18 Median Filter Example ......................................................................................................... 18 String Recogniser Example ................................................................................................... 19 Complex Counter Example: .................................................................................................. 20 Data-path and Control ......................................................................................................... 21 Combinational Lock Example ............................................................................................... 21

Finn Andersen, 2012

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Combinational Logic De Morgan’s Law  
  
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   Can be used to find complement of function:

Logic Gates

Canonical Form Ways of representing logic functions in Boolean algebra. Sum of Products form: Can then be reduced to minimal form by factorising and cancelling.

Finn Andersen, 2012

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Product of Sums form:

Alternate canonical representation: Sum of products

Minterms e.g. 2 = 010 = A’BC’

Product of sums

“Don’t care” terms

Min and Maxterms Forms of Boolean algebra, where: • •

Minterms are in a sum of products format Maxterms are in a product of sums format

Indexing 1 assigned to direct form, 0 assigned to complemented form. • • •

ABC’ = 110 = minterm #6 = m6 A’ + B + C = 011 = maxterm #3 = M3 Complement of minterm is corresponding maxterm: M5 = a’+ b +c’ = (ab’c)’ = m5’

Functional Equivalence Use minterms to represent logic 1. •

Direct form variables = logic 1         

Use maxterms to represent logic 0. •

Direct form variables = logic 0

                  Both forms represent the logic function. Finn Andersen, 2012

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Definitions Implicant – Single element or group of elements that can be combined to form a grouping Prime Implicant – Implicant that can’t be combined with another to form a larger grouping Essential Prime Implicant – Only implicant that covers a certain single element.

Logic Minimization Karnaugh Maps For product of sums

For sum of products, group the 0’s in the grid and treat 0’s of the variables as active.     0010 . .     

Quine-McCluskey Method Used to systematically find prime implicants. Alternative to K-maps (groups adjacent)

Finn Andersen, 2012

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