1. Top-down modular design

2. Decoders n-to-2n decoder: logic network with n inputs and 2n outputs.
One output is active for each of the 2n input combinations  each minterm
Decoder  minterm generator
Most common use:
Memory selection

3. Decoders
Parallel decoders (a) active-high (b) active-low

4. Decoders Parallel is expensive Problem extending for larger n
2n decoders require n-input ANDs: fan-in
Alternative: (c) alternate structure

5. Decoders: (a) parallel (b) tree

6. Decoders Tree decoders – constant fan-in: 2-input gates throughout
Compare the number of gates required for (a) and (b)
Dual tree decoders
n inputs divided into 2 groups j and k;
n = j+k
j-input decoder  2j outputs
k-input decoder  2k outputs
Array of 2j * 2k = 2n 2-input ANDs

7. Decoders
Why decision tree: @a
@ a
@ a
@ a
@a @a
@ a @ a
@a @a @a @a

8. Decoders: (c) dual tree
n=4
j=k=2

9. Implementing logic functions using Decoders
Since , each output can be considered a maxterm.

10. Logic functions using Decoders

11. Logic functions using Decoders
Active-high decoder with OR gate (a)
Active-low decoder with NAND gate (b)

12. Logic functions using Decoders
Active-high decoder with NOR gate (c)
Active-low decoder with AND gate (d)

13. Decoders
Active-high decoder with enable (a) Symbol (b)
Function:

14. Decoders
3  8 decoder using two 2  4 decoders with enable

15. Decoders
4  16 decoder using 2  4 decoders with enable

16. Encoders
Opposite of decoder: one output code (binary) for each input; assumes one input active at a time
n inputs; s outputs
n ≤ 2s
s ≥ log2n; usually s = log2n
4:2 encoder with
exclusive inputs:
functional diagram

17. Encoders
K-Maps of the 4:2 encoder outputs

18. Encoders
Logic diagram and Truth table of the 4:2 encoder

19. Encoders Functional diagram and Truth table of the 4:3 encoder
Outputs Zero unless exactly
one line is active high

20. Encoders
K-Maps of the outputs of the 4:3 encoder

21. Encoders
Logic diagram of the 4:3 encoder

22. Encoders 4:2 priority encoder No input active  EO = 1
At least one input active  GS = 1
If more than one input active  output the one with highest priority
3 > 2 > 1 > 0
Useful in resource management request / acknowledge circuits in computers

23. Encoders
K-Maps of the 4:2 priority encoder outputs

24. Encoders
Logic diagram and Truth table of the 4:2 priority encoder

25. Multiplexers Connects one of the inputs to the output
Needs log2n select lines

26. Multiplexers
Easy to realize in CMOS

27. Multiplexers
Question: we want a 8:1 MUX using 4:1s as building blocks

28. Demultiplexers Connects the input to one of the outputs
Needs log2n select lines

29. Demultiplexers
Where is it useful?

30. Adders Often realized as “bit slice”s Bit slice: Half Adder
Bit slice: module that handles one bit position
Can be replicated (arrayed)
Array handles n bits (the whole number)
Bit slice: Half Adder
Given two input bits, produces sum and carry

31. Adders Bit slice: Full Adder
Given two input bits and carry in, produces sum and carry

32. Adders
Pseudo parallel (ripple carry) Adder

33. Adders
Two-bit “parallel” Adder

34. Adders Four-bit “parallel” Adder
What is the difficulty with n-bit parallel adders?

35. Adders Parallel Adder carry logic: growing impractically
Generate Propagate

36. Adders
Expanding the carry expressions yields
or, generalized:

37. Adders
When there is carry-in, it becomes c0, and renumbering the terms yields:
Can form groups of 4:

38. Adders Using the group carries:
Generate and propagate are from one bit so far
Can extend the idea to groups of bit positions

39. Adders The group carry, generate and propagate are:
The groups can be adjoining or overlapping e.g.

40. Adders Subtraction is addition Two-s complement
Bitwise invert B and set the carry-in c0=1

41. Adders
Overflow detection

42. Adders
Overflow detection

43. Comparators

44. Comparators
Example: 2-bit comparator

45. Comparators
K-Maps of the comparator outputs

46. Comparators
Example: MIS 4-bit magnitude comparator

47. Comparators
4-bit magnitude comparator function table

48. Comparators
Cascading 4-bit comparators to obtain a 16-bit comparator