1 Tutorial: ITI1100 Dewan Tanvir Ahmed SITE, UofO

2 Today’s topics Comparators Multiplier

3 Comparators A comparator compares a two n-bit binary values to determine which is greater or if they are equal Consider the simple 1-bit comparator to illustrate the design It is possible to extend the design for multi-bit numbers

4 Equality Comparator XNOR X Y Z X Y Z Design a logic circuit which will compute F0 = (A = B)

5 2-bit Equal Comparator Truth Table b1b0a1a0F b1b0a1a0F

6 Solution You can show,

7 N-bit Equal Comparator

8 Not Equal Comparator Design a logic circuit which will compute F = (A <> B) F = (A = B) i.e. Just invert our Equal Comparator circuit

9 Magnitude Comparator Design a logic circuit which will compute F2 = (A>B) F1 = (A<B) Let’s develop a truth table for 2-bits

10 2-bit Magnitude (unsigned) Comparator Truth Table b1b0a1a0F2F b1b0a1a0F2F

11 You can show

12 1-bit Comparator x y x>yx=yx<y xy x>y x=y x<y CMP

13 8-bit comparator xy x>y x=y x<y CMP x n >y n x n =y n x n <y n

14 1 bit comparators X>Y only if Xi=1, Yi=0 X<Y only if Xi=0, Yi=1 X=Y only if Xi=Yi=0 or Xi=Yi=1

15 1 bit comparator with propagated inputs

16 N bit comparator If: X = Yin is active then the numbers are equal so far If X>Yin or X<Yin is active, that value is simply passed through; This corresponds to the case where we have checked the high-order bits and already know which value is larger.

17 Comparators (computer intelligence?) Let's build a comparator circuit for two 4-bit positive binary numbers. ENABLE A3 A2 A1 A0 B3 B2 B1 B0 A>B A=B A<B Nine inputs, three outputs (three 512 entries truth tables?)

18 Designing Comparators Functionally 1.Build a one-bit comparator A>B : AB' A=B : A'B' + AB A<B : A'B B 0 1 A 0 A=B A<B 1 A>B A=B OR What? A B A>B A=B A<B

19 Designing Comparators Functionally 2.Add an enable line A=B A B A>B Enable

20 Build a four-bit Comparator (from four one-bit ones) Not bad

21 Combinational Multiplier Basic Concept multiplicand multiplier 1101 (13) 1011 (11) * (143) Partial products product of 2 4-bit numbers is an 8-bit number

22 Multiplication Example:

23 Multiplication Example:

24 Multiplication Example:

25 Multiplication Example:

26 Multiplication Example:

27 Multiplication Example: M x N-bit multiplication –Produce NM-bit partial products –Sum these to produce M+N-bit product

28 General Form Multiplicand: Y = (y M-1, y M-2, …, y 1, y 0 ) Multiplier: X = (x N-1, x N-2, …, x 1, x 0 ) Product:

29 16X16 Mult. Dot Diagram Each dot represents a bit

30 Combinational Multiplier Partial Product Accumulation A0 B0 A0 B0 A1 B1 A1 B0 A0 B1 A2 B2 A2 B0 A1 B1 A0 B2 A3 B3 A3 B0 A2 B1 A1 B2 A0 B3 A3 B1 A2 B2 A1 B3 A3 B2 A2 B3 A3 B3 S6 S5 S4 S3S2 S1S0 S7

31 Partial Product Accumulation Note use of parallel carry-outs to form higher order sums 12 Adders, if full adders, this is 6 gates each = 72 gates 16 gates form the partial products total = 88 gates!

32 Another Representation of the Circuit Building block: full adder + and 4 x 4 array of building blocks

33 Parallel Binary Multiplier + Y P C Y X CO PO X

34 X + Y PC Y X CO PO One-Bit Multiplier Cell

35 With J multiplier bits and K multiplicand bits – need JxK AND gates and (J-1) K-bit adders to produce J+K bits 2-bit by 2-bit Binary Multiplier

36 With J=3 (A 0 A 1 A 2 ) multiplier bits, K=4 (B 3 B 2 B 1 B 0 ) multiplicand bits – need 12 (JxK) AND gates, 2 (J-1) 4-bit(K-bit) adders to produce 7(J+K) bits 4-bit by 3-bit Binary Multiplier

37 Thank You!