11.1. Unit 11. Adders & Arithmetic Circuits

Transcription

1 . Unit s & Arithmetic Circuits

2 .2 Learning Outcomes I understand what gates are used to design half and full adders I can build larger arithmetic circuits from smaller building blocks

3 ADDER.3

4 (+) Register.4 Intro Addition is one of the most common operations performed by computer systems We can use adders to build larger components like the counter to the right Every clock cycle, the value Q (let's say 4-bits: Q[3:]), feeds back to the adder circuit which adds to the value and the register captures that new value on the next clock edge The sequence on Q on each clock cycle would be:,, 2, 3, 4 Could you design what's inside the adder block? How would you do it? REET CLK Q = curr Q + = next Q

5 .5 Intro What if we had to add AN two 4-bit numbers, [3:] and [3:]? Do we have the techniques to build such a circuit directly? es and no No. Not with K-maps since there are 8-inputs es. We could use sum of minterms but that would take a long time, but it could be done + = =

6 .6 Intro Idea: Build a circuit that performs one column of addition and then use 4 instances of those circuits to perform the overall 4-bit addition Let's start by designing a circuit that adds 2-bits: and that are in the same column of addition + = = Half

7 .7 Addition Half s Addition is done in columns Inputs are the bit of, Outputs are the um Bit and Carry-Out ( ) Design a Half- (HA) circuit that takes in and and outputs and Use the truth table to find the gate implementation + Half = = um

8 .8 Problem With Half s We d like to use one adder circuit for each column of addition Problem: No place for Carry-out of half adder to connect to the next olution Redesign adder circuit to include an additional input for the carry Half + = = Half

9 .9 Addition s Add a Carry-In input( ) New circuit is called a (FA) = + = C out

10 . Addition s Find the minimal 2-level implementations for Cout and

11 . Logic = xor xor Cin Recall: OR is defined as true when ODD number of inputs are true exactly when the sum bit should be Cout = + Cin + Cin Carry when sum is 2 or more (i.e. when at least 2 inputs are ) Circuit is just checking all combinations of 2 inputs

12 .2 Addition s () Use for each column of addition +

13 .3 Addition s (2) Connect bits of top number to inputs +

14 .4 Addition s (3) Connect bits of bottom number to inputs = + =

15 .5 Addition s (4) Be sure to connect first to = + =

16 .6 Addition s (5) Use for each column of addition = + =

17 .7 Addition s (6) Use for each column of addition + = =

18 .8 Addition s (7) Use for each column of addition + = =

19 .9 Addition s (8) Use for each column of addition = + =

20 .2 Performing ubtraction To subtract Flip bits of Add - = = + C C C C out out in out

21 .2 4-bit s We can create a component to perform 4-bit addition A 3 A 2 A A + B 3 B 2 B B = A = B = A 3 B 3 A 2 B 2 A B A B 4-bit 3 2

22 .22 Device vs. ystem Labels When using hierarchy (i.e. building blocks) to design a circuit be sure to show both device and system labels Device Labels: ignal names used inside the block Placeholders to indicate which input/output is which to the outside user ystem labels: ignal names used outside the block Actual signals from the circuit being built Can have the same name as the device label if such a signal name exists out the outside level Analogy: Formal and Actual parameters. a and b are like device labels and indicate the names used inside a block. 2. x and y are like system labels and represent the actual values to be used. int div(int a, int b) { int s = a/b; return s; } int main() { int x=, y=2; int s = div(x,y); } Device Labels: Indicate which input/output is which inside the bock. ystem Labels: Actual signals from the circuit being built B3 B2 B B A3 A2 A A C4 4-bit C (GND)

23 EERCIE.23

24 .24 Building an 8-bit Use (2) 4-bit adders to build an 8-bit adder to add =[7:] and = [7:] and produce a sum, =[7:] and a carry-out, C8. Label the inputs and outputs and make appropriate connections B3 B2 B B A3 A2 A A B3 B2 B B A3 A2 A A C8 C4 4-bit Binary C C4 4-bit Binary C

25 .25 Adding Many Bits ou know that an FA adds + + Ci A B Use FA and/or HA components to add 4 individual bits: A + B + C + D olution: 4 bits could yield sums from 2. o we need 3 bits of output (2,,) Be sure that bits you connect to a HA or FA are all from the same column (weight) Half D Half 2 C

26 .26 Adding 3 Numbers Add [3:] + [3:] + Z[3:] to produce F[?:] using the adders shown plus any FA and HA components you need bit olution: Adding (3) 4-bit numbers yields a sum of at most 45 = which requires 6 bits of output (F[5:]) CA CB CA Z3 Z2 Z Z Be sure the bits you connect to the same adder column have the same significance/weight F5 Half F4 CB 4-bit F3 F2 F F

27 .27 Mapping Algorithms to HW Wherever an if..then..else statement is used usually requires a mux if(a[3:] > B[3:]) Z = A+2 else Z = B+5 B[3:] A[3:] A[3:] B[3:] Circuit Circuit Comparison Circuit I I A>B Z[3:]

28 .28 Mapping Algorithms to HW Wherever an if..then..else statement is used usually requires a mux if(a[3:] > B[3:]) Z = A+2 else Z = B+5 B[3:] A[3:] A[3:] B[3:] I I I I A>B Comparison Circuit Circuit Z[3:]

29 .29 / ubtractor If sub == Else Z = [3:]-[3:] Z = [3:]+[3:]

30 3 2 C.3 / ubtractor Go back and optimize the muxes by determining what logic function they are actually performing If sub == Z = [3:]-[3:] UB Else Z = [3:]+[3:] UB i Bi 2 3 A A A2 A3 C4 4-bit Binary Z Z 2 B B B2 B3 Z2 Z3 3 UB

31 B3 B2 B B A3 A2 A A 3 2 C.3 Another Example Design a circuit that takes a 4-bit binary number,, and two control signals, A5 and M and produces a 4-bit result, Z, such that: Z = + 5, when A5,M =, Z =, when A5,M =, Z =, when A5,M =, 4-bit Input A5 M B3 B2 B B d d d d 2 3 C4 4-bit Binary Z Z Z2 A5 M M Z3 M