8Binary adderBinary adder that produces the arithmetic sum of binary numbers can be constructed with full adders connected in cascade, with the output carry from each full adder connected to the input carry of the next full adder in the chainNote that the input carry C0 in the least significant position must be 0.
10Binary Adder For example to add A= 1011 and B= 0011 subscript i:Input carry: CiAugend: AiAddend: BiSum: SiOutput carry: Ci+1
11Binary SubtractorThe subtrcation A – B can be done by taking the 2’s complement of B and adding it to A because A- B = A + (-B)It means if we use the inveters to make 1’s complement of B (connecting each Bi to an inverter) and then add 1 to the least significant bit (by setting carry C0 to 1) of binary adder, then we can make a binary subtractor.
13Adder SubtractorThe addition and subtraction can be combined into one circuit with one common binary adder (see next slide).The mode M controls the operation. When M=0 the circuit is an adder when M=1 the circuit is subtractor. It can be don by using exclusive-OR for each Bi and M. Note that 1 ⊕ x = x’ and 0 ⊕ x = x
15Checking OverflowNote that in the previous slide if the numbers considered to be signed V detects overflow. V=0 means no overflow and V=1 means the result is wrong because of overflowOverflow can be happened when adding two numbers of the same sign (both negative or positive) and result can not be shown with the available bits. It can be detected by observing the carry into sign bit and carry out of sign bit position. If these two carries are not equal an overflow occurred. That is why these two carries are applied to exclusive-OR gate to generate V.
16Magnitude ComparatorIt is a combinational circuit that compares to numbers and determines their relative magnitudeThe output of comparator is usually 3 binary variables indicating: A>BA=BA<BFor example to design a comparator for 2 bit binary numbers A (A1A0) and B (B1B0) we do the following steps:
17ComparatorsFor a 2-bit comparator we have four inputs A1A0 and B1B0 and three output E ( is 1 if two numbers are equal) G (is 1 when A > B) and L (is 1 when A < B) If we use truth table and KMAP the result isE= A’1A’0B’1B’0 + A’1A0B’1B0 + A1A0B1B0 + A1A’0B1B’0or E=(( A0 ⊕ B0) + ( A1 ⊕ B1))’ (see next slide)G = A1B’1 + A0B’1B’0 + A1A0B’0L= A’1B1 + A’1A’0B0 + A’0B1B0A0ComparatorEA1GB0LB1
18Magnitude ComparatorHere we use simpler method to find E (called X) and G (called Y) and L (called Z)A=B if all Ai= BiAi Bi XiIt means X0 = A0B0 + A’0B’0 andX1= A1B1 + A’1B’1If X0=1 and X1=1 then A0=B0 and A1=B1Thus, if A=B then X0X1 = 1 it meansX= (A0B0 + A’0B’0)(A1B1 + A’1B’1) since (x ⊕ y)’ = (xy +x’y’)X= ( A0 ⊕ B0)’ ( A1 ⊕ B1)’ = (( A0 ⊕ B0) + ( A1 ⊕ B1))’It means for X we can NOR the result of two exclusive-OR gates
19Magnitude Comparator A>B means A1 B1 Y1 ------------ 0 0 0 0 1 0 if A1=B1 (X1=1) then A0 should be 1 and B0 should be 0A0 B0 Y0For A> B: A1 > B1 or A1 =B1 and A0 > B0It means Y= A1B’1 + X1A0B’0 should be 1 for A>B
20Magnitude Comparator For B>A B1 > A1 or A1=B1 and B0> A0 z= A’1B1 + X1A’0B0The procedure for binary numbers with more than 2 bits can also be found in the similar way. For example next slide shows the 4-bit magnitude comparator, in which(A= B) = x3x2x1x0(A> B) = A3B’3 + x3A2B’2 + x3x2A1B’1+ x3x2x1A0B’0(A< B) = A’3B3 + x3A’2B2 + x3x2A’1B1+ x3x2x1A’0B0
22DecoderIs a combinational circuit that converts binary information from n input lines to a maximum of 2n unique output lines For example if the number of input is n=3 the number of output lines can be m=23 . It is also known as 1 of 8 because one output line is selected out of 8 available lines:3 to 8decoderenable
24Decoder with Enable Line Decoders usually have an enable line,If enable=0 , decoder is off. It means all output lines are zeroIf enable=1, decoder is on and depending on input, the corresponding output line is 1, all other lines are 0See the truth table in next slide
25Truth table for decoder E a2 a1 a0 D7 D6 D5 D4 D3 D2 D1 D00 x x x1……………………………………….……………………………………..
27Major application of Decoder Decoder is use to implement any combinational cicuits ( fn )For example the truth table for full adder is s (x,y,z) = ∑ ( 1,2,4,7)and C(x,y,z)= ∑ (3,5,6,7). The implementation with decoder is:
28EncoderEncoder is a digital circuit that performs the inverse operation of a decoderGenerates a unique binary code from several input lines.Generally encoders produce2-bit, 3-bit or 4-bit code. n bit encoder has 2n input lines2 bit encoder
292-bit encoderIf one of the four input lines is active encoder produces the binary code corresponding to that lineIf more than one of the input lines will be activated or all the output is undefined. We can consider don’t care for these situations but in general we can solve this problem by using priority encoder.
302-bit Priority EncoderA priority encoder is an encoder circuit that includes priority function.It means if two or more inputs are equal to 1 at the same time, the input having higher subscript number, considered as a higher priority. For example if D3 is 1 regardless of the value of the other input lines the result of output is 3 which is 11.If all inputs are 0, there is no valid input. For detecting this situation we considered a third output named V. V is equal to 0 when all input are 0 and is one for rest of the situations of TT.
312-bit Priority EncoderBy using TT and K-map we get following boolean functions for 4-input (or 2-bit) priority encoder:X = D2 + D3Y = D3 + D1D’2V= D0 + D1 + D2 + D3See next two slides for K-maps and the logic circuit of 2-bit priority encoder
34MultiplexerIt is a combinational circuit that selects binary information from one of the input lines and directs it to a single output lineUsually there are 2n input lines and n selection lines whose bit combinations determine which input line is selectedFor example for 2-to-1 multiplexer if selection S is zero then I0 has the path to output and if S is one I1 has the path to output (see the next slide)
37Boolean function Implementation Another method for implementing boolean function is using multiplexerFor doing that assume boolean function has n variables. We have to use multiplexer with n-1 selection lines and1- first n-1 variables of function is used for data input2- the remaining single variable ( named z )is used for data input. Each data input can be z, z’, 1 or 0. From truth table we have to find the relation of F and z to be able to design input lines. For example : f(x,y,z) = ∑(1,2,6,7)
40Three-State GatesThree state gates exhibit three states instead of two states. The three states are:high : 1Low : 0High impedance : In that state the output is disconnected which is equal to open circuit. In the other words in that state circuit has no logic significant. We can have AND or NAND tree-state gates but the most common is three-state buffer gate
41Three-State GatesWe may use conventional gates such as AND or NAND as tree-state gates but the most common is three-state buffer gate.Note that buffer produces transfer function and can be used for power amplification. Three state buffer has extra input control line entering the bottom of the gate symbol (see next slide)