CS 105 DIGITAL LOGIC DESIGN Chapter 4 Combinational Logic 1

Outline  4.1 Introduction.  4.2 Combinational Circuits.  4.3 Analysis Procedure.  4.4 Design Procedure.  4.5 Binary Adder-Subtractor.  4.6 Decimal Adder.  4.7 Binary Multiplier.  4.9 Decoders.  4.10 Encoders.  4.11 Multiplexers. 2

4.1 Introduction (1-2)  Logic circuits for digital systems may be combinational or sequential.  Consists of logic gates whose outputs at any time are determined from only the present combination of inputs.  Performs an operation that can be specified logically by a set of Boolean functions. Combinational Circuit 3

4.1 Introduction (2-2) 4  Employs storage elements in addition to logic gates.  Their outputs are a function of the inputs and the state of the storage elements.  Because the state of the storage elements is a function of previous inputs, the outputs of a sequential circuit depend not only on present value of inputs, but also on past inputs. Sequential Circuit

5 4.2 Combinational Circuit (1-2) Input Variables Consists of: Logic Gates Output Variables Transforms input data into required output data. Combinational circuits n inputs m outputs Block diagram

6 4.2 Combinational Circuit (2-2)  n input variables  2 n binary input combinations.  Each possible combination  one possible combination output.  Combinational circuit can be specified with truth table.  Combinational circuit can be described by m Boolean functions.  Each output function is expressed in terms of the n input variables. Standard Combination Circuits  Adders, subtractors, comparators, decoders, encoders and multiplexers

7 4.3 Analysis Procedure (1-4)  Determine the function that the circuit implements from a logic diagram.  Circuit’s function can be determined by either Boolean function or truth table. Steps  Make sure that it is combinational not sequential.  No memory elements.  No feedback path (feedback path: a connection from the output of one gate to the input of a second gate that forms part of the input to the first gate).  Obtain Boolean function or the truth table.

8 4.3 Analysis Procedure (2-4) Boolean function

9 4.3 Analysis Procedure (4-4) Truth Table

Design Procedure (1-7) Steps  State the problem.  From the specifications of the circuit, determine the required number of inputs and outputs and assign a symbol to each.  Derive the truth table that defines the required relationship between inputs and outputs.  Obtain the simplified Boolean functions for each output as a function of the input variables.  Draw the logic diagram and verify the correctness of the design

Design Procedure (2-7) Example  Design a circuit that converts binary coded decimal (BCD) to the excess-3 code for the decimal digits. Inputs Outputs  BCD (4 bits).  4 inputs.  Symbols: A, B, C, D.  Ex-3 (4 bits).  4 outputs.  Symbols: w, x, y, z.

Design Procedure (7-7) Logic Diagram

4-5Binary Adder-Subtractor (1-20) 13 Binary Adder-Subtractor  Is a combinational circuit that performs the arithmetic operations of addition and subtraction with binary numbers.

14 Half adder  Is a combinational circuit that performs the addition of two bits = 0 ; = 1 ; = 1 ; 1+ 1 = 10 Elementary Operations Truth Table  two input variables  x, y.  two output variables.  C (output carry), S (least significant bit of the sum). 4-5Binary Adder-Subtractor (2-20)

15 Half adder  S = x'y+xy'  C = xy Simplified Boolean Function (Sum of Products) Logic Diagram (Sum of Products) 4-5Binary Adder-Subtractor (3-20)

16 Half adder S=x  y C = xy Simplified Boolean Function (XOR and AND gates) Logic Diagram (XOR and AND gates) Logic Diagram (XOR and AND gates) 4-5Binary Adder-Subtractor (4-20)

17 Functional Block: Full-Adder  It is a combinational circuit that performs the arithmetic sum of three bits (two significant bits and previous carry).  It is similar to a half adder, but includes a carry-in bit from lower stages.  Two half adders can be employed to implement a full adder. Inputs & Outputs  Three input bits:  x, y : two significant bits  Z : the carry bit from the previous lower significant bit.  Two output variables:  C (output carry), S (least significant bit in sum). 4-5Binary Adder-Subtractor (5-20)

18  For a carry-in (Z) of 0, it is the same as the half-adder:  For a carry- in (Z) of 1: Z0000 X Y C S Z1111 X Y C S Functional Block: Full-Adder Operations 4-5Binary Adder-Subtractor (6-20)

 Add two BCD‘s:  9 inputs: two BCD's and one carry-in.  5 outputs: one BCD and one carry-out.  Design approaches  Use unary full Adders.  A truth table with 2 9 entries  Each input digit does not exceed 9.  The output sum connot be greater than 9.  e.g =19, the 1 in sum being an input carry.  The output of the binay sum must be represented in BCD. BCD Adder 4-6Decimal Adder (1-4)

BCD Adder Truth Table 4-6Decimal Adder (2-4)

BCD Adder Logic Diagram 4-6Decimal Adder (4-4)

22 Multiplication of binary numbers is performed in the same way of decimal numbers. Example  Consider the multiplication of two 2-bit numbers.  Multiplicand bits are B 0 and B 1.  Multiplier bits are A 0 and A 1.  The product is C 3 C 2 C 1 C Binary Multiplier (1-4)

 The partial product can be implemented with AND gates.  The two partial products are added with two half-adder (HA) circuits. Logic Diagram 4-7Binary Multiplier (2-4)

J-Bit by K-Bit Binary Multiplier  For J multiplier bits and K multiplicand bits to produce J + K bits, we need :  J x K AND gates.  (J-1) K-bit adders. Example (4-Bit by 3-Bit Multiplier)  Multiplicand : B 3 B 2 B 1 B 0  Multiplier: A 2 A 1 A 0  12 AND gates  2 four-bit adders.  Produces product of 7 bits 4-7Binary Multiplier (3-4)

Example (4-Bit by 3-Bit Multiplier) 4-7Binary Multiplier (4-4)

4-9Decoder (1-16)  Discrete quantities of information are represented in digital systems by binary codes.  A binary code of n bits is capable of representing up to 2 n distinct elements of coded information.  Is a combinational circuit that converts the binary information from n input lines to a maximum of 2 n unique output lines.  If the n-bit coded information has unused combinations, the decoder may have fewer than 2 n outputs.  Called n-to-m-line decode, where m <= 2 n minterms of n input variables. Decoder

4-9Decoder (2-16)  Inputs = 3.  Outputs = 8 (minterms) Example Consider three-to-eight-line decoder circuit Truth Table Example: Binary – to –octal decoder Example: Binary – to –octal decoder ONLY one output can be active at any time

4-9Decoder (3-16) Logic Diagram

4-9Decoder (4-16) Generates decoder minterms in their complemented form NAND gates

4-9Decoder (5-16)  A decoder with one or more enable (E) inputs.  Control the circuit operation.  E =0, Decoder is disabled.  E =1, Decoder is enabled.  A circuit that receives information from a single Line and directs it to one of 2 n possible output lines. Demultiplixers

4-9Decoder (6-16) Demultiplixers Design a two-to-four-line decoder with an enable input. Truth table D3D3 D2D2 D1D1 D0D0 BAE 0000 XX Uncomplemented output

4-9Decoder (7-16) Demultiplixers Design a two-to-four-line decoder with an enable input. Logic Diagram

4-9Decoder (8-16) Demultiplixers Design a two-to-four-line decoder with an enable input constructed with NAND gates. Truth table D3D3 D2D2 D1D1 D0D0 BAE 1111 XX Complemented output

4-9Decoder (9-16) Demultiplixers Logic Diagram Design a two-to-four-line decoder with an enable input constructed with NAND gates.

4-9Decoder (10-16) Demultiplixers Design a 4-to-16 decoder. Design a 4-to-16 decoder using two 3-to-8 decoders.

4-9Decoder (12-16) Demultiplixers Design a 5-to-32 line decoder using four 3-to-8 line decoders with enable inputs and a 2-to-4 line decoder. D 0 – D 7 D 8 – D 15 D 16 – D 23 D 24 – D 31 A3A4A3A4 A0A1A2A0A1A2 2-4-line Decoder 3-8-line Decoder E E E E

4-9Decoder (13-16) Combinational Logic Implementation  Each output = minterm.  Implementing Boolean function (expressed in sum of minterms) by using:  A decoder.  An external OR gate.  Any combinational circuit with n inputs and m outputs can be implemented with an n-to-2 n -line decoder and m OR gates.

4-9Decoder (14-16) Combinational Logic Implementation Design a full adder using a decoder.

4-10Encoder (1-7)  Is a digital circuit that performs the inverse operation of a decoder.  An encoder has 2 n (or fewer) input lines and n output lines. Example Design an octal-binary encoder

4-10Encoder (2-7) Example Design an octal-binary encoder

4-10Encoder (3-7) Limitations on previous example  If two inputs are active simultaneously, the output produces an undefined combination. E.g. if D 3 and D 6 are 1 simultaneously, the output of the encoder will be 111. Resolve this ambiguity, establish an input priority to ensure. D6 will be the higher priority.  Output with all 0's is generated when: All the inputs are 0 D 0 is equal to 1. Resolve by providing one more output to indicate whether at least one input is equal to 1.

4-10Encoder (4-7) Priority Encoder  Is an encoder circuit that includes the priority function.  Resolve the ambiguity of illegal inputs.  if two or more inputs are equal o 1 at the same time. the input having the highest priority will take precedence.

4-10Encoder (5-7) Priority Encoder Design an four –to - two priority encoder

4-11Multiplexer (1-15)  is a combinational circuit that selects a binary information from one of many input lines and directs it to a single output line.  The selection of a particular input line is contro1led by a set of selection lines. 2 n input lines and n selection lines whose bit combinations determine which input is selected.  Also called a data selector, since it selects one of many inputs and steers the binary information to the output line.  The size of a multiplexer is specified by the number 2 n of its data input lines and the single output line.

4-11Multiplexer (2-15) Example Design a 2 –to-1 line MUX  Data Inputs = 2 1 =1  Selection Input= 1  Output = 1 YS0S0 I0I0 0 I1I1 1 Function Table

4-11Multiplexer (3-15) Example Design a 4 –to-1 line MUX  Data Inputs = 2 2 =4  Selection Input= 2  Output = 1 Function Table

4-11Multiplexer (4-15) Example Design a 4 –to-1 line MUX

4-11Multiplexer (5-15) Notes  2 n – to – 1 MUX can be implemented using decoder: Decode selection input lines. n (selection input lines) – to - 2 n decoder. Adding the 2 n input lines to each AND gate. OR all AND gates. An enable input (an option).  Multiplexers may have an enable input to control the operation of the unit When the enable input is in the inactive state, the outputs are disabled. When it is in the active state, the circuit functions as a normal multiplexer.

4-11Multiplexer (6-15) Example Design an 8–to-1 line MUX using a 3-to-8 line decoder.

4-11Multiplexer (7-15) Multiple bit Selection  Multiplexer circuits can be combined with common selection inputs to provide multiple-bit selection logic.  E.g. quadruple 2-to-1 line MUX.

4-11Multiplexer (8-15) Quadruple 2-to- 1 -line multiplexer.

4-11Multiplexer (9-15) Boolean Function Implementation  Boolean function with n variables can be implemented with a multiplexer that has: n-1 selection inputs. 2 n-1 data inputs.  The first n-1 variables connected to the selection inputs of the MUX.  The remaining single variable of the function is used for the data inputs. If the single variable is denoted by z, each data input of the MUX will be z,z’,1 or 0.

4-11Multiplexer (10-15) Example Implment the following function using a MUX F(A,B,C) =  (1,2,6,7) Implment the following function using a MUX F(A,B,C) =  (1,2,6,7) Fig Implment the following function using a MUX F(A, B, C, D) =  (1, 3, 4, 11, 12, 13, 14, 15) Implment the following function using a MUX F(A, B, C, D) =  (1, 3, 4, 11, 12, 13, 14, 15)

4-11Multiplexer (12-15) Three-State Gates  A multiplexer can be constructed with three-state gates.  Output state: 0, 1, and high-impedance (open circuit)  When the control input =1 : The output is enabled. The gate behave like a conventional buffer, with the output is equal to normal input.  When the control input=0 : The output is disabled The gate goes to a high-impedance state, regardless the value of the normal input.

4-11Multiplexer (13-15) Example Two -to-one-line multiplexer with three state gates

4-11Multiplexer (14-15) Example Four-to-one-line multiplexer using decoder with three state gates

4-11Multiplexer (15-15) Example Four-to-one-line multiplexer with three state gates  The control inputs to the buffers determine which one of the four normal inputs I 0 to I 3 will be connected to output line.  No more than one buffer may be in the active state at any given time.  The connected buffers must be controlled so that only 1 thee-state buffer has access to the output while all other buffers are maintained in a high-impedance state.  One way to ensure that no more than one control input is active at any given time is to use a decoder.  When E=0  high impedance state (all buffers are disable)  When E=1  one of the three buffers will be active.