Digital System and Binary Numbers

Digital Systems:
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 Digital systems are used in:

1. Communication
2. Business transaction
3. Traffic Control
4. Medical treatment
5. Internet

The signals in digital systems use just two discrete values: a binary digit. Binary digit
called a bit, has two values: 0 or 1.
Example: The decimal digits a through 9 are represented in digital system with a code
of four bits (e.g. is represented by 0111, 8 is represented by 1000, 9 is represented by
1001).

Binary Numbers

• Decimal number:

         The decimal number system is said to be of base, or radix, '10' because it uses 10 digits
          (0, 1, 2, 3, 4, 5, 6, 7, 8, 9).

Example:
The decimal number 245 may be written as
2x102+4x101+5x100
where 2, 4, and 5 are the coefficients. 

• Binary Number System: 

         The coefficients of the binary number have only two possible values: '0' or '1'.

Example:

(1001)2 its equivalent decimal number is : 1x23+0x22+0x21+1x20 = 8 + 0 + 0 + 1 = 9
where 1, 0, 0, and 1 are the coefficients. 

• Octal number system (Base 8):

         The octal number system is said to be of base, or radix, '8' because it uses digits
   (00, 01, 02, 03, 04, 05, 06, 07, 10, 11, 12, 13, 14, 15, 16, 17). 

• Hexadecimal number system (Base 16):

          It uses 16 digits : (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F). The letter A, B, C, D, E,
  and F are used for the digits 10,11, 12, 13, 14, and 15 respectively.

Example:
(124)16 = 1x162 + 2x161 + 4x160= 256 + 32 + 4 = 292

NUMBER BASE CONVERSION

Decimal to Binary conversion

1. Divide the number with base '2'.
2. Take the remainder 0 or 1 as a coefficient.
3. Take the quotient and repeat the division.

   

Decimal to Octal conversion

Binary to Octal conversion

 Binary to Hexadecimal

Octal to Hexadecimal conversion

Complements:


The meaning of complement is something required to make a thing complete. For example, salsa complements tortilla chips, beer complements pizza, an ice cream cone complements a hot summer day, and apple sauce complements pork chops. A key concept to explore is how two things complement each other. For example, when a piece of pizza is removed from a whole pizza the piece complements what is left behind and vice versa. Each of the 4 following complements use the same concept except in different bases and what is considered a complete number in that base.

• 1’s Complement: 

         The 1’s complement finds whatever is needed to make an entire set of 1’s. This is shown in the

•  2’s ComplementFinding the 8 digit 2’s complement of 01101100

• 9’s ComplementThe 9’s complement finds whatever is needed to make an entire set of 9’s. This is shown in the    

• 10’s ComplementFinding the 5 digit 10’s complement of 1357 

      

Basic Logic Gate

         A logic gate performs a logical operation on one or more logic inputs and produces a single logic output. The logic normally performed is Boolean logic and is most commonly found in digital circuits. Logic gates are primarily implemented electronically using diodes or transistors, but can also be constructed using electromagnetic relays (relay logic), fluidic logic, pneumatic logic, optics, molecules, or even mechanical elements.

       In electronic logic, a logic level is represented by a voltage or current, (which depends on the type of electronic logic in use). Each logic gate requires power so that it can source and sink currents to achieve the correct output voltage. In a pure logic diagram logic levels and power supply connections do not exist and are not shown, but in a full logic circuit diagram, documenting the implementation of the logic, these are required.


      All other types of Boolean logic gates (i.e., AND, OR, NOT, XOR, XNOR) can be created from a suitable network of NAND gates. Similarly all gates can be created from a network of NOR gates. Historically, NAND gates were easier to construct from MOS technology and thus NAND gates served as the first pillar of Boolean logic in electronic computation.


     Truth table is a table that describes the behaviour of a logic gate or any combination of logic gates. It lists the value of the output for every possible combination of the inputs and can be used to simplify the number of logic gates and level of nesting in an electronic circuit. In general the truth table does not lead to an efficient implementation; a minimization procedure, using Karnaugh maps, the Quine–McCluskey algorithm or an heuristic algorithm is required for reducing the circuit complexity.

Truth table & Symbols

 

 

32x8 FIFO

FIFO is nothing but simple queue like we standing in cinema theaters.one who standing first will get ticket last guy may or may not...this is concept.in technical 32x8 fifo is 32 locations with each location can accommodate 8 bit data...in this writeptr and readptr plays very important role...operation starts when write enables data stores into reg and startes incriment writeptr..after completion write operation we have to enable read operation..when read is enabled readptr starts incrimenting and extract data from register....

module fifo2 (dout,enb1, wrtenb, redenb, clk, rst, out,wrtptr,redptr, empty,full);
input enb1;
input [7:0] dout;
input wrtenb;
input redenb;
input clk;
input rst;   
output [4:0] wrtptr;
output [4:0] redptr;
output [7:0] out;
output full;
output empty;
reg [7:0] out;
wire full;
wire empty;
reg [7:0] dreg [0:31] ;
reg [4:0] wrtptr;
reg [4:0] redptr;
always@(posedge clk or negedge rst)
    begin
        if(~rst)
            begin   
            wrtptr<=5'b00000;
            redptr<=5'b00000;
           
            end   
        else  if(enb1)
            if(wrtptr==5'b11111)
                begin
                    wrtptr<=5'b00000;
                   
                end
           
         if(wrtenb==1&&redenb==0)
           
                begin
                   
                     dreg[wrtptr]<=dout;
                    wrtptr<=wrtptr+1;
                end
                if(redptr==5'b11111)
                    begin
                        redptr<=5'b00000;
                       
                        end
                   
              if(wrtenb==0&& redenb==1)
                    begin
                   out <=dreg[redptr];
                  redptr<=redptr+1;
               
                end
            end
       
            assign  full=(wrtptr==5'b11111)?1'b1:1'b0;
            assign empty=(redptr==5'b11111)?1'b1:1'b0;
       endmodule
You could download file 32x8fifo.v and testbench.tb here

3x8 Decoder

A decoder is a device which does the reverse operation of an encoder, undoing the encoding so that the original information can be retrieved. A slightly more complex decoder would be the n-to-2ⁿ type binary decoders. These type of decoders are combinational circuits that convert binary information from 'n' coded inputs to a maximum of 2ⁿ unique outputs. We say a maximum of 2ⁿ outputs because in case the 'n' bit coded information has unused bit combinations, the decoder may have less than 2ⁿ outputs. We can have 2-to-4 decoder, 3-to-8 decoder or 4-to-16 decoder. We can form a 3-to-8 decoder from two 2-to-4 decoders (with enable signals).


module dec (x, z);
input [2:0] x;
output [7:0] z;
reg [7:0] z;
always @ (x)
    begin
            case(x)
                3'b000 : z=8'b00000001;
                3'b001 : z=8'b00000010;
                3'b010 : z=8'b00000100;
                3'b011 : z=8'b00001000;
                3'b100 : z=8'b00010000;
                3'b101 : z=8'b00100000;
                3'b110 : z=8'b01000000;
                3'b111 : z=8'b10000000;
                default:z=8'bxxxxxxxx;
            endcase
                    end
endmodule

You could download file dec.v and testbench.tb here