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Representing Decimal Numbers
- When calculations are carried out electronically they will usually be in binary or twos complement notation, but the result will very probably need to be displayed in decimal form.
- A binary number with its bits representing values of 1, 2, 4, 8, 16 etc. presents problems.
- It would be better if a particular number of binary bits could represent the numbers 0 to 9, but this doesn’t happen in pure binary, a 3 bit binary number represents the values 0 to 7 and 4 bit represents 0 to 15.
- What is needed is a system where a group of binary digits can represent the decimal numbers 0-9, or ten times those values, 10-90, etc.
- To make this possible, binary codes are used that have ten values, but where each value is represented by the 1s and 0s of a binary code.
- These special ‘half way’ codes are called BINARY CODED DECIMAL or BCD. There are several different BCD codes, but they have a basic similarity.
- Each of the ten decimal digits 0 to 9 is represented by a group of 4 binary bits, but in codes the binary equivalents of the 10 decimal numbers do not necessarily need to be in a consecutive order.
- Any group of 4 bits can represent any decimal value, so long as the relationship for that particular code is known.
- In fact any ten of the 16 available four bit combinations could be used to represent 10 decimal numbers, and this is where different BCD codes vary.
There can be advantages in some specialist applications in using some particular variation of BCD.
For example it may be useful to have a BCD code that can be used for calculations, which means having positive and negative values, similar to the twos complement system, but BCD codes are most often used for the display of decimal digits.
The most commonly encountered version of BCD binary code is the BCD_{8421} code.
In this version the numbers 0 to 9 are represented by their pure binary equivalents, 4 bits per decimal number, in consecutive order.
BCD Codes
- BCD_{8421}code is so called because each of the four bits is given a ‘weighting’ according to its column value in the binary system.
- The least significant bit (lsb) has the weight or value 1, the next bit, going left, the value 2.
- The next bit has the value 4, and the most significant bit (msb) the value 8, as shown in Table 1.
- So the BCD_{8421} code for the decimal number 6_{10} is 01108421. Check this in Table 1.
Table1 | ||||
MSB | BCD_{8421} | LSB | ||
Decimal | 8 | 4 | 2 | 1 |
0 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 1 |
2 | 0 | 0 | 1 | 0 |
3 | 0 | 0 | 1 | 1 |
4 | 0 | 1 | 0 | 0 |
5 | 0 | 1 | 0 | 1 |
6 | 0 | 1 | 1 | 0 |
7 | 0 | 1 | 1 | 1 |
8 | 1 | 0 | 0 | 0 |
9 | 1 | 0 | 0 |
1 |
- For numbers greater than 9 the system is extended by using a second block of 4 bits to represent tens and a third block to represent hundreds etc.
24_{10} in 8 bit binary would be 00011000 but in BCD_{8421}is 0010 0100.
992_{10} in 16 bit binary would be 00000011111000002 but in BCD_{8421}is 1001 1001 0010.
- Therefore BCD acts as a half way stage between binary and true decimal representation, often preparing the result of a pure binary calculation for display on a decimal numerical display.
- Although BCD can be used in calculation, the values are not the same as pure binary and must be treated differently if correct results are to be obtained. The facility to make calculations in BCD is included in some microprocessors.
One of the main drawbacks of BCD is that, because sixteen values are available from four bits, but only ten are used, there are several redundant values whichever BCD system is used. This is wasteful in terms of circuitry, as the fourth bit (the 8s column) is under used.
Display Decoder/Drivers
- Depending on the type of display some further code conversion may also be needed.
- One popular type of decimal display is the 7 segment display used in LED and LCD numerical displays, where any decimal digit is made up of 7 segments arranged as a figure 8, with an extra LED or LCD dot that can be used as a decimal point, as shown in Fig 1. These displays therefore require 7 inputs, one to each of the LEDs a to g (the decimal point is usually driven separately).
- Therefore the 4 bit output in BCD must be converted to supply the correct 7 bit pattern of outputs to drive the display.
- The four BCD bits are usually converted (decoded) to provide the correct logic for driving the 7 inputs of the display by integrated circuits such as the HEF4511B BCD to 7 segment decoder/driver from NXP Semiconductors and the 7466 BCD to 7 segment decoder.
Table2 | |||||||||||
BCD Input | 7 Segment Output | Display | |||||||||
2^{3} | 2^{2} | 2^{1} | 2^{0} | a | b | C | d | e | F | g | |
0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 2 |
0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 3 |
0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 4 |
0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 5 |
0 | 1 | 1 | 0 | 0* | 0 | 1 | 1 | 1 | 1 | 1 | 6 |
0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0* | 0 | 7 |
1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 |
1 | 0 | 0 | 1 | 1 | 1 | 1 | 0* | 0 | 1 | 1 | 9 |
1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | Blank |
1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | Blank |
1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | Blank |
1 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | Blank |
1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | Blank |
1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | Blank |
Notice that the 4 bit input to the decoder illustrated in Table 2 can, in this case, be in either BCD_{8421}or in 4 bit binary as any binary number over 9 will result in a blank display.
Alternative BCD Codes
- Although BCD_{8421}is the most commonly used version of BCD, a number of other codes exist using other values of weighting. Some of the more common variations are shown below.
- The weighting values in these codes are not randomly chosen, but each has particular merits for specific applications.
- Some codes are more useful for displaying decimal results with fractions, as with financial data. With others it is easier to assign positive and negative values to numbers.
For example with Excess 3 code, 3_{10} is added to the original BCD value and this makes the code ‘reflexive’, that is the top half of the code is a mirror image and the complement of the bottom half. Other codes are designed to improve error detection in specific systems. Some of these less common BCD codes are shown in Table 3.
Table 3 | |||||
Decimal | 7421 | 5421 | 5211 | 2421 | Excess 3 |
0 | 0000 | 0000 | 0000 | 0000 | 0011 |
1 | 0001 | 0001 | 0001 | 0001 | 0001 |
2 | 0010 | 0010 | 0010 | 0010 | 0101 |
3 | 0011 | 0011 | 0101 | 0011 | 0110 |
4 | 0100 | 0100 | 0111 | 0100 | 0111 |
5 | 0101 | 1000 | 1000 | 0101 | 1000 |
6 | 0110 | 0110 | 1001 | 0110 | 1001 |
7 | 1000 | 0111 | 1011 | 0111 | 1010 |
8 | 1001 | 1011 | 1101 | 1110 | 1100 |
9 | 1010 | 1100 | 1111 | 1111 | 1100 |
Gray Code
- Binary codes are not only used for data output.
- Another special binary code that is extensively used for reading positional information on mechanical devices such as rotating shafts is Gray Code.
- This is a 4 bit code that uses all 16 values, and as the values change through 0-15_{10} the code’s binary values change only 1 bit at a time, (see Table 4).
- The binary values are encoded onto a rotating disk (Fig. 3) and as it rotates the light and dark areas are read by optical sensors.
As only one sensor sees a change at any one time, this reduces errors that may be created as the sensors pass from light to dark (0 to 1) or back again.
- The problem with this kind of sensing is that if two or more sensors are allowed to change simultaneously, it cannot be guaranteed that the data from the sensors would change at exactly the same time.
- If this happened there would be a brief time when a wrong binary code may be generated, suggesting that the disk is in a different position to its actual position.
- The one bit at a time feature of Gray Code effectively eliminates such errors.
Notice also that the sequence of binary values also rotates continually, with the code for 15 changing back to 0 with only 1 bit changing. With a 4 bit coded disk as illustrated in Fig. 3, the position is read every 22.5° but with more bits, greater accuracy can be achieved.
Table 4 | |
Decimal | Gray Code |
0 | 0000 |
1 | 0001 |
2 | 0011 |
3 | 0010 |
4 | 0110 |
5 | 0111 |
6 | 0101 |
7 | 0100 |
8 | 1100 |
9; | 1101 |
10 | 1111 |
11 | 1110 |
12 | 1010 |
13 | 1011 |
14 | 1001 |
15 | 1000 |
AUTHORS
1.Bunty B. Bommera
2.Dakshata U. Kamble[/vc_column_text][/vc_column][/vc_row]