Excess-33-excess [1] [2] [3] or excess-3 binary code often abbreviated as XS-33XS [1] or X3 [4] [5] or Stibitz code [1] [2] after George Stibitzwho built a relay-based adding machine in [6] [7] is a self-complementary binary-coded decimal BCD code and numeral system. It is a biased representation. Excess-3 code was used on some older computers as well as in cash registers and hand-held portable electronic calculators of the s, among other uses.

Biased codes are a way to represent values with a balanced number of positive and negative numbers using a pre-specified number N as a biasing value. Biased codes and Gray codes are non-weighted codes.

In excess-3 code, numbers are represented as decimal digits, and each digit is represented by four bits as the digit value plus 3 the "excess" amount :. To encode a number such asone simply encodes each of the decimal digits as above, giving, Excess-3 arithmetic uses different algorithms than normal non-biased BCD or binary positional system numbers. After adding two excess-3 digits, the raw sum is excess For instance, after adding 1 in excess-3 and 2 in excess-3the sum looks like 6 in excess-3 instead of 3 in excess In order to correct this problem, after adding two digits, it is necessary to remove the extra bias by subtracting binary decimal 3 in unbiased binary if the resulting digit is less than decimal 10, or subtracting binary decimal 13 in unbiased binary if an overflow carry has occurred.

In 4-bit binary, subtracting binary is equivalent to adding and vice versa. The primary advantage of excess-3 coding over non-biased coding is that a decimal number can be nines' complemented [1] for subtraction as easily as a binary number can be ones' complemented : just by inverting all bits.

This works because, after adding two digits, an "excess" value of 6 results in the sum. Because a 4-bit integer can only hold values 0 to 15, an excess of 6 means that any sum over 9 will overflow produce a carry out. Another advantage is that the codes and are not used for any digit. A fault in a memory or basic transmission line may result in these codes.

It is also more difficult to write the zero pattern to magnetic media. From Wikipedia, the free encyclopedia. For the experimental aircraft, see Douglas XS-3 Stiletto. Written at Karlsruhe, Germany. Taschenbuch der Nachrichtenverarbeitung in German 1 ed. Taschenbuch der Nachrichtenverarbeitung in German. Berlin, Germany: Springer Verlag. Arithmetic Operations in Digital Computers. Decimal Computation 1 ed. Retrieved Decimal Computation 1 reprint ed. Krieger Publishing Company.

At least some batches of this reprint edition were misprints with defective pages — Informations- und Kommunikationstechnik in German. Berlin, Germany. Archived from the original on Prerequisite — Number System and base conversions. Gray Code system is a binary number system in which every successive pair of numbers differs in only one bit. It is used in applications in which the normal sequence of binary numbers generated by the hardware may produce an error or ambiguity during the transition from one number to the next.

For example, the states of a system may change from 3 to 4 as- — — — Therefore there is a high chance of a wrong state being read while the system changes from the initial state to the final state.

This could have serious consequences for the machine using the information.

## Code Converters – Binary to Excess 3, Binary to Gray and Gray to Binary

The Gray code eliminates this problem since only one bit changes its value during any transition between two numbers. The truth table for the conversion is- To find the corresponding digital circuit, we will use the K-Map technique for each of the gray code bits as output with all of the binary bits as input.

K-map for — K-map for — K-map for — K-map for —. Corresponding minimized boolean expressions for gray code bits — The corresponding digital circuit —. Converting gray code back to binary can be done in a similar manner. Truth table. Using K-map to get back the binary bits from the gray code — K-map for — K-map for — K-map for — K-map for —.

Corresponding digital circuit —. This article is contributed by Chirag Manwani. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above. Writing code in comment? Please use ide. Prerequisite — Number System and base conversions Gray Code system is a binary number system in which every successive pair of numbers differs in only one bit. Improved By : KuntalSarkar1dhananjaykajla.

Load Comments.Next up in our digital electronics coursewe will take a look at binary code converters. We will look at different types of code converters.

By the end of this post, you will be able to design most of the important code converters, including a few important ones.

All the types of codes have some defining property in them. All of these codes are essentially used to help us encode decimal numbers in formats that can be understood by a digital system.

For a binary codeeach bit has a value that depends on its position in the number. Similarly, for gray codeits specialty lies in the fact that it requires each subsequent number to differ from its predecessor by only 1 bit.

You can notice this when we derive the truth table below. We will see these specialties in detail in this post. We can inter-convert between these different types of codes using special combinational logic circuits known as code converters.

These different types of codes give us flexibility in designing transmission paths, memory storage, etc. The following bits of the gray code can be obtained by EXORing the corresponding binary bit and the preceding binary bit. Based on the above equations, we can plot the following circuit diagram for a 3-bit binary to gray code converter using EX-OR logic gates.

Note: Notice how each subsequent gray code number differs with its predecessor by only one bit. The process is similar to the one we saw above.

The MSBs are going to be equal. The subsequent gray code bit will be obtained by EXORing the corresponding binary bit with the preceding binary bit. The equations above indicate the presence of three EXOR gates. Therefore the simple combinational circuit for 4-bit binary to gray code converters is as shown below. To convert from gray to binary, a slightly different approach from the one we saw above is used. The MSBs are always equal.

The next binary bit is obtained by EXORing the corresponding gray code bit with the preceding binary bit. An excess 3 code, as can be predicted from its name, is an excess of three of the binary number.

Yes, the number is written in binary format, and that can be a source of confusion. Think of it this way. You have a normal number system. However, your friend wants to be unique and says that for him, a six will be equal to your three. The representation is the same. However, the values differ by three.

Hence, from the equations above we can design the following combinational logic circuit for 3-bit binary to excess 3 code converter circuit. Following our footsteps from the designing of 3-bit binary to excess 3 code converters, we will first draft a truth table for the 4-bit version.

Using Kmaps, we will solve for the output terminals. You might end up with different equations than the ones in this post.Karnaugh map method or K-map method is the pictorial representation of the Boolean equations and Boolean manipulations are used to reduce the complexity in solving them. As it is evaluated from the truth table method, each cell in the K-map will represent a single row of the truth table and a cell is represented by a square.

The cells in the k-map are arranged in such a way that there are conjunctions, which differs in a single variable, are assigned in adjacent rows.

The K-map method supports the elimination of potential race conditions and permits the rapid identification.

### Design of BCD to Excess-3 Code Converter Circuit

By using Karnaugh map technique, we can reduce the Boolean expression containing any number of variables, such as 2-variable Boolean expression, 3-variable Boolean expression, 4-variable Boolean expression and even 7-variable Boolean expressions, which are complex to solve by using regular Boolean theorems and laws. It will look like see below image. The possible min terms with 2 variables A and B are A. The following table shows the positions of all the possible outputs of 2-variable Boolean function on a K-map.

When we are simplifying a Boolean equation using Karnaugh map, we represent the each cell of K-map containing the conjunction term with 1. After that, we group the adjacent cells with possible sizes as 2 or 4.

In case of larger k-maps, we can group the variables in larger sizes like 8 or The groups of variables should be in rectangular shape, that means the groups must be formed by combining adjacent cells either vertically or horizontally. Diagonal shaped or L-shaped groups are not allowed.

The following example demonstrates a K-map simplification of a 2-variable Boolean equation. Here the lower right cell is used in both groups. After grouping the variables, the next step is determining the minimized expression.

By reducing each group, we obtain a conjunction of the minimized expression such as by taking out the common terms from two groups, i. For a 3-variable Boolean function, there is a possibility of 8 output min terms.

The general representation of all the min terms using 3-variables is shown below. A typical plot of a 3-variable K-map is shown below. It can be observed that the positions of columns 10 and 11 are interchanged so that there is only change in one variable across adjacent cells. This modification will allow in minimizing the logic. Up to 8 cells can be grouped in case of a 3-variable K-map with other possibilities being 1,2 and 4.

The largest group size will be 8 but we can also form the groups of size 4 and size 2, by possibility. In the 3 variable Karnaugh map, we consider the left most column of the k-map as the adjacent column of rightmost column. So the size 4 group is formed as shown below. So the group of size 4 is reduced as the conjunction Y. To consume every cell which has 1 in it, we group the rest of cells to form size 2 group, as shown below. The 2 size group has no common variables, so they are written with their variables and its conjugates.

In this equation, no further minimization is possible. There are 16 possible min terms in case of a 4-variable Boolean function. The general representation of minterms using 4 variables is shown below. A typical 4-variable K-map plot is shown below. It can be observed that both the columns and rows of 10 and 11 are interchanged.The Web This site.

Karnaugh Maps offer a graphical method of reducing a digital circuit to its minimum number of gates. The map is a simple table containing 1s and 0s that can express a truth table or complex Boolean expression describing the operation of a digital circuit.

The map is then used to work out the minimum number of gates needed, by graphical means rather than by algebra. Karnaugh maps can be used on small circuits having two or three inputs as an alternative to Boolean algebra, and on more complex circuits having up to 6 inputs, it can provide quicker and simpler minimisation than Boolean algebra. The shape and size of the map is dependent on the number of binary inputs in the circuit to be analysed.

The map needs one cell for each possible binary word applied to the inputs. Notice that this edge numbering does not follow the normal binary counting sequence, but uses a Gray Code sequence where only one bit changes from one cell to the next.

This is an important feature of Karnaugh maps; get the sequence wrong and the map will not work! The input labels are written at the top left hand corner, divided by a diagonal line. The top and left edges of the map then represent all the possible input combinations for the inputs allocated to that edge.

For example, in the 3 input map b in Fig. Because example b in Fig. This map is therefore rectangular rather than square to cover the 8 possible combinations available from 3 inputs. As an example, Table 2. This results in a Boolean equation for the un-simplified circuit:. This table will serve to show the process of transferring the data from Table 2.

The process is shown step by step in Fig. From Table 2. In Table 2. Finally, in Table 2. All the truth table rows that produced a logic 1 have now been entered into the map and those lines that produced a logic 0 can be ignored, so the remaining three cells are left blank. Later it will be shown that these blank cells can be useful when mapping larger circuits, but for now the map is ready for simplification.

Simplifying Karnaugh Maps Circuit simplification in any Karnaugh map is achieved by combining the cells containing 1 to make groups of cells. In grouping the cells it is necessary to follow six rules. This helps make smaller groups as large as possible, which is an advantage in finding the simplest solution.

Map a follows rules 2, 3 and 4 and shows three groups containing 8, 4 and 2 cells. This will simplify the circuit being produced, but it is not optimum. Map b shows an improvement, still with 3 groups but they now contain 8, 4 and 4 cells. This map takes advantage of rule 5 by joining the 2 cells ringed in green in Map a with the top two cells in the blue group, see Map b to form a group of 4 ringed in cyan instead of a group of 2.

The map now conforms to all 6 rules. Sometimes however there may be a single cell that cannot be joined with other groups, as shown in map d.

Rule 3 prohibits diagonal grouping so there is no alternative other than to leave a group of 1. This is permissible, but in map dwhich represents a four input circuit, the simplified Boolean equation will contain an un-simplified expression relating to the single cell, which will have all four possible terms e.

Using the Karnaugh map rules on the three input map created from Table 2. The next task is to simplify the original Boolean equation for this circuit:. Converting the two groups in the Karnaugh map to Boolean expressions is done by discovering which input or inputs A, M or C does NOT change within each group.

Therefore the only input that does not change in the blue group is M, so the Boolean expression for the blue group is simply M. Looking at the green group of 2, A does not change but MC changes from 01 to Latest Projects Education. Home Forums Education Homework Help. JavaScript is disabled. For a better experience, please enable JavaScript in your browser before proceeding. Thread starter The Rock Start date Sep 25, Search Forums New Posts.

### Code Converters – Binary to/from Gray Code

I need to convert Excess 3 to BCD. What are my inputs in the truth tables so that i can do the K Maps? Do i need to put the don't cares in the Excess 3 Code column? Plzz help! Scroll to continue with content. Georacer Joined Nov 25, 5, If your input and output are 4 bits long, you could make 4 4-input Boolean functions that will take the Excess 3 number and produce one digit of the output each.

**BCD to Excess-3 code converter**

You will end up with a medium-sizes combinatorial circuit, which can be implemented either with simple gates or with MUXs. Is that clear? Post a conversion truth table if you need more help. Can u post the conversion truth table for me? Its still vague for me By the question, i think we need to implement it using K maps for each input used n construct the circuit as per the K map!

Your thinking this correct. You started the truth table correctly yourself, you just need to finish it: Rich BB code :.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

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I'm taking computer science courses and some digital design knowledge is required, so I'm taking digital design Image above is representing the conversion process of 8,4,-2,-1 to binary using K-map Karnaugh map. For,they could be expressed as 8,4,-2,-1 as, And for,can still be expressed as in 8,4,-2,-1 form as I understand the excess-3 to binary code conversion provided from my textbook example mm15 are marked as 'X' since excess-3 were used to express only The 4 given inputs can therefore represent only values from interval from 0 to 9.

Learn more. K-map karnaugh map 8,4,-2,-1 to binary code conversion Ask Question. Asked 3 years, 9 months ago. Active 3 years, 9 months ago. Viewed 7k times. I have no idea why,are marked as 'X'.

Is there something I'm missing? Kit Ostrihon 2 2 gold badges 9 9 silver badges 33 33 bronze badges. Minjae Minjae 21 1 1 silver badge 7 7 bronze badges. Try cs. This site is for actual practical programming, not theory. MarcB Thanks for your advise. Minjae Checking with Wikipedia: the 9 is the largest value of BCD, that is why there is no coding for 10, 11 and The solution corresponds with that definition.

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