Design And Simulation Of A Balanced Gray Code To Binary Enco

Design and Simulation of a Balanced Gray Code to Binary Encoder in Logisim

You will need Logisim to complete this project assignment. Further information about LogiSim is at . If you have not done so already, download and install Logisim 2.7.1 from .

The objective of this project is to reinforce your understanding of binary codes, combinational logic design, and logic simulation. You must design a combinational logic circuit that accepts a four-bit Balanced Gray code input and creates a four-bit binary output that represents the same hexadecimal value. Then, you need to test your design thoroughly in Logisim and document your work in a short report.

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Gray codes are a class of binary numeral systems where consecutive numbers differ in only one bit. This property makes them valuable in applications such as mechanical encoders and asynchronous systems, reducing transition errors and ensuring signal stability. Specifically, Balanced Gray codes extend this concept by ensuring that each bit position transitions equally often across a full cycle, which can be instrumental for certain hardware and communication systems requiring uniform transition distribution.

The core challenge in this project involves translating a Balanced Gray code into its corresponding binary (standard binary) representation. The detailed encoding for hexadecimal digits (0 through F) using 4-bit Balanced Gray code is provided in the project overview. For example, hexadecimal value 0 corresponds to 0000 in binary and the respective Gray code it maps from, while value 1 corresponds to 0001 and so forth, up to F (hexadecimal) which maps to 1111. The task is to design a circuit that performs this translation accurately.

In practical terms, you will build a combinational logic circuit using Logisim. Your design will feature four inputs, representing the 4-bit Balanced Gray code, and four outputs representing the equivalent binary code. You will utilize Logisim’s Pin devices to input the Gray code bits, and the output bits will connect via Splitters to a Hex Digit Display to show the final hexadecimal digit visually. This setup allows direct interaction and observation of the circuit’s behavior for all input combinations.

The logic functions governing each output bit are based on the properties of Gray to binary conversion. The least significant binary bit (Y0) can be derived as an XOR of all the Gray code bits, which is true if and only if there is an odd number of ones among the input bits. This is because XOR effectively computes parity and is standard in Gray-to-binary conversions.

The more significant bits (Y1, Y2, Y3) are obtained via a combination of XOR operations that encapsulate the rules of Gray and binary coding. Specifically, Y1 can be expressed as X3 XOR X2 XOR X1 XOR X0, Y2 involves XORing X3 with the sum of XORs of other bits, and Y3 typically depends on the most significant Gray code bit combined with certain functions of the other bits, following the encoding table provided. Although simplification of logic functions is optional, constructing truth tables based on the Gray-to-binary mappings can help derive accurate Boolean expressions for each output bit.

Additional considerations involve ensuring proper circuit assembly in Logisim. The Pin devices are set up for input, and the output bits are routed through Splitters to match the 4-bit binary input of the Hex Digit Display device. As part of testing, all 16 combinations of input Gray codes need to be simulated using Logisim's poke and simulate features to verify accurate hexadecimal outputs corresponding to each Gray code input.

Furthermore, understanding the underlying principles of Gray code transitions, including the balanced property, helps inform the design process. Balanced Gray codes maintain uniform transition counts for each bit, which minimizes signal bias and potential errors—properties desirable in high-reliability systems. The conversion circuit thus acts as a practical demonstration of how such codes can be effectively translated into binary representations for display or further digital processing.

In summary, this project involves designing a combinational circuit that translates 4-bit Balanced Gray code inputs into binary outputs, testing the circuit thoroughly via Logisim, and generating related documentation. This exercise consolidates knowledge in digital logic design, Boolean algebra, and digital simulation, illustrating key concepts through practical application.

References

  • Hwang, K. (2011). Digital Logic and Computer Design. Cengage Learning.
  • Brown, S. D., & Vranesic, Z. G. (2009). Fundamentals of Digital Logic with VHDL Design. McGraw-Hill.
  • Millman, J., & Grabel, A. (1987). Microelectronics (2nd ed.). McGraw-Hill.
  • Stallings, W. (2015). Computer Organization and Architecture. Pearson.
  • Logisim User Guide and Reference Manual. (n.d.). Available at https://github.com/redsheep00/logisim-evolution.
  • Kowalski, et al. (2008). Design of Gray Code Encoders for High-Speed Digital Systems. IEEE Transactions on Very Large Scale Integration (VLSI) Systems, 16(2), 236–245.
  • Sutherland, J. (2010). Introduction to Logic Design. Pearson.
  • Rabaey, J. M., Chandrakasan, A., & Nikolić, B. (2003). Digital Integrated Circuits. Prentice Hall.

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