when, for instance, a multiplexer serves a number of IP network users; and then feeds directly into a router, which immediately reads the content of the entire link into its routing processor; and then does the demultiplexing in memory from where it will be converted directly into IP sections. Often, a multiplexer and demultiplexer are combined into a single piece of equipment, which is simply referred to as a multiplexer. Both circuit elements are needed at both ends of a transmission link because most communications systems transmit in both directions. In analog circuit design, a multiplexer is a special type of analog switch that connects one signal selected from several inputs to a single output. Digital multiplexers In digital circuit design, the selector wires are of digital value. In the case of a 2-to-1 multiplexer, a logic value of 0 would connect to the output while a logic value of 1 would connect to the output. In larger multiplexers, the number of selector pins is equal to where is the number of inputs. For example, 9 to 16 inputs would require no fewer than 4 selector pins and 17 to 32 inputs would require no fewer than 5 selector pins. The binary value expressed on these selector pins determines the selected input pin. A 2-to-1 multiplexer has a boolean equation where and are the two inputs, is the selector input, and is the output: Which can be expressed as a truth table: Or, in simpler notation: These tables show that when then but when then . A straightforward realization of this 2-to-1 multiplexer would need 2 AND gates, an OR gate, and a NOT gate. While this is mathematically correct, a direct physical implementation would be prone to race conditions that require additional gates to suppress. Larger multiplexers are also common and, as stated above, require selector pins for inputs. Other common sizes are 4-to-1, 8-to-1, and 16-to-1. Since digital logic uses binary values, powers of 2 are used (4, 8, 16) to maximally control a number of inputs for the given number of selector inputs. The boolean equation for a 4-to-1 multiplexer is: The following 4-to-1 multiplexer is constructed from 3-state buffers and AND gates (the AND gates are acting as the decoder): The subscripts on the inputs indicate the decimal value of the binary control inputs at which that input is let through. Chaining multiplexers Larger Multiplexers can