State Diagram

State Diagram

This behavior can be seen from another point of view. When a data bit enters the shift register (in stage s₁). the output bits are determined not only by the data bit in s₁. but by the two previous data bits already in stages s₃ and s₂. There are four possible combinations of the two previous bits (in s₃ and s₂): 00, 01, 10, and 11. We shall label these four states as a, b, c, and d, respectively, as shown in Fig. a.

 

When the previous two bits are 01 (s₃ = 0_, s₂ = 1), the state is b, and so on. The number of states is equal to 2 ^{N –1}.

A data bit 0 or 1 generates four different outputs, depending on the encoder state. If the data bit is 0. the encoder output is 00, 10, 11, or 01. depending on whether the encoder state is simililarly if the data bit is 1. the encoder output is 11, 01, 00, or 10. depending on wether the encoder state is a, b, c, or d. This entire behavior can be concisely expressed by the state diagram shown in Fig. b. This is a four-state directed graph used to uniquely represent the input-output relation of this encoder. We use solid lines when the input bit is 0 and dashed lines when the input bit is 1. For instance, when the encoder is in stale a. and we input 1, the encoder output is 11 (Sashed line). The encoder now goes to state b for the next data bit because at this point the previous two bits become s₃ = 0 and s₂ = 1 . Similarly when the encoder is in state a and the input is 0, the output is 00 (solid line), and the encoder remains in state a. Note that the encoder cannot go directly from state a to states c or d. From any given states, the encoder can go to only two states directly by inputting a single data bit. This is an extremely important observation, which will be used later. The encoder goes from state a to state b (when the input is l),or to state a (when the input is 0), and so on. The encoder cannot go from a to c in one step. It must go from a to b to c, or from a to b to d to c and so on. We can

also verify these facts from the code tree. Figure b contains the complete information of the code tree.