Logic Equations for Control Signals

The input of the main control is the opcode field (bits 31:26) of an instruction. It is referred as Op6, where 6 is for 6 bits.

The implementation of the main control is simple. It includes a decoder and a logic unit implementing the following logic equations: 
  RegDst   = R-type
  RegWrite = ¬(sw + beq + bne + j)
  ExtOp    = ¬(andi + ori + xori)
  ALUSrc   = ¬(R-type + beq + bne) 
  MemRead  = lw
  MemtoReg = lw
  MemWrite = sw

The most common type of decoder has an n-bit input and 2n outputs, where only one output is asserted for each input combination. This decoder translates the n-bit input into a signal that corresponds to the binary value of the n-bit input. The outputs are thus usually numbered, say, 
Out0, Out1, ..., Out2n-1
If the value of the input is i, then Outi will be true and all other outputs will be false. The following figure shows a 3-bit decoder and the truth table. This decoder is called a 3-to-8 decoder since there are 3 inputs and 8 (23) outputs.