Best Demodulator and Error Rate

• For this signal set the best way to decide which of the two possible signals was transmitted is to:
  • Compute the average of the received signal over each bit period.
  • If the average is positive, decide that a positive pulse (and, hence, a ``1'' bit) was transmitted.
  • If the average is negative, decide that a negative pulse (and, hence, a ``0'' bit) was transmitted.
• This strategy is optimal in the sense that it leads to the fewest possible wrong decisions.
• Wrong decisions: decide ``1'' when actually ``0'' was transmitted or decide ``0'' when actually ``1'' was transmitted.
• For the received signal in Figure 2.30 the averages for each of the bit periods are:
  
  $$1.0672 \; 0.4552 \; -0.5043 \; 1.0938 \; -1.1555 \; -1.0485 \; 1.4041 \; -0.9178 \; -0.6448 \; -0.8376$$
  
  

• For this example, all decisions are correct!
• It can be computed that for the ratio of signal strength and noise level in the example above, on average 8 errors occur per 10,000 transmitted bits.
• We say, the bit error probability is $8 \cdot 10^{-4}$.