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For example, we know that schemes who are off to signal one bit value and transmit something for the other are in almost all cases power-wise inefficient, really. The medium of transmission is an electromagnetic wave – be it the radio interface of your phone, be it the field between the wires in a twisted pair, or be it the optical fiber for >= 100 Gbit/s links. And these have a phase, which allows us to transmit, say, amplitudes of -0.5/+0.5 instead of 0.0/1.0, and get the same "distance" between noisy received symbols at the receiver. However, the average power used by the first scheme is \$0.5^2=\frac14\$, whereas in the second case it's \$\frac12(0^2+1^2)=\frac12\$. This BPSK (binary phase-shift keying) vs OOK (on-off keying) example serves to illustrate that there's beauty in making things symmetrical – and then, you lose the "bit that has lower energy" argument altogether.

For example, we know that schemes who are off to signal one bit value and transmit something for the other are in almost all cases power-wise inefficient, really. The medium of transmission is an electromagnetic wave – be it the radio interface of your phone, be it the field between the wires in a twisted pair, or be it the optical fiber for >= 100 Gbit/s links. And these have a phase, which allows us to transmit, say, amplitudes of -0.5/+0.5 instead of 0.0/1.0, and get the same "distance" between noisy received symbols at the receiver. However, the average power used by the first scheme is \$0.5^2=\frac14\$, whereas in the second case it's \$\frac12\left(0^2+1^2\right)=\frac12\$. This BPSK (binary phase-shift keying) vs OOK (on-off keying) example serves to illustrate that there's beauty in making things symmetrical – and then, you lose the "bit that has lower energy" argument altogether.

Now, you come along and say, you want to optimize for power, so the higher-amplitude symbols should be occurring less often than the lower-amplitude ones. Turns out that under that condition, each symbols does no longer carry 10 bits; 10 bits per symbol is the maximum you can get across with 2¹⁰=1024 symbols, and that happens when you choose the probabilities of all the symbols identically. So, to transmit the same, say, 1 million bits, where in the equidistributed scheme you needed 100 thousand symbols, you now need more. How much more depends on how exactly you shape the probability¹.

Now, you come along and say, you want to optimize for power, so the higher-amplitude symbols should be occurring less often than the lower-amplitude ones. Turns out that under that condition, each symbols does no longer carry 10 bits; 10 bits per symbol is the maximum you can get across with 2¹⁰=1024 symbols, and that happens when you choose the probabilities of all the symbols identically. So, to transmit the same, say, 1 million bits, where in the equidistributed scheme you needed 100 thousand symbols, you now need more. How much more depends on how exactly you shape the probability¹.

The job of channel coding is then to take these equally likely bits and find a transmission scheme that is optimal for the end-to-end system - typically optimal as in least bit error rate for a given transmit power, or least power needed for a fixed bit error rate. There can be many other parameters to take into consideration, but these are the main things we usually look at when we optimize channel coding for long-haul high-rate communications, which use the most power.

The job of channel coding is to then take these equally likely bits and find a transmission scheme that is optimal for the end-to-end system – typically optimal as in least bit error rate for a given transmit power, or least power needed for a fixed bit error rate. There can be many other parameters to take into consideration, but these are the main things we usually look at when we optimize channel coding for long-haul high-rate communications, which use the most power.