# Line coding for a modulation order that is not power of 2

In case a symbol encodes a non-integer number of bits, how is done the bit-symbol pairing?

For example, we have a 3-levels FSK with bit rate of 1Mbps and a sequence 01011001. What is the symbol period? What is the sequence of symbols?

Note: My guess is that the actual bit period and symbol period are the same. Hence, the use of 3-levels only helps to increase somehow the robustness of the error detection (similar to Hamming coding or 8b/10b encoding). Thus, we apply a bipolar AMI (Alternate Mark Inversion) to the bit sequence to obtain {0,+1,0,-1,+1,0,0,-1}. Finally, in the receiver, we demodulate every symbol 0 as a bit 0, and every symbol +1 and -1 as a bit 1.

• There is no universal mapping scheme; it varies from design to design at the whims of the designer. For what it's worth, 3-level FSK typically only carries one bit per symbol. It can be accompanied with a data encoding scheme that attempts to ensure good properties in the resulting modulated signal (DC balance, avoiding long runs at a single transmit frequency, etc.). – Jason R Nov 20 '13 at 14:34
• $3^2$ is slightly larger than $2^3$, and so you can map $3$ bits onto $2$ 3-FSK signal. So, divide your bit sequence into 3-bit sets, encode each set into 2 3-FSK symbols, and away we go. – Dilip Sarwate Nov 21 '13 at 2:55
• Thank you @JasonR. It seems that a 3-FSK modulation scheme with an AMI encoding will help in the carrier recovery. However, I find hard to understand why it is worthy to decrease the symbol distance in order to have a better carrier recovery. For me, it does not seem a good trade-off, what do you think? – tashuhka Nov 21 '13 at 9:43
• Very nifty solution, @DilipSarwate. Using your idea, I can actually get closer to the Shannon limit. – tashuhka Nov 21 '13 at 9:45