# 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.). 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. 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? Nov 21 '13 at 9:43
• Very nifty solution, @DilipSarwate. Using your idea, I can actually get closer to the Shannon limit. Nov 21 '13 at 9:45

Another reason to encode to $$3$$ level is that the occupied bandwidth is less. This is often driven by an emission mask constraint for transmitter certification. To satisfy this constraint, the tradeoff is made on symbol distance. It is however a costly tradeoff affecting receiver sensitivity because the multi level scheme is done post detection where every factor of $$2$$ increase in throughput costs $$6$$ dB in sensitivity. Other schemes like QAM cost only $$3$$ dB in sensitivity when doubling the number of bits per signaling element. With the multilevel FSK, you do however retain some of the advantages of an FM strategy in terms of the CIR performance of the system.