I just start learning PSK. I am confuse on the order of bit sequence.
Why 000 is not at phase=0 and 111 is beside 000 at phase=-pi/4?
I saw most examples are not in numberical order (000, 001, 010, etc.) and most of them have different order. I watched this youtube tutorial. It said mapping of state to bit is flexible. Is there any rules in drawing the bit map with phase?
The YouTube tutorial is correct in stating that the mapping from octal to constellation symbol is flexible but this statement is nonetheless very misleading because any choice other than the Gray code order mentioned in Dan Boschen's answer leads to poorer performance with respect to bit error rate. In 8PSK, the most likely demodulation error maps the transmitted symbol into one of its nearest neighbors and so it makes sense that as we go around the circle, the labels of neighboring symbols must differ in only one bit, i.e. the three-bit labels must occur in Gray code order, not in natural binary order. As pointed out in the other answer, natural binary order puts 000 and 111 next to each other so that a single symbol error will result win 3 bit errors. In fact, it is even worse than that; 011 and 100 are also nearest neighbors! Also, nearest neighbors 001 and 010 differ in two bits, as do 100 nd 101 etc,
All that being said, the OP should keep in mind that if the measure of performance is the symbol error rate, then the labels don't matter at all, while if the individual bit error rates are of any interest at all, then the labels matter a lot. The three bits suffer different bit error rates, and the bit errors are not independent events; details can be found in this answer.
The mapping is usually done with "Gray Coding" such that there is only one bit change between the closest spaced symbols. Under the presence of increasing noise, a symbol error will first be between one symbol and its nearest neighbors. By Gray coding, if there is such a symbol error this would result in only one bit error (in contrast consider the worst case of mapping the bit pattern "000" next to "111"; if in error we would get 3 bit errors).
Other than that the bit patterns are just that: patterns, you can associate whatever pattern you want with no consequence beyond the Gray Coding described above.
All comments and answers to this question are correct.
But there is another reason why n-PSK constellations best not place all-zeros symbol right on 0 phase symbol:
1. Wireless channels may or may not need equalization
There are some wireless communications channels that for certain wireless channels do not need equalization.
Example
Terrestrial Trunked Radio (TETRA) Voice plus Data (V+D) Part 2: Air Interface (AI)
TETRA with QAM mapping plus specific setting explained in the ETSI standard 300 392-2 (TETRA Air Interface)
QUOTE
... The sub-carrier approach is used because the low symbol rate in each sub-carrier gives the modulation inherent resistance to time dispersion hence avoiding the need for an adaptive equalizer ...
END OF QUOTE
TETRA Air Interface standard available Free download here.
2. Better be safe than sorry
But more often than not, in wireless communications, equalization is needed.
In the same TETRA standard equalization is defined so it can used when/if needed.
After all wireless channels are often time variant.
So in order to keep BER below certain threshold the Transmitter-Receiver connection has to be re-assessed and adapt to whatever is happening to the wireless channel in use.
Each transmitted symbol S corresponds to a sequence of bits.
When BER suffers degradation, hopefully before disconnection, it’s most of times because the wireless channel has changed 'too fast' or 'too much'.
We assume here no malfunction on neither Transmitter nor Receiver equipment.
No equipment operation mis-use either.
3. Probe wireless channel all the time
So when assessing a wireless channel in order to satisfy the required end-to-end bandwidth, the transfer function of the channel as well as channel noise, and interference, all have to be measured, and it's done using known sequences.
Known in advance to both Transmitter and Receiver.
In binary the easiest symbol test sequence would be 010101.. or 000111000111...
But in a constellation, to measure BER we cannot just check amplitude, we have to check phase.
It's called Bit Error Ratio but to measure BER one cannot just check single bits. BER is measured using groups of bits, at least symbol to symbol.
Let's say S0 S1 S0 S1 .. is the symbol test sequence probing channel before sending any payload. Or through a different logic channel.
So the key point answering this question is as follows :
For a Receiver better assess phase problems, placing all-zeros symbol
On 0 phase is not as helpful as assigning test sequence S0 symbol away from zero phase, but right on the minimum phase resolution.
S0 is chosen such as S0 phase is 360/n degree (2*pi/n rad) or or 90+360/n degree ; n being total amount of symbols of nPSK constellation in use.
So S1 is 180 degree shifted from S0 plus again 360/n.
Again S1 phase is S0 minimum phase plus 180.
Note that some wireless communications may switch constellations, within same nPSK type, varying amount of points along ring, increasing or decreasing PSK ring radius, adding a second PSK ring, or even changing to QAM either to adapt to wireless channel changes and a satisfy required end-to-end bandwidth.
S0 S1 S0 S1 .. antipodal makes it easy find out channel noise in both phase and amplitude.
A small processing capacity Receiver while in the initial stage may just copy back to transmitter.
Modern smart phones can keep multiple radio channels on full payload while simultaneously measuring BER telling not just to nearest tower but to multiple near towers, all at the same time, about the quality of each wireless channel to each nearby tower. It's called soft hand-over.
GSM 2G 2nd generation had hard hand-over : each mobile was connected to only one tower during active conversation or on-going data, although other towers could tell distance by signal received from that mobile.
Mobile Operators are obliged by law to know who they deliver service to (annonymous wireless service for major carriers is illegal) and they need to know the locations of all active mobile phones to supply the agreed end-to-end bandwidth that in turn implies best possible QoS Quality of Service is being satisfied.
