Hi Gullik,
Using Carson's rule, minimum orthogonal shift 4-FSK looks like it
occupies slightly less bandwidth than minimum orthogonal shift 2-FSK
(but slightly less != half the bandwidth).
The minimum frequency shift between adjacent FSK tones for them to be
orthogonal is half of the symbol rate (eg 2400 symbols/sec = 1200 Hz
frequency shift between adjacent tones).
The deltaF (aka deviation) used in Carson's rule is half the frequency
shift between the two tones _furtherest apart in frequency_ (not the
frequency shift between two adjacent tones).
So, let;
n = number of tones (binary exponential: 2-FSK, 4-FSK, 8-FSK, etc)
sr = symbol rate (baud)
Frequency shift between adjacent tones (Hz)
(also the "highest modulating frequency" for Carson's rule eg
...10101010... @ 9600 baud = 4800 Hz)
fs = sr / 2
Peak deviation (Hz)
deltaF = (fs * (n - 1)) / 2
Bandwidth - Carson's rule (Hz)
BW = 2 (fs + deltaF)
or
BW = sr ( 1 + ((n - 1) / 2) )
Spectral efficiency (bits/s/Hz)
ŋ = 1 / ( BW / bits_per_symbol )
Thus;
*n-FSK**
* *bits/symbol**
* *Spectral Efficiency (bit/s/Hz)**
*
2
1
0.667
4
2
0.8
8
3
0.667
16
4
0.471
According to Carson's rule, _4-FSK looks like it is right in the "sweet
spot" for spectral efficiency_. Also, the reduced symbol rate vs 2-FSK
should help improve resistance to multipath (even before channel
equalisation is considered).
73 ZL2WRW
Ross Whenmouth
On 05/12/17 12:01, freetel-codec2-requ...@lists.sourceforge.net wrote:
Subject:
Re: [Freetel-codec2] Codec2 repeater
From:
Gullik Webjorn <gullik.webj...@corevalue.se>
Date:
05/12/17 12:00
To:
freetel-codec2@lists.sourceforge.net
Hmm, if we think of a modulation index around 1, Carsons rule says Bt
= 2*(deltaF + fm ).
If it is the modulation index that governs SNR in the recovered
baseband, the same index
will occur at 1/2 deltaF if we reduce fm by 2, i.e. symbol rate.
Thus 4FSK would require half the swing ( for same Beta ) and that
would imply half deltaF.
We could reduce the bandwith to half, and gain 3 dB of reduced noise.
Of course there is
also a loss since our baseband must be discriminated with twice as
many levels.
So, IS 4-FSK more power efficient than 2-FSK? Commercial FM has a
Large Beta. ( to exploit SNR )
Gullik / SM6FBD
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