why GMSK use a gaussian filter instead of raised cosine filter?

I'm wondering why GMSK use a gaussian filter instead of raised cosine filter as a pulse shaping filter? Is there special reason?

GMSK doesn't just use a Gaussian filter instead of a raised-cosine filter. It uses a Gaussian filter on the phase, before applying it to the modulator. This makes it a nonlinear operation.

When a raised-cosine filter is applied to some modulated signal, it is applied after modulation, as a linear operation.

So there's a whole lot of convenient rules about filters, ISI, "optimal" signal processing, etc., that get thrown out the window with GMSK, because all those rules are predicated on linear operations. But the advantages are worth it, sometimes.

Bottom line: GMSK can be transmitted at a constant amplitude, which means it can be amplified by cheaper, more-efficient class C (or possible E) amplifiers. That makes the increased ISI and decreased ease of analysis worth it.

As to "why Gaussian" -- probably because someone tried it, and it worked good. Good solid theoretical explanations are hard to come by when you stick nonlinearities into the mix -- if there is one, it's probably hand-wavy, and along the lines that a Gaussian filter is everywhere-continuous, unlike a raised-cosine, and so it make the phase transitions everywhere-continuous, too.

• A Gaussian pulse has the minimum time-bandwidth product but this assumes a true Gaussian pulse which extends to +/- infinity so not actually realizable but approximated. The intention in using the Gaussian pulse is the optimization of maximum power in the main lobe. We could choose any other pulse shaping of phase and still have the constant amplitude feature so ultimately it is the time-bandwidth that is getting optimized. Commented Apr 6, 2021 at 23:24

(Assuming the context is FSK signals, where the pulse shape is applied to the phase, instead of the amplitude as in linear modulation).

The main reason is that the sideband and out-of-band emissions of GMSK are lower than those produced with a raised cosine filter. See for example this plot from Wikipedia: https://en.wikipedia.org/wiki/File:GMSK_PSD.png

This property is especially important for mobile communications. Since spectrum is so limited, it has to be used very efficiently. Users may use channels that are adjacent to each other, with little or no guard band between them. Any power emitted outside the band assigned to a channel resuls in degraded SINR for other users in the cell.

GMSK requires a more complex receiver, because it reduces bandwdith at the cost of introducing ISI. However, in many scenarios this is an acceptable tradeoff, since bandwidth is more valuable than receiver simplicity.

• Respectfully, I'm not yet convinced this is true @MBaz, have you compared directly to a raised cosine where we can trade complexity (filter length) for sideband supression? I haven't looked at it side by side recently but intuitively feel we could easily exceed the sideband levels you show in your plot (and with a much flatter in band spectrum which would ultimately suggest more efficiency in terms of spectral utilization). Commented Apr 6, 2021 at 23:19
• @DanBoschen No worries, this is a good point. I did a simulation sometime comparing MSK and GMSK; I will dig it up and post the results. May take a few days, though.
– MBaz
Commented Apr 7, 2021 at 0:18
• GMSK is more efficient than MSK (no sharp transitions); what about RC filtered BPSK vs GMSK? Commented Apr 7, 2021 at 0:21
• Oh! Since the question was about GMSK, I assumed the comparison was to other types of FSK. In fact, BPSK (or any other linear modulation) with an RC filter is superior to GMSK and to any other constant-envelope signaling. As you know, a constant envelope signal sacrifices bandwidth to gain energy efficiency.
– MBaz
Commented Apr 7, 2021 at 0:49
• Yes I see, very good that all makes sense to me now (this is in the context of FSK only and I assume that was also the OP’s intention)- thank you. Commented Apr 7, 2021 at 4:21