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if we talk about a modulation, we mention a linear and nonlinear modulation. GMSK is nonlinear modulation, but it can be derived from a linear modulation as MSK, PAM and we talk about a linear approximation.

  1. As I understood, the linear approximation, it is a representation a phase of a signal as a sum of two phases, right? Write as the sum of PAM signals?
  2. How does a block scheme of the linear approximation look? I have read some researches and in the most of them it was represented as a quadrature modulation structure. Does it mean I can represent GMSK as QPSK and Gaussian filter?
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The Laurent Decomposition is the approximation for GMSK as a sum of PAM signals.

This technique is most useful for GMSK with partial response signaling, which is when we overlap subsequent pulses to send more date in less time which serves to increase the spectral efficiency at the expense of introducing intentional inter-symbol interference (ISI).

This does not simplify the transmitter, as compared to the very simple implementation I have detailed here with a Gaussian filter and NCO (which supports partial response signaling directly), but the Laurent decomposition shines in the receiver implementation for partial response GMSK. This is because the optimum receiver would need to correlate to every possible phase trajectory over the duration of symbol overlap, which creates a trellis of possibilities, with complexity increasing by the number of symbols that are overlapped. A Viterbi decoder is an efficient but still computationally intensive operation for implementing such an optimum receiver. The use of the Laurent Decomposition simplifies the Viterbi decoder by reducing the number of states in the trellis.

More details on Laurent Decomposition for GMSK can be found here. and the original paper by Pierre A. Laurent is:

"Exact and Approximate Construction of Digital Phase Modulations by Superposition of Amplitude Modulated Pulses (AMP)", IEEE Transactions on Communications, Vol. COM-34 No 2 February 1986. pp 150-160

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  • $\begingroup$ you wrote, it can reduce the number of states in the trellis of the decoder. Could you provide more details? $\endgroup$
    – FrimHart64
    Mar 22, 2022 at 7:40
  • $\begingroup$ i dont understand how laurent decomposition , modulator, can reduce the number of states $\endgroup$
    – FrimHart64
    Mar 22, 2022 at 7:42
  • $\begingroup$ See this link and specifically on page 76, what he refers to as the "Laurent Expansion". The point is each successive term of the Laurent Decomposition has less and less energy, so even if you use just the first term (which is the linear result you are looking for, you get most of the energy (suboptimum but the optimum linear result). etd.lib.metu.edu.tr/upload/1057348/index.pdf $\endgroup$ Mar 22, 2022 at 8:08
  • $\begingroup$ i have read about it, but how it helps to reduce number of states in the trellis? GMSK is only modulation technique at the transmitter side. $\endgroup$
    – FrimHart64
    Mar 22, 2022 at 8:44
  • $\begingroup$ I am referring to demodulation of GMSK in a receiver. As I noted, I don't see any real advantage of using this in the transmitter compared to the (simple) implementation I gave with an NCO and Gaussian filter. We can implement the transmitter as I suggested, and then (and especially if we use partial response signaling) we can implement a suboptimum receiver using the Laurent Decomposition. Otherwise you would use the Viterbi algorithm as an efficient approach to correlate to every possible combination of overlapping symbols (in the receiver) which is the trellis I refer to. $\endgroup$ Mar 22, 2022 at 8:53

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