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I implement a GMSK modulation scheme presented here (p 64 in pdf and copied below)

GMSK modulation format

GMSK mod format part 2

As gaussian density function I use the standard Maltab function gaussfir and convolve the rect-function with it.

My rect function:

a = randi([0,1],N,1);                       % Random data,  N x 1
ak = 2*a-1;                                 % NRZ data: +/- 1, N 
ak_1 =  kron(ones(length(ak), 1), 0);  
ak_1(1) = ak(end);
for i = 1: length(ak)-1
    ak_1(i+1) = ak(i);
end

for i = 1 : length(ak)
    akp = (-1)^i .* ak.*ak_1;    % (-1)^k * d_k * d_{k-1} 
end
ak_rect = kron(akp,ones(M,1));    % M*N x 1

I have a doubt in the realisation the following equation:

enter image description here

My attempt:

nrz_ov_f = conv(H,ak_rect,'full');
nrz_ov_f = nrz_ov_f/max(abs(nrz_ov_f));
%integrate to get phase information
phi = filter(1,[1,-1],nrz_ov_f*Ts);  % Ts = Tb/M; M = 4- oversampling
phi = phi *0.5*pi/2;  % Tb = 2

In the equation there is a product with a_k. Does it mean I have to implement phi as follow:

phi  = phi *ak
% phi = ak * pi/2 * cumsum(phi)

?

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1 Answer 1

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The phase function is simply the Gaussian shape accumulated, and using the filter function as the OP has done is a reasonable approach to making an accumulator (digital integrator). The scaling should be such that the waveform transitions from $0$ to $\pi/2$ radians over the duration of one symbol period.

Below shows the expected waveforms of the frequency vs time (normalized Gaussian function) and phase vs time (in radians over one symbol duration) for one GMSK symbol.

GMSK freq and phase vs time

The above was oversampled with 100 samples per symbol to show the underlying function. If we were to sample at 8 samples per symbol the results would appear as in the plot below:

8 samples per symbol

The MATLAB code for this was:

Ts= 1; 
B = 1; 
sigma = sqrt(log(2))/(2*pi*B*Ts);
fs = 8;  % samples per symbol
t = [-1:1/fs:1-1/fs];
rect = ones(1,fs);
h = 1/(sqrt(2*pi)*sigma*Ts) .* exp(-t.^2./(2*sigma^2*Ts^2)); 
g = conv(rect/fs, h);
ph = (pi/2*Ts) cumsum(g)/fs;

As to further details into the significance that the phase transition over $\pi/2$ radians over one symbol period (either in the positive or negative direction), please refer to this post.

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  • $\begingroup$ do you assume that the equation ( precoded data, rect) has an error? I have seen more than one reseach papers where the same equition with rect pulse was used $\endgroup$
    – FrimHart64
    Commented Apr 26, 2022 at 5:28
  • $\begingroup$ @FrHart64 Yes, I see - I think that was a bad assumption on my part; I do see that it is consistently everywhere shown as the NRZ pulses themselves that are passed through the Gaussian filter. I'll remove that comment about it being an error. $\endgroup$ Commented Apr 26, 2022 at 5:40
  • $\begingroup$ how to implement correctly the phase-equation given in the pdf? Did i do it correctly? $\endgroup$
    – FrimHart64
    Commented Apr 26, 2022 at 5:52
  • $\begingroup$ @FrHart64 I gave my code for how I did the phase-equation and the verification plots; importantly you want to confirm that one symbol will transition in phase from 0 to pi/2 as I showed. Does yours match what I showed? $\endgroup$ Commented Apr 26, 2022 at 5:59
  • $\begingroup$ @FrHart64 I also redid the plots (and the code) to show the result with the rect. $\endgroup$ Commented Apr 26, 2022 at 6:34

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