Transceiver52M: Add 4 samples-per-symbol Laurent pulse shape

When 4 samples-per-symbol operation is selected, replace the
existing pulse approximation, which becomes inaccurate with
non-unit oversampling, with the primary pulse, C0, from the
Laurent linear pulse approximation.

Pierre Laurent, "Exact and Approximate Construction of Digital Phase
  Modulations by Superposition of Amplitude Modulated Pulses", IEEE
  Transactions of Communications, Vol. 34, No. 2, Feb 1986.

Octave pulse generation code for the first three pulses of the
linear approximation are included.

Signed-off-by: Thomas Tsou <tom@tsou.cc>

1 files changed, 83 insertions, 0 deletions

diff --git a/Transceiver52M/laurent.m b/Transceiver52M/laurent.m
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+%
+% Laurent decomposition of GMSK signals
+% Generates C0, C1, and C2 pulse shapes
+%
+% Pierre Laurent, "Exact and Approximate Construction of Digital Phase
+%   Modulations by Superposition of Amplitude Modulated Pulses", IEEE
+%   Transactions of Communications, Vol. 34, No. 2, Feb 1986.
+%
+% Author: Thomas Tsou <tom@tsou.cc>
+%
+
+% Modulation parameters
+oversamp = 16;
+L = 3;
+f = 270.83333e3;
+T = 1/f;
+h = 0.5;
+BT = 0.30;
+B = BT / T;
+
+% Generate sampling points for L symbol periods
+t = -(L*T/2):T/oversamp:(L*T/2);
+t = t(1:end-1) + (T/oversamp/2);
+
+% Generate Gaussian pulse
+g = qfunc(2*pi*B*(t - T/2)/(log(2)^.5)) - qfunc(2*pi*B*(t + T/2)/(log(2)^.5));
+g = g / sum(g) * pi/2;
+g = [0 g];
+
+% Integrate phase 
+q = 0;
+for i = 1:size(g,2);
+    q(i) = sum(g(1:i));
+end
+
+% Compute two sided "generalized phase pulse" function
+s = 0;
+for i = 1:size(g,2);
+    s(i) = sin(q(i)) / sin(pi*h);
+end
+for i = (size(g,2) + 1):(2 * size(g,2) - 1);
+    s(i) = sin(pi*h - q(i - (size(g,2) - 1))) / sin(pi*h);
+end
+
+% Compute C0 pulse: valid for all L values
+c0 = s(1:end-(oversamp*(L-1)));
+for i = 1:L-1;
+    c0 = c0 .* s((1 + i*oversamp):end-(oversamp*(L - 1 - i)));
+end
+
+% Compute C1 pulse: valid for L = 3 only!
+%   C1 = S0 * S4 * S2
+c1 = s(1:end-(oversamp*(4)));
+c1 = c1 .* s((1 + 4*oversamp):end-(oversamp*(4 - 1 - 3)));
+c1 = c1 .* s((1 + 2*oversamp):end-(oversamp*(4 - 1 - 1)));
+
+% Compute C2 pulse: valid for L = 3 only!
+%   C2 = S0 * S1 * S5
+c2 = s(1:end-(oversamp*(5)));
+c2 = c2 .* s((1 + 1*oversamp):end-(oversamp*(5 - 1 - 0)));
+c2 = c2 .* s((1 + 5*oversamp):end-(oversamp*(5 - 1 - 4)));
+
+% Plot C0, C1, C2 Laurent pulse series
+figure(1);
+hold off;
+plot((0:size(c0,2)-1)/oversamp - 2,c0, 'b');
+hold on;
+plot((0:size(c1,2)-1)/oversamp - 2,c1, 'r');
+plot((0:size(c2,2)-1)/oversamp - 2,c2, 'g');
+
+% Generate OpenBTS pulse
+numSamples = size(c0,2); 
+centerPoint = (numSamples - 1)/2;
+i = ((0:numSamples) - centerPoint) / oversamp;
+xP = .96*exp(-1.1380*i.^2 - 0.527*i.^4);
+xP = xP / max(xP) * max(c0);
+
+% Plot C0 pulse compared to OpenBTS pulse
+figure(2);
+hold off;
+plot((0:size(c0,2)-1)/oversamp, c0, 'b');
+hold on;
+plot((0:size(xP,2)-1)/oversamp, xP, 'r');