Theory and Practice of Modified Frequency Modulation Synthesis by Lazzarini and Timoney: https://core.ac.uk/download/pdf/29701...

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MATLAB/Octave code:

pkg load signal
fs = 48000;
tt = 0:1/fs:10;

fc = 110;
fm = 220*sqrt(2);

oc = 2*pi*fc;
om = 2*pi*fm;

%a = linspace(0,1,floor(length(tt)/2));
%a = 21 * [a, 1, a(end:1:1)];
a = 21 * linspace(1,0,length(tt));

%xx = cos(oc * tt + a .* cos(om * tt));
xx = exp(a) .* cos(oc * tt) .* exp(a .* cos(om * tt));

figure(1);
specgram(xx,4096,fs);
axis([0 10 0 3000])
figure(2)
plot(tt,xx);
soundsc(xx,fs,16)

0:00 Introduction
0:30 Standard FM
2:22 Bessel functions
2:48 Euler's formula
4:12 Modified FM
5:08 Amplitude normalization
6:35 Waveshaping/AM interpretation
8:13 Harmonic examples
10:33 Inharmonic examples
12:50 Exponential envelopes