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Solving modulation equations [Under GSM > GMSK modulation]

In this article, we will solve the modulation equations from earlier article so that we could plot it and get insight into mathematics behind.

Let us start with Convolution of Gaussian function h(t) with rectangular pulse of width T.

 g(t) = h(t) * rect( t ) T

g(t) can be written as:

 . t+T/2 . g(t) = ∫ h(u) du . t-T/2 .

and further be evaluated as

 g(t) = 1 ( erf( t+T/2 ) - erf( t-T/2 ) ) 2T δT√2 δT√2

Error function (erf) is defined as below:

. . x . . .
erf(x) =

 2 √π

e

-t2

dt

. . -∞ . . .
Integral of erf is:

erf(ax+b) dx =

 1 a

(ax+b) erf(ax+b) +

 1 a.√π

e

-(ax+b)2

gnuplot has function erf, so it is possible to plot equations involving erf. Next step is Integral:

 . t' . G(t') = πh . ∫ g(u) du . -∞ .

Using above formula, we can evaluate it as below:

G(t') = πh . (

 t+T/2 2T

erf(

 t+T/2 δT.√2

) +

 δ √2π

e

-(

 t+T/2 δT.√2

)2

-

 t-T/2 2T

erf(

 t-T/2 δT.√2

) -

 δ √2π

e

-(

 t-T/2 δT.√2

)2

)

Now φ(t') and x(t') can be plot (with gnuplot).

In next article, we will take a small example of 1 bit change.

References: GSM book by Mouly and Pautet, Wolfram online integrator, and Handbook of mathematical functions by Abramowitz and Stegun