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GMSK modulation

GSM GMSK modulation [Under GSM]

In this article, we would illustrate GSM GMSK modulation. Below are the equations specified in 3GPP TS 45.004:2.

Differential encoding is used; it means rather than every bit value, difference in consecutive bits is encoded as +1 or -1 (+1 is when bits are same, -1 otherwise), refer αi below. Along with equations, corresponding waveforms are shown for an example of one differential encoding of "+1".

 Gaussian function with BT=0.3, T=48/13 μs 

 
h(t) =  

       1       
δT . (2π)
 
  e 
  -t2  
2T2

 
 
δ = 
  ____  
  = 0.44168 with BT=0.3
ln(2)
2πBT


gsm-8f-1st.svg
 Convolution 

g(t) = h(t) * rect(  t  )
T


rect(  t  ) =   1   if |t| <   T 
T T 2


rect(  t  ) =  0    if |t| >   T 
T 2


gsm-8f-2nd.svg
 Integral 

. t' .
G(t') = πh . g(u) du
. -∞ .


φ(t') = n . t'-iT .
Σ  ( αiπh . g(u) du )
i=1 . -∞ .


αi = { +1, -1 }
h = 0.5

gsm-8f-3rd.svg
 Phase shift of carrier 

____
x(t') = (
2Ec
T
) . cos(2πf0t' + φ(t') + φ0)


gsm-8f-4th.svg


As you notice in above diagram, (modulation) phase shift increases and saturates after 2T. Above plots are drawn with gnuplot, more about it in later article.

In next article, we will take an interesting case of Frequency Correction Burst (FB) transmitted over FCCH.

References: GSM book by Mouly and Pautet.

© Copyright Samir Amberkar 2013

. GSM Index » Case of Frequency Correction Burst