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 α
i 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
δ =
|
____
|
= 0.44168 with BT=0.3
|
|
√ln(2) |
|
2πBT
|
|
|
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
|
|
|
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
|
|
Phase shift of carrier
|
|
|
____
|
|
x(t') =
|
√
|
(
|
|
) . cos(2πf0t' + φ(t') + φ0)
|
|
|
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
