Mathematical introduction to GFSK

MSK

      Here let’s consider the basic modulation and demodulation techniques used in Transmitter & Receiver. Generally, Bluetooth implements a very basic and popular digital modulation, say GMSK.

      Let’s begin from the very beginning and go through every detail about the configuration of MSK. MSK is the abbreviated for Minimum Frequency Shift-Keying. As a special form of FSK, MSK also denotes binary signals with two different frequencies, and :

      But there is another constrain anyhow. In order to enable and simplify the demodulation, it is desired that the two different signals when sending '0' and '1' are orthogonal to each other, mathematically:

       Here T denotes the time to transmit one bit information.

       To make the first part zero, should be equal to. Subsequently, when k=1, will have the minimum value, from which the name MSK comes.

       On this occasion,

       In another equation, we can definite a new parameter:

       To make the second part null,

       By now, it is easy to see:

       With the knowledge above, we can get:

Hereis the information of the kth bit, '+1' or '-1', . For example, when data '1' is send, :

          Further more, in order  to keep the phase in the connection continuous,should satisfy:

          When setting, there will be a very interesting conclusion thatis eitheror, and is by no means equal to 0

      Though another conclusion is ambiguous, but it is true. If readers are interested in it, playing the trick will be very useful.

From the expand of equation for MSK, we can easily get an intuition on how to make up a device to implement it.

And specially,

Where

is bipolar digital data.

          And as the result of one conclusion mentioned above,

          Which means that can never be changed in time T, but in time 2T. So this is the reason to use a time delay device in the implementation configuration.

          To make the specification more clearly, a graph is presented below:

        

           With no prove and process, MSK signal’s PDF can be formulized as

           It is clear that MSK's main lobe is denser than binary PSK, shrinking into a narrower band. It is more suitable for transmission.

GMSK

Anyhow, on some application occasion (especially the communication), MSK also seems too wide in bandwidth for transmission and economy's sake. Such occasions may requires the decay outside the main lobe reaching up to 70-80dB. To solve such a severe requirement, a pre-filter locating before ordinary MSK Module will promise, which is called Gaussian Lowpass Filter. Gaussian LPF should meet the criteria below:

1.      Narrow bandwidth and transition band should be as narrow as possible;

2.      With short impulse response;

3.      Keep the impulse response area constant.

A usually used Gaussian Lowpass Filter can be described with its transfer function as blow:

With inverse Fourier Transform, its impulse response can be achieved:

Usually, its 3dB bandwidth  is multiplied with the duration of sending one bit to measure the characteristics of Gaussian LPF.

What should be pointed out is that, though Gaussian LPF makes the system satisfactory with the spectrum requirement, it achieves its goal with cost of satisfying BER. The narrower the transition band is, the better the spectrum becomes, and consequently the worse the BER gets. As a thumb rule of experiment, When, BER doesn’t decline a lot, also the bandwidth is somewhat acceptable.

 GFSK

           MSK and GMSK all require that h=0.5, which limits the modulation too much. An alternative technique is to use GFSK. GFSK don't limit h to exact 0.5, any h between 0.4 to 0.7 will do.