GSM Physical Layer Modulation GSM uses Gaussian-Fitered Minimum Shift Keying (GMSK) as it's modulation scheme. Before the GMSK can be explained, some fundamentals of Minimum Shift Keying (MSK) must be known.

 MSK Signal Pulse MSK uses changes in phase to represent 0's and 1's, but unlike most other keying schemes we have seen in class, the pulse sent to represent a 0 or a 1, not only depends on what information is being sent, but what was previously sent. The pulse used in MSK is the following [1]: where if a '1' was sent if a '0' was sent Right from the equation we can see that depends not only from the symbol being sent (from the change in the sign), but it can be seen that is also depends on which means that the pulse also depends on what was previously sent. To see how this works let's work through an example. Assume the data being sent is 111010000, then the phase of the signal would fluctuate as seen in Figure 1. If it assumed that h = 1/2, then the figure simplifies. The phase can now go up or down by increments of pi/2, and the values at which the phase can be (at integer intervals of Tb) are {-pi/2, 0, pi/2, pi} [1] The above example now changes to the graph in figure 2. The figure illustrates one feature of MSK that may not be obvious, when a large number of the same symbol is transmitted, the phase does not go to infinity, but rotates around 0 phase.

 Signal Constellation So what does the signal constellation of MSK look like. Taking the equation for the pulse and using the trigonometric identity for a sum in a cosine we get [1]: It turns out that the function above can be simplified into the following [1]: where and Thus the equations for s1 and s2 depend only on andwith each taking one of two possible values. Therefore there are 4 different possibilities [1]: Now that the signal space has been defined byand, and the range of values for s1 and s2 have been determined, the signal constellation can be drawn. See Figure 3 for the signal constellation [1].

 Advantages of MSK Even though the derivation of MSK was produced by analyzing the changes in phase, MSK is actually a form of frequency-shift-keying (FSK) with (where f1 and f2 are the frequencies used for the pulses). MSK produces an FSK with the minimum difference between the frequencies of the two FSK signals such that the signals do not interfere with each other [1]. MSK produces a power spectrum density that falls off much faster compared to the spectrum of QPSK. While QPSK falls off at the inverse square of the frequency, MSK falls off at the inverse fourth power of the frequency. Thus MSK can operate in in a smaller bandwidth compared to QPSK [1].

 GMSK Even though MSK's power spectrum density falls quite fast, it does not fall fast enough so that interference between adjacent signals in the frequency band can be avoided. To take care of the problem, the original binary signal is passed through a Gaussian shaped filter before it is modulated with MSK. Frequency Response The principle parameter in designing an appropriate Gaussian filter is the time-bandwidth product WTb. Please see figure 4 for the frequency response of different Gaussian filters. Note that MSK has a time-bandwidth product of infinity [1]. As can be seen from Figure 4, GMSKs power spectrum drops much quicker than MSK's. Furthermore, as WTb is decreased, the roll-off is much quicker.

 Time-Domain Response Since lower time-bandwidth products produce a faster power-spectrum roll-off, why not have a very small time-bandwidth product. It happens that with lower time-bandwidth products the pulse is spread over a longer time, which can cause intersymbol interference. Please see Figure 5 for the time-domain response of the Gaussian filter [1]. Therefore as a compromise between spectral efficiency and time-domain performance, an intermediate time-bandwidth product must be chosen.

 GSM Specifics In the GSM standard a time-bandwidth product of 0.3 was chosen as a compromise between spectral efficiency and intersymbol interference. With this value of WTb, 99% of the power spectrum is within a bandwidth of 250 kHz, and since GSM spectrum is divided into 200 kHz channels for multiple access, there is very little interference between the channels [2]. The speed at which GSM can transmit at, with WTb=0.3, is 271 kb/s. (It cannot go faster, since that would cause intersymbol interference) [2].

 References S. Haykin, Communication Systems, 4th Edition, New York: John Wiley & Sons, Inc., 2001, pp. 387-399. J.G. Sempere, "An overview of the GSM system by Javier Gozalvez Sempere," [Online document], April 1998, Available HTTP:  http://www.comms.eee.strath.ac.uk/~gozalvez/gsm/gsm.html