A special case of FSK called Gaussian minimum-shift keying (GMSK) is used in the GSM cellular radio and PCS systems to be described later. In a minimum-shift system, the mark and space frequencies are separated by half the bit rate, that is:

ƒm − ƒs = 0.5 ƒb ; where

ƒm = frequency transmitted for mark (binary 1)
ƒs = frequency transmitted for space (binary 0)
ƒb = bit rate

If we use the conventional FM terminology, we see that GMSK has a deviation each way from the center (carrier) frequency, of δ = 0.25 ƒb which corresponds to a modulation index of

mƒ = δ/ƒm = 0.25ƒb/ƒb = 0.25

The word Gaussian refers to the shape of a filter that is used before the modulator to reduce the transmitted bandwidth of the signal. GMSK uses less bandwidth than conventional FSK, because the filter causes the transmitted frequency to move gradually between the mark and space frequencies.

With conventional FSK the frequency transition is theoretically instantaneous, and in practice as rapid as the hardware allows, producing sidebands far from the carrier frequency.

The GSM cellular radio system uses GMSK in a 200-kHz channel, with a channel data rate of 270.833 kb/s. Calculate:

(a) the frequency shift between mark and space
(b) the transmitted frequencies if the carrier (center) frequency is exactly 880 MHz
(c) the bandwidth efficiency of the scheme in b/s/Hz

(a) The frequency shift is ƒm − ƒs = 0.5 ƒb = 0.5 × 270.833 kHz = 135.4165 kHz

(b) The shift each way from the carrier frequency is half that found in (a) so the maximum frequency is ƒmax = ƒc + 0.25 ƒb = 880 MHz + 0.25 × 270.833 kHz = 880.0677 MHz and the minimum frequency is ƒmin = ƒc − 0.25 ƒb = 880 MHz − 0.25 × 270.833 kHz = 879.93229 MHz

(c) The GSM system has a bandwidth efficiency of 270.833/200 = 1.35 b/s/Hz, comfortably under the theoretical maximum of 2 b/s/Hz for a two-level code.

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