PULSE DRIVEN GAUSSIAN MINIMUM SHIFT KEYING*

0-7803-4902-498$1000 (c) 1998 EEE PULSE DRVEN GAUSSAN MNMUM SHFT KEYNG* Russell l% Rhodes Kenneth J Hetling Massachusetts nstitute Of Technology Lincoln Laboratory 244 wood St Lexington, MA 02173 Abstract A modification to the classical GiklSK waveform is presented The new waveform, called Pulse Driven GMSK (PDGMSK), differs from the classical de$nition m that it converges to SDPSK or DPSK for large jilter bandwidths This allows a trade oj spectral eficiency for demodulator complexity After describing the salient features of PDGMSK, some of its spectral properties are given t is shown how the bandwidth of the modulated signal remains nearly constant as the symbol rate changes Some performance results of using PDGMSK with the simplified demodulator structures are given 1 ntroduction Due to bandwidth constraints, the use of Continuous Phase Modulation (CPM) combined with partial response signaling is becoming more common in the design of communication systems The advantage of such systems is increased spectral efficiency t is generally accept ed, however, that, as phase transitions are spread over several symbol periods, the complexity of the receiver will increase f the channel under consideration is of fixed bandwidth, such as an FDMA satellite link, increased spectral efficiency is required only at high symbol rates A low rate data stream, therefore, could operate in the same bandwidth yet employ a modulation which requires less receiver complexity *This work was sponsored by the Department of the Army under contract F19628-95-COO02 A modulation which can offer this trade of complexity versus spectral efficiency is Gaussian Minimum Shift Keying (GMSK) [1] Through the judicious choice of the modulation parameters, the bandwidth, complexity and performance of the modulation can be traded n the limit, it can be shown that GMSK converges to Minimum Shift Keying (MSK) and, therefore, a simple MSK demodulator can recover the data stream [2] Even less complex than MSK, however, is Differential Phase Shift Keying (DPSK) and Symmetric Differential Phase Shift Keying (SDPSK) A variation of GMSK, called Pulse Driven GMSK (PDGMSK), is thus presented which converges to SDPSK and DPSK thereby allowing very simple demodulators at the lower symbol rates At high symbol rates, the spectral efficiency of the modulation is increased so that, alt bough the receiver complexity increases, the required bandwidth remains nearly equal to that of the low rate signal 2 Pulse Driven Gaussian Minimum Shift Keying 21 Classical GMSK GMSK modulation belongs to a class of signals known as Continuous Phase Modulation (CPM) which are described by [1] s(t) = ~2ET cos(2tfct + @(t, d)) (1) with the information sequence contained in (j(t, d) = 2~h ~ aiq(t W) (2) where T is the length of a symbol interval and j~ is the center frequency The information symbols are rep- co

0-7803-4902-498$1000 (c) 1998 EEE resented by the Qi E {+1, +3,, +(J4 1)} with M 22 Pulse Driven GMSK even The modulation index, given by h, can vary from symbol to symbol but will usually take on a value of h = $ since this will simplify a receiver and still achieve good performance [3] The distinguishing characteristics of these modulation schemes are established by the phase transition function q(t), This transition function is normally defined as the integral of the instantaneous frequency offset, g(t), so that As previously mentioned, GMSK becomes MSK for a sufficiently large filter bandwidth n order to even further simplify receiver design, lhowever, it is desirable for the modulation to resemble SDPSK and DPSK For this reason, a variation on GMSK is proposed nstead of using an NRZ sequence to excite the filter in Figure 1, a Return-To-Zero (RTZ) sequence will be used The pulse width of the sequence will be of constant duration and independent oft he symbol rate The length of this pulse, TP, is defined as the length of the symbol duration q(t) = t g(t)dt (3) co for the the highest symbol rate of interest Additionally, The pulse g(t) is normalized so that q(m) = ~ For GMSK, g(t) is defined as the response of a Gaussian filter to a rectangular pulse of width T A binary GMSK signal can, therefore, be generated by frequency modulating the response of a Gaussian filter to a Non-Returnto-Zero (NRZ) sequence as shown in Figure 1 A critical 2h rr Figure 1: Generation of GMSK Using FM property of GMSK is that the FM modulator will produce a constant envelope signal This is highly useful since constant envelope signals tend to perform better in non-linear environments such as a saturated satellite amplifier The primary parameter in a GMSK system is the By-product of the Gaussian filter f the 13T-product of the filter is sufficiently large, then the NRZ sequence will pass unfiltered into the frequency modulator and the system will become an MSK system As the BTproduct decreases, the frequency pulses associated with the data become extended in time so that the phase pulses overlap from symbol to symbol Some examples oft his are given in Figure 2 The effect of this stret thing in time of the phase transitions is a compression in the spectrum of the signals thereby making the GMSK modulation more spectrally efficient than instantaneous phase change techniques such as DPSK or Quadrature Phase Shift Keying (QPSK) this is the rate where the BT-prcJduct of the modulator filter is specified At this rate, therefore, the waveform is identical to classical GMSK As the symbol rate, R,!,,! decreases, the input sequence becomes a series of pulses as shown in Figure 3 For convergence to SDPSK, a modulation index of h = } is recluired 3in GMSK Freauenw Pulses 06,,, ~3 ~T=loo~ : --- -! 05,+ - :+;;:, J A21 l jy, BRk i+--- - 04 \\:::: ; ;!\ ;; 03,,! 1::::,,!, <, 1 02,,!,,,, 01 1 J +!J ;,, \v!::;; n -3-25 -2-15 -1-05 0 05 1 15 2 25 3 Time m; GMSK Phaae Tmiectories 06 c!!--: 05 0,1 - n -3-25 -2-15 -1-05 #e 05 1 1,5 2 25 3,,!,, Ul1222T i Figure 2: g(t) and q(t) For GMSK Note that the bandwidth of the Gaussian filter remains constant for all symbol rates f the bandwidth of the filter is sufficiently large with respect to the pulse

0-7803-4902-498$1000 (c) 1998 EEE rate, the pulses will pass unfiltered into the frequency modulator and the system will become SDPSK =1i3#tFH- R = 112T n p -lflp -&p n nnn 3 Mathematical Derivation The pulse shapes for PDGMSK can be derived using the block diagram of Figure 1 The message signal for the frequency modulator m(t) is the convolution of the information sequence with the Gaussian filter All of the sequences in Figures 3 and 4 can be represented by (4) where? 1 0 1 0 0 ltp R = l4tp l-l n n u u -J HP q,+ Figure 3: PDGMSK Bipolar RTZ Pulses (5) is a rectangular pulse of widt h TP For the bipolar waveform, the ai are from the set { l,, +1} For the unipolar waveform, the a~ are from the set {O, +2} The (unrealizable) impulse response of a Gaussian filter is defined by For convergence to DPSK, a different variation is required n this case, the modulation index remains at h = ~ while the input signal changes to a Unipolar RTZ sequence as shown in Figure 4 Once again, the filter bandwidth and pulse width remain constant and are specified at the highest symbol rate Notice that a phase change will occur only when a logic 1 is sent Additionally, the phase change caused by a logic 1 is n as opposed to ~ in classical GMSK n the limiting case, as the symbol duration increases, the phase change becomes instantaneous relative to the symbol rate and the modulation, therefore, becomes DPSK = ; -JUL R = l2tp ~P (+) h(t) = 1 (6) @QTP e The parameter a determines the 13T-product of the filter and is given by (7) TP is the length of a symbol period at the highest symbol rate Using these definitions, the message signal m(t) becomes co (8) where the symbol period is given by Tb The instantaneous frequency, g(t), becomes g(t) = * ~ m J M) (9) P [ (2 %)) - (2 B@m where Q( ) is the standard Q-~unction [4] The -phase trajectory of the signal, q(t), follclws directly from Equation 3 Finally, the signal at the output of the frequency modulator is Figure 4: FDGMSK Unipolar RTZ %lses Tr ~ Cw(t - w))) (lo) i= cm where E is the energy of the symbol with period Tb

0-7803-4902-498$1000 (c) 1998 EEE 4 Phase Trajectory Examples 41 Bipolar Driven Waveforms Bipolar Driven PDGMSK Phase Tree r ( The effect of decreasing symbol rate on the modulation can be clearly seen by examining phase trees associated with the different rates Consider, for example, the phase tree for Bipolar PDGMSK with BTP = $ and R = ~ which is plotted in Figure 5 Each path in the tree represents the phase of the modulated signal for a particular bit sequence The initial bit state in the figure is [0, O, O] The initial phase is normalized to zero at the start of the first bit For comparison, in Figure 6, o T 2T 3T,me it-t SipOlar Driven PDGMSK Phase TrW 4pi :,,, Figure 6: Bipolar PDGMSK Phase Tree, R = & shown in Figure 8, is even simpler since the phase transitions are nearly complete after only one third of the symbol period This implies thi~t a simple DPSK demodulator should be able to decode this waveform with only a slight performance loss Figure 5: Bipolar PDGMSK Phase Tree, R = * A second observation about Figure 8 is that, for each bit, the phase contribution is either O or m This implies a constantly increasing phase throughout the transmission The spectral ramifications of this are discussed in the next section the phase tree for R = & mode is plotted Since the same absolute phase trajectory is used as in Figure 5, the transitions in Figure 6 are nearly complete prior to the next bit period The receiver required for this signal, therefore, will be less complex than that required for Figure 5 42 Unipolar Driven Waveforms As previously mentioned, the unipolar version of PDGMSK will resemble DPSK as the symbol rate decreases This observation is also easy to discern by examining phase trees such as those plotted in Figures 7 and 8 n these examples, the filter bandwidth is chosen so that BTP = ~ and the phase trajectory extends over approximately two symbol periods when R = $ The phase tree is thus considerably simpler than that of Figure 5 and, therefore, so will the corresponding demodulator structure When R = &, the phase tree, Figure 7: KJnipolar PDGMSK Phase T!ce@, R = & time

0-7803-4902-498$1000 (c) 1998 EEE 4P 3pi UniPOlar Driven PDGMSK Phase Tree J ~,,?,, Bipolar PDGMSK Spectral Densities :,,, y<,, La 2pi,, { ;,~ pi J,,,,, 0 [ : 0 T 2T 3T 4T 5T time Figure 8: Unipolar PDGMSK Phase Tree, R = & Figure 9: Bipolar PDGMSK Spectrums 5 Spectral Characteristics As mentioned in Section 22, the By-product for PDGMSK is defined at the highest symbol rate and both the H -product and the pulse width of the driving waveform remain constant for all symbol rates As the symbol duration increases, therefore, the pulse, TP, appears like an impulse relative to the symbol duration and the signal becomes SDPSK The spectrum of the signal, therefore, is expected to widen relative to the symbol rate Since, however, the symbol rate has decreased, the absolute bandwidth of the signal is not expected to dramatically change As an example, some spectra of Bipolar PDGMSK are plotted in Figure 9 using BTP = ~ The frequency axis is normalized to the symbol duration R = ~ Although the sidelobes of the low rate waveform are larger, the lower symbol rate keeps the absolute bandwidth approximately the same as that of the more bandwidth efficient waveform operating at a higher rate The spectra for the unipolar driven waveforms are more complex This is due to the generating waveforms of Figure 4 For example, Figure 10 shows the normalized spectra of unipolar driven waveforms using the parameters of Figures 7 and 8 Since the symbol duration equals the pulse width, the spectrum looks similar to the classical GMSK spectrum with 13T = ~ Notice, however, the entire spectrum is shifted in frequency Since a(t) is either O or 2, the phase in s(t) always changes in a positive direction Alternatively, the instantaneous frequency offset given by Equation 9 is always non-negative The spectrum, therefore, is shifted in a positive direction Unipolar PDGMSK Spectral Densities 10( 1 Figure 10: Unipolar PDGMSK Spectrums5 As the bit period increases with respect to TP, the spectra become more complex (Once again, the asymmetry of a(t) causes an asymmetric spectrum Notice how the sidelobes begin to increase sharply on one side as the waveform becomes more like DPSK As with the bipolar modulation, this increase is not significant since the lower symbol rate will keep this energy in-band 6 Performance t can be shown that the optimal coherent CPM demodulator uses Maximum Likelihood Sequence Es-

0-7803-4902-498$1000 (c) 1998 EEE timation (MLSE) which can be implemented by the Viterbi algorithm [3] Performance attained by using an MLSE receiver can be calculated using minimum distance techniques and will not be discussed [1] nstead, results will be given for the utilization of DPSK demodulators for Unipolar PDGMSK Results for SDPSK demodulators with Bipolar PDGMSK are similar to DPSK Additionally, due to space limitations, derivations for the BER expressions will not be given and only the simulation results will be presented, :::, J --9-- R=,(Tp) 1] Simulated Results for a Coherent DPSK Receiver Figure 12: Noncoherent DPSK Receiver Results is about 40 db for R = #- This is reduced signifi- P cantly when R = ~ and near full response signaling occurs As with the coherent de:modulator, the performance degradation decreases with R until the Noncoherent DPSK results are achieved, 7 Conclusion Figure 11: Coherent DPSK Receiver Results Demodulation with Coherent DPSK assumes knowledge of the carrier phase Therefore, the inphase matched filter output is sampled and the bit decisions, based upon these samples, are differentially decoded Simulations were performed using this type of receiver for a Unipolar PDGMSK signal The BTP-product of the signal was BTP = ~ so that the received signal had phase trees like those shown in Figures 7 and 8 The results are shown in Figure 11 along with a reference curve for Coherent DPSK Perfect receiver synchronization and timing are assumed For the classical GMSK waveform, where R = &, there is about 28 db performance degradation T~is degradation decreases as the pulse width narrows with respect to the symbol duration until, as expected, Coherent DPSK performance is achieved f the absolute phase is not known, the information bit stream can still be recovered using a Noncoherent DPSK demodulator Simulations results for this type of demodulation, under the same conditions as above, are plotted in Figure 12 n this case, the degradation Two different versions of PDCrMSK were introduced as a means of trading bandwidth efficiency for receiver complexity Unlike classical GMSK which tends towards MSK, PDGMSK converges to SDPSK or DPSK as the symbol rate decreases t was shown how the absolute spectral content of the waveform remains nearly constant as the symbol rate changes This feature, combined with its constant modulated envelope, make PDGMSK well suited for non-linear constant bandwidth channels such as an FDMA satellite link 8 References [1] J Anderson, T Aulin, and C Sundberg Digital Phase Modulation Plenum Press, 1986 [2] S Pasupathy Minimum shift keying: A spectrally efficient modulation EEE Communications iwagazinc July 1979 [3] C Sundberg Continuous phase modulation EEE Communications Magazine April, 1986 [4] J Wozencraft and 1 Jacobs Principals of Communications John Wiley and Sons, 1965