Constrained Envelope Continuous Phase Modulation Tommy Svensson Department of Signals and Systems Communication Systems Group Chalmers University of Technology SE-412 96 G¨ oteborg, Sweden Email: tommy.svensson@s2.chalmers.se Arne Svensson Department of Signals and Systems Communication Systems Group Chalmers University of Technology SE-412 96 G¨ oteborg, Sweden Email: arne.svensson@s2.chalmers.se Abstract— We develop a coded modulation system with con- tinuous phase and a small controlled envelope variation. We call the system Constrained Envelope Continuous Phase Modulation (ceCPM) and show that, there is a gain for ceCPM over CPM under the same spectrum constraint. With normalized amplitude varying in the interval {1 ± 0.2} the gain for ceCPM is up to 2.5 dB over CPM at symbol error probability 10 -3 . We also show that ceCPM performs well with reduced state sequence detection (RSSD). I. I NTRODUCTION The continuous phase property of Continuous Phase Mod- ulation (CPM) makes it possible to define digital modulation schemes with a narrow spectral main lobe and small spectral side lobes by using a smooth phase pulse [1]. The constant envelope property of CPM also allows the use of non-linear amplifiers, which have lower cost and significantly higher power efficiency than linear amplifiers, since they can work in a saturated mode. However, the constant envelope restricts the signal space and thus, the minimum Euclidean distance at the receiver, which has to be compensated for by larger transmitted RF power. A basic question is, how large is this penalty in the required RF power for the constant envelope property? In this paper, we answer the question above by designing a coded modulation system with a small controlled envelope variation and comparing the performance with CPM. The proposed scheme, which we call Constrained Envelope Contin- uous Phase Modulation (ceCPM), has both continuous phase and continuous envelope variation, making the power spectrum to be well confined. This coded modulation system provides a gain over CPM. The gain becomes larger as the envelope is allowed to fluctuate more. We show ceCPM systems having a gain up to 2.5 dB over CPM at symbol error probability 10 −3 while still satisfying the same power spectrum requirement. Another important property of ceCPM is that ceCPM can be described by the same trellis as CPM, thus the number of states are not increased and as we show, the proposed system also performs well with Reduced State Sequence Detection (RSSD) [2]. The proposed modulation system challenges the power amplifier designers to design amplifiers with a local linearity close to the high efficiency working point. By using our proposed modulation scheme with such an amplifier, both high power efficiency at the transmitter and low RF power requirement at the receiver could be achieved at potentially a low amplifier cost. In the literature there are some work devoted to finding good modulation systems with somewhat restricted envelope, e.g. multi-amplitude CPM [3] and [4, p. 199], staggered modulation formats (e.g. OQPSK see [4, p. 194]) and recently in the EDGE modulation system for evolution of GSM, see [5] for an overview of EDGE and [6] for a definition of the modulation format. In the EDGE modulation the envelope variation is quite large and linearization techniques are necessary. In [7] a digital linearization technique for EDGE is proposed and an IQ- plot of the EDGE signal space is given. The EDGE envelope signal has a normalized envelope varying approximately in the interval [1.56, 0.23], i.e. almost 17 dB dynamic range. We illustrate ceCPM signals that improve in power effi- ciency more than 1.4 dB at symbol error probability 10 −4 over CPM, while still satisfying the same spectrum requirement and with as small envelope variation as in the interval [1.1, 0.9], i.e. less than 1.8 dB dynamic range. Such a small envelope variation should simplify a linearization unit. We define ceCPM, discuss the state-description and receiver structures, and give expressions for the Euclidean distance, auto-correlation function, power spectral density and maxi- mum/minimum amplitude. With the framework for ceCPM established, we compare the performance of ceCPM and CPM by numerically maximize the minimum Euclidean distance of the systems under a spectrum constraint defined by a spectrum mask that makes the systems to be more than twice as bandwidth efficient as GSM, and we verify the performance gain for ceCPM by simulations. Derivation of equations and much more details can be found in [8]. II. CONTINUOUS PHASE MODULATION The baseband signal in CPM is defined as s(t, α)=  2E T e jφ(t,α) , (1) where T is the length of a symbol interval, E is the symbol energy and α is the transmitted M -ary information symbol sequence with symbols taken from the set α i ∈ {±1, ±3,..., ±(M − 1)}. Note that the amplitude is constant