Wireless Pers CommunDOI 10.1007/s11277-008-9509-y

An All Digital Implementation of Constant Envelope:Bandwidth Efficient GMSK Modem using AdvancedDigital Signal Processing Techniques

Arjun Ramamurthy · fredric j. harris

© Springer Science+Business Media, LLC. 2008

Abstract Gaussian Minimum Shift Keying (GMSK) has been the most common modula-tion format belonging to the class of partial response Continuous Phase Modulation (CPM)scheme. It is primarily adopted in the GSM standards (B = 0.3) for land mobile radio commu-nication systems because of its high bandwidth efficiency and constant envelope modulationcharacteristics. The focus of this paper is the design of the demodulator wherein we demon-strate an all digital implementation of sub-optimal synchronization techniques for a GMSKmodem based on two Laurent Amplitude modulation pulse (AMP) streams approximationrepresenting the matched filter. In this all digital implementation, we perform a joint estima-tion of the symbol timing and carrier offset wherein the symbol timing is performed usinginterpolation techniques.

Keywords GMSK synchronization · O-QPSK · Joint timing and carrier recovery ·Polyphase matched filter

1 Introduction

For several years Continuous Phase Modulation (CPM) has been a well studied class ofmodulation characterized with high performance metrics, primarily efficient spectrum uti-lization and power efficiency. CPM schemes can be broadly classified into two categories,namely full response and partial response depending upon whether the modulation frequencypulse is of single symbol duration of longer. Minimum Shift Keying (MSK) has been thepopular example of a full response spectrally efficient modulation scheme [12]. Within theclass of partial response CPM scheme, Gaussian Minimum Shift Keying (GMSK) is the most

A. Ramamurthy (B)San Diego State University, 6450, Sequence Dr., San Diego, CA 92121, USAe-mail: arjun.ramamurthy@motorola.com

f. j. harrisDepartment of Electrical and Computer Engineering, San Diego State University, 5500 Campanile Drive,San Diego, CA 92182-1309, USAe-mail: fred.harris@sdsu.edu

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A. Ramamurthy, f. j. harris

{a}n

p(t-nT)

a g(t-nT )n b

a (-1,+1)ε n

Gaussian Filter h(t)

FrequencyModulator

Data NRZ Pulse Train

πfc

s(t)

Fig. 1 GMSK transmitter (CPM representation) Redrawn from [1]

common example for its high spectral efficiency and constant envelope modulation charac-teristics. It is an h = 0.5 partial response continuous phase modulation scheme derived fromMSK with the addition of baseband Gaussian filtering applied to the identically and inde-pendently distributed (i.i.d) random rectangular pulse shaped input signal prior to frequencymodulation of the carrier. As indicated in [1], it is important to emphasize that although theacronym GMSK was assigned to the term Gaussian-filtered MSK in [2], the modulation actu-ally described in this reference applies to Gaussian filtering of rectangular pulses at basebandas shown in Fig. 1 [1], i.e. prior to modulation onto the carrier, and, hence, it does not destroythe constant envelope property of the resulting modulation. It is assumed that the frequencypulse g(t) in Fig. 1 is the result of a convolution (filtering) operation performed between therectangular pulse p(t) and the Gaussian filter h(t) as illustrated in Eqs. 1–3.

Although g(t) of Eq. 1 (or Eq. 4) appears to have a “Gaussian-looking” shape, we empha-size that the word Gaussian in the GMSK refers to the impulse response h(t) of the filterthrough which the input rectangular pulse train is passed and not the shape of the resultingfrequency pulse g(t).

g(t) = p(t)∗h(t),∗ is convolution operation (1)

p(t) = 1, 0 ≤ t ≤ Ts (2)

h(t) = 1/(2πσ 2)1/2. exp(−t2/2σ 2), σ 2 = ln2/(2π B)2 (3)

Alternatively, as illustrated in [1: Eq. 2.8.52], the GMSK frequency pulse is the differenceof the two time displaced (by Ts seconds) Gaussian probability integrals (Q functions), i.e.where B is the single sided 3-dB bandwidth of the Gaussian filter h(t) as shown in Fig. 2.

g(t) = 12TS

[Q

(K ·

(t

TS− 1

))− Q

(K ·

(t

TS

))]

K = 2πBTS√ln(2)

Q(x) = ∫ ∞x

1√2π

exp(− y2

2

)dy

(4)

It is a common practice to refer to the product of the Gaussian filter 3dB bandwidth andthe coded symbol period as the BTs factor, commonly used to vary the spectral (bandwidth)efficiency of the modulated signal. However, in practice it is only the 3dB bandwidth (B) ofthe Gaussian filter that controls the spectral occupancy as shown in Fig. 2. Smaller values of Bresult in lesser spectral bandwidth occupancy (Fig. 2) but greater ISI (Figs. 3–5) resulting inhigher BER. Hence, depending upon the application, a particular value of B is selected result-ing in a compromise between spectral efficiency and BER performance. Viterbi equalizationor trellis demodulation [3] techniques are normally used to compensate for the increased ISIdue to lower B values (or equivalently lower BTs product). The Gaussian shape provides anadditional advantage of reduced sidelobe levels. The constant envelope of the GMSK signalreduces spectral re-growth and signal distortion due to any amplifier nonlinearity.

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An All Digital Implementation of Constant Envelope

Fig. 2 Impulse response and Frequency response of Gaussian filter h(t), and Gaussian filtered frequency pulseg(t) with B = 0.5, 0.3, 0.25

Fig. 3 Eye diagram I-Q representation of GMSK signal with B = 0.5

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Fig. 4 Eye diagram I-Q representation of GMSK signal with B = 0.3

Fig. 5 Eye diagram I-Q representation of GMSK signal with B = 0.25

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An All Digital Implementation of Constant Envelope

{a }n

p(t-nT)

a g(t-nT )n b

a (-1,+1)ε n

Gaussian Filter h(t)

Integrator

Data NRZ Pulse Train

π

fc

s(t)

cos(--)

cos(--)

-π/2

Fig. 6 GMSK transmitter (CPM representation) Redrawn from [1]

2 Modulator

The GMSK modulator can be implemented as a FSK modulator or as an offset QPSKmodulator [3]. It is a common practice to adopt the I-Q implementation to generate theGMSK signal resulting in a representation equivalent to staggered I-Q modulation format.Subsequently, in addition to the intentional and controlled ISI introduced in the GMSKsignaling format, there exists correlation between adjacent symbols on the quadrature chan-nels. As a result, the conventional receiver synchronization techniques, involving the carrierand symbol timing recovery needs to be modified to accommodate this cross talk betweenadjacent symbols on the I and Q channels.

There are various optimal and suboptimal methods used to estimate the carrier offset andsymbol timing to achieve successful signal detection and decoding at the demodulator. Wefollow the offset quadrature implementation approach as shown in Fig. 6. From Eq. 4 itcan bee seen that the Q function is doubly infinite in extent and it’s a common practice totime truncate the GMSK frequency pulse resulting in finite ISI. It has been shown in [4–6]that g(t) is truncated to 4 symbol intervals and 3-symbols intervals for B=0.25 and B=0.3respectively.

The GSM application adopts the B=0.3 considering ISI only from adjacent neighbors(say, for symbol αn, onlyαn−1 and αn+1 introduce ISI) as shown in [6]. Hence, in practicalGMSK implementations, an I-Q representation (Fig. 6) with an approximation as shown inEq. 5 [1:Eq. 2.8.54] is adopted where L is chosen based on the value of B. Based upon this,in our computer simulations we have illustrated the modulation-demodulation technique forB = 0.3.

g(t) = 1

2TS

[Q

(K ·

(t

TS− 1

))− Q

(K ·

(t

TS

))]

where − (L − 1)TS

2≤ t ≤ −(L + 1)

TS

20 Elsewhere (5)

3 Channel Impairments

The signal received from the channel arrives with unknown amplitude, time delay, frequencyshift, phase shift and noise. This is shown in Eq. 6

r(t) = Re{A(t) . R(t − τ) . ejθ(t−τ) . ejωc(t−τ) . ej�ωt + n(t) . ejωc(t−τ)} (6)

We will assume that the unknown channel attenuation A(t), varies slowly with respectto the modulation signal and is removed by an AGC loop. If A(t) introduces frequency

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Fig. 7 Signal degradation due to channel impairments

Fig. 8 First and Second Laurent AMP streams with B = 0.5

dependent distortion, the received signal will have to be processed by an equalizer. Weassume this is done and can ignore interaction between the various loops (AGC, carrier,timing and equalizer). Thus the signal delivered to the demodulator can be described as aknown signal, s(t) with unknown values of parameters namely, frequency offset (�ω), phaseoffset (�ϕ) and time delay (τ ) as shown in Eq. 7 and illustrated in Fig. 7

r(t) = s(t : �ω,�ϕ, τ) (7)

4 Demodulator

The demodulator can be implemented in a number of different optimum and sub-optimummethods as indicated in [1]. We use the Laurent’s sub-optimal two pulse approximationmethod in our simulations. Laurent [7] described a representation of CPM signal as a super-position of phase shifted amplitude modulation pulse (AMP) streams. In [1], a detailedinterpretation of the key results of this paper has been conducted for both, the exact andapproximate AMP representation of GMSK signal. In our computer simulations of the demod-ulator we have used only the first two Laurent AMP streams (matched filter) because the firstand second AMP component corresponding to the pulse streams C0[n] and C1[n] contains thesignificant fraction of 0.991944 and 0.00803 of the total energy [1]. Figures 8–10 demonstratethe first two pulse stream AMP representation for different values of B.

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An All Digital Implementation of Constant Envelope

Fig. 9 First and Second Laurent AMP streams with B = 0.3

Fig. 10 First and Second Laurent AMP streams with B = 0.25

In this all digital implementation, we perform a joint estimation of the symbol timing andcarrier offset. The carrier phase synchronization can be performed based on the MAP and/orML phase estimation techniques using data aided/non-data aided or decision directed con-figurations. We perform the carrier phase synchronization using the non-data aided methodas detailed in [8]. The symbol timing is performed using an interpolation technique [9].

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Fig. 11 Three samples, early,punctual, late samples on positiveand negative correlation.Redrawn from [9]

5 Symbol Timing Estimation—Recovery

In modern digital receivers the sampling is performed with a fixed clock i.e. the receivedsignal is not synchronized with the incoming data symbols. Hence, subsequent to the ana-log-to-digital conversion the timing adjustment is performed using interpolation techniques.In this method, the phase alignment occurs not by moving the sample instances to the correctposition in the time waveform, but rather by interpolation of the matched filter samples fromthe collected sample locations to the desired sample locations. This is achieved by designinga polyphase matched filter with an increased sample rate using multirate signal processingtechniques whose filter coefficients are aligned with the data samples. This matched filteroperating at an increased sample rate is then re-sampled to attain the filter response at theoriginal sample rate with successive time offsets of 1/M, 2/M, 3/M etc, to form a bank of Mfilters matched to different time offsets between the input sample location and the envelopeof the received waveform. The timing recovery process determines one of the M possiblepaths required to align the filter with the signal time offset.

We emphasize that the M paths (or phases) of the polyphase matched filter correspond tothe acceptable time granularity of the re-sampling process. Also, for a fixed roll-off factor α,the width of the eye openings decrease with an increase in the number of levels in the eyediagram, and for a fixed number of levels in the eye diagram, the width of the eye openingalso decreases as the roll-off factor α, decreases [9].

We have an intuitive sense that timing information in a modulated signal resides in signallevel transitions. The transition detector seeks the midpoint of the matched filter output as ittransitions between different modulation states. The timing recovery process must ascertainif the clock sample position is at the correct position or needs to be advanced or retardedrelative to the input time waveform. A derivative operator is used to detect the signal tran-sitions i.e. the correlation function of the correct matched filter (selected from the bank ofM polyphase matched filters) would have a zero derivative at the peak. This has been moti-vated from the idea of the early and late gate output values used in the conventional analogtiming recovery systems. However, in addition to the information provided by the derivativefunction, the timing recovery process also needs to know a conditional piece of informationto reliably decide the direction of movement with respect to the received time waveform.This conditional additional information represents the sign of the sample of the correlationfunction which is folded into the observable error parameter in the timing recovery process.If the sample is positioned prior to the peak, a positive sign represents positive slope and anegative sign represents negative slope as illustrated in Fig. 11 [9]. However, if the sampleis located ahead of the peak the reverse is true.

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An All Digital Implementation of Constant Envelope

MF

PA

DMF

IndexSelection LF

1:2

Tanh

Z-1

Tanh

Z-1

I

Q

Q

I

MF

PA

DMF

IndexSelection LF

1:2

Tanh

Z-1

Tanh

Z-1

I

Q

Q

I

PA LF

C0 Path

C1 Path

DDS

2 samples/symbol *

Fig. 12 Joint symbol timing and carrier recovery block diagram (MF: Matched Filter, DMF: DerivativeMatched Filter, LF: Loop Filter, PA: Phase Accumulator, DDS: Direct Digital Synthesizer)

Fig. 13 Eye diagram at the output of the combined (C0[n] and C1[n]) matched filter

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Fig. 14 Transition diagram and constellation plot at the output of the combined (C0[n] and C1[n]) matchedfilter

Furthermore, zero crossing points can also be used to align the symbol timing during thetiming recovery process. At this point the derivative function would hold a maximum value.Using a simple sine wave as a model, it is easy to visualize that the midpoint of the transitioncorresponds to the location of the maximum value of the derivative. Thus the transition detec-tor often seeks the time location for which the magnitude of the average slope is a maximum.As shown in Fig. 11 this time corresponds to the midpoint of the symbol interval and alsocoincides with the center of mass of the zero-crossings in the eye-diagram.

In sampled data systems, the derivative of the bank of polyphase filters, say, when testingsegment (k) is derived using polyphase segments (k − 1) and (k + 1) as shown in Eq. 8.

ý(n + k/M) = y(n) . hk+1(n) − y(n) . hk−1(n)

= y(n) . [hk+1(n) − hk−1(n)]= y(n) . hk(n) (8)

Since the input signal is collected at two samples/symbol, one sample, the one with aneven index for instance, is chosen as a data sample, and the other is the non data samplesometimes called the timing sample. The data sample is the one delivered to the detector. Ifthe address pointer tries to cross the address boundaries, the address wraps circularly and theinput sample identified as the data sample is switched to the odd index as detailed in [9].

These parameters can be estimated independently in a specified order, or can be estimatedwith aid from previous estimators in the sequential chain, or can be estimated concurrentlybut independently, or concurrently and cooperatively [11]. The most common practice isconcurrent independent acquisition or concurrent aided acquisition.

Another common practice involves switching between modes, initializing unaided andthen operating in the aided mode to improve noise immunity. A severely distorted wave-shape may defeat the acquisition of the timing parameters by the timing recovery PLL.

To reliably demodulate a received signal, in addition to estimation of symbol timing,the carrier phase of the receiver must be synchronized to the carrier phase of the receivedsignal. Phase lock loops operating in a closed loop mode provides the appropriate controlto the carrier phase recovery process. In this paper we illustrate through computer simula-tions an all digital implementation of joint symbol and phase estimation—recovery scheme

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An All Digital Implementation of Constant Envelope

Fig. 15 Symbol timing using polyphase matched filter

Fig. 16 PLL error and input-output phase plots

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Fig. 17 Comparison of input-output phase profile and GMSK I-Q signal

in the presence of non-ideal channel, carrier offset and noise. Figure 12 demonstrates theblock diagram for the joint estimation of symbol timing and carrier recovery used in oursimulations.

6 Computer Simulations

Figures 13–17 illustrate computer simulations using eye diagrams and constellation plotsrepresenting the signal at various stages of the demodulator. Figure 13 presents the eye dia-grams of the offset in-phase and quadrature components at 2-samples per symbol formedby the I and Q matched filters. Figure 14 shows the transition diagram and the constellationdiagram of the ordered pairs formed by I and Q matched filters. Figure 15 shows the contentof the timing recovery phase accumulator and the quantized output, the index pointing to thebranch of the polyphase matched filter.

Figure 16 presents the carrier phase error, the input to the carrier PLL, and the input phaseprofile of the received signal carrier and the phase profile of the receiver’s locally gener-ated direct digital synthesizer. Figure 17 presents the time and phase aligned versions of thephase and quadrature components of the base band signal formed at the modulator and thenestimated at the receiver.

7 Conclusions

This paper has described a GMSK modem using the sub-optimal Laurents AMP approxi-mation method for matched filter processing and implemented with the Simon ML receiver

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An All Digital Implementation of Constant Envelope

architecture for carrier offset and symbol timing estimation and recovery. We further havedemonstrated the use of polyphase matched filter in the timing recovery process to support anall DSP based implementation of a receiver [9]. A full exposition of DSP based receiver struc-tures for a larger class of efficient modulation formats related to staggered I-Q modulationcan be found in [10].

References

1. Simon, M. K. (2003). Bandwidth efficient digital modulation with application to deep space communi-cations, Wiley-Interscience.

2. Murota, K., Kinoshita, K., & Hirade, K. (1981). Spectrum efficiency of GMSK land mobile radio Inter-national Conference on Communications, 2, pp. 23.8.1–23.8.5, June 14–20.

3. Consultative Committee for Space Data Systems (2003). Bandwidth efficient modulations, CCSDS 413.0-G-1. Green Book.

4. Kaleh, G. K. (1989). Simple coherent receivers for partial response continuous phase modulation IEEEJournal on Selected Areas in Communications, 7(9), 1427–1436.

5. Hodges, M. R. L. (1990). The GSM radio interface British Telecom Technological Journal, 8(2).6. Haspeslagh, J. et al. (1990) A 270 Kb/s 35-mW modulation IC for GSM cellular radio hand held terminals.

IEEE Journal on Solid State Circuits, 25(12), 1450–1457.7. Laurent, P. A. (1986). Exact and approximate construction of digital phase modulations by superposition

of amplitude modulated pulses. IEEE Transactions on Communications, COM-34(2), 150–160.8. Vassallo, E., & Visintin, M. (2002). Carrier phase synchronization for GMSK signals. International

Journal on Satellite Communications, 20, 391–415 (2002). doi:10.1002/sat.729.9. harris, f. j. (2004). Multirate signal processing for communication systems, Prentice Hall PTR.

10. Ramamurthy, A. (2006). Synchronization methods for efficient modulation formats related to staggeredI-Q modulation, Masters Thesis, Spring 2006, San Diego State University, San Diego.

11. harris, f. j. Manuscript on Synchronization for digital communications, available from author, San DiegoState University (unpublished manuscript).

12. Pasupathy, S. (1979). Minimum shift keying: A spectrally efficient modulation. IEEE CommunicationMagazine, 17(4), 14–22.

Author Biographies

Arjun Ramamurthy received his B.E in Electronics and Communica-tion from National Institute of Engineering, University of Mysore, Indiain 2001. He worked as a DSP Engineer at Deptartment of Embeddedand Product Engineering, Wipro Technologies, Bangalore, India until2004. He received his M.S in Electrical Engineering from San DiegoState University in 2006 and is currently working as Sr. Software Engi-neer at the Advanced Technology Group, Motorola Inc, San Diego. Hisinterests include in Multirate Signal Processing, Modern Wireless Com-munication, Video Coding algorithms, Embedded software developmentfor multimedia applications.

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fedric j. harris hold the CUBIC Signal Processing Chair of the Com-munication Systems and Signal Processing Institute at San Diego StateUniversity where since 1967 I have taught courses in Digital Signal Pro-cessing and Communication Systems. I hold a number of patents ondigital receiver and DSP technology and lecture throughout the worldon DSP applications. I consult for organizations requiring high perfor-mance, cost effective DSP solutions.

I have written over 140 journal and conference papers, the most wellknown being my 1978 paper “On the use of Windows for HarmonicAnalysis with the Discrete Fourier Transform”. I am the author of thebook “Multirate Signal Processing for Communication Systems” andI have contributed to a number of other books on DSP applicationsincluding the Source Coding chapter in Bernard Sklar’s 1988 book,Digital Communications and the Multirate FIR Filters for Interpola-tion and Resampling and the Time Domain Signal Processing with the

DFT chapters in Doug Elliot’s 1987 book Handbook of Digital Signal Processing.In 1990 and 1991 I was the Technical and then the General Chair of the Asilomar Conference on Signals,

Systems, and Computers and was Technical Chair of the 2003 Software Defined Radio Conference and of the2006 Wireless Personal Multimedia Conference. I became a Fellow of the IEEE in 2003, cited for contribu-tions of DSP to communications systems. In 2006 I received the Software Defined Radio Forum’s “IndustryAchievement Award”. I am the Co-Editor-in-Chief of the Elsevier DSP Journal.

The spelling of my name with all lower case letters is a source of distress for typists and spell checkers.A child at heart, I collect toy trains and old slide-rules.

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