An All Digital Implementation of Constant Envelope Bandwidth Efficient GMSK Modem using Advanced Digital Signal Processing Techniques Arjun Ramamurthy, IEEE Student,
[email protected] fredric j. harris, IEEE Fellow,
[email protected] San Diego State University
Abstract: Gaussian Minimum Shift Keying (GMSK) has been the most common modulation 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 communication systems because of its high bandwidth efficiency and constant envelope modulation characteristics. The focus of this paper is the design of the demodulator wherein we demonstrate an all digital implementation of sub-optimal synchronization techniques for a GMSK modem based on two Laurent Amplitude modulation pulse (AMP) streams approximation representing the matched filter. In this all digital implementation, we perform a joint estimation of the symbol timing and carrier offset wherein the symbol timing is performed using interpolation techniques. Introduction: GMSK is an h=0.5 partial response continuous phase modulation scheme derived from MSK with the addition of baseband Gaussian filtering applied to the identically and independently distributed (i.i.d) random rectangular pulse shaped input signal prior to frequency modulation of the carrier. As indicated in [1], it is important to emphasize that although the acronym GMSK was assigned to the term Gaussianfiltered MSK in [2], the modulation actually described in this reference applies to Gaussian filtering of rectangular pulses at baseband as shown in Figure 1 [1], i.e. prior to modulation onto the carrier, and, hence, it does not destroy the constant envelope property of the resulting modulation. It is assumed that the frequency pulse g(t) in Figure 1 is the result of a convolution (filtering) operation performed between the rectangular pulse p(t) and the Gaussian filter h(t) as illustrated in Eqns 1-3. Although g(t) of Eqn 1 (or Eqn 4) appears to have a “Gaussian-looking” shape, we emphasize that the word Gaussian in the GMSK refers to the impulse response h(t) of the filter through which the input rectangular pulse train is passed and not the shape of the resulting frequency pulse g(t). g(t) = p(t)*h(t), * is convolution operation (1) p(t) = 1, 0 t Ts (2) h(t) = 1/(2 2)1/2.exp(-t2/2 2), 2=ln2/(2 B)2 (3) Alternatively, as illustrated in [1: Eqn 2.8.52], the GMSK frequency pulse is the difference of 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
g (t ) =
Q( x) =
t t 1 [Q ( K ⋅ ( − 1)) − Q( K ⋅ ( )) 2TS TS TS (4) BTS K = 2π ln(2) ∞ x
1 y2 exp(− ) dy 2 2π
p(t-nT) { a n} a n ε (-1,+ 1)
π Ga ussian Filter h(t)
Data NRZ Pulse Train
fc
Frequenc y Modula tor
s(t)
a n g(t-nTb)
Figure 1. GMSK transmitter (CPM representation) Redrawn from [1] in Figure 2. It is a common practice to refer to the product of the Gaussian filter 3dB bandwidth and the 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) of the Gaussian filter that controls the spectral occupancy as shown in Figure 2. Smaller values of B result in lesser spectral bandwidth occupancy (Figure 2) but greater ISI (Figures 3-5) resulting in higher BER. Hence, depending upon the application, a particular value of B is selected resulting in a compromise between spectral efficiency and BER performance. Viterbi equalization or trellis demodulation [3] techniques are normally used to compensate for the increased ISI due to lower B values (or equivalently lower BTs product). The Gaussian shape provides an additional advantage of reduced sidelobe levels. The constant envelope of the GMSK signal reduces spectral re-growth and signal distortion due to any amplifier nonlinearity. Modulator: The GMSK modulator can be implemented as a FSK modulator or as an offset QPSK modulator [3]. It is a common practice to adopt the I-Q implementation to generate the GMSK signal resulting in a representation equivalent to staggered I-Q modulation format. Subsequently, in addition to the intentional and controlled ISI introduced in the GMSK signaling format, there exists correlation between adjacent symbols on the quadrature channels. As a result, the conventional receiver synchronization techniques, involving the carrier and symbol timing recovery needs to be modified to accommodate this cross talk between adjacent symbols on the I and Q channels.
Figure 2. Impulse response and Frequency response of Gaussian filter h(t), and Gaussian filtered frequency pulse g(t) with B = 0.5,0.3,0.25
Figure 3. Eye diagram I-Q representation of GMSK signal with B = 0.5
Figure 5. Eye diagram I-Q representation of GMSK signal with B = 0.25
Figure 4. Eye diagram I-Q representation of GMSK signal with B = 0.3
There are various optimal and suboptimal methods used to estimate the carrier offset and symbol timing to achieve successful signal detection and decoding at the demodulator. We follow the offset quadrature implementation approach as shown in Figure 6. From Eqn 4 it can bee seen that the Q function is doubly infinite in extent and it’s a common practice to time 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 = 0.3 respectively. The GSM application adopts the B = 0.3 considering ISI only from adjacent neighbors (say, for symbol n, only n-1 and n+1 introduce ISI) as shown in [6]. Hence, in practical GMSK implementations, an I-Q representation (Figure 6) with an approximation as shown in Eqn 5 [1:Eqn 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 for B = 0.3.
g (t ) =
1 t t [Q( K ⋅ ( − 1)) − Q( K ⋅ ( )) 2TS TS TS (5)
TS T ≤ t ≤ −( L + 1) S 2 2 0 Elesewhere
where − ( L − 1)
p(t-nT) { an} a n ε (-1,+ 1)
c os(--)
π Ga ussian Filter h(t)
Data NRZ Pulse Tra in
computer simulations an all digital implementation of joint symbol and phase estimation - recovery scheme in the presence of non-ideal channel, carrier offset and noise. Figure 10 demonstrates the block diagram for the joint estimation of symbol timing and carrier recovery used in our simulations. Figure 13 illustrates the phase accumulator and the polyphase matched filter index selection plots.
s(t) Integra tor
a n g(t-nTb )
fc π/2
-
c os(--)
Figure 6. GMSK transmitter (CPM representation) Redrawn from [1] Demodulator: The demodulator can be implemented in a number of different optimum and sub-optimum methods as indicated in [1]. We use the Laurent’s suboptimal two pulse approximation method in our simulations. Laurent [7] described a representation of CPM signal as a superposition of phase shifted amplitude modulation pulse (AMP) streams. In [1], a detailed interpretation of the key results of this paper has been conducted for both, the exact and approximate AMP representation of GMSK signal. In our computer simulations of the demodulator we have used only the first two Laurent AMP streams (matched filter) because the first and second AMP component corresponding to the pulse streams C0[n] and C1[n] contains the significant fraction of 0.991944 and 0.00803 of the total energy [4]. Figures 7-9 demonstrate the first two pulse stream AMP representation for different values of B. In this all digital implementation, we perform a joint estimation of the symbol timing and carrier offset. The carrier phase synchronization can be performed based on the MAP and/or ML phase estimation techniques using data aided/non-data aided or decision directed configurations. We perform the carrier phase synchronization using the non-data aided method as detailed in [8]. The symbol timing is performed using an interpolation technique [9]. In this method, the phase alignment occurs not by moving the sample instances to the correct position in the time waveform, but rather by interpolation of the matched filter samples from the collected sample locations to the desired sample locations. This is achieved by designing a polyphase matched filter with an increased sample rate using multirate signal processing techniques whose M-path filter coefficients are aligned with the offset data samples. 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 received signal. Phase lock loops operating in a closed loop mode provides the appropriate control to the carrier phase recovery process. In this paper we illustrate through
Figure 7. First and Second Laurent AMP streams with B=0.5
Figure 8. First and Second Laurent AMP streams with B=0.3
Figure 9. First and Second Laurent AMP streams with B=0.25
I MF
C0 Path
Index Selection
PA
LF
1:2
Q
I DMF
2 samples/symbol
* C1 Path
Z-1
Tanh
Z-1
Tanh
Q
I MF Index Selection
PA
LF
1:2
Q
Z-1
Tanh I
DMF
Z-1
DDS
PA
Tanh
Q
LF
Figure 10. Joint symbol timing and carrier recovery block diagram (MF: Matched Filter, DMF: Derivative Matched Filter, LF: Loop Filter, PA: Phase Accumulator, DDS: Direct Digital Synthesizer) Computer simulations: Figures 11-15 illustrate computer simulations using eye diagrams and constellation plots representing the signal at various stages of the demodulator. Figure 11 presents the eye diagrams of the offset inphase and quadrature components at 2-samples per symbol formed by the I and Q matched filters. Figure 12 shows the transition diagram and the constellation diagram of the ordered pairs formed by I and Q matched filters. Figure 13 shows the content of the timing recovery phase accumulator and the quantized
Figure 11. Eye diagram at the output of the combined (C0[n] and C1[n]) matched filter
Figure 12. Transition diagram and constellation plot at the output of the combined (C0[n] and C1[n]) matched filter
Figure 13. Symbol timing using polyphase matched filter
output, the index pointing to the branch of the polyphase matched filter. Figure 14 presents the carrier phase error, the input to the carrier PLL, and the input phase profile of the received signal carrier and the phase profile of the receiver’s locally generated direct digital synthesizer. Figure 15 presents the time and phase aligned versions of the phase and quadrature components of the base band signal formed at the modulator and then estimated at the receiver.
further have demonstrated the use of polyphase matched filter in the timing recovery process to support an all DSP based implementation of a receiver [9]. A full exposition of DSP based receiver structures for a larger class of efficient modulation formats related to staggered I-Q modulation can be found in [10]. References [1] Marvin. K. Simon, Bandwidth Efficient Digital Modulation with Application to Deep Space Communications, WileyInterscience, 2003. [2]
K. Murota, K. Kinoshita, and K. Hirade, “Spectrum efficiency of GMSK land mobile radio” International Conference on Communications, vol.2, pp. 23.8.1-23.8.5, June 1420, 1981.
[3]
Consultative Committee for Space Data Systems, Bandwidth Efficient Modulations, CCSDS 413.0-G-1, Green Book, 2003.
[4]
G. K. Kaleh, “Simple coherent receivers for partial response continuous phase modulation” IEEE Journal on Selected Areas in Communications,vol. 7, no. 9, pp. 1427– 1436, December 1989.
[5]
M. R. L. Hodges, “The GSM radio interface” British Telecom Technological Journal, vol. 8, no. 2, January 1990.
[6]
J. Haspeslagh et al., “A 270 Kb/s 35-mW modulation IC for GSM cellular radio hand held terminals” IEEE Journal on Solid State Circuits, vol. 25,no. 12, pp. 1450–1457, December 1990.
[7]
P.A. Laurent, “Exact and approximate construction of digital phase modulations by superposition of amplitude modulated pulses,” IEEE Transactions on Communications, vol. COM-34, no.2, pp. 150-160, February 1986.
[8]
E.Vassallo, M. Visintin, “Carrier phase synchronization for GMSK signals”, Int. J. Satell. Commum.2002, 20:391-415 (DOI: 10.1002/sat.729).
[9]
fredric. j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 2004.
[10]
Arjun Ramamurthy, “Synchronization Methods for Efficient Modulation Formats Related to Staggered I-Q Modulation”, Masters Thesis, Spring 2006, San Diego State University, San Diego.
Figure 14. PLL error and input-output phase plots.
Figure 15. Comparison of input-output phase profile and GMSK I-Q signal Conclusions This paper has described a GMSK modem using the sub-optimal Laurents AMP approximation method for matched filter processing and implemented with the Simon ML receiver architecture for carrier offset and symbol timing estimation and recovery. We
