A Digital Implementation of Multi-h CPM Modem

Kulikov G. V., Unger A. U., Suhanov P. G.

Abstract

In this paper, an all-digital design for multi-h

continuous-phase modulated signals modem is proposed. The receiver of the modem performs joint demodulation, symbol timing and carrier synchronization, using the Viterbi-type algorithms. The developed algorithms may be applied to a wide class of continuous-phase modulations. 1. Introduction

Continuous-phase modulations (CPMs) have been

extensively studied in the literature for many years. CPM is a signaling method with good power and bandwidth efficiency [1]. CPM was initially conceived for satellite applications, because of its very attractive property of constant envelope, but it found its first important practical application in wireless communication. An interesting generalization of CPM is the so-called multi-h modulation, which differs from the ordinary single-h format in that it uses a set of modulation indices in a cyclic manner [2]. Cyclically changing indexes leads to delayed merging of neighboring phase trellis paths and, ultimately, to improved error performance [3]. In spite of its favourable features, current applications of CPM are limited to a few simple schemes because of implementation complexity and synchronization problems.

In this paper we describe a design of a digital multi-h CPM modem. In addition to the proper modulator/demodulator the modem contains all necessary synchronization circuits.

This paper is organized as follows. After this introduction, Section 2 introduces signal model. In Section 3 a digital implementation of multi-h CPM modulator is discussed. Section 4 covers the algorithms used for joint demodulation, symbol timing and carrier synchronization of multi-h CPM. 2. Signal model

The received waveform is composed of signal plus noise

),()),(2cos(2)( 0 tnattfTEtr nc +++= ϕψπ r (1)

where E is the signal energy per symbol, T is the signaling interval, fc is the carrier frequency, φ0 is the phase shift introduced by the channel, and ),( nat rψ is the information-bearing phase defined as

,)(2),(0∑=

−=n

kkkn kTtqahat πψ r (2)

with hk modulation index over the symbol interval, ),...,,( 10 nn aaaa =r is the data sequence transmitted up

to time nT, and q(t) is the phase response of the modulator. The pulse g(t) = dq(t)/dt is the frequency response and is nonzero over (0, LT), L being the correlation length. The data symbols nar are independent, equally likely, and belong to the alphabet {±1, ±3,…, ±(M − 1)}. The index hk cyclically changes from symbol to symbol with a period of K, but only one index is used during one symbol interval. The interval KT is called superbaud period. It is normally assumed that the set of modulation index is rational with a common denominator hi = 2pi/q, i = 1…K.

In [1], it is shown that (2) may be rearranged as follows:

TntnTatat nnn )1(,),(),( +<≤+= θθψ rr (3)

where θn is the phase state

)2mod(0

ππθ ∑−

=

=Ln

iiin ha (4)

and ),( nat rθ is defined as

.)(2),(1

∑+−=

−=n

Lniiin iTtqhaat πθ r (5)

A correlative state vector is defined as

).,...,,( 121 −+−+− nLnLn aaa

It can be shown [1], that there are ML-1 distinct correlative states and q phase states. Hence, given hi and q(t), it appears that ),( nat rψ is specified by the nth

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symbol an, the phase state θn and the correlative state. It follows that the CPM modulator may be viewed as a trellis encoder with qML-1 states in cascade with a mapper which associates the waveform (1) with the encoder output.

PHASE STATE ROM

PHASE BRANCH

ROM

C

T

I(t)

M

m · 1/T

αn

In the following discussion we consider binary 1REC (that is, rectangular frequency response of length 1T) multi-h CPM signals. 3. Modulator implementation

The most general way of implementation a CPM transmitter is to use the basic formula for modulated signal into quadrature components. These are stored in sampled and quantized form in look-up tables. The quadrature structure can be derived by rewriting the CPM waveform as

)2sin()()2cos()()( tftQtftIts cc ππ −= (6)

where,

)],(sin[)()],(cos[)(

n

n

attQattIr

r

ψψ

==

Figure 1 shows the I-generator structure using ROM (read only memory) to store I(t) and Q(t). All possible I(t) and Q(t) shapes over one symbol interval are addressed by the phase state θn, the current input symbol an and the current modulation index hn. The phase states θn are generated by a sequential machine consisting of a phase state ROM and a delay of length one symbol interval T. The phase state machine is actually implemented as an up/down counter, with the phase state serial number νn related to the phase values by the equation

)2mod()2( ππνθqnn = (7)

Similarly, the present modulation index serial number is generated by counter M limited to K − 1.

The I(t) and Q(t) shapes are swept by a counter C at a speed m/T, where m is the number of stored samples per symbol interval. The present symbol an, the phase state serial number νn, the present modulation index serial number and the counter C output are combined to form the addresses to obtain the samples of I(t) and Q(t) in the ROMs.

Fig. 1. I - generator block diagram

The I(t) ROM size is 2qKmmq bits, where mq is the

number of bits per sample. The Q(t) ROM is of course the same size. For example, for multi-h CPM with hi = (4/8, 6/8) the ROM size is 8192 bits assuming m = 16 and mq = 16.

The samples of I(t) and Q(t) are multiplied by carrier in-phase and quadrature component samples, respectively, and are fed to the adder. This results in multi-h CPM signal generation, since carrier frequency is chosen as an integer multiple of the symbol rate (1/T). In case of m is equal to the number of samples stored in carrier components ROMs, the resampling of I(t) and Q(t) is not needed. 4. Joint demodulation, symbol timing and carrier synchronization of multi-h CPM

For reception of the data transferred by multi-h CPM signal we use Viterbi [4] demodulator. Based on the received signal r(t), 0 ≤ t ≤ nT, the receiver has to determine the correct path through the phase trellis. To this end, it applies the Viterbi algorithm, which requires the computation of metrics between the received signal r(t) and the references over each symbol interval. In case of the phase shift of the received signal relative to the initial phases of the references the magnitudes of the computed metrics decrease what leads to the decrease of the noise stability. For the purpose to increase the noise stability we propose to densify the basic phase trellis by additional phase states. Such a multiple-phase trellis is a collection of simple phase trellises shifted relative to each other in a certain value Δφd. Hence, using the Viterbi demodulator with multiple-phase trellis allows increasing the probability of the coincidence of the correct path through the phase trellis with one of the additional phase trellises. This effect is identical to the self-synchronization of the demodulator. The idea of

the multiple-phase trellis is described in more detail in [5].

Thus, Viterbi demodulator with multiple-phase trellis is capable of coherent detection of multi-h CPM signal. Let us consider such a demodulator as a unit cell of the entire multi-h receiver. The receiver also requires symbol timing and superbaud synchronization circuits. To obtain the information suitable for synchronization we use time delay discriminator, which consists of the two lateral channels formed by time shift in +τ and −τ concerning the central (decoding) channel. It was found that the values of the time metrics derivatives may be used as synchronization characteristics. Figure 2 shows the block diagram of such a receiver joint with symbol timing and superbaud synchronization.

Fig. 2. Block diagram of multi-h CPM receiver

The central channel contains a multiple-phase trellis. Lateral channels consist of a Viterbi demodulator with a single phase trellis coincident with the phase trellis Δφd|max of the central channel with the maximal metrics derivative. The difference between metrics derivatives of the two lateral channels is used for input data stream manipulation. For noise stability increase integrators with buffers, included in front of the maximum choice block and in front of the sequence modifier are used. 5. Conclusion

We have investigated an all-digital multi-h CPM modem. The proposed concept of the multiple-phase trellis turned out a very useful method for symbol timing and carrier synchronization. It is worth noting that the developed algorithms are equally applicable for a wide class of CPM.

The efficiency of the entire modem was tested using FPGA-based digital device, designed for Moscow State Institute of Radioengineering, Electronics and Automation (Technical University) R&D. References [1] J. B. Anderson, T. Aulin, and C. E. Sundberg, Digital Phase Modulation, Plenum Publishing Company, New York, 1986. [2] J. B. Anderson, and D. P. Taylor, “A bandwidth-efficient class of signal-space codes”, IEEE Trans. Inform. Theory, vol. IT-24, Nov. 1978, pp. 703–712. [3] T. Aulin, and C. E. Sundberg, “On the optimum Euclidean distance for a class of signal space codes”, IEEE Trans. Inform. Theory, vol. IT-28, Jan. 1982, pp. 43–55. [4] Viterbi, A. J., and J. K. Omura, Principles of Digital Communication and Coding, McGraw-Hill, New York, 1979. [5] G. V. Kulikov, P. G. Suhanov, “Research on carrier phase self-synchronization of the Viterbi demodulator with multiple-phase trellis”, Scientific bulletin, MIREA, vol. 16(1), 2009, pp. 35-41.