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Research Letters in Communications
Volume 2008 (2008), Article ID 235761, 4 pages
http://dx.doi.org/10.1155/2008/235761
Research Letter

Wiener's Loop Filter for PLL-Based Carrier Recovery of OQPSK and MSK-Type Modulations

Dipartimento di Elettronica e Informazione, Politecnico di Milano, Piazza Leonardo da Vinci, 32, 20133 Milano, Italy

Received 7 April 2008; Accepted 28 May 2008

Academic Editor: J. S. Mandeep

Copyright © 2008 Arnaldo Spalvieri and Maurizio Magarini. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This letter considers carrier recovery for offset quadrature phase shift keying (OQPSK) and minimum shift keying-type (MSK-type) modulations based on phase-lock loop (PLL). The concern of the letter is the optimization of the loop filter of the PLL. The optimization is worked out in the light of Wiener's theory taking into account the phase noise affecting the incoming carrier, the additive white Gaussian noise that is present on the channel, and the self-noise produced by the phase detector. Delay in the loop, which may affect the numerical implementation of the PLL, is also considered. Closed-form expressions for the loop filter and for the mean-square error are given for the case where the phase noise is characterized as a first-order process.

1. Introduction

Offset quadrature phase-shift keying (OQPSK) and minimum shift keying-type (MSK-type) modulations play a prominent role in optical and radio transmission systems. The renewed interest in such modulations for application to optical and 60 GHz radio transmissions is demonstrated by the recent contributions [1, 2]. As is known, MSK-type signals can be regarded as approximated forms of OQPSK when the first term of the Laurent's decomposition dominates the series [3]. This suggests the extension of synchronization algorithms usually employed for OQPSK to MSK-type signals [4].

In carrier recovery for OQPSK and MSK-type formats, it is worth taking into account the self-noise that affects the phase detector, a noise that is not present in quadrature amplitude modulation (QAM) and phase-shift keying (PSK) formats. The design of a prefilter that mitigates the effects of the self-noise in an open-loop synchronizer is considered in [5]. In this letter, we focus on the design of the loop filter of closed-loop carrier recovery based on the phase-lock loop (PLL). The Wiener's approach, studied in [6] for QAM modulation formats, is extended here to the case where self-noise is present, enabling application to OQPSK and MSK-type formats.

2. System Model

The complex envelope of the continuous-time received signal is𝑟(𝑡)=𝑒𝑗𝜃(𝑡)𝑖𝑎𝑖𝑢(𝑡𝑖𝑇)+𝑤(𝑡),(1)where 𝑗 is the imaginary unit, 𝑎𝑘=±1 for 𝑘 even and 𝑎𝑘=±𝑗 for 𝑘 odd, 𝜃(𝑡) is the phase noise, 𝑢(𝑡) is the impulse response of the shaping filter, 𝑇 is the bit repetition interval, and 𝑤(𝑡) is the complex additive white Gaussian noise with power spectral density 𝑁0. The model given by (1) holds exactly for OQPSK and MSK, while it provides a close approximation for many MSK-type signals of practical interest, as for instance the popular GMSK with 𝐵𝑇=0.3, when the impulse response 𝑢(𝑡) is the first term of the Laurent decomposition.

The received signal is filtered through the matched filter 𝑢(𝑡), sampled at the time instants 𝑘𝑇, and filtered through the zero forcing (ZF) prefilter with frequency response𝐶𝑒𝑗2𝜋𝑓𝑇=2𝑇𝑘|||𝑈|||𝑓𝑘/2𝑇2,(2)where 𝑈(𝑓) is the Fourier transform of 𝑢(𝑡). The 𝑘th sample of the signal at the output of the prefilter has the form𝑥𝑘=𝑒𝑗𝜃𝑘𝑖𝑎𝑖𝑣𝑘𝑖+𝑛𝑘,(3)where the samples of the impulse response {𝑣𝑘} satisfy the Nyquist condition in the form 𝑣0=1 and 𝑣2𝑘=0, for 𝑘0. The sequence {𝑛𝑘} is the complex additive Gaussian noise with power spectral densityΨ𝑛𝑒𝑗2𝜋𝑓𝑇=𝑁0𝑇|||𝐶𝑒𝑗2𝜋𝑓𝑇|||2|𝑘𝑈𝑘𝑓𝑇|2.(4)

The 𝑘th sample at the output of the phase detector is𝑒𝑘𝑎=𝑘𝑥𝑘𝑒𝑗̂𝜃𝑘𝜃=sin𝑘̂𝜃𝑘+𝜆𝑘𝜃cos𝑘̂𝜃𝑘+𝜉𝑘,(5)where the superscript denotes complex conjugation, {𝑥} extracts the imaginary part of 𝑥, and ̂𝜃𝑘 is the 𝑘th phase sample produced by the carrier recovery loop. The sequence {𝑒𝑘} is filtered and used to produce the carrier estimate by a number controlled oscillator (NCO). The block diagram of the system is reported in Figure 1. The even and the odd samples of the Gaussian noise {𝜉𝑘} come from the imaginary and the real part of {𝑛𝑘}, respectively, and therefore are uncorrelated. The power spectral density of {𝜉𝑘} isΨ𝜉𝑒𝑗2𝜋𝑓𝑇=14Ψ𝑛𝑒𝑗2𝜋𝑓𝑇+14Ψ𝑛𝑒𝑗2𝜋(𝑓1/2𝑇)𝑇,(6)where the factor 1/4 comes from the folding (factor 1/2) and from the projection of the complex noise over one dimension (factor 1/2).

235761.fig.001
Figure 1: Block diagram of the system.

The 𝑘th sample of the self-noise is𝜆𝑘=𝑖𝑎𝑘𝑎𝑘2𝑖1𝑣2𝑖+1.(7)In what follows we use the 𝑧-transform to represent sequences, where 𝑧1 is the unit delay, that is, 𝑇. The 𝑧-spectrum of {𝜆𝑘} isΨ𝜆(𝑧)=𝑖𝑣22𝑖+1𝑘𝑣22𝑘+1𝑧2𝑘1.(8)Note that the power spectrum of self-noise depends on the odd samples of {𝑣𝑘}. When |𝑘𝑈(𝑓𝑘/𝑇)|2 satisfies the condition of not having spectral zeros, as it happens for instance with GMSK, it is possible to design the prefilter in such a way that the condition 𝑣𝑘=𝛿𝑘 is satisfied (𝛿𝑖 denotes the Kronecker delta). This design of the prefilter induces noise enhancement. However, at high signal-to-noise ratio, noise enhancement can be accepted, enabling the approach of [5]. Also note that, when 𝑈(𝑓) is the square root of a Nyquist filter, as it often happens with OQPSK, we obtainΨ𝜉𝑁(𝑧)=02𝐸𝑏,(9)where 𝐸𝑏=|𝑢(𝑡)|2𝑑𝑡 is the energy per bit.

3. Derivation of the Optimal Transfer Function

For small phase error, the phase detector can be linearized, ̂̂𝜃sin(𝜃𝜃)𝜃, ̂cos(𝜃𝜃)1, leading to the error polynomial𝐸(𝑧)=Θ(𝑧)Θ(𝑧)+Λ(𝑧)+Ξ(𝑧).(10)The open-loop transfer function is𝐺(𝑧)=Θ(𝑧)=𝐸(𝑧)𝑘=𝐾𝑔𝑘𝑧𝑘,(11)where 𝐾>0 is the delay of the loop. The number of poles of 𝐺(𝑧) is the order of the loop, while the number of poles of 𝐺(𝑧) at 𝑧=1 is the type of the loop. In the analysis of the loop, it is useful to introduce the polynomial𝑌(𝑧)=Θ(𝑧)+Λ(𝑧)+Ξ(𝑧).(12)We assume that {𝜃𝑘}, {𝜆𝑘}, and {𝜉𝑘} are independent random sequences, therefore the 𝑧 -spectrum of {𝑦𝑘} isΨ𝑦(𝑧)=Ψ𝜃(𝑧)+Ψ𝜆(𝑧)+Ψ𝜉(𝑧).(13)

The closed-loop transfer function is𝐻(𝑧)=𝐺(𝑧)=1+𝐺(𝑧)Θ(𝑧)𝑌(𝑧).(14)The optimal 𝐻(𝑧) is obtained from the Wiener-Hopf equations. Following the steps of [6], we get𝐻(𝑧)=𝛼2𝑧1𝑃(𝑧)1𝑃1Ψ𝜃(𝑧)𝐾,(15)where the notation [𝑋(𝑧)]𝑗𝑖=𝑗𝑘=𝑖𝑥𝑘𝑧𝑘 is used. The real number 𝛼2 is computed by Szego's formula, and 1𝑃(𝑧) is the causal monic and minimum phase transfer function which results from the spectral factorization𝛼1𝑃(𝑧)2𝑧1𝑃1=Ψ𝑦1(𝑧).(16)

4. Approximate Solution

We observe that in many cases of practical interest, the bandwidth of the loop filter is small compared to the bit frequency, therefore we aim to approximate the spectrum of self-noise and the spectrum of channel noise in the low-frequency region. The spectrum of self-noise is well approximated in the low-frequency region byΨ𝜆(𝑧)=1𝑧1𝛾2(1𝑧),(17)with 𝛾2=𝑘=0(2𝑘+1)2𝑣22𝑘+1. The spectrum of the additive channel noise {𝜉𝑘} is approximated in the low-frequency region by (9). Using (9) and (17) in (13) and performing the spectral factorization, we obtain𝐻(𝑧)=𝑄(𝑧)+𝛼2𝛾2𝑧1𝐾1+𝑄(𝑧),(18)and the mean square error (MSE)𝐸𝜃𝑖̂𝜃𝑖2=1𝛼2+𝐾1𝑘=1𝑞2𝑘𝛼22𝛾2𝑁02𝐸𝑏𝛼2𝛾22+2𝛾2𝑞1𝛿𝐾1,(19)where 1+𝑄(𝑧)=(1𝑃(𝑧))1.

5. Case Study

Assume that the phase noise is the sum of random phase walk plus white phase. The phase noise spectrum isΨ𝜃𝛾(𝑧)=21𝑧1(1𝑧)+𝛾0.(20)The spectral factorization is performed by root finding. The result is1𝑃(𝑧)=1𝑧11𝑧1𝑧11𝑧2𝑧1,(21)𝑄(𝑧)=1𝑧1𝑧2𝑧11𝑧1𝑧1+1𝑧21𝑧1𝑧2𝑧1,(22)where the two roots are𝑧𝑖=𝑥𝑖𝑥2𝑖42,𝑖=1,2,(23)with𝑥1,2𝛾=2+0+𝑁0/2𝐸𝑏2𝛾2±𝛾0+𝑁0/2𝐸𝑏2𝛾22𝛾2𝛾2.(24) Putting (22) in (18) and using (14), one gets𝐻(𝑧)=1𝑧1𝑧2+𝑧1𝑧2𝑧𝐾1𝑧1𝑧11𝑧2𝑧1,(25)𝐺(𝑧)=1𝑧1𝑧2+𝑧1𝑧2𝑧𝐾1𝑧1(1𝑧1𝑧2𝑧1+𝐾1𝑘=11𝑧1𝑧2+𝑧1𝑧2𝑧𝑘).(26)Note that with 𝐾=1 and without self-noise, one would obtain an optimal type-I order 1 loop [6]. The presence of self-noise has increased the order of the optimal loop inducing a pole at 𝑧=𝑧1𝑧2. For 𝐾>1, a type-I order 𝐾 loop is obtained. Using (22) and substituting 𝛼2=𝑧1𝑧2/𝛾2 in (19), we get for the MSE𝐸𝜃𝑖̂𝜃𝑖2=𝛾21𝑧1𝑧2+2𝑧1+2𝑧2𝑧1𝑧2+𝛾42𝑧1𝑧2(𝐾1)1𝑧1𝑧2+𝑧1𝑧22𝑁02𝐸𝑏.(27)

6. Numerical Results

We derive numerical results for OQPSK with raised-cosine shaping and roll-off 0.1 and GMSK with 𝐵𝑇=0.3. The spectrum of phase noise that affects the free-running oscillator used for the analysis is characterized by 𝛾2=5105 and 𝛾0=104. Figure 2 shows analytical results and simulation results for the MSE versus 2𝐸𝑏/𝑁0 for 𝐾=1 and 𝐾=32 with OQPSK. Note that, while analytical results have been obtained assuming that no decision errors affect the phase detector (5), in the simulations the phase detector makes use of decisions in place of true data. At low SNR simulation results deviate from the theory due to decision errors. The best MSE is given by the optimum 𝐻(𝑧) obtained without approximations using the numeric explog method [7]. From the figure we see that the use of the loop filter we propose gives a benefit of about 2 dB on the floor compared to the classical order-I type 1 loop when 𝐾=1. When 𝐾>1, the benefit diminishes. Also observe that the performance of the approximated transfer function derived in Section 5 is close to the optimal one. Figure 3 reports the performance of GMSK. The benefits obtained by our method are similar to those shown in Figure 2. In this case, the performance of the approximated transfer function is virtually the same as that of the optimal one.

235761.fig.002
Figure 2: MSE versus 2𝐸𝑏/𝑁0 for OQPSK with roll-off 0.1.
235761.fig.003
Figure 3: MSE versus 2𝐸𝑏/𝑁0 for GMSK with 𝐵𝑇=0.3.

7. Conclusions

The main novel results of the letter are the optimal loop filter and the MSE for carrier recovery based on discrete-time PLL for OQPSK and MSK-type modulation formats. Compared to previous literature, self-noise that affects the phase detector and delay in the loop are taken into account in the design of the loop filter. Closed-form expressions for the loop filter and for the MSE are given for the case where the phase noise is modelled as a first-order process.

Acknowledgment

This work is supported by MIUR-FIRB Integrated System for Emergency (InSyEme) project under the grant RBIP063BPH.

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