# Wiener's Loop Filter for PLL-Based Carrier Recovery

Authors: Arnaldo Spalvieri; Maurizio Magarini
Journal: Research Letters in Communications
 (2008)

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## 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.


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## Body

## 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 is(1)r(t)=ejθ(t)∑iaiu(t−iT)+w(t),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
filteru∗(−t),
sampled at the time instants kT, and filtered through the zero forcing (ZF)
prefilter with frequency response(2)C(ej2πfT)=2T∑k|U(f−k/2T)|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 form(3)xk=ejθk∑iaivk−i+nk,where the samples of the impulse
response {vk} satisfy the Nyquist condition in the form v0=1 and v2k=0,
for k≠0.
The sequence {nk} is the complex additive Gaussian noise with
power spectral density(4)Ψn(ej2πfT)=N0T|C(ej2πfT)|2|∑kU∗(f−kT)|2.Thekth sample at the output of the phase detector
is(5)ek=ℑ{ak*xke−jθ^k}=sin(θk−θ^k)+λkcos(θk−θ^k)+ξk,where the superscript * denotes complex conjugation, ℑ{x} extracts the imaginary part of x, and θ^k 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} is(6)Ψξ(ej2πfT)=14Ψn(ej2πfT)+14Ψn(ej2π(f−1/2T)T),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).Figure 1
Block diagram of the system.Thekth sample of the self-noise is(7)λk=∑iℑ{ak*ak−2i−1}v2i+1.In what follows we use the z-transform to represent sequences, where z−1 is the unit delay, that is, T.
The z-spectrum of {λk} is(8)Ψλ(z)=∑iv2i+12−∑kv2k+12z−2k−1.Note that the power spectrum of
self-noise depends on  the odd samples of {vk}.
When |∑kU∗(f−k/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 obtain(9)Ψξ(z)=N02Eb,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,sin(θ−θ^)≈θ−θ^, cos(θ−θ^)≈1,
leading to the error polynomial(10)E(z)=Θ(z)−Θ^(z)+Λ(z)+Ξ(z).The open-loop transfer function is(11)G(z)=Θ^(z)E(z)=∑k=K∞gkz−k,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 polynomial(12)Y(z)=Θ(z)+Λ(z)+Ξ(z).We assume that {θk}, {λk}, and {ξk} are independent random sequences, therefore
the z -spectrum of {yk} is(13)Ψy(z)=Ψθ(z)+Ψλ(z)+Ψξ(z).Theclosed-loop transfer function is(14)H(z)=G(z)1+G(z)=Θ^(z)Y(z).The optimal H(z) is obtained from the Wiener-Hopf equations.
Following the steps of [6], we get(15)H(z)=α2(1−P(z))[(1−P(z−1))Ψθ(z)]K∞,where the notation [X(z)]ij=∑k=ijxkz−k is used. The real number α2 is computed by Szego's formula, and 1−P(z) is the causal monic and minimum phase transfer
function which results from the spectral factorization(16)(1−P(z))α2(1−P(z−1))=Ψy−1(z).


## 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 by(17)Ψλ(z)=(1−z−1)γ2(1−z),with γ2=∑k=0∞(2k+1)2v2k+12. 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
obtain(18)H(z)=[Q(z)+α2γ2z−1]K∞1+Q(z),and the mean square error
(MSE)(19)E{(θi−θ^i)2}=1α2+∑k=1K−1qk2α2−2γ2−N02Eb−(α2γ22+2γ2q1)δK−1,where 1+Q(z)=(1−P(z))−1.


## 5. Case Study

Assume that the phase noise is the sum of random phase
walk plus white phase. The phase noise spectrum is(20)Ψθ(z)=γ−2(1−z−1)(1−z)+γ0.The spectral factorization is
performed by root finding. The result is(21)1−P(z)=1−z−1(1−z1z−1)(1−z2z−1),(22)Q(z)=(1−z1−z2)z−11−z−1(1+z1z21−z1−z2z−1),where the two roots
are(23)zi=xi−xi2−42,i=1,2,with(24)x1,2=2+γ0+N0/2Eb2γ2±(γ0+N0/2Eb2γ2)2−γ−2γ2.
Putting (22) in (18) and using (14), one
gets(25)H(z)=(1−z1−z2+z1z2)z−K(1−z1z−1)(1−z2z−1),(26)G(z)=(1−z1−z2+z1z2)z−K(1−z−1)(1−z1z2z−1+∑k=1K−1(1−z1−z2+z1z2)z−k).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 MSE(27)E{(θi−θ^i)2}=γ2(1z1z2+2z1+2z2−z1z2−4)+γ2z1z2(K−1)(1−z1−z2+z1z2)2−N02Eb.


## 6. Numerical Results

We derive numerical results for OQPSK with
raised-cosine shaping and roll-off0.1 and GMSK with BT=0.3.
The spectrum of phase noise that affects the free-running oscillator used for
the analysis is characterized by γ−2=5⋅10−5 and γ0=10−4.
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 exp−log 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.Figure 2
MSE versus2Eb/N0 for OQPSK with roll-off 0.1.Figure 3
MSE versus2Eb/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.



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*Source: 235761-2008-06-11.xml*