2. Feedforward carrier recovery using DSP
Fig. 1. Schematic of feedforward carrier recovery using DSP
Fig. 2. Processing unit block diagram
It is crucial to note that although the filtering process reduces the effect of shot noise, it will inherently introduce an error on the phase estimation since
φ_{estPU1} is used as a common phase estimate for all the samples in the block. There is an evident tradeoff here: a longer filter (larger
Nb) reduces the shot noise more but the phase estimate is then common on more samples, thus reducing the phase estimate accuracy on each sample. Conversely, a shorter filter (ideally
Nb = 1, i.e., no filtering at all) allows better following of the phase noise in the absence of shot noise, but will perform poorly in the presence of it. An optimal block size can be determined as will be shown later. In order to decode the differentially encoded data, the phase threshold operator T extracts the i
^{th} symbol quadrant:
Ti=⌊(arg{Zi}−φestPU1)(π2)⌋,
where the ⌊
X⌋ operator eliminates fractional part of
X. Note that this detection scheme is not equivalent to the worseperforming differential detection scheme [
5
5
.
W. C.
Lindsey
and
M. K.
Simon
, Telecommunication Systems Engineering (
PrenticeHall Inc.
,
1973
).
]. Synchronous CD is still employed here where the decoding of the data is performed on the basis of comparison of consecutive quadrant numbers, not pairwise comparison of sample phases. The operator C decodes the data such that
d_{i}
=(
T_{i}

T
_{i1})
_{mod4} ;∀
i = 1..
Nb1. To decode the last symbol quadrant in PU
_{1}, the first symbol quadrant from PU
_{2} and its phase estimate (i.e.,
T
_{1PU2},
φ_{estPU2} when decoding
d_{NbPU1}) are used.
d_{Nb}
=(
T
_{1PU2} 
T
_{NbPU1} +
A(
φ
_{estPU2} 
φ
_{estPU1}))
_{mod4} where
A(
x) = 1 if
x >
π/4, 1 if
x < 
π/4 and 0 otherwise. This rather complex operation needed to decode the last symbol in each PU is explained by the fact that the differential encoding employed dictates that only
Nb1 decoded symbols can be extracted from
Nb samples. To decode the last symbol, the quadrant number of the first sample in the next PU is required.
The BER of a differentially encoded QPSK signal with noisy phase reference can be derived in a similar manner as described in [
6
6
.
M. K.
Simon
, “
On the BitError probability of differentially encoded QPSK and offset QPSK in the presence of carrier synchronization
,”
IEEE Transactions on Comm.
54
,
806
–
812
(
2006
).
[CrossRef]
, Eq. (
12)], although the phase estimation error on two consecutive samples (
ε
_{1} ,
ε
_{2}) are not assumed to be identical here:
where γ_{b}
= E_{b}
/N
_{0} is the electrical signal to noise ratio (SNR) per bit, E_{b}
is the energy per bit and N
_{0} is the singlesided power spectral density of the shot noise. P
_{Δφ}{ε} is the probability density function (PDF) of the RV Δφ, the phase estimation error.
Note that the simplification taken in Eq. (
2) is made under the assumption that
γ_{b}
≫1 and a small phase estimation error. These conditions are easily met in the range of parameters (SNR>6dB and beat LW<2MHz) considered in this paper. There is a factor of approximately 2 when Eq. (
2) is compared to the BER expression for the gray coded case (i.e., no differential encoding) [
6
6
.
M. K.
Simon
, “
On the BitError probability of differentially encoded QPSK and offset QPSK in the presence of carrier synchronization
,”
IEEE Transactions on Comm.
54
,
806
–
812
(
2006
).
[CrossRef]
, Eq. (
14)]. This factor originates from differential encoding where any error in a symbol is manifested twice through differential decoding, to the firstorder approximation. From Eq. (
2) it is seen that in order to evaluate
P_{e}
it is necessary to obtain
PΔ
φ, the distribution of Δ
φ.
3. Distribution of the phase estimation error  Δφ
The phase estimation error associated with the scheme presented in
Fig. 2 is defined by:
To see how Δφ is distributed, start by considering the (·)^{4} operation:
For high SNR, all terms containing the shot noise of third order and higher can be neglected because
o(
n
^{3})≪
n
^{2}. Subsequent simulation results and analytical considerations will confirm the validity of this assumption for high SNR values. Substituting Eq. (
4) into Eq. (
3), while making this approximation, yields:
We consider first the phase estimation error in the absence of shot noise. Recall that although shot noise is not considered at first, the filtering operation necessary for optimal phase tracking in the presence of shot noise introduces an error on the phase estimation. We set to investigate this error before introducing the shot noise. The phase estimation error in this case is given by:
Eq. (
6) may be simplified by noting that the laser phase noise is a Wiener process [
7
7
.
J.
Salz
, “
Modulation and detection for Coherent Lightwave Communications
,”
IEEE Comm. Mag.
24
,
38
–
49
(
1986
).
[CrossRef]
] characterized by a zero mean white Gaussian frequency noise
δ~N(0,
σδ2=2
π·2Δ
v/B
_{r}) where 2Δ
v and B
_{r}
are the beat LW of the transmitter and LO laser, and symbol rate, respectively. The frequency noise is independent of data modulation and shot noise. The instantaneous phase
φ_{k} may then be written as
φk=∑q=1kδq, where
δ_{q}
is the carrier frequency noise at time
q =
p/B
_{r}
. The phase difference within a time interval of n/B
_{r}
is then given by:
Hence,
where
B(Nb) can be derived through an example for the case of Nb= 4. B(4)=1 + exp{j·4δ
_{2}}(1 + exp{j·4δ
_{3}}(1 + exp{j·4δ_{4}
4})). Assuming that ∣4∑p=1wδp∣≪1,∀w = 2..4, one may use the fact that 1 + j·x ≅ e
^{j·x} for x≪1 to approximate B(4) ≅ 4+ j·4(3δ
_{2}+2δ
_{3}+δ
_{4}) ≅ 4·exp{j·(3δ
_{2}+2δ
_{3} + δ
_{4})}. In general, the approximated expression for B(Nb) is given by:
both valid assuming ∣4Nb−1∑p=0Nb−2(p+1)δNb−p∣≪1, which can be shown to be valid within the range of SNR of interest. Hence, Δφ_{k}
= φ_{k}
φ_{1}
arg{B(Nb)}/4 ≅ φ_{k}
φ
_{1}θ(Nb). Noting that Δφ_{k}
is a linear combination of independent identically distributed (iid) Gaussian RVs, which may be written conveniently in a matrix notation:
The variance of Δφ is the sum of the variances of the independent, identicallydistributed RVs, so that
We proceed to incorporate the shot noise contribution to the distribution of Δ
φ. Eq. (
5) can be rewritten as
Noting that the phase of a complex Gaussian white noise is uniformly distributed as arg{ρ} ~ U(0,2π) , any other arbitrarily distributed angle can be lumped into the phase of the shot noise without affecting its statistical attributes. Assuming ∣4Nb∑w=1Nbρw∣≪1 and noting that arg{1 + x e
^{j·arg{x}}} ≅ Im{x} for x ≪ 1, this relation may be written as:
As the shot noise and phase noise are independent, the shot noise contribution to the variance of Δφ is additive. To determine this contribution, the distribution of ρ_{w}
is to be established. Let ρ = n·(1 + 1.5·n)·e^{jη}
where η∈(0,2π) represents an angle with an arbitrary PDF, independent of the angle of n (note that ρ_{w}
is a random sample of ρ). E{n} = E{n_{x}
+ j · n_{y}
} = 0 and E{n_{x}
n_{y}
} = 0, from independence of shot noise quadratures. E{ne
^{jη}
(1+1.5·n)} = E{e
^{jη}
}·E{n(1+1.5n)}=1.5E{e
^{jη}
}E{n
^{2}}=1.5E{e
^{jη}
}E{nx2ny2+2jn_{x}n_{y}
}=0, since E{nx2} = E{ny2}. Also, var{ne
^{jη}
(1 + 1.5n)} = 2E{n^{3}}E{cos(arg{n})} + E{n^{2}} +1.5^{2}
E{n^{4}}·n^{2} is a centralchisquare distributed with two degrees of freedom. arg{n} is uniformly distributed: E{cos(arg{n})} = 0. From all the above, var{ρ} = E{n^{2}}+1.5^{2}
E{n^{4}} = σn2(1 + 4.5σn2). Noting that Im{ρ} is one of the quadratures of ρ, its variance is half of that of ρ. Hence, var{Im{ρ}} = var{ρ}/2 = σn2(1+4.5σn2)/2.
Even though the term Im{
ρ_{w}
} contains a second order shot noise term, which becomes nonGaussian distributed, the central limit theorem (CLT) may be applied to obtain an approximation of the distribution of Δ
φ as a Gaussian [
8
8
.
G.
Casella
and
R. L.
Berger
, Statistical Inference, 2
^{
nd
}
Ed. (
Pacific Grove, CA: Thomson Learning
,
2002
).
]:
where
σδ2 is associated with the beat LW (
σδ2=2
π·2Δ
v/B
_{r}). For a QPSK signal in the complex baseband representation with normalized symbol power the complex noise variance may be written in terms of SNR (
γ
_{b}
) as
σn2=1/(2
γ_{b}
). Eq. (
11) may be equivalently represented using either of these parameters. Subsequent plots use
γ_{b}
where the proper substitution into Eq. (
11) is made when necessary.
Special care should be taken when invoking the CLT, since at high SNR levels the block size
Nb which determines the number of summands reduces. Presence of heavy tails might render the CLT approximation invalid beyond first order. However, as the SNR increases, even though
Nb becomes smaller, the significance of the 2
^{nd} order shot noise is diminished and the distribution of Δ
φ approaches Gaussian anyway. To verify the validity of this approach, a series of 5.5·10
^{11} samples following the distribution of RV Δ
φ as defined in Eq. (
10) was generated using several computers. The beat LW, SNR and block size used were 2MHz and 13.5dB and 8, respectively. The PDF of the obtained series (generated PDF) was compared to a Gaussian PDF defined by Eq. (
11), using the same SNR, LW and block size.
Figure 3 presents the two PDFs and the associated BERs as these are accumulated under the integral in Eq. (
2) as a function of the integration variable. As seen in
Fig. 3, the tails of the generated PDF are somewhat wider compared to the Gaussian PDF. However, by observing the respective BER curves, it is seen that this tails’ widening does not significantly affect the final BER; i.e. the difference in BER in both cases is negligible (approximately 5%). Note that the series of generated samples must be long enough to allow for enough events at the tails. It is observed on
Fig. 3 that the series used is indeed long enough since the BER curve for the generated PDF case levels off at roughly Δ
φ = 0.325 where the generated PDF still has enough samples to validate this test. Similar results are obtained for a beat LW of 600KHz, SNR of 13dB and
Nb of 15 (parameters which also achieve an approximate BER of 10
^{9}). When lower SNR values are considered, the second order noise becomes more significant. However, a higher shot noise level requires tighter filtering, thus
Nb is increased. This in turn improves the accuracy of the Gaussian approximation since the number of summed terms is now increased. The above explanation does not imply that the actual phase estimation error variance is better approximated at a lower SNR, but simply justifies the use of a Gaussian approximation of the phase estimation error PDF at the SNR range under consideration.
Fig. 3. Top: Generated and approximated PDFs of Δφ, Bottom: Accumulation of BER integral for the two PDFs
As can be seen from Eq. (
11), the contribution of the phase noise to the variance of Δ
φ increases with increasing
Nb while the impact of shot noise is reduced. Clearly, the expression for the variance of the phase estimation error obtained analytically reflects the tradeoff discussed in section 2. Using Eq. (
11), it is possible to find the optimal
Nb which gives minimal standard deviation (std) of phase estimation error and hence, smallest BER (considering Eq.
2):
Fig. 4. log(std(Δφ)) vs. SNR and block size with optimal Nb superimposed. Left: Beat LW of 600kHz, Right: 2MHz.
4. Comparison with MonteCarlo simulation
Fig. 5. Comparison of var(Δφ) from MC simulation and analytical expression
Fig. 6. MC simulation and approximated analytical BER. Left: Beat LW of 600kHz, Right: 2MHz.
5. Conclusion
In this paper, an estimate of the BER for the QPSK feedforward carrier recovery scheme using DSP suggested in [
3
3
.
D.
LyGagnon
,
S.
Tsukamoto
,
K.
Katoh
, and
K.
Kikuchi
, “
Coherent detection of Optical Quadrature Phaseshift keying signals with carrier phase estimation
,”
J. Lightwave Technol.
24
,
12
–
21
(
2006
).
[CrossRef]
] was obtained analytically through a series of approximations.
The DSP phase estimation scheme was presented in detail. A 4fold phase ambiguity associated with this detection scheme was resolved by using differential encoding. It was also determined that shot noise filtering is needed to reduce the effect of shot noise on the phase tracking performance. However, the filtering process itself introduces an error in phase noise tracking. A tradeoff between these two factors is to be addressed where the variable controlling this tradeoff is the PU block size which determines both the width of the shot noise filter and number of samples which share the same phase estimate. Through a series of approximations it was shown that the phase estimation error can be modeled as a zero mean Gaussian RV. The phase estimation error variance was shown to be associated with the beat LW, electrical SNR and block size. Extensive simulation results show that the phase estimation error approximation to Gaussian is viable.
To optimize the system performance (i.e., balance between shot noise filtering and phase noise tracking) the variance of approximated PDF for the phase estimation error was minimized with respect to the block size, thus obtaining an optimal block size at a given SNR and LW. The values obtained from MC simulations and the analytical expression for the variance of phase estimation error are in excellent agreement.
The analytical approximation allows prediction of the system performance (i.e., BER), for varying parameters. It was observed that the DSP receiver scheme introduces a small power penalty at a BER level of 10^{9} , compared to the ideal (no phase estimation error) case. The need to phaselock the LO to the carrier’s phase is alleviated, dramatically reducing the complexity of CD reception. Using results obtained in this paper, an intuitive understanding of the design tradeoffs is obtained and optimization may be carried out without reverting to timeand resource consuming MonteCarlo simulations.