Constellation recovery and impairment evaluation through minimization of the blind EVM

Steven A. Jacobs; Robert A. Marsland

doi:10.1364/OE.24.021708

1. Introduction

Leading-edge optical transmitters, coupled with coherent optical receivers, utilize polarization multiplexing and complex modulation formats to obtain high bit rates and spectral efficiency. Before these devices can be integrated into an optical communication network, their performance must be carefully tested and characterized. This testing is usually performed in a back-to-back configuration with a short length of fiber, for which chromatic dispersion and PMD are negligible. Even for this simplified configuration, substantial processing is required to recover transmitted symbols from the electric field data provided by the coherent optical receiver [1–6]. Conventionally, this processing is performed in a sequential manner. First, the clock frequency and phase are evaluated so that the field can be obtained at the symbol-center times. Then, polarization demultiplexing is performed to reverse the polarization rotation induced by transmission through the fiber. Next, the carrier phase is removed, leaving the transmitted symbol constellation. Finally, the constellation impairments introduced by the transmitter, which include In-phase (I) and Quadrature (Q) DC offsets, I-Q gain imbalance, and I-Q phase error, are evaluated for each polarization to verify that they are within acceptable limits. While this sequential processing is generally successful, it raises a number of significant questions: Is the resulting constellation the best that could be obtained? If not, then in what sense does the constellation characterize the transmitter performance? Could the presence of impairments affect the quality of the polarization demultiplexing? Could the earlier polarization demultiplexing interfere with the later evaluation of the impairments? Finally, does the presence of multiple impairments interfere with their accurate individual evaluation? The last two questions become particularly important for standards bodies, which must specify exactly how these impairments are to be measured before they can set performance limits. Motivated by these questions, we present here a new method for recovering the constellation and evaluating transmitter impairments: minimization of the blind error-vector magnitude (EVM). In this approach, the distorted symbols provided by the receiver are represented as functions of a set of parameters that characterize the distortions introduced by the transmitter and the fiber. Values of these parameters that minimize the EVM of the recovered constellation are obtained and then used to define the transmitter impairments.

We note that an adaptive linear filter, such as the filter obtained from the Constant Modulus algorithm or its variants [3], is also widely used to recover the transmitted constellation. However, unlike blind EVM minimization, this technique removes the transmitter impairments without directly providing their values. Moreover, it is not clear if the impairments can be accurately extracted from the filter: the filter may convolve multiple impairments or mix impairments between polarizations in a non-reversible manner.

The EVM is widely considered to be one of the best measures of received constellation quality [7,8], and is second only to the Bit Error Ratio (BER) as a measure of overall system performance. Unlike the BER, the EVM can be evaluated even when the true values of the transmitted symbols are unknown. In addition, the BER can be reduced to its minimum possible value (zero) even for constellations that have significant residual impairments. This latter drawback can be overcome by noise-loading the signal, but this adds complexity to the impairment measurement process. For these reasons, we have elected to base our constellation recovery and impairment evaluation method on EVM minimization, rather than BER minimization.

When the actual transmitted symbols are unknown, the “true” value of a symbol is taken to be its closest match in the ideal constellation. We will refer to the EVM evaluated in this manner as the blind EVM, while the EVM evaluated from a known symbol sequence will be referred to as the true EVM (also called the nondata-aided and data-aided EVM [7]). In general, the blind EVM underestimates the true EVM, although the difference becomes negligible when the BER < ~10⁻² [7]. Because the EVM is arguably the best measure of constellation quality, constellation recovery and impairment evaluation by minimizing the blind EVM produces, by definition, the best possible results and therefore the most reliable characterization of transmitter performance. For example, because all the parameters governing constellation recovery and transmitter impairments are evaluated simultaneously, there is no risk of interference or confusion between parameters, which can occur when parameters are evaluated sequentially. These advantages do not come without a cost: blind EVM minimization is significantly slower than conventional recovery methods. For example, processing a set of 4000 QPSK symbols in both polarizations requires about 4 seconds of run-time on a standard desktop PC, and processing a similar set of QAM16 symbols requires about 28 seconds. This may, however, reflect the performance of the particular numerical optimization methods employed, rather than an inherent limitation of the method. In any case, if too slow for widespread application, blind EVM minimization can serve as a standard for defining and measuring transmitter impairments and for judging the performance of faster recovery and impairment evaluation methods. We note that parameter estimation from minimization of the true EVM has been employed successfully for impairment evaluation in wireless transmission [9], and for evaluating parameters characterizing an SOA [10].

For simplicity, in the following we will assume that x-y and I-Q skew at the transmitter is negligible, and that clock recovery has already been performed on the field data prior to blind EVM minimization. In addition, to mitigate the effect of laser phase noise, field data will be processed in blocks with duration well below the coherence time of the signal laser and local oscillator [11]. Finally, we will focus on dual polarization QPSK and QAM16 formats. Extending the method to handle transmitter skew, clock recovery, phase noise estimation, and other modulation formats is straightforward, but involves complications that will be addressed in a subsequent paper [12].

2. Definition of recovery and transmitter parameters and the blind EVM

Let $I_{k}^{(x)} + i Q_{k}^{(x)}$ and $I_{k}^{(y)} + i Q_{k}^{(y)}$ represent the true symbols in the x and y polarizations to be transmitted at the center of the k^th time slot. We assume that the corresponding symbols produced by the transmitter are given by

(\begin{matrix} T_{k}^{(x)} \\ T_{k}^{(y)} \end{matrix}) = (\begin{matrix} S_{I}^{(x)} (I_{k}^{(x)} - μ_{I}^{(x)}) + e^{i θ^{(x)}} S_{Q}^{(x)} (Q_{k}^{(x)} - μ_{Q}^{(x)}) \\ S_{I}^{(y)} (I_{k}^{(y)} - μ_{I}^{(y)}) + e^{i θ^{(y)}} S_{Q}^{(y)} (Q_{k}^{(y)} - μ_{Q}^{(y)}) \end{matrix})

where S_I and S_Q denote the in-phase and quadrature gains, μ_I and μ_Q denote the DC offsets, and θ denotes the I-Q phase angle. The superscript on each of these parameters denotes the polarization to which it applies. We will refer to these parameters collectively as the transmitter parameters. After transmission through the fiber, detection by the coherent optical receiver, and clock recovery, the received symbols are given by

(\begin{matrix} Z_{k}^{(x)} \\ Z_{k}^{(y)} \end{matrix}) = \exp (- 2 π i ν t_{k} - i ϕ) U (\begin{matrix} T_{k}^{(x)} \\ T_{k}^{(y)} \end{matrix})

where ν is the heterodyne frequency,

t_{k}

is the center of the k^th time slot, φ is the optical phase, and

U = (\begin{matrix} e^{i η_{1}} \cos τ & - e^{i η_{2}} \sin τ \\ e^{- i η_{2}} \sin τ & e^{- i η_{1}} \cos τ \end{matrix})

is the unitary matrix representing the polarization rotation induced by the fiber. The parameters τ, η₁, and η₂ in Eq. (3) are the angles that characterize this rotation; in any case, an arbitrary unitary matrix can be parameterized in this form (apart from an overall scalar phase factor). We will refer to the

Z_{k}^{(x, y)}

as the symbol data, and we will refer to {ν, φ, τ, η₁, η₂} as the recovery parameters. Inserting Eqs. (3) and (1) into Eq. (2), we solve for the true symbols in terms of the symbol data and the recovery and transmitter parameters. We obtain

I_{k}^{(x, y)} = μ_{I}^{(x, y)} - (A_{k}^{(x, y)} \cos τ + B_{k}^{(x, y)} \sin τ) / (S_{I}^{(x, y)} \sin θ^{(x, y)})

Q_{k}^{(x, y)} = μ_{Q}^{(x, y)} + (C_{k}^{(x, y)} \cos τ + D_{k}^{(x, y)} \sin τ) / (S_{Q}^{(x, y)} \sin θ^{(x, y)})

where

A_{k}^{(x, y)} = Z_{k r}^{(x, y)} \sin {(k - 1) ψ + ϕ \mp η_{1} - θ^{(x, y)}} + Z_{k i}^{(x, y)} \cos {(k - 1) ψ + ϕ \mp η_{1} - θ^{(x, y)}}

B_{k}^{(x, y)} = \pm Z_{k r}^{(y, x)} \sin {(k - 1) ψ + ϕ \pm η_{2} - θ^{(x, y)}} \pm Z_{k i}^{(y, x)} \cos {(k - 1) ψ + ϕ \pm η_{2} - θ^{(x, y)}}

C_{k}^{(x, y)} = Z_{k r}^{(x, y)} \sin {(k - 1) ψ + ϕ \mp η_{1}} + Z_{k i}^{(x, y)} \cos {(k - 1) ψ + ϕ \mp η_{1}}

D_{k}^{(x, y)} = \pm Z_{k r}^{(y, x)} \sin {(k - 1) ψ + ϕ \pm η_{2}} \pm Z_{k i}^{(y, x)} \cos {(k - 1) ψ + ϕ \pm η_{2}}

The subscripts “r” and “i” on the symbol data denote the real and imaginary parts, respectively. We have assumed t_k = (k – 1)P where P is the symbol period (inverse of the baud rate), and we have introduced the heterodyne frequency parameter ψ = 2πνP.

We assume that the transmitted symbols form a square QAM constellation with M symbols, normalized so that the largest symbol has a squared magnitude of unity. Because the constellation is square, the ideal constellation points will be ± a_j ± ia_m, where a_j, a_m (1 ≤ j, m ≤ ½ M^1/2 ≡ M_R) are real and positive, and any combination of + and – is allowed. The blind EVM is defined to be

E V M_{B} = \sqrt{\frac{1}{2} [{(E V M_{B}^{(x)})}^{2} + {(E V M_{B}^{(y)})}^{2}]}

where

E V M_{B}^{(x, y)} = {\frac{1}{N} \sum_{k = 1}^{N} [\underset{1 \leq j \leq M_{R}}{m i n} {(I_{k}^{(x, y)} \pm a_{j})}^{2} + \min_{1 \leq m \leq M_{R}} {(Q_{k}^{(x, y)} \pm a_{m})}^{2}]}^{1 / 2}

and N is the number of symbols being processed. The minima in Eq. (7) will be attained when the ± signs in the terms in the parentheses are chosen to be the opposite of the sign of

I_{k}^{(x, y)}

or

Q_{k}^{(x, y)}

. We then have

E V M_{B}^{(x, y)} = {\frac{1}{N} \sum_{k = 1}^{N} [\underset{1 \leq j \leq M_{R}}{m i n} {(| I_{k}^{(x, y)} | - a_{j})}^{2} + \min_{1 \leq m \leq M_{R}} {(| Q_{k}^{(x, y)} | - a_{m})}^{2}]}^{1 / 2}

For a given set of symbol data, the blind EVM becomes a function of the recovery and transmitter parameters. We evaluate these parameters by finding the values that minimize the blind EVM, or equivalently, minimize its square. We take the resulting values of $I_{k}^{(x, y)} + i Q_{k}^{(x, y)}$ as the impairment-free recovered symbols in the x, y polarizations.

We note that an alternative definition of the blind EVM, evaluated prior to the removal of the transmitter impairments, is also in widespread use. To prevent confusion with this alternative definition, we will in the following refer to Eqs. (6) or (8) as the blind impairment-free EVM (abbreviated to blind IF-EVM). Most transmission systems employ some form of post-receiver digital signal processing to remove the transmitter impairments. As a result, it is expected that the blind impairment-free EVM will better correlate with the receiver OSNR penalty. Our adoption of the IF-EVM should facilitate comparison of EVM and impairment measurement across different transmission system implementations.

3. Minimization of the blind IF-EVM

The blind IF-EVM is a continuous function of the recovery and transmitter parameters, but due to the presence of the min function in its definition, it will not necessarily have continuous partial derivatives. Hence, the minimization method to be employed cannot be gradient-based. Standard Matlab has a multivariate minimization function, called fminsearch, that is not gradient-based, and so would appear to be perfect for our application. This function performs minimization using a version of the Nelder-Mead simplex method [13]. All it requires is a starting point for the minimization process. However, Eq. (6) has a large number of local minima (mostly, but not exclusively, associated with the heterodyne frequency parameter ψ), and unless the starting point is sufficiently close to the global minimum, fminsearch will converge to a sub-optimal local minimum.

To obtain a good initial value for ψ, we approximately remove the modulation in the x-polarized symbol data (following the Viterbi and Viterbi algorithm [14]) by taking its fourth power. We then compute the FFT of the fourth power of the data, and find the frequency that produces the largest FFT component. We then divide this frequency by 4 to obtain the corresponding heterodyne frequency in the original data. In the left panel of Fig. 1, we show the amplitude of the FFT of the fourth power of the x-polarized symbol data for a typical set of QPSK data (32 GBd), while in the right panel, we show a result for typical QAM16 data (32 GBd). In both cases, the determination of the peak is obvious. We have found that the initial heterodyne frequency obtained in this manner, which we will denote by ν₀, is always within 50 MHz of the frequency that minimizes the blind IF-EVM. This initial heterodyne frequency is obtained using all the symbol data – it is evaluated prior to the division of the data into smaller blocks.

Fig. 1 The FFT amplitude versus frequency of the 4th power of typical symbol data. Left: QPSK. Right: QAM16.

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To obtain good starting values for the other parameters, and to improve the starting value for ψ, we use generalized simulated annealing (GSA), a global minimization procedure that avoids getting trapped in local minima [15]. Following the implementation in [15], we update the recovery and transmitter parameters one at a time using a 1-dimensional visiting distribution. For this purpose, we use the visiting distribution in [15] with parameter q_V = 1.7. We also use the acceptance probability in [15] with parameter q_A = 0. In addition, we impose upper and lower bounds on the parameters: if a new value for a parameter provided by the visiting distribution is outside of these bounds, the visiting distribution is called again until a value within the bounds is obtained. We use upper and lower bounds 2π(ν₀ ± Δν)P for the heterodyne frequency parameter, where Δν = min{4/(NP), 50 MHz} (note that 1/NP is the frequency spacing in the FFT used to obtain ν₀). We use upper and lower bounds ± 90° for the other recovery parameters {φ, τ, η₁, η₂}. The bounds for the transmitter parameters are 60° $\leq θ^{(x, y)} \leq$ 120°, $- 0.1 \leq μ_{I, Q}^{(x, y)} \leq 0.1$ , and $σ^{(x, y)} / 2 \leq S_{I, Q}^{(x, y)} \leq 2 σ^{(x, y)}$ , where

σ^{(x, y)} = \sqrt{\frac{1}{N} \sum_{k = 1}^{N} {| Z_{k}^{(x, y)} |}^{2}} .

These bounds are based on judgments made by the authors on the typical performance variation of a transmitter. If transmitter performance outside of these ranges is expected, the parameter bounds should be adjusted accordingly.

The GSA process is continued until a point in the multi-dimensional parameter space is obtained with $E V M_{B}^{(x, y)} < E_{0}$ , where E₀ is an EVM threshold that ideally should be set just below the level reached by the deepest sub-optimal local minimum of the blind IF-EVM. This choice ensures that the starting point for fminsearch will be in the region of attraction of the global minimum [16]. Because detailed information about the local minima of the blind IF-EVM is generally not available, we employ experimentation to find the proper choice for E₀. If it is set too high, GSA will terminate with a starting point not close enough to the global minimum, and fminsearch will converge to a sub-optimal local minimum. This condition can be easily detected by examining the resulting constellation, which will be grossly distorted. For example, the two left-hand panels of Fig. 2 show the QAM16 constellations resulting from local minima, while the right-hand panel shows the constellation at the global minimum.

Fig. 2 Distorted QAM16 constellations at local minima of the blind IF-EVM (two left-hand panels) and the constellation at the global minimum (right).

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On the other hand, if E₀ is set too low, more time than necessary will be spent in GSA, which is substantially slower than fminsearch. Thus, the largest value of E₀ that consistently produces distortion-free constellations on multiple sets of data after blind IF-EVM minimization will be the best choice. To obtain this desired value, E₀ is first set to a large value (for example, E₀ = 0.4). The resulting constellations obtained from blind IF-EVM minimization for several sets of data are examined. If they are free from the gross distortions illustrated in Fig. 2, then this initial value of E₀ is taken to be the desired value. If distortions are present, E₀ is reduced in steps until distortion-free constellations are consistently obtained. For QPSK data, we have found that a value of E₀ = 0.3 works well, while for QAM16 data, we use E₀ = 0.13. Not surprisingly, the more complex modulation format has deeper local minima, necessitating the reduction in E₀.

To mitigate the effect of laser phase noise, a data set of N symbols is broken up into blocks of size $N_{b} = ⌈ P_{b} / P ⌉$ , where P_b is the block duration, chosen to be well below the coherence time of the system lasers. Results from individual blocks are averaged to provide results for the data set. Following [17], we generally use (70 MHz)⁻¹ ≤ P_b ≤ (30 MHz)⁻¹. GSA is performed only for the first block of symbol data, and the resulting parameters are used as the starting point for the fminsearch function. For subsequent blocks, the optimized parameters from the previous block are used as the starting point, with the exception of ψ and φ. Because of the time-dependent laser phase noise, these parameters can vary from block to block. Hence, a small-scale random search over these two parameters is performed: 500 pairs of values of ψ and φ are generated, randomly and uniformly distributed between their upper and lower bounds and the blind IF-EVM is evaluated for each pair. In this search, we use upper and lower bounds ψ_prev ± 2πP × 10MHz for ψ, where ψ_prev is the optimizing value of ψ from the previous block; the upper and lower bounds for φ remain at ± 90°. The values for these two parameters resulting in the lowest blind IF-EVM in this search are used with the other optimized parameters from the preceding block of data as the starting point for fminsearch. The convergence tolerances for this minimization function can be set by the user. We use a tolerance of 10⁻⁶ for the relative change in the parameter vector between iterations, and a tolerance of 10⁻⁸ for the relative change in the value of the blind IF-EVM. That is, fminsearch will terminate when an iteration results in a relative change in the parameter vector below 10⁻⁶ and a relative change in the EVM below 10⁻⁸. In addition, to assure convergence, fminsearch is called a second time, starting from its claimed minimum.

4. Performance of blind IF-EVM minimization

In this section, we assess the performance of blind IF-EVM minimization for constellation recovery and impairment evaluation. First, in section 4.1, we consider data in the QPSK format. We then consider data in the QAM16 format in section 4.2.

4.1 Data in the QPSK format

To begin our assessment, we use simulated dual-polarization QPSK data, for which the properties are known exactly. We first generate a set of N = 4000 symbols for the x-polarization $I_{k}^{(x)} + i Q_{k}^{(x)}$ , where $I_{k}^{(x)}$ and $Q_{k}^{(x)}$ are random variates taking the values $\pm 1 / \sqrt{2}$ with equal probability. In the same manner, we generate a set of N symbols for the y-polarization. Using the transmitter parameters given in Table 1, we then generate the transmitted symbols using Eq. (1), but add zero-mean complex-valued Gaussian noise:

Table 1. Transmitter parameters used for simulated QPSK data

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(\begin{matrix} T_{k}^{(x)} \\ T_{k}^{(y)} \end{matrix}) = (\begin{matrix} S_{I}^{(x)} (I_{k}^{(x)} - μ_{I}^{(x)}) + e^{i θ^{(x)}} S_{Q}^{(x)} (Q_{k}^{(x)} - μ_{Q}^{(x)}) \\ S_{I}^{(y)} (I_{k}^{(y)} - μ_{I}^{(y)}) + e^{i θ^{(y)}} S_{Q}^{(y)} (Q_{k}^{(y)} - μ_{Q}^{(y)}) \end{matrix}) + σ (\begin{matrix} n_{k}^{(x)} \\ n_{k}^{(y)} \end{matrix}) .

The real and imaginary parts of $n_{k}^{(x, y)}$ are random variates with a standard normal distribution (i.e., mean zero, unit variance). The signal-to-noise ratio (SNR) associated with the value of σ in Table 1 is 23.1 dB.

Using the recovery parameters and symbol period in Table 2, we generate the simulated symbol data from Eqs. (10), (2), and (3). However, instead of using a constant phase, we simulate laser phase noise by replacing the constant φ with a time-dependent Brownian motion process

ϕ_{k} = \sqrt{2 π λ P} \sum_{j = 1}^{k} B_{j} 1 \leq k \leq N

where λ = 100 KHz is the convolved linewidth of the signal laser and the local oscillator and the B_j are standard normal random variates. All random variates in the simulation are generated using appropriate Matlab functions.

Table 2. Recovery parameters and symbol period used for simulated QPSK data

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Blind IF-EVM minimization using the method of section 3 is performed on this simulated data with block duration P_b = (30 MHz)⁻¹. This process of data generation and minimization is repeated 100 times, all using the parameters in Tables 1 and 2, but with different values for the random deviates. The statistics for the run-time and blind IF-EVM over these 100 trials are shown in Table 3.

Table 3. Statistics for the run-time and blind IF-EVM for simulated QPSK data

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The tight distribution for the blind IF-EVM demonstrates that the minimization process was successful for every trial, and required about 4 seconds of run-time.

From the transmitter parameters, we define the I-Q gain imbalances

G^{(x, y)} = \max {S_{I}^{(x, y)} / S_{Q}^{(x, y)}, S_{Q}^{(x, y)} / S_{I}^{(x, y)}},

the DC offset magnitudes

M_{D C}^{(x, y)} = \sqrt{{(μ_{I}^{(x, y)})}^{2} + {(μ_{Q}^{(x, y)})}^{2}},

and the I-Q phase errors

Φ^{(x, y)} = | 90 ° - θ^{(x, y)} |

for each polarization.

While the blind IF-EVM minimization process can identify two data streams with a phase difference of (approximately) 90°, it cannot tell which of them should be assigned to the I transmission and which to the Q. To eliminate this ambiguity, we employ the max function in defining the gain imbalances in Eq. (12). The DC offset magnitude Eq. (13) can be viewed as normalized with respect to the maximum magnitude of the ideal constellation, which by assumption is unity.

In Table 4, we compare the exact values of the transmitter impairments defined above with the values obtained from the transmitter parameters provided by blind IF-EVM minimization for the simulated QPSK data.

Table 4. Transmitter impairments from blind IF-EVM minimization of simulated QPSK data

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With this simulated data, all transmitter impairments are reproduced with high accuracy and small variation, although the gain imbalances and phase errors appear to be slightly underestimated.

With all other parameters fixed at their values in Tables 1 and 2, we performed a sweep over $θ^{(y)}$ from 61° to 119° in steps of 1°. Each trial in this sweep contained 4000 symbols. The resulting values for the I-Q phase error, obtained from blind IF-EVM minimization, are shown in Fig. 3 (left panel), along with the exact values. The right panel of the figure shows the blind IF-EVM for the two polarizations.

Fig. 3 The I-Q phase error (left) and blind IF-EVM (right) versus I-Q phase angle (y-pol) for simulated QPSK data.

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For all values of $θ^{(y)}$ , the method provided an accurate value for the I-Q phase error, although the accuracy degraded slightly as $θ^{(y)}$ deviated further from 90°.

We turn next to measured QPSK data provided by a commercial telecom equipment manufacturer. The lasers used to produce this data were known to have higher phase noise, so a smaller block duration of P_b = (70 MHz)⁻¹ was used during blind IF-EVM minimization. A total of 32 trials were obtained, all at 32.166 GBd. For each trial, approximately 2500 symbols of data were collected. For the first 10 trials, the transmitter was free of intentional impairments. For the next 22 trials, the bias voltages on the transmitter were adjusted to produce an I-Q gain imbalance for the y-polarization of about 20%; the gain imbalance for the x-polarization remained nominally at 1. In Fig. 4, we show the I-Q gain imbalances, while in Fig. 5 (left panel) we show the blind IF-EVM for the x and y polarizations obtained after blind IF-EVM minimization. The right-hand panel of Fig. 5 shows the recovered y-polarized constellation for trial 26.

Fig. 4 Gain imbalance evaluated from blind IF-EVM minimization on measured QPSK data.

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Fig. 5 The blind IF-EVM obtained on measured QPSK data (left). The recovered constellation of the y-polarization on trial 26 (right).

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Blind IF-EVM minimization was successful for all trials. The method provided low-noise estimates of the gain imbalance, and accurately detected the intentional increase in the gain imbalance of the y-polarization that started at trial 11. Because the true value of this impairment was not known, however, the absolute accuracy of the gain imbalance provided by blind IF-EVM minimization could not be determined.

4.2 Data in the QAM16 format

We now turn to dual-pol data in the QAM16 format, starting again with simulation. Using the parameters given in Tables 1 and 2, we generate simulated data from Eqs. (10), (2), and (3) but with $I_{k}^{(x, y)}, Q_{k}^{(x, y)}$ taking on the values $\pm 1 / \sqrt{18}, \pm 3 / \sqrt{18}$ with equal probability. With the QAM16 format, the SNR associated with the value of σ in Table 1 is 20.5 dB. We perform 100 simulated trials at 32 GBd, each trial having 4000 symbols; for each trial, we recover the constellation and evaluate the transmitter impairments using blind IF-EVM minimization. The run-time and EVM statistics over these trials are shown in Table 5, while the transmitter impairment statistics are provided in Table 6.

Table 5. Statistics for the run-time and blind IF-EVM for simulated QAM16 data

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Table 6. Transmitter impairments from blind IF-EVM minimization for simulated QAM16 data

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For this more complicated format, the method continued to produce accurate and tightly distributed values for the transmitter impairments, although the QAM16 run-time increased 7-fold, from about 4 seconds for QPSK to about 28 seconds. The impairment statistics closely followed the corresponding QPSK results; however, the standard deviation of the I-Q phase error increased by about 36%. The slight under-estimation of the gain imbalances and I-Q phase errors persisted.

The increase in run-time in switching from QPSK to QAM16 is associated mostly with the increase in run-time required by GSA, which, as we noted in section 3, must meet a much smaller EVM threshold for QAM16 before termination. However, GSA is performed only on the first block of data, and for subsequent blocks, there is a smaller difference in run-time between the two formats. This is demonstrated in Fig. 6, where we show the average run-time for the two formats as a function of the total number of symbols in a simulated data sample. Because of the increased GSA processing time for the first block, the run-time for QAM16 is shifted upward relative to QPSK, but the slopes of the two curves are comparable.

Fig. 6 Total run-time for processing simulated QPSK and QAM16 data versus number of symbols.

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Next, the sweep over $θ^{(y)}$ was repeated. The results are shown in Fig. 7: the left panel displays the I-Q phase errors, along with the exact values, while the right panel displays the blind IF-EVM. Comparison with Fig. 3 demonstrates that the performance of blind IF-EVM minimization with QAM16 data is virtually identical to its performance with QPSK data.

Fig. 7 The I-Q phase error (left) and blind IF-EVM (right) versus I-Q phase angle (y-pol) for simulated QAM16 data.

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Finally, we consider the performance of blind IF-EVM minimization on measured QAM16 data. The data was generated using a Tektronix OM5110 Multi-Format Optical Transmitter operating at 32 GBd. The transmitter was fed by a Tektronix AWG70001A Arbitrary Waveform Generator. The data content was obtained from a 2¹⁵-1 PRBS. The output from the transmitter was sent to an EDFA for amplification and the optical signal from the amplifier went to a Tektronix OM4106 Optical Modulation Analyzer (OMA). The OMA output was sampled and digitized by a Tektronix DPO 73304D oscilloscope (50 GSa/sec). The raw electric field data was filtered with a root-raised-cosine filter with an optical bandwidth of 32 GHz prior to clock recovery. Twenty trials were obtained, with each trial consisting of approximately 4000 symbols. Blind IF-EVM minimization was performed on the symbol data using a block duration P_b = (30 MHz)⁻¹. To improve the realism of the test, no attempt was made to optimize the transmitter configuration, but no intentional transmitter impairments were introduced.

The blind IF-EVM for the twenty trials is shown in Fig. 8 (left panel), along with the recovered x-polarization constellation of trial 10 (right panel). The I-Q gain imbalance and phase error are shown in Figs. 9(a) and 9(b), respectively. Although the transmitter was free from intentional impairments, we see that the y-polarization had a gain imbalance of about 17% and a phase error of about 3°.

Fig. 8 The blind IF-EVM obtained in 20 trials of measured QAM16 transmission (left). The recovered constellation of the x-polarization in trial 10 (right).

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Fig. 9 The I-Q gain imbalance (a) and I-Q phase error (b) evaluated from blind IF-EVM minimization in twenty trials of measured QAM16 transmission.

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We note that although the x-polarization had almost no impairments and the y-polarization had significant impairments, the blind IF-EVM of the y-polarization was smaller than the EVM of the x-polarization. This result suggests that the blind IF-EVM minimization process correctly evaluated and removed the y-polarization impairments. To further support this suggestion, in Fig. 10(a) we compare the blind IF-EVM of the y-polarization obtained from blind IF-EVM minimization and the blind EVM from the OMA, where the latter EVM was evaluated under the assumption that no transmitter impairments were present (the blind EVM was obtained by disabling data synchronization). The EVM obtained from blind IF-EVM minimization, which included impairment evaluation and removal, was significantly smaller than the result from the OMA. In Fig. 10(b), we compare the blind EVM of the x-polarization, where the transmitter impairments were smaller. In this case, there is a much smaller difference in the EVM.

Fig. 10 Comparison between the blind IF-EVM of the y-polarization (a) and x-polarization (b) obtained from blind IF-EVM minimization and the blind EVM from the OMA. The latter was evaluated under the assumption that no transmitter impairments were present.

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The increase in run-time required by the QAM16 format may be a significant barrier to the use of blind IF-EVM minimization. The GSA is the most time-consuming part of this minimization, which accounts for about 90% of the total run-time with 4000 symbols. Hence, if a faster method could be found to provide a starting point for fminsearch within the region of attraction of the global minimum, the run-time could be sharply reduced. One simple approach would be to reduce the number of symbols used in the GSA portion of the minimization, which is currently performed with the first block of data. Alternatively, a genetic algorithm or one of its variants could be used in place of simulated annealing [18]. Another possibility would be to use an OMA to provide initial estimates of the parameters, which might obviate the need for GSA entirely. Finally, we note that some reduction in run-time could be obtained by implementing the minimization in a lower-level language (Fortran or C + + , for example), rather than Matlab®.

5. Conclusions

We have presented a new method, blind IF-EVM minimization, for constellation recovery and transmitter impairment evaluation of dual-polarization optical signals with complex modulation formats. The minimization is performed using the Matlab® fminsearch function, starting with an initial set of parameters obtained with generalized simulated annealing. Because the blind IF-EVM is widely considered one of the best measures of constellation quality, the new method produces, by definition, the best possible constellation and therefore the most accurate characterization of transmitter performance. Using simulated data, for which the transmitter impairments are known precisely, the method was shown to be accurate and robust, even when multiple transmitter impairments were present. The method was also successfully tested on measured QPSK and QAM16 data. For the former, the method correctly evaluated an intentional I-Q gain imbalance that was produced by modifying the transmitter bias voltages of the y-polarized transmission. For the latter, the method indicated the presence of an unintended I-Q gain imbalance and phase error, producing an EVM significantly smaller than what was obtained with an OMA which did not correct for these impairments. Blind IF-EVM minimization was substantially slower than conventional constellation recovery methods, requiring on average about 4 seconds to process 4000 symbols of QPSK data, and about 28 seconds to process the same amount of QAM16 data. The ability of the method to accurately evaluate multiple transmitter impairments from noisy data suggests that it would be advantageous for use as a standard for defining and measuring transmitter impairments and for judging the performance of faster methods. In this role, the longer run-time of the method would not inhibit its utility.

An extension of the method to handle clock recovery, I-Q and x-y skew at the transmitter, and to perform phase noise estimation and removal, is in preparation [12].

References and links

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	Mean	Std Dev	Min	Max
Run-time^a	3.8 sec	7.7 sec	2.2 sec	80.4 sec
EVM_B (x-pol)	8.7%	0.3%	8.0%	9.7%
EVM_B (y-pol)	9.3%	0.3%	8.6%	10.5%

Impairment	Exact	Mean	Std Dev	Min	Max
G (x-pol)	1.333	1.328	0.002	1.322	1.332
G (y-pol)	1.333	1.327	0.002	1.321	1.334
M_DC (x-pol)	0.040	0.040	7.2e-04	0.037	0.041
M_DC (y-pol)	0.030	0.030	0.001	0.028	0.034
Φ (x-pol)	18.000°	17.697°	0.096°	17.447°	17.939°
Φ (y-pol)	27.000°	26.530°	0.103°	26.255°	26.777°

	Mean	Std Dev	Min	Max
Run-time^a	28.4 sec	13.6 sec	10.8 sec	118.5 sec
EVM_B (x-pol)	8.3%	2.1%	7.9%	9.2%
EVM_B (y-pol)	8.8%	2.2%	8.4%	9.6%

Impairment	Exact	Mean	Std Dev	Min	Max
G (x-pol)	1.333	1.324	0.003	1.318	1.333
G (y-pol)	1.333	1.324	0.003	1.318	1.333
M_DC (x-pol)	0.040	0.040	7.0e-04	0.038	0.042
M_DC (y-pol)	0.030	0.030	9.2e-04	0.028	0.032
Φ (x-pol)	18.000°	17.527°	0.136°	17.116°	17.865°
Φ (y-pol)	27.000°	26.318°	0.136°	25.888°	26.638°

	Mean	Std Dev	Min	Max
Run-time^a	3.8 sec	7.7 sec	2.2 sec	80.4 sec
EVM_B (x-pol)	8.7%	0.3%	8.0%	9.7%
EVM_B (y-pol)	9.3%	0.3%	8.6%	10.5%

Constellation recovery and impairment evaluation through minimization of the blind EVM

Abstract

1. Introduction

2. Definition of recovery and transmitter parameters and the blind EVM

3. Minimization of the blind IF-EVM

4. Performance of blind IF-EVM minimization

4.1 Data in the QPSK format

4.2 Data in the QAM16 format

5. Conclusions

References and links

Cited By

Figures (10)

Tables (6)

Equations (18)

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