Digital coherent receiver based transmitter penalty characterization

David J. Geisler; John E. Kaufmann

doi:10.1364/OE.24.029734

1. Introduction

For free-space inter-satellite or space-to-ground optical communications links, where receivers are signal-power-starved, it is important to design transmitters and receivers that can operate as close as practically possible to theoretical limits [1]. Characterizing the sensitivity of optical communications systems enables determining an end-to-end system penalty with respect to theoretical limits. Bit-error rate (BER) measurements are a common means for assessing system penalty. However, it is difficult to determine the division of this penalty between the transmitter and receiver. A quantitative optical transmitter signal waveform metric that isolates which portion of the end-to-end system penalty can be assigned to the transmitter helps to quantify how close a transmitter is to the theoretical ideal. A method for accurately assessing the transmitter penalty also allows the transmitter designer to identify waveform defects that could impair link performance, helps ensure interoperability with receivers, and can accurately predict BER performance with a particular receiver implementation.

Methods commonly used to evaluate transmitter waveforms include eye masks [2], Q-factor [3], or constellation diagrams [4]. Compliance to eye mask standards is a semi-qualitative method that can be useful for identifying transmitter waveform defects but is not necessarily useful for accurately predicting end-to-end BER performance. The Q-factor is commonly used to predict BER performance under the assumption that the receiver’s noise statistics are Gaussian. However, optical receivers that use optical pre-amplification have noise statistics that are not Gaussian, particularly in the low signal-to-noise ratio (SNR) regime. Likewise, constellation diagrams provide a semi-qualitative visual assessment of transmitter modulation imperfections, but do not provide a direct evaluation of a BER performance penalty.

This work introduces a different approach that is based on a rigorous, computationally based methodology. The result is a single numerical metric that quantifies how an imperfect transmitter waveform impacts BER performance in a “golden” (i.e., ideal) receiver compared to a perfect waveform. That number, when expressed in dB units, can be entered in a link budget as an effective power penalty for that transmitter. If desired, the transmitter penalty can be subdivided further through analysis of the individual transmitter components.

The basic functions of the measurement system using a “golden” receiver are depicted in Fig. 1. The “golden” receiver consists of hardware and software including a coherent receiver front end, analog-to-digital converters, a “golden” baseband demodulator, and computation of waveform metrics. This approach uses a hardware coherent receiver front end to perform a linear frequency down-conversion of the optical signal field from the transmitter under test (TUT) to the electrical domain, preserving both optical amplitude and phase information. The optical signals are assumed to be sufficiently strong so as to be considered noise-free. The transmitter waveform data from the coherent receiver is digitized and processed off-line.

Fig. 1 “Golden” receiver based waveform characterization block diagram

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Post-processing involves passing the transmitter waveform data through a software-defined “golden” baseband demodulator that can, in principle, replicate perfectly the demodulation functions of a theoretically optimal receiver. Free-space links do not exhibit temporal dispersion or nonlinearities that are encountered in fiber. Therefore the optimal demodulator employs a matched filter. If desired, it is possible to incorporate nonideal receiver characteristics, such as nonideal matched filtering, in software to assess transmitter waveform penalty against a specific receiver implementation.

A major advantage of implementing a software-based receiver, compared to a real optical hardware receiver to perform the same tasks, is that the imperfections of a hardware receiver are completely avoided. With full amplitude and phase information available, the optimal demodulator can be defined in software to demodulate any waveform, whether it contains amplitude, frequency, or phase modulation. Demodulation statistics are accumulated and used to compute analytically the BER of the golden receiver against the actual waveform output of the TUT. The analysis in this paper can be extended to include polarization multiplexed modulation formats.

Section 2 motivates the general approach to transmitter waveform characterization based on a matched-filter detection of the signal. Section 3 provides an overview of the waveform characterization technique applied to a particular optical return-to-zero differential phase-shift keying (RZ-DPSK) transmitter [5]. Section 4 presents laboratory results on the characterization of the transmitter described in Section 3. Section 5 concludes the paper.

2. Generalized waveform analysis methodology

The baseband demodulator in Fig. 1 generates the signal data that is used to compute the waveform quality metrics. The inputs to the demodulator are digitized in-phase (I) and quadrature-phase (Q) signal samples from the A/D device. The outputs of the demodulator are a set of decision variables that are generated once per symbol interval. The values of these variables are used to perform an analytic computation of receiver BER. Deviations of the computed BER from the theoretical optimum can be translated directly into a transmitter waveform penalty.

2.1 Optimal baseband demodulation

This section motivates the use of an optimal baseband demodulator architecture using matched-filter signal detection. The analysis in this section continues to carry the continuous time representation of the signals here, even though the actual processing occurs on digitized samples of the signals. The paper uses the results of [6], including the notation. Assume the transmitter sends one of M equiprobable optical signals, _{$s_{m} (t)$}, in each signaling interval where _{$m = 1, 2, ..., M$}. The signal waveform presented to the coherent receiver in Fig. 1 can be represented as a complex waveform

s_{m} (t) = u_{m} (t) e^{j 2 π f_{c} t}

where _{$u_{m} (t)$} is the complex low-pass equivalent signal waveform that contains the modulation, and _{$f_{c}$} is the optical carrier frequency. Here, _{$u_{m} (t)$} is digitally constructed from the I and Q outputs of the coherent receiver in Eq. (1).

The optimal receiver implements a demodulator that maximizes the a posteriori probability of choosing the correct signal that was transmitted in each signaling interval in the presence of noise or other perturbations. It is assumed the signal is corrupted by additive white Gaussian noise (AWGN), which is used to model the statistics of optical domain amplified spontaneous emission (ASE) of an optical preamplifier. This also serves to model the noise statistics in the electrical domain output of a coherent optical receiver. Thus the results given here allow characterization of the TUT against a far-end receiver that employs either type of front-end architecture.

Mixing the signal down to baseband with the receiver local oscillator (LO) in the measurement setup of Fig. 1 set to frequency _{$f_{c}$}, yields the following low-pass equivalent signal, _{$r (t)$}, at the input to the baseband demodulator:

r (t) = e^{- j ϕ} u_{m}^{} (t)

where _$ϕ$ is a fixed but unknown phase term. This leads to two possible options for assessing the waveform quality of the TUT: 1) processing with a phase-coherent demodulator that estimates _$ϕ$, which is, in general, the globally optimal approach, or 2) processing with a phase-incoherent demodulator that assumes no knowledge of _$ϕ$.

The phase-coherent approach scores the TUT against an ideal phase-coherent receiver that recovers phase. Such an approach would be used for phase modulations such as BPSK or QPSK. Thus the software baseband demodulator would implement a phase-recovery algorithm as part of the baseband processing. In practice, a real receiver could implement a phase-locked loop for this purpose.

If the phase, _$ϕ$, can be treated as a known parameter, it is well known that the optimal baseband demodulator implements a bank of M matched filters [6], each with impulse response, _{$h_{m} (t)$}, given by

h_{m} (t) = u_{m}^{*} (T - t)

where _{$m = 1, 2, ..., M$} and _{$0 \leq t \leq T$} with T being the duration of one symbol. The outputs of the baseband filters are, in general, complex variables. The receiver’s decision logic then picks the matched filter output with the largest real part. If the M signals have unequal energies, then bias terms that account for the unequal energies are subtracted first from the filter outputs before picking the maximum output [7].

A performance benchmark is established by assuming the filters are matched to a perfect transmitter waveform. Thus a perfect transmitter will yield a zero waveform penalty when scored against a matched demodulator. Any non-perfect waveform will show degraded performance.

The phase-incoherent approach treats _$ϕ$ as unknown parameter that is not recovered or estimated. Additional processing must be carried out to remove the _$ϕ$ dependence. Section 3 will describe the details of differentially coherent detection of RZ-DPSK that does not require knowledge of _$ϕ$.

2.2 Calculation of error probability with a ‘golden’ receiver

The TUT is modulated with a random data sequence and a time history of waveform data is obtained from the A/D outputs. The time history must be sufficiently long to capture all possible realizations of the transmitter waveform from the modulating data sequence. In the phase-coherent demodulator the baseband waveform data from the TUT is passed through the bank of M software-defined matched filters. If, for some reason, the TUT is to be scored against a suboptimal receiver (e.g., using a non-matched filter), the software approach provides the flexibility to implement alternative demodulator realizations. The filter outputs contain all the essential information about the transmitter waveform that allows an analytical calculation of BER with a matched-filter demodulator when AWGN is present.

The flexibility of a software-defined golden receiver enables the inclusion of algorithms for transmission media, such as optical fiber, that introduce signal impairments [7]. It is also possible to match any predistortion applied in the TUT with DSP in the golden receiver. In this case, the predistorted golden receiver would be compared against a predistorted TUT to assess a transmitter waveform penalty. In this paper, a TUT is assessed against a golden receiver assuming transmission through free space without any predistortion.

Let the signaling intervals in the signal time series be indexed by k, with K being the total number of intervals. If waveform _{$u_{i} (t)$} is sent by the TUT in the kth signaling interval, _{$k T < t < (k + 1) T$}, a set of M decision variables are obtained by sampling the matched-filter outputs _{${{\tilde{U}}_{m, k}^{} |}_{m = 1}^{M}$} at _{$t_{k} = (k + 1) T$},

{\tilde{U}}_{m, k} (t_{k}) = u_{i} \otimes h_{m} |_{t = t_{k}}

where _$\otimes$ denotes convolution with the matched-filter as characterized in Eq. (3) and _{$t_{k}$} is the time at which the matched filter output is sampled.

To calculate BER, noise statistics are introduced at this point. Adding a noise term, _{$n_{m, k}$}, to each filter output, _{${\tilde{U}}_{m, k}$}, yields

U_{m, k} = {\tilde{U}}_{m, k} + n_{m, k} .

When using phase-coherent demodulation, the processing of the noisy signals is linear and thus the noise terms, _{$n_{m, k}$}, are jointly Gaussian random variables that are completely characterized by a covariance matrix. If the demodulation is phase-incoherent, the processing generally involves a nonlinearity that produces non-Gaussian statistics that can, in principle, be calculated.

Knowledge of the noise statistics in either the coherent or incoherent case allow computing a conditional bit-error probability, _{$P_{e, k}$}, in the kth signaling interval, conditioned on the noise-free matched filter outputs values _{${{\tilde{U}}_{m, k}^{} |}_{m = 1}^{M}$} in Eq. (4) and a priori knowledge of which signal waveform was actually transmitted by the TUT. It is assumed that a known data pattern is used to modulate the TUT and thus the sequence of transmitted symbols is known a priori. If waveform _{$u_{i} (t)$} was sent in this interval, the error probability is simply

P_{e, k} = 1 - \Pr (U_{i, k} > U_{j, k} for all j \neq i)

A set of K conditional error probabilities _{${P_{e, k} |}_{k = 1}^{K}$} is calculated via Eq. (6) for each of the K signaling intervals. The overall probability, _{$P_{e}$}, of receiver error over the duration of the entire waveform time history is then given by

P_{e} = \frac{1}{K} \sum_{k = 1}^{K} P_{e, k}

At any given error rate, the offset between _{$P_{e}$} as computed in Eq. (7) and the BER of the theoretically optimal demodulator with a perfect transmitter waveform translates into a waveform penalty for the TUT. In general, the penalty will be a function of the error rate. Application of this procedure to an RZ-DPSK transmitter is discussed next in Section 3.

3. Waveform characterization for an RZ-DPSK transmitter

This section describes the details of how a particular RZ-DPSK optical transmitter’s waveform has been characterized in the laboratory. Additionally, this narrative provides insight into how the general procedure described in Section 2 can be translated into practice. The RZ-DPSK transmitter has a unique multi-rate capability that has been described previously [5]. The utility of the TUT waveform characterization results will be demonstrated in Section 4 by examining them in the context of end-to-end link operation with a real companion RZ-DPSK receiver.

The transmitter uses 50% return-to-zero (RZ50) pulse shaping where the ideal RZ50 signal field envelope, _{$a (t)$}, is given by

a (t) = \cos [\frac{π}{4} \cos (\frac{2 π t}{T}) + \frac{π}{4}]

where _{$0 \leq t \leq T$} and T is the period of one RZ pulse. Next, a matched-filter demodulator based on this description of the RZ50 pulse is defined.

3.1 Calculation of error probability with a ‘golden’ receiver

The differential encoding employed in DPSK spans a period of two bits, or 2T. Thus a DPSK waveform can be treated as a signal of duration 2T for purposes of specifying matched filters. There are two possible waveforms sent during a 2T interval:

1. A digital ‘0’ is transmitted as two successive RZ pulses with a 0 radian phase shift between the two pulses.
2. A digital ‘1’ is transmitted as two successive RZ pulses with a _$π$ radian phase shift between the two pulses.

If we define an impulse response _{$h (t)$} as

h (t) = a (T - t)

a bank of two baseband matched filters, _{$h_{+} (t)$} and _{$h_{-} (t)$}, can be created for an ideal RZ-DPSK waveform with

h_{+} (t) = h (t - T) + h (t)

matched to a digital ‘0’ and

h_{-} (t) = h (t - T) - h (t)

matched to a digital ‘1’.

The two-bit waveforms corresponding to a digital ‘1’ and digital ‘0’, as described above, are orthogonal. Therefore, at the end of each bit interval, a perfect RZ-DPSK waveform will produce a non-zero output from only one of the two matched filters described by Eqs. (10) and (11). The other filter output will be zero.

Under the assumption of phase-incoherent processing and a noisy received signal, _{$r (t)$}, the optimal demodulator picks the symbol for whichever matched-filter magnitude-squared output, _{${| r (t) \otimes h_{+} (t) |}^{2}$} or _{${| r (t) \otimes h_{-} (t) |}^{2}$}, is larger at each sampling time, t_k.

The commonly used optical delay line interferometer (DLI) demodulator for DPSK, shown in Fig. 2, can be rigorously justified as being exactly equivalent to the optimal phase-incoherent demodulator with the two matched filters described by Eqs. (10) and (11) when the baseband pre-filter has the matched impulse response defined in Eq. (9). The delay, _$τ$, is equal to one bit period, T. The DLI demodulator architecture is assumed for the remainder of this paper.

Fig. 2 Optical DPSK delay-line demodulator. MF: matched filter.

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It is straightforward to show, using the results of [8], that with ideal optical pre-amplification, the bit-error rate, _{$P_{e}^{*}$}, of this ideal demodulator with a perfect RZ-DPSK waveform input is the well-known result given by

P_{e}^{*} = \frac{1}{2} e^{- \frac{E_{b}}{h ν}}

where _{$E_{b}$} is the average energy per bit, _$h$ is Planck’s constant, and _$ν$ is the frequency of the optical carrier. This result assumes the optical receiver includes a polarization controller and polarizer at the output of the optical preamplifier that passes only the polarization state of the received signal _{$r (t)$}. The BER for a two-polarization receiver can be found in [8], but is not used in this analysis.

Figure 3 shows a block diagram of the optimal baseband demodulator. The signal input to the baseband matched-filter is represented by the I and Q signals in complex form as

r (t) = I (t) + j Q (t)

and the complex signal output from the matched filter is

\tilde{r} (t) = h (t) \otimes r (t) .

The DLI has a delay matched to the RZ pulse period, T. The two outputs, _{${\tilde{r}}_{+} (t)$} and _{${\tilde{r}}_{-} (t)$}, of the DLI are then given by

{\begin{cases} {\tilde{r}}_{+} (t) = [\tilde{r} (t - T) + \tilde{r} (t)] \\ {\tilde{r}}_{-} (t) = [\tilde{r} (t - T) - \tilde{r} (t)] \end{cases} .

The photodetection process of the optical receiver can be modeled as the square-law operation depicted in Fig. 3,

{\begin{cases} {| {\tilde{r}}_{+} (t) |}^{2} = {[Re ({\tilde{r}}_{+} (t))]}^{2} + {[Im ({\tilde{r}}_{+} (t))]}^{2} \\ {| {\tilde{r}}_{-} (t) |}^{2} = {[Re ({\tilde{r}}_{-} (t))]}^{2} + {[Im ({\tilde{r}}_{-} (t))]}^{2} \end{cases},

where _{$Re (\cdot)$} and _{$Im (\cdot)$} denote the real and imaginary parts of the arguments within the parentheses.

Fig. 3 Software-based demodulation functions for RZ-DPSK

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The sampler in Fig. 3 samples the square-law outputs, _{${| {\tilde{r}}_{+} (t) |}^{2}$} and _{${| {\tilde{r}}_{-} (t) |}^{2}$}, at the end of every signaling interval, t_k, to generate the decision statistics, _{${\tilde{U}}_{+}$} and _{${\tilde{U}}_{-}$}. These two decision statistics are then used to compute a bit-error rate for the TUT and the “golden” RZ-DPSK demodulator.

3.2 Calculation of error probability with a ‘golden’ receiver

This section provides a rigorous derivation for calculating BER for an imperfect transmitter waveform and assessing waveform penalty. Waveform imperfections of the TUT can include signal envelope deviation from the ideal RZ50 pulse shape, temporal variations or drifts in phase (including laser phase noise) from bit to bit, modulation phase change between successive bits being not exactly π radians when a digital ‘1’ is transmitted, data patterning effects in either signal amplitude or phase, or nonlinear distortions induced in a high-power transmit amplifier.

Waveform imperfections create a mismatch between the signal and matched filter that can steer signal energy out of the “correct” DLI port and into the “wrong” port. Furthermore, with inter-symbol interference (ISI) the noise statistics at the two DLI output ports differ from those encountered with a perfect waveform. One outcome of this analysis is an exact BER expression that accounts for the effects of ISI and the exact non-Gaussian detection statistics with an optical preamplifier front end.

Suppose a ‘1’ is transmitted by the TUT and the decision statistics in the absence of noise have the following values:

{\begin{cases} {\tilde{U}}_{+} = a_{+}^{2} \\ {\tilde{U}}_{-} = a_{-}^{2} \end{cases} .

The BER computation then adds noise random variables with the appropriate statistics to _{${\tilde{U}}_{+}$} and _{${\tilde{U}}_{-}$} under the assumption that the noise arises from the ASE of an optical preamplifier. In the optical domain, the ASE at the output of the preamplifier can be modeled as a Gaussian random process with a one-sided spectral density, _{$N_{0}^{*}$}, per polarization mode of

N_{0}^{*} = n_{s p} h υ (G - 1),

where _{$n_{s p}$} is the spontaneous emission factor of the amplifier and _$G$ is its optical gain [9].

The square-law nonlinearity of the photodetection process generates two decision variables that are noncentral chi-squared random variables with two degrees of freedom and noncentrality parameters _{$λ_{+}$} and _{$λ_{-}$}, respectively,

{\begin{cases} λ_{+} = \frac{a_{+}^{2}}{2 σ^{2}} \\ λ_{-} = \frac{a_{-}^{2}}{2 σ^{2}} \end{cases}

where _$σ^{2}$ is the variance of the I and Q components of the down-converted ASE at the output of the baseband matched filter. If the spectral density of the I and Q noise components of _{$r (t)$} at the input to the baseband matched filter is white with one-sided spectral density, _{${\tilde{N}}_{0}$}, and _{$H (f)$} is the transfer function of the baseband matched filter, the variance, _$σ^{2}$, is given by

σ^{2} = {\tilde{N}}_{0}^{} \int_{0}^{\infty} {| H (f) |}^{2} d f .

When noise is added to the matched filter outputs per Eq. (5), the two DLI outputs will contain statistically independent noises when the outputs are sampled at the optimal time (at the end of each bit interval t = t_k). This is because the additive noise (representing EDFA ASE) entering the baseband matched filter in Fig. 3 is accurately modeled as white noise, and white noise processes in two consecutive and disjoint bit intervals are independent. The baseband matched filter that is defined by Eqs. (8) and (9) has an impulse response that is exactly one bit time (T seconds) long. Therefore, the matched filter outputs at sampling time t = t_k contain no noise contributions from the previous bit intervals ending at t = t_k-1, t = t_k-2, etc.

An error is made if _{$U_{+} > U_{-}$} when a ‘1’ is sent. The probability of that occurrence conditioned on the values _{$λ_{+}$} and _{$λ_{-}$} in Eq. (19) is denoted as _{$P_{e 1} (U_{+} > U_{-} | λ_{+}, λ_{-})$} and because of the independence of the noise components in U₊ and U_-, can be calculated using results found in [10],

P_{e 1} (U_{+} > U_{-} | λ_{+}, λ_{-}) = Q_{1} (\sqrt{\frac{λ_{+}}{2}}, \sqrt{\frac{λ_{-}}{2}}) - \frac{1}{2} \exp [- \frac{λ_{+} + λ_{-}}{4}] I_{0} (\frac{\sqrt{λ_{+} λ_{-}}}{4})

Here, _{$Q_{1} (\cdot)$} is the Marcum Q-function and _{$I_{0} (\cdot)$} is the modified Bessel function of order zero.

In the case where the signal waveform is perfect and a ‘1’ is sent, _{$λ_{+} = 0$} and _{$P_{e 1} (U_{+} > U_{-} | λ_{+}, λ_{-})$} reduces to

P_{e 1} (U_{+} > U_{-} | λ_{+}, λ_{-}) = \frac{1}{2} e^{- \frac{λ_{-}}{4}} .

It is desirable to express the BER as a function of the signal energy per bit, _{$E_{b}$}, which makes the results independent of data rate and allows the calculated BER to be compared to theoretical limits. In order to do so, the average signal waveform power, _{$P_{a v g}$}, is calculated first from the complex signal waveform input, _{$r (t)$}, to the baseband demodulator as follows:

P_{a v g} = \frac{1}{T_{d}} \int {| r (t) |}^{2} d t .

Here, _{$T_{d}$} is the duration of the waveform history used for the computation. _{$P_{a v g}$} does not need to be calibrated to absolute power units because arbitrary scaling constants of signal power will factor out in the BER calculation. The average signal energy per bit is then _{$E_{b} = P_{a v g} T$}. There may be variations in instantaneous power with respect to _{$P_{a v g}$}from bit to bit due to patterning effects, but these variations contribute to the waveform penalty that is calculated here.

Next, the decision variables _{${\tilde{U}}_{+}$} and _{${\tilde{U}}_{-}$} are normalized by _{$E_{b}$} such that they can be expressed as

{\begin{cases} {\tilde{U}}_{+} = k_{+} E_{b} \\ {\tilde{U}}_{-} = k_{-} E_{b} \end{cases}

where _{$k_{+}$} and _{$k_{-}$} are scaling factors.

For a perfect RZ50 DPSK waveform, it is straightforward to show that when a ‘1’ is sent,

λ_{-} = \frac{4 E_{b}}{h ν}

and consequently Eq. (22) reduces to the ideal DPSK result in Eq. (12).

If a ‘0’ is transmitted, an error is made if _{$U_{-} > U_{+}$}. A similar analysis yields the error probability _{$P_{e 0} (U_{-} > U_{+} | λ_{+}, λ_{-})$} conditioned on the values _{$λ_{+}$} and _{$λ_{-}$}:

P_{e 0} (U_{-} > U_{+} | λ_{+}, λ_{-}) = Q_{1} (\sqrt{\frac{λ_{-}}{2}}, \sqrt{\frac{λ_{+}}{2}}) - \frac{1}{2} \exp [- \frac{λ_{+} + λ_{-}}{4}] I_{0} (\frac{\sqrt{λ_{-} λ_{+}}}{4}) .

When the transmitter waveform is perfect, _{$P_{e 0}$}, as given by Eq. (26), also reduces to the ideal result in Eq. (12).

Since the probability of making an error on a particular bit depends on knowing the actual pair of _{${\tilde{U}}_{+}$} and _{${\tilde{U}}_{-}$} values that are obtained for that bit, the waveform penalty must be evaluated by computing _{$P_{e 1}$} or _{$P_{e 0}$} on a bit-by-bit basis.

3.3 Summary of waveform penalty calculation

The overall BER is obtained by averaging the series of computed bit-error rates at each bit over the entire time history of the signal. The following steps provide a summary of the procedure.

1. Determine the optimal sampling time for obtaining the decision statistics _{${\tilde{U}}_{+}$} and _{${\tilde{U}}_{-}$}. The optimal sampling time typically occurs at the time of maximum signal eye opening. An eye can be constructed from the difference signal _{${| {\tilde{r}}_{+} |}^{2} - {| {\tilde{r}}_{-} |}^{2}$}.
2. Calculate the noise variance, _$σ^{2}$, using Eq. (20).
3. Let _{${\tilde{U}}_{+, k}$} and _{${\tilde{U}}_{-, k}$} be the sampled DLI outputs obtained at the kth bit and compute the corresponding noncentrality parameters _{$λ_{+, k}$} and _{$λ_{-, k}$}, respectively, assuming an average signal power, _{$P_{a v g}$}, as computed in Eq. (23) with the decision statistics scaled as in Eq. (24).
4. If the kth transmitted bit is a ‘1’, compute the conditional bit error probability via Eq. (21) and set $P_{e, k} = P_{e 1} (U_{+, k} > U_{-, k} | λ_{+, k}, λ_{-, k})$
If the kth transmitted bit is a ‘0’, compute the conditional bit error probability via Eq. (26) and set
$P_{e, k} = P_{e 0} (U_{-, k} > U_{+, k} | λ_{+, k}, λ_{-, k})$
Perform these calculations for every bit in the time history and compile a series of K bit-error rate values _{${P_{e, k}}_{k = 1}^{K}$}.
5. Compute the overall bit-error rate via the averaging operation of Eq. (7).
6. Assess the transmitter waveform penalty as the offset in _{$P_{e}$}, as computed in Eq. (7), from the theoretical result _{$P_{e}^{*}$} of Eq. (12). In general, the offset will vary with the specific error rate of interest.

It should be pointed out that the software-based approach used here also allows waveform diagnostics to be implemented at intermediate steps in the processing chain of Fig. 3. These diagnostics may help in identifying specific waveform imperfections for a particular TUT and, in general, are easier to carry out in software than they would be in a hardware receiver. For example, bit patterning effects and non-ideal modulator extinction can be identified by recreating in software the time history of signal waveforms at intermediate points in the processing chain of Fig. 3. Clock jitter can be identified from eye diagrams generated from the difference signal _{${| {\tilde{r}}_{+} |}^{2} - {| {\tilde{r}}_{-} |}^{2}$}. ISI can be observed by comparing the relative magnitudes of the _{${\tilde{U}}_{+, k}$} and _{${\tilde{U}}_{-, k}$} decision statistics from bit to bit.

4. Coherent waveform characterization of DPSK transmitter experimental results

This section applies the coherent waveform characterization methodology introduced in Section 3 to a DPSK transmitter. The coherent waveform characterization methodology is demonstrated by correlating the calculated transmitter waveform penalty against the end-to-end BER performance of the DSPK transmitter with a hardware test bed receiver.

4.1 Experimental arrangement

Figure 4 shows the experimental arrangement. In particular, Fig. 4(a) shows the DPSK TUT, which uses a single Mach-Zehnder modulator (MZM) to generate burst-mode DPSK waveforms [5]. This is the same architecture proposed for the NASA Laser Communications Relay Demonstration [11]. The burst-mode transmitter supports duty cycles ranging from full-rate to 1/40-rate that yield channel data rates ranging from 2.88 Gb/s to 72 Mb/s. For optical transmission systems with extremely low duty cycles, it is also advantageous to determine a penalty due to a finite transmitter extinction ratio [1, 12].

Fig. 4 (a) Transmitter under test, (b) test bed receiver, and (c) waveform characterization experimental arrangement. MZM: Mach-Zehnder modulator EDFA: erbium-doped fiber amplifier. PC: polarization controller. LO: local oscillator. DSP: digital signal processing. Atten.: attenuator. FBG: fiber Bragg grating. DLI: delay-line interferometer.

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The laser in the TUT is a 1.55-μm distributed feedback (DFB) laser with a <2 MHz line width. The TUT uses the transmitter electronics and a single MZM to determine the pulse shape, the phase modulation, and the duty cycle for the burst-mode waveforms [13]. Here, a clock rate of 2.88 GHz is used to generate RZ50 pulses. In the transmitter electronics, it is possible to vary the duty cycle from 1 to 1/40 for a 16-dB dynamic range of channel rates. The modulated optical waveform then proceeds to a 0.5-W erbium-doped fiber amplifier (EDFA). The output data sequence consisted of repeated 65,824 bit frames with a 1024-bit header and 64,800-bit payload generated from a linear feedback shift register with a pattern length of 2¹⁶-1. Each frame was divided into bursts consisting of 176 bits.

Figure 4(b) shows the test bed receiver used to perform end-to-end BER measurements. After attenuation, the signal proceeds to two amplification stages that each consist of an EDFA with a 3.5-dB noise figure, and a 5-GHz fiber Bragg grating (FBG) filter. The cascade of two 5-GHz FBGs with Gaussian passbands yields a close approximation to a matched filter [1]. Next, the signal proceeds to a 2.88-GHz delay-line interferometer in which the phase modulation is converted to intensity modulation. After balanced detection, the receiver electronics perform clock recovery and produce a real-time BER.

Figure 4(c) shows the experimental arrangement used for waveform characterization. The deliberate choice to use separate signal and LO lasers enables the waveform penalty measurement to account for transmitter laser phase noise as part of the penalty. The signal and LO are first matched in polarization and then mixed together in a 90° optical hybrid. It is also necessary to adjust the LO laser frequency to within several GHz of the signal laser to ensure that there is sufficient detection bandwidth to receive the entire signal, which is centered about the beat frequency between the two lasers. After balanced detection and rf amplification, the in-phase (I) and quadrature-phase (Q) components of the optical waveform are acquired using a 20-GHz 40-GSample/s real-time oscilloscope. Next, offline digital signal processing (DSP) determines the transmitter waveform penalty.

The DSP chain, shown in Fig. 5, begins by performing front-end compensation on the digitized I and Q signals. This involves removing any timing skew due to different rf path lengths, and equalizing the amplitude levels of the two signals to compensate for differences in rf amplification. At this point, it is also possible to compensate for the residual phase error in 90° optical hybrids, which is typically less than 5°. Next, estimation of the residual beat frequency between the signal and LO lasers enables centering the signal about baseband. Here, since the signal and LO lasers were within 200 MHz of each other, frequency estimation was performed by determining the linear term in the unwrapped temporal phase of the digitized waveform. In cases where the signal and LO lasers have too large a frequency offset to be determined with a time domain approach, other techniques can be used [14].

Fig. 5 DPSK digital signal processing steps for waveform characterization.

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At this point, the compensated signals are passed on to the RZ50 matched filter. Next, software-based delay-line interferometer (DLI) and balanced photodiodes are used to emulate the process of demodulation and detection of the DPSK waveform. After performing clock recovery to identify the center of each of the DPSK symbols, the two balanced photodiode outputs are resampled and normalized to yield one sample per symbol taken at the center of each symbol eye. Finally, the waveform characterization penalty is calculated according to the methodology in Section 3.

4.2 Experimental results

Figure 6 shows a measured BER curve from the TUT to the test bed receiver for various duty cycles. At each power level, bits were evaluated until a minimum of 10 errors were counted. Duty cycles ranging from 1 to 1/8 exhibit similar BER performance and trend well with the DPSK theory curve at all BER levels. Duty cycles of 1/24 and 1/40 exhibit additional penalties at all BER levels and exhibit additional penalties at lower BER values. These additional penalties are due to nonlinearities, such as self-phase modulation, at the lower duty cycles due to the high peak powers that result after going through the power amplifier in the TUT. Specifically, the range of duty cycles adds an additional 16 dB to the peak power when comparing the lowest duty cycle to the highest duty cycle. Note that this waveform characterization technique will only account for the contributions of nonlinearities within the coherent receiver detection bandwidth.

Fig. 6 Measured DPSK BER curves at various duty cycles.

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Figure 7 shows the calculated BER penalty of the TUT and test bed receiver at all measured duty cycles as a function of BER. Duty cycles of 1, ½, and ¼ have nearly identical penalties that range between 2.5 and 2.7 dB over a BER range of 0.8 to 10⁻⁴. The 1/8 duty cycle result has an additional 0.1 to 0.2 dB additional penalty. Duty cycles of 1/24 and 1/40 have much higher penalties, especially at lower BER values, caused by the large peak powers.

Fig. 7 Measured end-to-end DPSK BER implementation penalty as a function of BER at various duty cycles.

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Figure 8 depicts the measured waveform characterization penalty for the transmitter under test at various duty cycles as a function of BER set point using our coherent technique. All measurements were taken using a high (>15 dB) SNR signal. Waveform characterization calculations used 375 bursts of 176 bits each for a total of 66,000 bits, which limited the range of high confidence BER set points to >1E-4. Note that all duty cycles show a nearly identical trend of change in penalty as a function of BER set point as can be seen for the overall system penalty in Fig. 7. Since the waveform characterization penalty only measures transmitter penalty, it may be inferred that the difference between the total system penalty in Fig. 7 and the transmitter penalty in Fig. 8, ~2.1 dB, is attributable to receiver imperfections.

Fig. 8 Measured DPSK waveform characterization penalty calculated as a function of BER set point at various duty cycles. WC: waveform characterization.

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5. Conclusion

This work presents a new rigorous and computationally based methodology that quantifies analytically how close an optical communications transmitter’s signal waveform is to the ideal. The result is a single numerical transmitter quality metric at a desired BER set point based on calculating its BER performance against a theoretically perfect matched-filter receiver. That number, when expressed in dB units, can be entered in a link budget as an effective power penalty for that transmitter. It produces a more accurate assessment of the BER-based penalty than can be obtained from eye masks, constellation diagrams, or calculations based on Q-factor.

The approach uses a hardware coherent receiver front end to perform frequency down-conversion of the optical signal from the TUT to the electrical domain, preserving both optical field amplitude and phase information. The waveform data at the baseband output of the coherent receiver is digitized and processed off-line. Post-processing involves passing the transmitter waveform data through a “software-defined” baseband demodulator that perfectly replicates a theoretically optimal matched-filter receiver. Demodulation statistics are accumulated and used to compute analytically the BER obtained from the TUT. Deviations from theoretical BER limits can then be assigned unambiguously as a penalty of the TUT.

We have validated this method experimentally by determining the waveform penalty of a RZ-DPSK transmitter [5]. This narrative provides insight into how the general procedure described in the first half of the paper can be translated into practice, and provides the framework for applying to other modulation formats and receiver architectures. The results in Section 4.2 demonstrate a strong correlation between the calculated transmitter waveform penalty and the end-to-end system BER measured in a hardware test bed receiver.

Funding

This material is based upon work supported under Air Force Contract No. FA8721-05-C-0002 and/or FA8702-15-D-0001. Any opinions, findings, conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the U.S. Air Force.

Acknowledgements

The authors wish to thank Clement Burton and Andrew Horvath for assisting with measurements in the lab, and P. Steven Bedrosian for help optimizing the data processing routines.

References and links

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Digital coherent receiver based transmitter penalty characterization

Abstract

1. Introduction

2. Generalized waveform analysis methodology

2.1 Optimal baseband demodulation

2.2 Calculation of error probability with a ‘golden’ receiver

3. Waveform characterization for an RZ-DPSK transmitter

3.1 Calculation of error probability with a ‘golden’ receiver

3.2 Calculation of error probability with a ‘golden’ receiver

3.3 Summary of waveform penalty calculation

4. Coherent waveform characterization of DPSK transmitter experimental results

4.1 Experimental arrangement

4.2 Experimental results

5. Conclusion

Funding

Acknowledgements

References and links

Cited By

Figures (8)

Equations (28)

Optics Express