Transmitter IQ mismatch compensation in coherent optical OFDM systems using pilot signals

Wonzoo Chung

doi:10.1364/OE.18.021308

1. Introduction

Coherent orthogonal frequency division multiplexing (CO-OFDM) is one of the most promising technologies for high-speed optical fiber communication systems, because of its robustness to intersymbol interference (ISI) distortion caused by chromatic dispersion (CD) and polarization mode dispersion (PMD) [1–3]. Unfortunately, OFDM is sensitive to non-ideal conditions, such as in-phase/quadrature (IQ) mismatch and phase noise in the transmitter (Tx) and in the receiver (Rx) front-end [4,5]. IQ mismatch is caused by the difference from the desired 90° phase shift and the amplitude imbalance between I and Q. IQ mismatch significantly degrades the performance of CO-OFDM systems and compensating for it in the time domain is not a trivial task.

Recent digital time domain compensation schemes for IQ mismatch in CO-OFDM have been based on the Gram-Schmidt orthogonalization procedure (GSOP) [4,6], which mitigates the IQ mismatch by removing the correlation between the distorted I and Q. However, a recent study [4] showed that although GSOP does work well for mitigating the effects of Rx IQ mismatch, this is not the case for Tx IQ mismatch, especially in the presence of carrier frequency offset, CD/PMD, and noise. This is because the correlation between I and Q of the received signals in the presence of CD/PMD distortion is generated by the CD/PMD distortion as well as by Tx IQ mismatch. Because Rx IQ mismatch can be completely removed by the GSOP in time domain and, once removed, the remaining signals become conventional CO-OFDM signals with Tx IQ mismatch, we may focus on compensation of Tx IQ mismatch separately. The Tx IQ mismatch can be modeled as inter carrier interference (ICI) in the frequency domain, and pilot-aided equalization methods for IQ mismatch compensation have been proposed, originally devised for direct-detection (DD) OFDM systems [7–9]. However, these methods require entire OFDM symbols to estimate the channel and ICI due to Tx IQ mismatch, which decreases data rate and impairs the channel tracking ability using pilot tones. In practical CO-OFDM systems, channel knowledge and pilot tones are essential to mitigate phase noise [5].

In this paper, we consider a digital domain Tx IQ mismatch compensation scheme for the equalization stage of an Rx. We propose a dedicated pilot structure and estimation method to separately estimate the channel distortion and the Tx IQ mismatch factor. Therefore, the proposed algorithm requires only a few pilot tones to acquire channel information and to equalize ICI due to Tx IQ mismatch. Hence, the proposed scheme can be used with the conventional OFDM pilot tone structure [5]. Using the estimated data, we develop a minimum mean square error (MMSE) compensation scheme that combines equalization with Tx IQ mismatch compensation.

2. Tx IQ mismatch in CO-OFDM systems

A CO-OFDM system consists of an OFDM Tx, an optical link, and an OFDM Rx as illustrated in Fig. 1 .

Fig. 1 Baseband equivalent of the CO-OFDM system.

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Let Wdenote the discrete Fourier matrix (DFT) and $s_{k} = {[s_{k} (0), s_{k} (1), \dots, s_{k} (N - 1)]}^{t}$ be the k ^th OFDM symbol vector. The discrete time domain signal is obtained by inverse DFT (IDFT),

x_{k} = W^{H} s_{k} .

The $x_{k}$ are converted from serial to parallel, a cyclic prefix of length $N_{p}$ is added, and the result is converted to an analog signal $x (t)$ . In the presence of Tx IQ mismatch, the distorted signal can be modeled as

x_{i q} (t) = G_{1} x (t) + G_{2} x^{*} (t),

where

G_{1} = \frac{1 + ϵ e^{j ϕ}}{2}

and

G_{2} = \frac{1 - ϵ e^{j ϕ}}{2}

are determined by the phase difference ϕ and the amplitude imbalance ϵ, and

{(\cdot)}^{*}

denotes the conjugation operation. Under this IQ mismatch condition, the discrete time domain signal in Eq. (1) becomes

x_{k} = G_{1} W^{H} s_{k} + G_{2} W s_{k}^{*} .

From the relationship between the DFT and IDFT,

\sum_{m = 0}^{N - 1} s_{k}^{*} (m) e^{- j 2 π \frac{n m}{N}} = \sum_{m = 0}^{N - 1} s_{k}^{*} (\mod (N - m, N)) e^{j 2 π \frac{n m}{N}},

and Eq. (3) can be written with only IDFT matrixes as

x_{k} = G_{1} W^{H} s_{k} + G_{2} W^{H} {\bar{s}}_{k}^{*},

where

{\bar{s}}_{k}^{*}

denotes the mirror image of

s_{k}^{*}

:

{\bar{s}}_{k}^{*} (n) = s_{k}^{*} (m o d (N - n), N)) .

At the Rx, assuming perfect carrier and timing synchronization, and after removing the cyclic prefix and performing the DFT, we obtain the received symbol vector in the following form, where circular matrix C is the channel effect and w _k is an additive white Gaussian noise vector:

r_{k} = W C x_{k} + w_{k} .

The frequency domain symbol vector is

\begin{matrix} r_{k} = G_{1} W C W^{H} s_{k} + G_{2} W C W^{H} {\bar{s}}_{k}^{*} + w_{k} = G_{1} Λ_{c} s_{k} + G_{2} Λ_{c} {\bar{s}}_{k}^{*} + w_{k} \\ = G_{1} Λ_{c} (s_{k} + \frac{G_{2}}{G_{1}} {\bar{s}}_{k}^{*}) + w_{k} \\ = Λ (s_{k} + G {\bar{s}}_{k}^{*}) + w_{k}, \end{matrix}

where

Λ_{c}

is the diagonal matrix consisting of the N-point DFT of the channel. We define

Λ : = G_{1} Λ_{c} and G : = \frac{G_{2}}{G_{1}} = \frac{1 - ϵ e^{j ϕ}}{1 + ϵ e^{j ϕ}} .

Note that Λis a diagonal matrix of dimension N, and G is a constant.

Equation (8) reveals that in the presence IQ mismatch, each subcarrier experiences interference from its mirror-image position subcarrier proportional to G and distortion from the scalar channel $Λ (n)$ . To restore the transmitted symbol perfectly, we must compensate for the negative effects of both the intercarrier interference and the channel distortion.

3. Proposed IQ mismatch compensation method

3.1 Pilot symbol design for estimating channel and Tx IQ mismatch

The first task is to estimate both Λ and G, which include N + 1 unknown variables. Because the channel $Λ_{c}$ is assumed to be generated by a time domain channel impulse response of at most length N_p, we may theoretically estimate Λ and G when the number of pilot tones is greater than N_p + 1 [10]. However, to simplify the estimation process and improve the estimation performance in the presence of noise, we propose a method of directly estimating Λ and G using pilots with the following structure.

We require that a subset of OFDM pilot subcarriers carry zero values and their mirror image position subcarriers also be pilot tones carrying non-zero pilot signals. This can be readily achieved by choosing a subset of sub-carriers asymmetric to the center frequency. For example, for an OFDM system with a block length of N, a subset of subcarriers indexed by $A = {M n | n = 1, \dots, ⌊ N / M ⌋}$ will have mirror image position subcarriers $\bar{A} = {M n + N - ⌊ N / M ⌋ M | n = 0, \dots, ⌊ N / M ⌋ - 1}$ with $A \cap \bar{A} = \emptyset$ as long as N is not divisible by M. We can build the desired pilot symbol by selecting $A \cup \bar{A}$ to be a pilot group, $\bar{A}$ to be all zeros, and A to be a sequence of pseudorandom signals. Using this rule, an OFDM symbol containing $2 ⌊ N / M ⌋$ pilot signals can be described as

p_{k} (n) = {\begin{matrix} 0 & i f n = M i + N - ⌊ N / M ⌋ M \\ p (i) & i f n = M i \\ s (m) & e l s e \end{matrix}

where

{p (i) | i = 1, \dots, ⌊ N / M ⌋}

is a sequence of pseudorandom signals and

{s (m)}

is a data sequence. Note that as M decreases the number of pilot tones increases and the channel estimation performance is enhanced, but data throughput is decreased.

With such pilot symbols, the received group of pilot signals A does not experience IQ mismatch distortion because the mirror imaged signals are zero. Furthermore, the received group of pilot signals $\bar{A}$ contains only interference signals,

r_{k} (n) = {\begin{matrix} Λ (n) s_{k} (n) + w_{k} (n) & n \in A \\ Λ (n) G s_{k}^{*} (\bar{n}) + w_{k} (n) & n \in \bar{A} \\ Λ (n) (s_{k} (n) + G s_{k}^{*} (\bar{n})) + w_{k} (n) & e l s e \end{matrix}

where

\bar{n}

is the index of the mirror image position of n:

\bar{n} : = m o d (N - n, N) .

We can estimate channel $Λ (n)$ for $n \in A$ using Eq. (11). Once channel estimation has taken place for the pilots in A, we can extend the channel estimation to all subfrequencies using standard interpolation methods, such as low-pass interpolation [10,11]. Once Λ is known, G can be estimated using the pilots in $\bar{A}$ using

\hat{G} = \frac{1}{| \bar{A} |} \sum_{n \in \bar{A}} \frac{r (n)}{Λ (n) s^{*} (n)} .

where

| \bar{A} |

is the number of elements in

\bar{A}

.

3.2 MMSE compensation

Because the mirror image index of a mirror-imaged index is the original index, i.e., $\bar{(\bar{n})} = n$ , we can restore $s (n)$ perfectly from a linear combination of $r (n)$ and $r^{*} (\bar{n})$ in the absence of noise as we will show later:

f_{Z F} (n) r (n) + g_{Z F} (n) r^{*} (\bar{n}) = f Λ (n) (s_{k} (n) + G s_{k}^{*} (\bar{n})) + g Λ^{*} (\bar{n}) (s_{k}^{*} (\bar{n}) + G s_{k} (n)) = s (n)

However, in the presence of noise, we consider the MMSE compensator (i.e., weights f and g that minimize the following MSE):

{(f (n), g (n))}_{M M S E} = {arg min}_{f, g} E {| f r (n) + g r^{*} (\bar{n}) - s (n) |}^{2}

= {arg min}_{f, g} {| f Λ (n) + g Λ^{*} (\bar{n}) G^{*} - 1 |}^{2} σ_{s}^{2} + {| f Λ (n) G + g Λ^{*} (\bar{n}) |}^{2} σ_{s}^{2} + σ_{w}^{2} ({| f |}^{2} + {| g |}^{2})

where we assume that each subcarrier has the same signal power and noise power, denoted by

σ_{s}^{2}

and

σ_{w}^{2}

, respectively.

Defining the noise to signal power ratio as

λ : = \frac{σ_{w}^{2}}{σ_{s}^{2}}

and considering the derivative of the quadratic cost function in Eq. (16) gives

[\begin{matrix} {| Λ (n) |}^{2} (1 + {| G |}^{2}) + λ & 2 Λ^{*} (n) Λ^{*} (\bar{n}) G^{*} \\ 2 Λ (n) Λ (\bar{n}) G & {| Λ (\bar{n}) |}^{2} (1 + {| G |}^{2}) + λ \end{matrix}] [\begin{matrix} f_{M M S E} (n) \\ g_{M M S E} (n) \end{matrix}] = [\begin{matrix} Λ^{*} (n) \\ Λ (\bar{n}) G \end{matrix}]

and the matrix equation yields

f_{M M S E} (n) = \frac{Λ^{*} (n) ({| Λ (\bar{n}) |}^{2} (1 - {| G |}^{2}) + λ)}{{| Λ (n) |}^{2} {| Λ (\bar{n}) |}^{2} {(1 - {| G |}^{2})}^{2} + λ (1 + {| G |}^{2}) ({| Λ (n) |}^{2} + {| Λ (\bar{n}) |}^{2}) + λ^{2}}

g_{M M S E} (n) = \frac{- Λ (\bar{n}) G ({| Λ (n) |}^{2} (1 - {| G |}^{2}) + λ)}{{| Λ (n) |}^{2} {| Λ (\bar{n}) |}^{2} {(1 - {| G |}^{2})}^{2} + λ (1 + {| G |}^{2}) ({| Λ (n) |}^{2} + {| Λ (\bar{n}) |}^{2}) + λ^{2}}

Note that in the absence of noise (i.e., when $λ = 0$ ), we have the following zero-forcing (ZF) solution that achieves perfect channel equalization and IQ mismatch compensation in Eq. (14):

f_{Z F} (n) = \frac{1}{Λ (n) (1 - {| G |}^{2})}

g_{Z F} (n) = \frac{- G}{Λ^{*} (\bar{n}) (1 - {| G |}^{2})}

4. Numerical Results

Standard Monte Carlo simulation was used to evaluate the performance of the proposed Tx IQ mismatch compensation scheme. We considered a 10-gigasample-per-second CO-OFDM system using 128 subcarriers with a cyclic prefix length of 14 samples. A pilot symbol with M = 3 in Eq. (10) was inserted every 10 OFDM symbols to estimate the channel and IQ mismatch. The Tx laser and LO laser line width were set to 10 kHz to exclude the effect of phase noise. CD was considered as the cause of ISI in the optical fiber channel, a standard single-mode fiber (SSMF).

To illustrate the performance of the proposed algorithm, we first considered a back-to-back system with $ϕ = 25^{°}$ and $ϵ = 0.7$ . Figure 2 shows that both the GSOP method and the proposed method successfully removed IQ mismatch in the absence of ISI. Figure 3 illustrates the case of CD distortion corresponding to 200 km of SSMF. The received constellation was rotated due to CD, and GSOP failed to compensate for the Tx IQ mismatch, while the proposed algorithm successfully mitigated the Tx IQ-mismatch. Figure 4 shows the output signal-to-noise ratio (OSNR) penalty for a target bit error rate (BER) of 10⁻³ versus Tx IQ phase imbalance for 200 km of SSMF in the case of conventional equalization only, equalization with pre-GSOP, and the proposed combined IQ compensation and equalization scheme. The performance of the proposed algorithm was superior to that of the other methods. Figure 5 compares MMSE equalizers and ZF equalizers in terms of OSNR penalty for 300km SMMF. MMSE equalizers perform better than ZF equalizers, but the difference is marginal. Considering the complexity of MMSE equalizers, ZF equalizers can be used as a good approximation of the optimal MMSE equalizers. Figure 6 illustrates the performance of the above three methods for CD distortion in up to 1000 km of SSMF as a function of the OSNR penalty (target BER of 10⁻³). The result indicates that the GSOP does not perform better than the equalization-only scheme for more than 200 km of SSMF.

Fig. 2 Back-to-back CO-OFDM with $ϵ = 0.7$ and $ϕ = 25^{°}$ IQ mismatch: (a) received signal, (b) equalization only, (c) GSOP compensation and equalization, and (d) proposed IQ compensation and equalization.

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Fig. 3 200-km SSMF CO-OFDM with $ϵ = 0.7$ and $ϕ = 25^{°}$ IQ mismatch: (a) received signal, (b) equalization only, (c) GSOP compensation and equalization, and (d) proposed IQ compensation and equalization.

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Fig. 4 OSNR penalty versus IQ mismatch (200-km SSMF, target BER = 10⁻³).

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Fig. 5 Performance compassion: MMSE and ZF equalizers (300-km SSMF, target BER = 10⁻³).

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Fig. 6 OSNR penalty versus CD (SSMF length) for 25° IQ imbalance (target BER = 10⁻³).

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7. Conclusions

This paper describes a joint Tx IQ mismatch compensation and equalization scheme. We designed a pilot structure to estimate both the channel and Tx IQ mismatch, and have proposed a MMSE compensation method. Numerical results demonstrated that the proposed algorithm can improve BER performance of a CO-OFDM system in the presence of Tx IQ mismatch.

Acknowledgement

This work was supported by Brain Korea Project 2010 and the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the MEST (NRF-2010-0015621).

References and links

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Transmitter IQ mismatch compensation in coherent optical OFDM systems using pilot signals

Abstract

1. Introduction

2. Tx IQ mismatch in CO-OFDM systems

3. Proposed IQ mismatch compensation method

3.1 Pilot symbol design for estimating channel and Tx IQ mismatch

3.2 MMSE compensation

4. Numerical Results

7. Conclusions

Acknowledgement

References and links

Cited By

Figures (6)

Equations (22)

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