WO2014060031A1

WO2014060031A1 - Method and apparatus for estimating channel coefficients of a mimo communications channel

Info

Publication number: WO2014060031A1
Application number: PCT/EP2012/070570
Authority: WO
Inventors: Fabian Nikolaus Hauske; Fabio PITTALA
Original assignee: Huawei Technologies Co.,Ltd.
Priority date: 2012-10-17
Filing date: 2012-10-17
Publication date: 2014-04-24

Abstract

The invention relates to a method (100) for estimating channel coefficients of a multi-input multi-output communications channel, the method comprising: receiving (101) a set of receive sequences at outputs of the multi-input multi-output communications channel responsive to a set of transmit sequences provided at inputs of the multi-input multi-output communications channel; and estimating (103) the channel coefficients based on the set of receive sequences and based on a predetermined characteristic of the set of transmit sequences, wherein the predetermined characteristic is such that frequency spectra of the transmit sequences are orthogonal with respect to each other by using zero frequency components alternating with respect to the frequency spectra of the transmit sequences.

Description

DESCRIPTION

Method and apparatus for estimating channel coefficients of a MIMO

communications channel

BACKGROUND

The present invention relates to a method and a channel estimating device for estimating channel coefficients of a multi-input multi-output (MIMO) communications channel.

Aspects of the invention relate to an optical receiver with a digital signal processing circuit for estimating channel coefficients of an MIMO channel, in particular to a 2x2 MIMO channel, based on x-polarized and y-polarized receive sequences.

In fiber optic transmission system, chromatic dispersion (CD), polarization mode dispersion (PMD) and other channel impairments accumulate during fiber transmission and cause severe inter-symbol-interference (ISI) which brings severe degradation to system.

In traditional fiber optic communications, the channel can be described as a 2x2 MIMO system where each polarization defines one dimension. Initialization of the equalizer and tracking of time-varying channel impairments can be performed by blind non-training- aided (NTA) acquisition or by training-aided (TA) acquisition employing a known training sequence (TS). Currently, the most adopted solution combines the constant modulus algorithm (CMA) for pre-convergence and the decision-directed least-mean-square (DD- LMS) to reduce the steady state error of the tap coefficients and to track channel variations. Unfortunately, such algorithms are influenced by the properties of the modulation format and suffer from a relatively slow convergence with potential sub- optimum acquisition and even failures. All these problems can be solved at the cost of slight bandwidth efficiency degradation by using frequency domain equalization (FDE) combined with training-based channel estimation. In fact, TA channel estimation allows instantaneous channel acquisition leading to immediate initialization and rapid tracking, apart from improved stability and reliability issues. The TS is repeated at a regular rate fast enough to track time-varying channel distortions. Overhead around 2% is sufficient in dual-stage FD equalizer (FDE) receivers for 28 GBaud transmission where a FD CD compensation is followed by an adaptive TA 2><2 MIMO FDE, which can be implemented by the zero-forcing (ZF) solution or by the minimum-mean-square-error (MMSE) criteria. The training sequence has to be designed to fulfill the optimum properties for a 2x2 Ml MO channel estimation as the optical channel allows transmission over two polarization modes. Ideally, the training sequences in both polarizations are orthogonal in time-domain (TD) within the length of the channel memory to obtain an optimum channel estimation.

In the first stage equalization 1005 of a typical coherent transponder 1000 the received signals in the X- and Y-polarization as described below with respect to Fig. 10 and Fig. 11 are individually compensated for CD in frequency domain using a fast Fourier

transformation (FFT) block 1015 and an inverse FFT (IFFT) block 1017. CD is efficiently compensated in between the FFT blocks 1015 and 1017 according to the representation depicted in Fig. 10 or in between the FFT blocks 1101 and 1105 according to the representation depicted in Fig. 11. The compensation CD function according to the functional block CD^"1 1103 depicted in Fig. 11 is:

where λ₀ is the signal wavelength, f_s is the sampling frequency, N is the FFT size, c is the speed of light, n is the tap number, and D_res is the residual chromatic dispersion (CD). The residual CD can accumulate over the complete transmission link and may consist of the contributions from various fiber types with different positive or negative fiber dispersion coefficients. Due to complexity reasons, only one FFT block using complex input is applied to each polarization as illustrated by the FFT block 1101 in Fig. 11 . The IFFT illustrated by the block 1005 in Fig. 11 is identical to the FFT 1101 although real and imaginary parts are swapped at input and output. From the above equation it becomes clear that the equalization function can be obtained analytical as long as the parameter of the residual CD is known and the equalizer length satisfies the memory requirement. No interpolation operation is necessary.

Blind NTA CD estimation algorithms are published in 'R. Andres Soriano, F. N. Hauske, N. Guerrero Gonzalez, Z. Zhang, Y. Ye, I. Tafur Monroy, "Chromatic Dispersion Estimation in Digital Coherent Receivers", Journal of Lightwave Technology, vol.29, no.1 1 , pp.1627- 1637, June 201 1 ' and 'M. Kuschnerov, F.N. Hauske, K. Piyawanno, B. Spinnler, M.S. Alfiad, A. Napoli, B. Lankl, " DSP for Coherent Single-Carrier Receivers", Journal of Lightwave Technology, vol.27, no.16, pp.3614-3622, Aug.15 2009' and described in the patent applications PCT/CN2009/072624 and PCT/CN2010/070866.

Polarization tracking, PMD compensation and residual CD compensation are done in the second stage equalizer 1003. Such second stage equalizer 1003 is usually implemented in time domain by using finite impulse response (FI ) filters

W_XY, W_YX, Ww arranged in butterfly structure as illustrated in Fig. 12.

The coefficients of the linear equalizer can be adapted by NTA methods based on gradient algorithms like CMA or DD-LMS. Unfortunately, filter update by CMA and DD-LMS is strongly dependent on the modulation format, which requires complex implementation with individual cost functions for each modulation and suffers from a relatively slow convergence with potential sub-optimum acquisition and even failures. Frequency domain equalization combined with TAchannel estimation does not experience convergence problems in terms of speed and singularity. In principle, the modulation of the training sequence and the payload data is independent from each other, which allows flexible switching of the data modulation format. In contrast to CMA/DD-LMS, the FD TA channel estimation can be performed on non-integer numbers of samples per symbols.

Training-based channel estimation is known from wireless communications, where fast channel tracking is required, in particular for mobile communications where each training sequence instantly leads to a full channel estimation, which comes at the cost of additional overhead widening the spectrum of the transmitted signal, usually smaller than 3 percent.

The TS for the 2x2 Ml MO system can be composed of two orthogonal blocks C1 , C2 as illustrated in Fig. 13. In the optimum case, both sequences are orthogonal to each other, which can be reached by different approaches. One such approach is the design of orthogonal non-zero sequences. Two sequences 1301 and 1303 with complex valued, non-zero constellation points as depicted in Fig. 13 are used that are orthogonal.

Orthogonality in time-domain (TD) can also refer to orthogonality in frequency-domain FD and vice versa. This scheme avoids zero signal amplitudes, which can cause problematic power fluctuations in the signal generation of the transmitter, during transmission and in the digital coherent receiver. So called CAZAC sequences are a prominent examples of such training sequences.

The length of each block C1 , C2 must at least cover two times the channel impulse response (CIR). This scheme requires only a single training slot. Similar to the method as described below with respect to Fig. 17, also two consecutive sequences can be applied, which refers to the "Alamouti-Scheme".

Properties of CAZAC training sequences are illustrated in Fig. 14 with respect to time- domain where Fig. 14a) and Fig. 14b) represent real and imaginary part of the first block C1 , 14c) and Fig. 14d) represent real and imaginary part of the second block C2 and Fig. 14e) and Fig. 14f) represent constellation diagram of the first block C1 and the second block C2. The CAZAC sequence is a 16-symbol CAZAC sequence. Frequency domain representation and correlation relations of a 16-symbol CAZAC training sequence are illustrated in Fig. 15, where Fig. 15a) and Fig. 15b) represent absolute value and angle of the first block C1 , Fig. 15c) and Fig. 15d) represent absolute value and angle of the second block C2 and Figures 14e), 14f), 14g) and 14h) represent the four cross correlations C1[(jo_k]C1 [(jo_k]*, C1 [oo_k]C2[oo_k]*, C2[oo_k]C1 [oo_k]* and

C2[oo_k]C2[oo_k]*, where asterisk denotes conjugate complex values.

With the aid of the received spectra R(C1[oo_k], C2[oo_k]) and the known transmitted spectra S(C1 [oo_k], C2[oo_k]) of the training sequence TS as illustrated in Fig. 16, the receiver estimates the channel according to the equation:

From the channel estimation, the FD filter taps can be calculated as

where and (-)^H denote the complex-conjugate (Hermitian) transpose and the inverse, respectively. o_n ² and o_s ² are the noise and signal powers which should be estimated at the receiver. The ZF filter function requires much less complexity for the computation of thefilter solution and does not require an estimation of o_n ² and o_s ², whereas the MMSE filter function requires more complexity for the computation of the filter solution, but it also provides the better performance. In the estimated channel matrix the phase relation between Hn[u)_k] and H₁₂[(jo_k] (and therefore H₂₁[oo_k] and H₂₂[oo_k]) is due to the orthogonality property of the TD CAZAC sequences.

To eliminate redundancy and to guaranty orthogonality between the channel coefficients at each frequency point, the CI is windowed. The complexity of the receiver can be further reduced by computing just Hn[u)_k] and H₂₂[oo_k] and their corresponding inverse discrete FFT hn[n] and h₂₂[n] according to the following equation:

where rect [n] takes 0 for N/4+1 mod(n-1 ,Ν) 3N/4 and 1 elsewhere.

The so called "TD Nulling" scheme due to its simplicity can be described by the steps: Taking a TS for one polarization signal and keeping the output of the second signal to zero; and repeating for the other polarization signal vice versa. However, it causes power fluctuations if the amplitude of the training signal (TS) is not carefully adjusted. Adjustment of amplitudes in the "active" (non-zero) polarization signal might cause problems in the modulator stage as the signal amplitude is increased by a factor of sqrt(2), i.e. the square root of 2, which can cause nonlinear clipping in the optical modulator. In a different approach as illustrated in Fig. 17, the training sequence for the 2x2 MIMO system can be composed of four independent blocks C1 , C2, C3, C4. In the simplest scheme the orthogonality is guaranteed by setting C2 and C3 to zero. The channel is then estimated by solving the following equation:

]

A disadvantage of this approach is that power fluctuations, e.g. zero power or additional measures such as complexity are required to avoid the power fluctuations resulting from digital 45 degree SOP rotation combined with amplitude scaling.

In summary, TA channel estimation is preferred to NTA channel estimation because of the following reasons. Different channel estimations can be realized with various filtering functions such as MMSE or ZF filtering function; this allows optimization for different channel conditions. With NTA filter acquisition, only the MMSE filter function can be achieved. Also timing recovery and carrier recovery can be supported by TSs to improve the performance. TA channel estimation has no problem with singularity, i.e. convergence to the same source. Furthermore, the repetition of the TS provides a practical connection point for independent parallelized processing of data frames. From the closed-form estimated channel transfer function optical performance and parameter monitoring can be achieved. In combination with the ZF filter function which refers to inverse of maximum- likelihood channel estimation precise estimation of CD, PMD, PDL and OSNR can be achieved. With NTA channel estimation using DD-LMS or CMA only the MMSE filter solution is available, which prevents precise PDL and OSNR estimation. However, the so called "classical" TA channel estimation has the following draw-backs: "TD Nulling" method suffers from implementation penalties due to power fluctuations, modulator saturation or additional complexity to generate the training sequence.

Furthermore, additional guard intervals are required in both slots, that is one for each polarization according to Fig. 17. A compact scheme as depicted in Fig. 13 cannot be implemented. SUMMARY

It is the object of the invention to provide a concept for an efficient channel estimation that avoids power fluctuations in the time-domain signal.

This object is achieved by the features of the independent claims. Further implementation forms are apparent from the dependent claims, the description and the figures.

The invention is based on the finding that applying an alternate Nulling structure in frequency domain avoids power fluctuations of the time domain signal but still enables optimal training aided channel estimation. In particular, orthogonal frequency domain training sequences using zero frequency components of odd and even coefficients for x- and y- polarization respectively can be used for channel estimation (=alternate FD

Nulling). With respect to conventional non-zero TA channel estimations, windowing the estimated channel coefficients is not required. This results in reduced complexity and no need of additional FFT/IFFT and windowing operations. The channel estimation based on training sequences can be used for fast initialization of the equalizer or for fast tracking of state of polarization (SOP) rotation. No singularity problem or convergence to suboptimum solution is seen. Given a sequence of length N, the maximum channel impulse response that can be estimated is N/2. This is the same capability as the non-zero single-slot scheme represented in Fig. 13. At the same time, zero-elements of the time domain signal are prevented. Equalization using the estimated channel transfer function can be performed either in time or in frequency domain. The equalizer is not limited to N/2 taps. The training aided alternate frequency domain Nulling channel estimation provides zero forcing and MMSE filter solution. Zero forcing solution is the best option for optical performance monitoring with respect to CD, PMD and PDL.

By applying such alternate Nulling frequency domain structure in optical transponders with digital coherent receiver, signal transmission with constant power more suitable for the generation and transmission in high-speed communications can be employed. As a result the digital compensation of channel impairments is significantly improved with respect to accuracy, robustness and speed as will be presented in the following. In order to describe the invention in detail, the following terms, abbreviations and notations will be used:

CD: chromatic dispersion,

PMD: polarization mode dispersion,

FD: frequency domain, TD: time domain,

IS I : inter-symbol-interference,

PDM: polarization division multiplexing,

(D)QPSK: (differential) quaternary phase shift keying

quadrature phase shift keying,

FFT: fast Fourier transform,

I FFT: inverse fast Fourier transform,

DSP: digital signal processing, ASIC: application specific integrated circuit,

ADC: analog/digital converter,

LO: local oscillator,

TA: training aided

NTA: non-training-aided, WDM: wavelength division multiplex, POLMUX-

QPSK: polarization-multiplexed quadrature phase shift keying,

BE : bit error rate,

OSNR: optical signal-to-noise ratio,

FIR: finite impulse response,

EQ: equalizer,

FO: frequency offset,

sps: samples per symbol,

FFW: feed forward,

FB: feed-back,

SOP: state of polarization,

polarization dependent loss,

DGD: differential group delay,

FEC: forward error correction,

CPE: carrier phase estimation,

I: in-phase,

Q: quadrature,

CAZAC: constant amplitude zero auto-correlation, PN: pseudo noise,

M: maximum length ZF: zero forcing,

MMSE: minimum mean square error,

MIMO: multi input multi output,

DAC: digital analogue converter,

TS: training sequence, CMA: constant-modulus algorithm,

DD: decision-directed,

LMS: least mean squares,

CI : channel impulse response.

According to a first aspect, the invention relates to a method for estimating channel coefficients of a multi-input multi-output communications channel, the method comprising: receiving 101 a set of receive sequences at outputs of the multi-input multi-output communications channel responsive to a set of transmit sequences provided at inputs of the multi-input multi-output communications channel; and estimating 103 the channel coefficients based on the set of receive sequences and based on a predetermined characteristic of the set of transmit sequences, wherein the predetermined characteristic is such that frequency spectra of the transmit sequences are orthogonal with respect to each other by using zero frequency components alternating with respect to the frequency spectra of the transmit sequences.

The method according to the first aspect provides an efficient channel estimation that avoids power fluctuations in the time-domain signal. Power fluctuations in the transmit signal can cause power ripples and overshoots due to the adaptive gain control of electrical and optical amplifier stages, which typically try to achieve a constant output power. Signal power fluctuations can furthermore, raise nonlinear distortions and cause resolution problems or clipping in the analogue/digital conversion in the receiver. Applying an alternate nulling structure in frequency domain avoids power fluctuations of the time domain signal but still enables optimal training aided channel estimation. In the

modulation stage, a laser signal used as carrier signal does not need to be switched off or does not need to be suppressed to zero amplitude. In a first possible implementation form of the method according to the first aspect, the predetermined characteristic is such that for each frequency of the frequency spectra of the transmit sequences only one of the transmit sequences has a non-zero frequency component. If only one of the transmit sequences has a non-zero frequency component, all other transmit sequences have zero frequency components, thereby realizing the desired frequency domain orthogonal characteristic. Channel estimation can be performed by using these orthogonal frequency domain training sequences. In a second possible implementation form of the method according to the first aspect as such or according to the first implementation form of the first aspect a distance between non-zero frequency components of a frequency spectrum of a transmit sequence corresponds to a number of inputs of the multi-input multi-output communications channel. Thus, only one of the Ml MO components is non-zero while the other Ml MO components are zero, thereby realizing the frequency domain orthogonal characteristic.

In a third possible implementation form of the method according to the first aspect as such or according to the first implementation form of the first aspect, the predetermined characteristic is such that the frequency spectra of the transmit sequences are block-wise orthogonal with respect to each other by using blocks of zero frequency components alternating with respect to the frequency spectra. By a block-wise orthogonal property, switching times between the different MIMO components can be reduced, as a whole data block is transmitted on one MIMO channel before switching to the next. In a fourth possible implementation form of the method according to the first aspect as such or according to any of the preceding implementation forms of the first aspect, the method further comprises: transforming one of the following sequences in an alternate nulling-orthogonal sequence of the predetermined characteristic: a CAZAC sequence, a maximum length, M, sequence, a PN sequence, a white noise sequence, a sequence generated by an Alamouti scheme, a spectrally shaped sequence, and a training sequence.

Different input sequences such as a CAZAC sequence, an M sequence, a PN sequence, a white noise sequence, a sequence generated by an Alamouti scheme, a spectrally shaped sequence and a training sequence can be used. However, their characteristic properties are not preserved by the transformation in FD orthogonal nulling sequences.

In a fifth possible implementation form of the method according to the first aspect as such or according to any of the preceding implementation forms of the first aspect, average power envelopes of the set of transmit sequences are constant.

That means, a laser can be used with average power envelope. The laser can be operated in a permanently on mode of transmission. It does not have to be switched off and its amplitude does not need to be suppressed to zero in a modulator.

In a sixth possible implementation form of the method according to the first aspect as such or according to any of the preceding implementation forms of the first aspect, the multi- input multi-output communications channel is a radio frequency channel comprising a multi-input antenna array and a multi-output antenna array.

The method can be applied in different communication systems, e.g. radio, optics, cable or others.

In a seventh possible implementation form of the method according to the first aspect as such or according to any of the first to the fifth implementation forms of the first aspect, the multi-input multi-output communications channel is a fiber optic channel transmitting an optical signal; and wherein the plurality of transmit sequences represent polarization modes of the optical signal. Optical transmission is improved when the method is applied to a MIMO system.

In an eighth possible implementation form of the method according to the seventh implementation form of the first aspect, the multi-input multi-output communications channel is a 2x2 MIMO optical channel transmitting an optical signal comprising a first polarization mode and a second polarization mode.

Optical transmission is improved when the method is applied to a 2x2 MIMO system, where a first input is configured for processing the x-polarization and a second input is configured for processing the y-polarization.

In a ninth possible implementation form of the method according to the eighth

implementation form of the first aspect, the predetermined characteristic is such that frequency spectra of the transmit sequences have zero frequency components in the first polarization mode at even frequency points and in the second polarization mode at odd frequency points or vice versa.

By that property, channel estimation can be used for fast initialization of the equalizer or for fast tracking of state of polarization rotation. In a tenth possible implementation form of the method according to the ninth

implementation form of the first aspect, the method comprises: down-sampling the receive sequences, in particular by a down-sampling factor 2 for providing down-sampled receive sequences removed from zero frequency components in their frequency spectra. The down-sampler removes zero-frequency components in the received frequency spectra.

In an eleventh possible implementation form of the method according to the tenth implementation form of the first aspect, the method comprises: Interpolating estimated channel coefficients obtained from the down-sampled receive sequences. The interpolator interpolates the removed zero-frequency components by intermediate frequency values in the received frequency spectra. In a twelfth possible implementation form of the method according to any of the seventh to the eleventh implementation forms of the first aspect, the estimating the channel coefficients comprises: determining a channel matrix Hxx[k] from a first polarization mode (X) input to a first polarization mode (X) output of the optical channel as F¾_,0[k]/S_x,0[k]; determining a channel matrix H_XY[k] from a first polarization mode (X) input to a second polarization mode (Y) output of the optical channel as R_x,e[k]/S_Y,e[k]; determining a channel matrix H_YX[k] from a second polarization mode (Y) input to a first polarization mode (X) output of the optical channel as RY_,0[k]/S_x,0[k]; and determining a channel matrix Ηγγ[ Ι from a second polarization mode (Y) input to a second polarization mode (Y) output of the optical channel as R_Y,e[k]/S_Y,e[k], wherein S_x[k] represents a spectrum of a transmit sequence of the first polarization mode (X), S_{X o}[k] represents non-zero frequency points of S_x[k] and S_{X e}[k] represents zero frequency points of S_x[k] and R_x[k] represents a nonzero spectrum of a receive sequence of the first polarization mode (X), S_Y[k] represents a spectrum of a transmit sequence of the second polarization mode (Y), S _,0[k] represents zero frequency points of S_Y[k] and S_{Y e}[k] represents non-zero frequency points of S_Y[k] and R_Y[k] represents a non-zero spectrum of a receive sequence of the second polarization mode (Y), and k represents the frequency points.

The estimation according to that formula is easy to implement. According to a second aspect, the invention relates to a channel estimating device for estimating channel coefficients of a multi-input multi-output communications channel, the channel estimating device comprising: a receiver configured to receive a set of receive sequences at outputs of the multi-input multi-output communications channel responsive to a set of transmit sequences provided at inputs of the multi-input multi-output communications channel; and a channel estimating circuit configured to estimate the channel coefficients based on the set of receive sequences and based on a

predetermined characteristic of the set of transmit sequences, wherein the predetermined characteristic is such that frequency spectra of the transmit sequences are orthogonal with respect to each other by using zero frequency components alternating witi respect to the frequency spectra of the transmit sequences. The channel estimating device according to the second aspect provides an efficient channel estimation that avoids power fluctuations in the time-domain signal. Applying an alternate nulling structure in frequency domain avoids power fluctuations of the time domain signal but still enables optimal training aided channel estimation.

According to a third aspect, the invention relates to an optical receiver, comprising: an optical front end for providing an x-polarized receive sequence and a y-polarized receive sequence at a first and a second output of a 2x2 multi-input multi-output communications channel responsive to an x-polarized transmit sequence and a y-polarized transmit sequence provided at a first and a second input of the 2x2 multi-input multi-output communications channel; and a digital signal processing circuit configured to estimate channel coefficients of the 2x2 multi-input multi-output communications channel based on the x-polarized and y-polarized receive sequences and based on a predetermined characteristic of the x-polarized and y-polarized transmit sequences, wherein the predetermined characteristic is such that frequency spectra of the x-polarized and y- polarized transmit sequences are orthogonal with respect to each other by using zero frequency components alternating with respect to the frequency spectra of the x-polarized and y-polarized transmit sequences.

The optical receiver according to the third aspect provides an efficient channel estimation that avoids power fluctuations in the time-domain signal. Applying an alternate nulling structure in frequency domain avoids power fluctuations of the time domain signal but still enables optimal training aided channel estimation.

According to a fourth aspect, the invention relates to a communication system, in particular an optical communication system with coherent receiver and digital signal processing for equalization, in particular with training sequences proving orthogonality by alternate nulling in frequency domain. The communication system comprises estimation means for selection and arrangement of orthogonal frequency components of a received spectrum and synthesis of orthogonal components for complete channel estimation, acquisition means for low complexity DA channel acquisition and parameter estimation means. When applying the method according to aspects of the invention, channel estimation can be performed by using nulling orthogonal frequency domain training sequences.

Windowing the impulse response of the estimated channel coefficients is not required. Because of the reduced complexity, no additional FFT/IFFT and windowing operations are required. The channel estimation based on training sequences can be used for fast initialization of the equalizer or for fast tracking of state of polarization (SOP) rotation. No singularity problem or convergence to suboptimum solution occurs. Equalization can be performed either in time or in frequency domain. The training-based channel estimation provides zero forcing and MMSE filter solution. Zero forcing solution is the best option for optical performance monitoring with respect to CD, PMD, PDL and OSNR. The method 100 shows good correlation properties for framing synchronization. There is no problem with power fluctuations.

A 100G transponder applying the method according to aspects of the invention shows improved parameter estimation performance with respect to OSNR, PDL and DGD.

Degraded channel estimation due to power fluctuations is avoided. A fast and robust initialization is reached and can be combined with subsequent non-training aided filter adaptation. In this case, the training-aided channel estimation can apply fault control to avoid non-convergence due to high nonlinearities/noise. In this case, convergence is limited due to FB CPE and DD-LMS. Non-convergence at large nonlinearity/noise is overcome by pre-setting TDEQ taps with TA channel estimation.

The methods described here are applicable in particular for long-haul transmission using 100-Gb/s polarization-multiplexed quadrature phase shift keying (POLMUX-QPSK) modulation, which is widely applied in products for long-haul optical transmission systems. POLMUX-QPSK modulation is often also referred to as CP-QPSK, PDM-QPSK, 2P-QPSK or DP-QPSK. Similarly, the method applies for other digital modulation formats being single polarization modulation, binary phase shift keying (BPSK) or higher-order quadrature amplitude modulation (QAM).

The methods described herein may be implemented as software in a Digital Signal Processor (DSP), in a micro-controller or in any other side-processor or as hardware circuit within an application specific integrated circuit (ASIC). The invention can be implemented in digital electronic circuitry, or in computer hardware, firmware, software, or in combinations thereof.

BRIEF DESCRIPTION OF THE DRAWINGS

Further embodiments of the invention will be described with respect to the following figures, in which:

Fig. 1 shows a schematic diagram of a method for estimating channel coefficients of a MIMO communications channel according to an implementation form;

Fig. 2 shows diagrams of a set of training sequences with orthogonally-nulling frequency spectra according to an implementation form; Fig. 3 shows diagrams of a set of training sequences with orthogonally-nulling frequency spectra according to an implementation form;

Fig. 4 shows diagrams of a set of training sequences with orthogonally-nulling frequency spectra according to an implementation form;

Fig. 5 shows a block diagram of a device for signal decomposition in odd and even points according to an implementation form;

Fig. 6 shows a block diagram of an optical transmission system with an optical receiver according to an implementation form;

Fig. 7 shows a diagram of channel coefficients of an optical channel with chromatic dispersion and optical/electrical filtering according to an implementation form; Fig. 8 shows a performance diagram illustrating bit error rate for channel estimation based on a method according to an implementation form;

Fig. 9 shows a performance diagram illustrating convergence speed for channel estimation based on a method according to an implementation form; Fig. 10 shows a block diagram of a conventional dual stage equalizer for data aided channel estimation;

Fig. 1 1 shows a block diagram of a conventional chromatic dispersion block in an equalizer of an optical receiver;

Fig. 12 shows a block diagram of a conventional FIR filter structure in an equalizerof an optical receiver; Fig. 13 shows a conventional training sequence scheme using time-domain non-zero sequences;

Fig. 14 shows a set of diagrams illustrating time domain representation and constellation of a conventional 16-symbol CAZAC training sequence;

Fig. 15 shows a set of diagrams illustrating frequency domain representation and correlation relations of a conventional 16-symbol CAZAC training sequence;

Fig. 16 shows a diagram illustrating receive spectra and transmit spectra of a training sequence used for conventional channel estimation; and

Fig. 17 shows a conventional training sequence scheme using orthogonal sequences in time-domain. DETAILED DESCRIPTION OF EMBODIMENTS

Fig. 1 shows a schematic diagram of a method for estimating channel coefficients of a MIMO communications channel according to an implementation form. The method 100 comprises receiving 101 a set of receive sequences at outputs of the multi-input multi-output communications channel responsive to a set of transmit sequences provided at inputs of the multi-input multi-output communications channel.

The method further comprises estimating 103 the channel coefficients based on the set of receive sequences and based on a predetermined characteristic of the set of transmit sequences, wherein the predetermined characteristic is such that frequency spectra of the transmit sequences are orthogonal with respect to each other by using zero frequency components alternating with respect to the frequency spectra of the transmit sequences. In an implementation form, the predetermined characteristic of the set of transmit sequences describes a so called orthogonal nulling frequency characteristic as described below with respect to Figures 2 to 4. In an implementation form, the set of transmit sequences is a set of training sequences. The channel estimation is based on the orthogonal-nulling frequency spectra of the adopted training sequences. As shown in Figures 2 to 4 described below, one polarization has all information distributed in the odd frequency points, whereas the other polarization has all information distributed in the even frequency points. In implementation forms, the method 100 is applied in all digital coherent transponders for optical communication.

Fig. 2 shows diagrams of a set of training sequences with orthogonally-nulling frequency spectra according to an implementation form. The set of training sequences can be applied in a method 100 as described with respect to Fig. 1 . Two training sequences Cx and Cy are depicted in frequency domain. Fig. 2a) illustrates the absolute value of the FFT of the first training sequence Cx and Fig. 2b) illustrates the absolute value of the FFT of the second training sequence Cy. The training sequences Cx and Cy have the predetermined characteristic that the frequency spectrum of the first training sequence Cx is orthogonal with respect to the frequency spectrum of the second training sequence Cy. Zero frequency components alternating with respect to the frequency spectra of the both training sequences Cx and Cy are used to obtain that frequency domain orthogonal nulling characteristic. In particular, the spectrum of the first training sequence Cx has zero points at even frequency bins 0, 2, 4, 6, 8, 10, 12, 14 and 16 while the spectrum of the second training sequence Cy has zero points at odd frequency bins 1 , 3, 5, 7, 9, 1 1 , 13, 15 and 17.

The frequency domain received signals R_x, R_y and the sent known signals S_x, S_y can be expressed as: = {r_x[l],r_x[2],r_x[3],...,r_x[K- l],r_x[K]} = Rx° + Rx^e

= {r_x[l],0,r_x[3],...,r_x[K-l],0} + {0,r_x[2],0,...,0,r_x[K]}

-- {r_y[l],r_y[2],r_y[3],...,r_y[K - l],r_y[K]} = R°' + R_y ^e'

= {r_y[l],0, r_y [3], ... , r_y [K - 1] ,0} + {0, r_y [2] ,0, ... ,0, r_y [K]}

{s_x[l],s_x[2],s_x[3], ...,s_x[K - l],s_x[K]} = S°' + S_x ^e'

= {s_x [1],0, s_x [3], ... , s_x [K - 1] ,0} + {0, s_x [2] ,0, ... ,0, s_x [K]}

{Sy[l],Sy[2],Sy[3], ...,Sy[K - l],Sy[K]} = S°' + S°'

= {Sy [1] ,0, Sy [3] , ... , Sy [ff ^~ 1] ,0} + {(J, Sy [2] ,0, ... ,0, Sy [K]}

V K E N : mod(A:,2) = 0

The transmission over a 2x2 MIMO optical channel can be described as:

However, the FD orthogonal-nulling property of the TSs implies that the spectra of the transmitted TSs in the x-pol. and y-pol. have zero value in the even and odd points, respectively. Therefore:

R_X R_X H_XX (S_X + S_X ) + Η_χγ (Sy + Sy ^

Ry ~\~ Ry Ηγ_χ (5 _χ ~\~ Ξ_χ ) "I" Hyy (S y ~\~ Sy

ei co'

where °x =0 and ^Jy = 0.

Fig.3 shows diagrams of a set of training sequences with orthogonally-nulling frequency spectra according to an implementation form. The set of training sequences can be applied in a method 100 as described with respect to Fig.1. Four training sequences Cx-i, Cx₂, Cx₃ and Cx₄ are depicted in frequency domain. Fig.3a) illustrates the absolute value of the FFT of the first training sequence Cx-i, Fig.3b) illustrates the absolute value of the FFT of the second training sequence Cx₂, Fig.3c) illustrates the absolute value of the FFT of the third training sequence CX₃ and Fig.3d) illustrates the absolute value of the FFT of the fourth training sequence CX4.

The training sequences Cxi, Cx₂, Cx₃ and Cx₄ have the predetermined characteristic that the frequency spectrum of the four training sequences Cxi, Cx₂, Cx₃ and Cx₄ are orthogonal with respect to each other. Zero frequency components alternating with respect to the frequency spectra of the four training sequences Cxi, Cx₂, Cx₃ and Cx₄ are used to obtain that frequency domain orthogonal nulling characteristic. In particular, the spectrum of the first training sequence Cxi has zero points at frequency bins 2, 3, 4, 6, 7, 8, 10, 1 1 , 12, 14, 15, 16, the spectrum of the second training sequence Cx₂ has zero points at frequency bins 1 , 3, 4, 5, 7, 8, 9, 1 1 , 12, 13, 15, 16, the spectrum of the third training sequence CX3 has zero points at frequency bins 1 , 2, 4, 5, 6, 8, 9, 10, 12, 13, 14, 16 and the spectrum of the fourth training sequence C¾ has zero points at frequency bins

I , 2, 3, 5, 6, 7, 9, 10, 1 1 , 13, 14, 15. That means that the spectrum of the first training sequence Cxi has non-zero points at frequency bins 1 , 5, 9, 13, the spectrum of the second training sequence Cx₂ has non-zero points at frequency bins 2, 6, 10, 14, the spectrum of the third training sequence CX3 has non-zero points at frequency bins 3, 7,

I I , 15 and the spectrum of the fourth training sequence Cx_t has non-zero points at frequency bins 4, 8, 12, 16.

Fig. 4 shows diagrams of a set of training sequences with orthogonally-nulling frequency spectra according to an implementation form. The set of training sequences can be applied in a method 100 as described with respect to Fig. 1. Two training sequences Cx and Cy are depicted in frequency domain. Fig. 4a) illustrates the absolute value of the FFT of the first training sequence Cx and Fig. 4b) illustrates the absolute value of the FFT of the second training sequence Cy.

The training sequences Cx and Cy have the predetermined characteristic that the frequency spectrum of the first training sequence Cx is orthogonal with respect to the frequency spectrum of the second training sequence Cy. Zero frequency components alternating blockwise with respect to the frequency spectra of the both training sequences Cx and Cy are used to obtain that frequency domain orthogonal nulling characteristic. In particular, the spectrum of the first training sequence Cx has zero points at frequency bins 4, 5, 6, 7, 8, 9, 13, 14, 15 while the spectrum of the second training sequence Cy has zero points at frequency bins 1 , 2, 3, 7, 8, 9, 10, 1 1 , 12, 16.

Fig. 5 shows a block diagram of a device for signal decomposition in odd and even points according to an implementation form. The FD orthogonal-nulling property of the TSs depicted in Figure 2 implies that the spectra of the transmitted training sequences in the x-polarization and y-polarization have zero value in the even and odd points, respectively. Therefore, as already desaibed above with respect to Fig. 2, the following equation system has to be solved:

(S_X + S_X ) + Η_χγ ^~\^~ Sy )

Ry ~\~ Ry Ηγ_χ (S _χ ~\~ Ξ_χ ) "I" Hyy (S y ~\~ Sy , where = 0 and >' = 0.

Due to the 0/0 operation, this equation system cannot be solved for all frequency points, which implies that the null elements need to be discarded by a down-sampling operation as depicted in Fig. 5.

Fig. 5 illustrates a down-sampling device 501 and an up-sampling device 503. The down- sampling device 501 comprises a first down-sampler 505 in a first signal path and a delay unit z^"1 507 and a second down-sampler 509 in a second signal path. An input signal Y[k] passes both, the first and the second signal paths and is transformed at the first signal path to a first down-sampled signal Y°[i] provided at a first output of the down-sampling device 501 and is transformed at the second signal path to a second down-sampled signal Y^e[i] provided at a second output of the down-sampling device 501 .

The up-sampling device 503 comprises a first up-sampler 51 1 and a delay unit z^"1 513 in a first signal path and a second up-sampler 515 in a second signal path. The first down- sampled signal Y°[i] provided at the first output of the down-sampling device 501 passes a first signal path of the up-sampling device 503 and is transformed to a first up-sampled signal Y°'[i]. The second down-sampled signal Y^e[i] provided at the second output of the down-sampling device 501 passes a second signal path of the up-sampling device 503 and is transformed to a second up-sampled signal Y^el[i]. The up-sampling device 503 further comprises an adder 517 for adding the first up-sampled signal Y°'[i] and the second up-sampled signal Y^el[i] to an up-sampled signal provided at an output of the up- sampling device 503.

The null elements in the spectra are discarded by first down-sampling the input signal Y[k] using the down-sampling device 501 and subsequently up-sampling the down-sampled signals Y°[i] and Y^e[i] using the up-sampling device 503. The up-sampled signal Y[k] corresponds to an interpolated input signal Y[k] without null elements in its spectrum. The resulting spectra are as follows:

R_x ={r_x[l],r_x[3],...,r_x[K-l]}

R_x = {r_y[2],r_y[4],...,r_y[K]}

S° = {s_x[l],s_x[3],...,s_x[K-l]}

S_x ^e ={s_y[2],Sy[4],...,s_y[K]}

V K E N ■ mod( 2) = 0

S_x[k] represents a spectrum of a training sequence of the first polarization mode (X), S_x°[k] represents non-zero frequency points of S_x[k] and S_x ^e[k] represents zero frequency points of S_x[k] and R_x[k] represents a non-zero spectrum of a receive sequence of the first polarization mode (X).

S_Y[k] represents a spectrum of a transmit sequence of the second polarization mode (Y), S_Y°[k] represents zero frequency points of S_Y[k] and S_Y ^e[k] represents non-zero frequency points of S_Y[k] and R_Y[k] represents a non-zero spectrum of a receive sequence of the second polarization mode (Y). k represents the frequency points. Then, the resulting estimated channel matrix reads as:

k' E{l,...,K/2]

Note that the estimated channel has length ' = K/2, which means reduction in spectrum resolution.

Fig.6 shows a block diagram of an optical transmission system with a coherent optical receiver 600 applying the method 100 as described with respect to Fig.1 according to an implementation form.

The coherent optical transmission system 602 comprises an optical sender 601 for providing an optical signal 650, an optical channel 609 for transmitting the optical signal 650 and a coherent receiver 600 for receiving a received optical signal 652 which corresponds to the optical signal 650 transmitted over the optical channel 609 and influenced by the optical channel 609. The optical sender 601 comprises a laser diode 603 for providing an optical carrier signal with a center frequency f_T and a given laser line-width 604. The optical sender 601 further comprises a QPSK modulator 605 for modulating the optical carrier signal with a user data signal to provide a modulated optical data signal. The optical sender 601 further comprises a multiplexer for multiplexing the modulated optical data signal with other modulated optical data signals to provide a multiplexed optical data signal. The multiplexed optical signal may be multiplexed according to a Wavelength Division

Multiplex (WDM) transmission system. The multiplexed optical signal corresponds to the optical signal 650 to be transmitted. The optical channel 609 comprises a plurality of amplifier stages and optical fibers for transmitting the optical signal 650. An output of the optical channel 609 is coupled to an input of the coherent receiver 600, such that the coherent receiver 600 receives the received optical signal 652 which corresponds to the optical signal 650 transmitted over the optical channel 609 at its input.

The coherent receiver 600 comprises a de-multiplexer 623, a polarization beam splitter (PBS) 625, two 6-port 90-degree optical hybrids 627, 629, two sets of balanced detectors 633, two sets of trans-impedance amplifiers (TIA) 635, four analog-digital converters (ADC) 637 and a digital signal processing device (DSP) 639, for example a digital signal processor or a micro-processor or any other processor which is able to perform digital signal processing.

The de-multiplexer 623 is coupled to the input port of the coherent receiver 600 and receives the received optical signal 652 at its input. The de-multiplexer 623 demultiplexes the received optical signal 652 into a plurality of demultiplexed optical signals following a plurality of receiving paths in the coherent receiver 600. Fig. 6 depicts only one of the plurality of receiving paths. In the following, one of these receiving paths is illustrated. The demultiplexed optical signal following one receiving path is provided to the polarization beam splitter 625 which splits the signal into its X-polarized and its Y-polarized signal components. The X-polarized signal component is provided to a first input, which is a signal input, of the first 6-port 90-degree optical hybrid 627 and the Y-polarized signal component is provided to a first input, which is a signal input, of the second 6-port 90- degree optical hybrids 629. A second input, which is a LO input, of the first 6-port 90- degree optical hybrid 627 receives a Local Oscillator signal from a laser diode 631 providing the Local Oscillator signal having a center frequency f_B. The same Local

Oscillator signal is provided to a second input, which is a LO input, of the 6-port 90-degree optical hybrid 629.

The 90° Optical Hybrids 627, 629 comprise two inputs for signal and LO and four outputs mixing signal and LO. The 90° Optical Hybrids 627, 629 deliver both amplitude and phase of signal, amplify signal linearly and are suitable for both homodyne and heterodyne detection.

The six-port 90° Optical Hybrids 627, 629 comprise linear dividers and combiners interconnected in such a way that four different vectorial additions of a reference signal (LO) and the signal to be detected are obtained. The levels of the four output signals are detected by balanced receivers 633. By applying suitable baseband signal processing algorithms, the amplitude and phase of the un-known signal can be determined. For optical coherent detection, each of the six-port 90° optical hybrids 627, 629 mixes the incoming signal with the four quadrature states associated with the reference signal in the complex-field space. Each of the optical hybrids 627, 629 then delivers the four light signals to two pairs of balanced detectors 633 which detect a respective optical signal and provide a corresponding electrical signal to the succeeding set of trans-impedance amplifiers 635, one trans-impedance amplifier for each pair of balanced detectors 633. The electrical signals amplified by the trans-impedance amplifiers 635 are analog-digitally converted by the set of A/D converters 637 and then provided as digital signals 654 to a digital signal processing 639. The digital signal processing may be implemented as software on a Digital Signal Processor (DSP) or on a micro-controller or as hardware circuit within an application specific integrated circuit (ASIC). In addition, to limit the power consumption associated with inter-chip communication, both the ADCs 637 and digital signal processing 639 may be preferably integrated on a single-chip.

The digital signal processing 639 implements the method 100 as described with respect to Fig. 1 . Fig. 6 also depicts a channel estimating device 600 for estimating channel coefficients of a multi-input multi-output communications channel 609. The channel estimating device 600 comprises a receiver 657 configured to receive a set of receive sequences 652 at outputs of the multi-input multi-output communications channel 609 responsive to a set of transmit sequences 650 provided at inputs of the multi-input multi-output communications channel 609. The channel estimation device 600 further comprises a channel estimating circuit 639 configured to estimate the channel coefficients based on the set of receive sequences 652 and based on a predetermined characteristic of the set of transmit sequences 650, wherein the predetermined characteristic is such that frequency spectra of the transmit sequences 650 are orthogonal with respect to each other by using zero frequency components alternating with respect to the frequency spectra of the transmit sequences 650 as described above with respect to Fig. 1 .

Fig. 6 also illustrates an optical receiver 600, comprising an optical front end 657 and a digital signal processing circuit 639. The optical front end 657 is configured for providing an x-polarized receive sequence and a y-polarized receive sequence at a first and a second output of a 2x2 multi-input multi-output communications channel 609 responsive to an x-polarized transmit sequence and a y-polarized transmit sequence provided at a first and a second input of the 2x2 multi-input multi-output communications channel 609.

The digital signal processing circuit 639 is configured to estimate channel coefficients of the 2x2 multi-input multi-output communications channel 609 based on the x-polarized and y-polarized receive sequences and based on a predetermined characteristic of the x- polarized and y-polarized transmit sequences, wherein the predetermined characteristic is such that frequency spectra of the x-polarized and y-polarized transmit sequences are orthogonal with respect to each other by using zero frequency components alternating with respect to the frequency spectra of the x-polarized and y-polarized transmit sequences. Fig. 7 shows a diagram of channel coefficients of an optical channel with chromatic dispersion and optical/electrical filtering according to an implementation form. The length of the estimated frequency domain channel vectors is halved with respect to the training sequence spectra length. Each frequency point k' has a direct correspondence with the frequency points k of the training sequence spectra. Fig. 8 shows a performance diagram illustrating bit error rate for channel estimation based on a method according to an implementation form. The first curve 801 shows back-to- back bit error rate over a single series of estimated channel coefficients, while the second curve 802 shows back-to-back bit error rate over an averaged number of ten estimated channel coefficients.

The system performance illustrated in Fig. 8 refers to channel estimation based on exemplary orthogonal-nulling training sequences which are employed in 100G PDM- QPSK optical transmission. Equalizing the channel by using the estimated filter taps already provides reasonable BE , filter convergence also in highly noisy scenarios. BER performance rapidly improves by averaging the channel estimations as relatively short training sequences are applied.

Fig. 9 shows a performance diagram illustrating convergence speed for channel estimation based on a method according to an implementation form. The first curve 903 shows a minimum required OSNR at BER=10^"3 via number of averages over channel estimations. The second curve 902 shows a mean required OSNR at BER=10^"3 via number of averages over channel estimations. The third curve 901 shows a maximum required OSNR at BER=10^"3 via number of averages over channel estimations. OSNR performance rapidly improves by averaging the channel estimations. Extending the length of the training sequences would further enhance the convergence speed of the channel estimation at the cost of larger overhead for a given repetition rate of the training sequences. From the foregoing, it will be apparent to those skilled in the art that a variety of methods, systems, computer programs on recording media, and the like, are provided.

The present disclosure also supports a computer program product including computer executable code or computer executable instructions that, when executed, causes at least one computer to execute the performing and computing steps described herein.

The present disclosure also supports a system configured to execute the performing and computing steps described herein. Many alternatives, modifications, and variations will be apparent to those skilled in the art in light of the above teachings. Of course, those skilled in the art readily recognize that there are numerous applications of the invention beyond those described herein. While the present inventions has been described with reference to one or more particular embodiments, those skilled in the art recognize that many changes may be made thereto without departing from the spirit and scope of the present invention. It is therefore to be understood that within the scope of the appended claims and their equivalents, the inventions may be practiced otherwise than as specifically described herein.

Claims

CLAIMS:

1 . Method (100) for estimating channel coefficients of a multi-input multi-output communications channel, the method comprising: receiving (101 ) a set of receive sequences at outputs of the multi-input multi-output communications channel responsive to a set of transmit sequences provided at inputs of the multi-input multi-output communications channel; and estimating (103) the channel coefficients based on the set of receive sequences and based on a predetermined characteristic of the set of transmit sequences, wherein the predetermined characteristic is such that frequency spectra of the transmit sequences are orthogonal with respect to each other by using zero frequency components alternating with respect to the frequency spectra of the transmit sequences.

2. The method (100) of claim 1 , wherein the predetermined characteristic is such that for each frequency of the frequency spectra of the transmit sequences only one of the transmit sequences has a non-zero frequency component.

3. The method (100) of claim 1 or claim 2, wherein a distance between non-zero frequency components of a frequency spectrum of a transmit sequence corresponds to a number of inputs of the multi-input multi-output communications channel.

4. The method (100) of claim 1 or claim 2, wherein the predetermined characteristic is such that the frequency spectra of the transmit sequences are block-wise orthogonal with respect to each other by using blocks of zero frequency components alternating with respect to the frequency spectra.

5. The method (100) of one of the preceding claims, further comprising: transforming one of the following sequences in an alternate nulling-orthogonal sequence of the predetermined characteristic: a constant amplitude zero auto-correlation, CAZAC, sequence, a maximum length, M, sequence, a pseudo noise, PN, sequence, a white noise sequence, a sequence generated by an Alamouti scheme, a spectrally shaped sequence, a training sequence, TS.

6. The method (100) of one of the preceding claims, wherein average power envelopes of the set of transmit sequences are constant.

7. The method (100) of one of the preceding claims, wherein the multi-input multi- output communications channel is a radio frequency channel comprising a multi-input antenna array and a multi-output antenna array.

8. The method (100) of one of claims 1 to 6, wherein the multi-input multi-output communications channel is optical fiber optic channel transmitting an optical signal; and wherein the plurality of transmit sequences represent polarization modes of the optical signal.

9. The method (100) of claim 8, wherein the multi-input multi-output communications channel is a 2x2 MIMO fiber optic channel transmitting an optical signal comprising a first polarization mode (X) and a second polarization mode (Y).

10. The method (100) of claim 9, wherein the predetermined characteristic is such that frequency spectra of the transmit sequences have zero frequency components in the first polarization mode (X) at even frequency points and in the second polarization mode (Y) at odd frequency points or vice versa.

1 1 . The method (100) of claim 10, comprising: down-sampling the receive sequences, in particular by a down-sampling factor 2 for providing down-sampled receive sequences removed from zero frequency

components in their frequency spectra.

12. The method (100) of claim 1 1 , comprising: Interpolating estimated channel coefficients obtained from the down-sampled receive sequences.

13. The method (100) of one of claims 8 to 12, wherein the estimating (103) the channel coefficients comprises: determining a channel matrix Hxx[k] from a first polarization mode (X) input to a first polarization mode (X) output of the optical channel as Rx,₀[k]/S_x,₀[k]; determining a channel matrix H_XY[k] from a first polarization mode (X) input to a second polarization mode (Y) output of the optical channel as Rx,_e[k]/S_Y,_e[k]; determining a channel matrix H_YX[k] from a second polarization mode (Y) input to a first polarization mode (X) output of the optical channel as RY,₀[k]/S_x,₀[k]; and determining a channel matrix Ηγγ[ Ι from a second polarization mode (Y) input to a second polarization mode (Y) output of the optical channel as R_Y,_e[k]/S_Y,_e[k], wherein

S_x[k] represents a spectrum of a transmit sequence of the first polarization mode (X), S_{X o}[k] represents non-zero frequency points of S_x[k] and S_{X e}[k] represents zero frequency points of S_x[k] and R_x[k] represents a non-zero spectrum of a receive sequence of the first polarization mode (X),

S_Y[k] represents a spectrum of a transmit sequence of the second polarization mode (Y), S_{Y o}[k] represents zero frequency points of S_Y[k] and S_{Y e}[k] represents non-zero frequency points of S_Y[k] and R_Y[k] represents a non-zero spectrum of a receive sequence of the second polarization mode (Y), and k represents the frequency points.

14. Channel estimating device (600) for estimating channel coefficients of a multi-input multi-output communications channel (609), the channel estimating device (600) comprising: a receiver (657) configured to receive a set of receive sequences (652) at outputs of the multi-input multi-output communications channel (609) responsive to a set of transmit sequences (650) provided at inputs of the multi-input multi-output

communications channel (609); and a channel estimating circuit (639) configured to estimate the channel coefficients based on the set of receive sequences (652) and based on a predetermined characteristic of the set of transmit sequences (650), wherein the predetermined characteristic is such that frequency spectra of the transmit sequences (650) are orthogonal with respect to each other by using zero frequency components alternating with respect to the frequency spectra of the transmit sequences (650).

15. Optical receiver (600), comprising: an optical front end (657) for providing an x-polarized receive sequence and a y- polarized receive sequence at a first and a second output of a 2x2 multi-input multi-output communications channel (609) responsive to an x-polarized transmit sequence and a y- polarized transmit sequence provided at a first and a second input of the 2x2 multi-input multi-output communications channel (609); and a digital signal processing circuit (639) configured to estimate channel coefficients of the 2x2 multi-input multi-output communications channel (609) based on the x-polarized and y-polarized receive sequences and based on a predetermined characteristic of the x- polarized and y-polarized transmit sequences, wherein the predetermined characteristic is such that frequency spectra of the x-polarized and y-polarized transmit sequences are orthogonal with respect to each other by using zero frequency components alternating with respect to the frequency spectra of the x- polarized and y-polarized transmit sequences.