In this paper, we propose a novel multiple-input multiple-output (MIMO) equalization algorithm for optical space division multiplexing (SDM) systems impaired by mode-dependent loss (MDL). The proposed algorithm is a least mean square (LMS)-like algorithm derived on the basis of the alternating direction method of multipliers (ADMM) framework and consists of an iterative filtering process and filter tap update process. The filtering process is implemented in the frequency domain using the overlap-save technique and the filter update process is a gradient descent-based algorithm which tries to minimize the mean squared error of the equalizer output. The computational complexity of the proposed algorithm increases in proportion to the number of iterations and is the same as that of conventional LMS-based MIMO equalization when the number of iterations is one. We evaluate, through simulations, the bit error rate (BER) performance of the proposed algorithm for a coupled multi-core fiber channel model with significant MDL. The simulation results show that increasing the number of iterations improves the BER performance significantly compared to the conventional LMS-based equalization algorithm. The proposed algorithm offers a good tradeoff between BER performance and computational complexity and enhances the MDL tolerance for SDM systems.

The time-varying phase noise (PN) due to a local oscillator of a base station and a set of user equipment (UE) is a major impairment to the backhaul and access links employing millimeter-wave bands. This paper presents the bit error rate (BER) performance of a pilot symbol assisted (PSA)-extended Kalman filter (EKF) phase noise compensation (PNC) method that combines PSA PNC in the time domain and iterative PNC using the EKF in the 3rd Generation Partnership Project (3GPP) PN model for orthogonal frequency division multiplexing (OFDM) backhaul links. We first derive the discrete-time transfer function of the power spectral density based on the z-transform for time-domain 3GPP PN signal generation employing a small number of low-pass filter taps, and show the validity of the generated PN signal based on a comparison to the continuous-time transfer function that is specified by the 3GPP Technical Report. We then show that when using PSA-EKF PNC with a low-density parity-check (LDPC) code and the coding rate of 8/9, the loss in the required received signal-to-noise ratio (SNR) satisfying the BER of $10^{-8}$ in 3GPP PN Models 1 and 2 compared to that without PN is only approximately 2.3 dB and 0.6 dB, respectively.

This paper proposes an iterative phase noise compensation (PNC) method using an extended Kalman filter (EKF) that employs pilot symbol assisted (PSA) PNC for computing the initial values in the EKF for orthogonal frequency division multiplexing radio backhaul links. In the PSA PNC stage of the proposed method, the phase noise is estimated and compensated using frequency division multiplexing (FDM) pilot symbols that are multiplexed into the data symbols. Based on the decision symbols after the PSA PNC, the time-varying phase noise estimate and variance in the time domain are tracked and updated using the EKF iteratively. The proposed method achieves a low computational complexity because low-density parity-check (LDPC) decoding is conducted only once after the phase noise in the received signal is suppressed by PSA-EKF PNC. Computer simulation results show that the loss in the required received signal-to-noise power ratio (SNR) satisfying the bit error rate (BER) of 10-8of the proposed method compared to that without phase noise is suppressed to within approximately 1.2 dB for 256QAM with LDPC coding at the coding rate of 8/9.

This paper presents the bit error rate (BER) performance of frequency domain equalization (FDE) using low-density parity-check (LDPC) coding for single-carrier line-of-sight (LOS)-multiple-input multiple-output (MIMO) multiplexing. We use partial LDPC coding with double Gray mapping and collaborative decoding, which is suitable for a high coding rate such as 0.9. Computer simulation results show that partial LDPC coding decreases the required received signal-to-noise power ratio (SNR) at the bit error rate (BER) of 10-7by approximately 1.0 dB compared to that for full LDPC coding for 256QAM in a Rummler fading channel. We show that the required received SNR at the BER of 10-7using partial LDPC coding is decreased by more than 6.6 dB compared to that without LDPC coding even for the deep notch depth of -20dB regardless of the relationship between the notch frequencies in the direct and cross links for 2×2 LOS-MIMO in a Rummler fading channel. Therefore, we conclude that FDE using partial LDPC coding with double Gray mapping and collaborative decoding is effective in achieving a low BER for single-carrier LOS-MIMO in microwave radio backhaul channels.

Typical construction of CRC concatenated polar codes append the CRC bits at the end of information bits and then feeds the resulting vector as input to the polar encoder. This construction comes with a caveat that CRC check cannot be performed till all information bits are decoded by the polar decoder. Hence it becomes impossible to do an early termination (ET) of successive cancellation decoder in the case of decoding failure, thereby causing latency and unnecessary power consumption associated with the decoding of bits that follow the first error bit. While interleaving of CRC codeword prior to polar encoding can make ET possible, but it can potentially jeopardize the undetected error rate (UER) of the code. In this paper, a novel construction of interleaved-CRC assisted polar codes is proposed aimed at high ET gain while controlling the rise in UER. Exemplary simulation results are provided.

This paper presents the bit error rate (BER) performance of frequency domain equalization (FDE) using cyclic-shifted code division multiplexing (CDM) pilot signals for single-carrier line-of- sight (LOS) - multiple-input multiple-output (MIMO) multiplexing. We use different cyclic-shift resources of the same Zadoff-Chu sequence for transmission-stream-specific pilot signals that are essential for estimating the channel response for FDE and phase noise in LOS-MIMO. Computer simulation results show that the required received signal-to-noise power ratio satisfying the BER of 10-4by averaging approximately 4 channel responses after despreading the pilot signal in the frequency domain is decreased by approximately 2.0 dB compared to that without averaging of the channel response for 256QAM, 1024QAM, and 4096QAM in 2×2 and 4×4 LOS-MIMO. Through the simulation results, we show that CDM-based pilot signal multiplexing with different cyclic shifts and the frequency domain averaging of the estimated channel response are effective in achieving accurate channel estimation with a short delay and a low pilot-overhead loss for single-carrier LOS- MIMO.

The authors analyse and investigate the effectiveness of frequency domain equalisation (FDE) for single‐carrier (SC) transmission over a line‐of‐sight multiple‐input multiple‐output (LOS‐MIMO) channel with frequency selective fading. For this purpose, they propose a LOS‐MIMO system model employing SC‐FDE techniques together with a data frame structure having a cyclic prefix and postfix (CPP) overhead. The channel capacity of the proposed system is derived taking into account system design parameters, such as symbol rate and roll‐off factor, as well as multipath fading channel parameters. An efficient channel estimation method is also proposed for implementing and simulating the proposed system. Signature curves estimated in simulations are consistent with those expected from the theoretical capacity analysis. The simulation results show that the SC‐FDE approach to the LOS‐MIMO system can, with small CPP overhead, effectively reduce intersymbol and interstream interference.

Puncturing and shortening are widely used methods employed to design arbitrary rate codes from a mother code of given rate. In the context of polar codes, the encoding process has to be completely redone based on the codelength and coding rate for any given shortening/puncturing pattern which makes the implementation of a rate-compatible polar code system very complex. This happens as reliability ordering of the polarized bit-channels gets altered post shortening/puncturing; thus making it necessary to recompute the bit-channel reliabilities tailored to the new reduced codelength followed by sorting them again (in other words, a complete reconstruction). It has been noted before that a bit reversal shortening (BRS) scheme has a special property: it does not require the said reordering of bit-channels, which makes the encoder implementation significantly simple and pragmatic. While this property is extremely intriguing both theoretically and practically, no detailed study on this subject is available in literature. In this paper, we show the relation of BRS with prior art, highlight the said property across wide range of code parameters and then delve deep into what leads up to that property. We observe that like all other schemes, BRS also changes the reliability order of indices, but in a very localized manner. Consequently, the non-frozen set for bit-reverse shortened polar code is nearly same as that of the mother code. Extensive simulation of polar codes is provided to validate the findings.

Recently, we proposed a high-order circular quadrature amplitude modulation (QAM) constellation scheme with a high low-density parity-check (LDPC) coding rate that is suitable for partial LDPC coding associated with parallel double Gray mapping for microwave mobile backhaul. This paper presents the bit error rate (BER) performance of circular QAM when using a combination of Wiener filter based pilot symbol assisted (PSA) and phase locked loop (PLL) phase noise (PN) estimation and compensation methods. Computer simulation results show that the required received signal-to-noise power ratio (SNR) at the BER of 10-6using the PSA and PLL PN estimation and compensation is decreased by approximately 0.3, 0.6, and 1.3 dB for the pilot symbol insertion interval of 25, 50, and 100 symbols, respectively, compared to that using only the PSA PN estimation for circular 1024QAM. Accordingly, we show that the PLL PN estimation is effective in decreasing the pilot symbol overhead of PSA PN estimation for circular QAM, which is robust against time-varying PN.

In this paper, we present a novel phase error estimation scheme for low-density parity-check (LDPC) coded high-order quadrature amplitude modulation (QAM) transmission systems, and we provide a theoretical analysis of its performance. The scheme uses known pilot symbols which are assumed to be placed at equally spaced positions, and phase errors occurred at these pilot symbol positions are first estimated. They are then interpolated over the other positions by using a Wiener filter (WF), and estimation accuracy is next refined by using an LDPC code-aided phase-locked loop (PLL). We present an efficient implementation of this two-stage, Pilot-Symbol-Assisted and Code-Aided (PSA/CA) scheme and evaluate its performance by means of power spectrum analysis. We determine the mean square error at each step of the PSA/CA scheme, and we find an optimal value of the noise equivalent loop bandwidth of the PLL under the assumption that local oscillator phase noise can be modeled as a single-pole single-zero model. We show through simulations that the PSA/CA scheme has a very high phase noise tolerance comparable to that of state-of-the-art phase estimation schemes and provides a computationally efficient alternative.

In this paper, we present an efficient systematic encoding algorithm for quasi-cyclic (QC) low-density parity check (LDPC) codes that are related to cyclic maximum-distance separable (MDS) codes. The algorithm offers linear time complexity, and it can be easily implemented by using polynomial multiplication and division circuits. We show that the division polynomials can be completely characterized by their zeros and that the sum of the respective numbers of their zeros is equal to the parity-length of the codes.

A novel decoder architecture is presented for quasi- cyclic low-density parity-check (QC-LDPC) codes. The architecture implements a min-sum algorithm and uses feedback shift-registers to store all the messages exchanged within the algorithm. This shift-register-based min-sum (SRMS) decoder does not require complex interconnections between processors and registers and is amenable to a simple, high throughput hardware implementation. A codec with the SRMS decoder for high-rate codes of length 4095 bits has been fabricated with a 0.15 mum, 1.5 V CMOS technology. The decoder has a gate count of 693 K gates and achieves a throughput of 155.52 Mbps (STM-1) using high-order QAM schemes. Measurement results show that a microwave radio system employing this LDPC codec exhibits good error performance.

This paper shows that several attractive classes of quasi-cyclic (QC) low-density parity-check (LDPC) codes can be obtained from affine planes over finite fields. One class of these consists of duals of one-generator QC codes. Presented here for codes contained in this class are the exact minimum distance and a lower bound on the multiplicity of the minimum-weight codewords. Further, it is shown that the minimum Hamming distance of a code in this class is equal to its minimum additive white Gaussian noise (AWGN) pseudoweight. Also discussed is a class consisting of codes from circulant permutation matrices, and an explicit formula for the rank of the parity-check matrix is presented for these codes. Additionally, it is shown that each of these codes can be identified with a code constructed from a constacyclic maximum distance separable code of dimension$2$. The construction is similar to the derivation of Reed–Solomon (RS)-based LDPC codes presented by Chenand DjurdjevicExperimental results show that a number of high rate QC-LDPC codes with excellent error performance are contained in these classes.

This paper shows that several attractive classes of quasi-cyclic (QC) low-density parity-check (LDPC) codes can be obtained from affine planes over finite fields. One class of these consists of duals of one-generator QC codes. Here, for codes contained in this class, the exact minimum-distance and a lower bound on the multiplicity of the minimum-weight codewords are presented. It is shown that the lower bound on the multiplicity provides a very accurate indication of the bit error performance at moderate and high signal-to-noise ratios, and thus error-floor behavior can be easily predicted. Also discussed is a class consisting of codes from circulant permutation matrices. An explicit formula for the rank of the parity-check matrix is presented for these codes. Furthermore, it is shown that each of these codes can be identified as a code constructed from a constacyclic maximum distance separable (MDS) code in a similar manner to the RS-based LDPC codes presented by Chen et al. and Djurdjevic et al. Experimental results show that a number of high rate QC-LDPC codes with excellent error performance are contained in these classes

Three algebraic soft-decision decoding algorithms are presented for binary Bose-Chaudhuri-Hocquengham (BCH) codes. Two of these algorithms are based on the bounded distance (BD)+1 generalized minimum-distance (GMD) decoding presented by Berlekamp (1984), and the other is based on Chase (1972) decoding. A simple algebraic algorithm is first introduced, and it forms a common basis for the decoding algorithms presented. Next, efficient BD+1 GMD and BD+2 GMD decoding algorithms are presented. It is shown that, for binary BCH codes with odd designed-minimum-distance d and length n, both the BD+1 GMD and the BD+2 GMD decoding algorithms can be performed with complexity O(nd). The error performance of these decoding algorithms is shown to be significantly superior to that of conventional GMD decoding by computer simulation. Finally, an efficient algorithm is presented for Chase decoding of binary BCH codes. Like a one-pass GMD decoding algorithm, this algorithm produces all necessary error-locator polynomials for Chase decoding in one run.

We describe an efficient algorithm for successive errors-and-erasures decoding of BCH codes. The decoding algorithm consists of finding all necessary error locator polynomials and errata evaluator polynomials, choosing the most appropriate error locator polynomial and errata evaluator polynomial, using these two polynomials to compute a candidate codeword for the decoder output, and testing the candidate for optimality via an originally developed acceptance criterion. Even in the most stringent case possible, the acceptance criterion is only a little more stringent than Forney's (1966) criterion for generalised minimum distance (GMD) decoding. We present simulation results on the error performance of our decoding algorithm for binary antipodal signals over an AWGN channel and a Rayleigh fading channel. The number of calculations of elements in a finite field that are required by our algorithm is only slightly greater than that required by hard-decision decoding, while the error performance is almost as good as that achieved with GMD decoding. The presented algorithm is also applicable to efficient decoding of product RS codes.