In this paper we study a realistic setup for phase retrieval, where the signal of interest is modulated or masked and then for each modulation or mask a diffraction pattern is collected, producing a coded diffraction pattern (CDP) [CLM13]. We are interested in the setup where the resolution of the collected CDP is limited by the Fraunhofer diffraction limit of the imaging system.
We investigate a novel approach based on a geometric quantization scheme of phase-less linear measurements into (one-bit) coded diffraction patterns, and a corresponding recovery scheme. The key novelty in this approach consists in comparing pairs of coded diffractions patterns across frequencies: the one bit measurements obtained rely on the order statistics of the un-quantized measurements rather than their values . This results in a robust phase recovery, and unlike currently available methods, allows to efficiently perform phase recovery from measurements affected by severe (possibly unknown) non linear, rank preserving perturbations, such as distortions. Another important feature of this approach consists in the fact that it enables also super-resolution and blind-deconvolution, beyond the diffraction limit of a given imaging system.
In this work we analyze the problem of phase retrieval from Fourier measurements with random diffraction patterns. To this end, we consider the recently introduced PhaseLift algorithm, which expresses the problem in the language of convex optimization. We provide recovery guarantees which require O(log^2 d) different diffraction patterns, thus improving on recent results by Candes et al. [arXiv:1310.3240], which require O(log^4 d) different patterns.
This paper investigates the design and performance of delayed bit-interleaved coded modulation (DBICM) with low-density parity-check (LDPC) codes. For Gray labeled square $M$-ary quadrature amplitude modulation (QAM) constellations, we investigate the optimal delay scheme with the largest spectrum efficiency of DBICM for a fixed maximum number of delayed time slots and a given signal-to-noise ratio. When analyzing the capacity of DBICM, we find two important properties: the capacity improvement due to delayed coded bits being mapped to the real and imaginary parts of the transmitted symbols ar…
We study quantization of log-likelihood ratios (LLR) in bit-interleaved coded modulation (BICM) systems in terms of an equivalent discrete channel. We propose to design the quantizer such that the quantizer outputs become equiprobable. We investigate semi-analytically and numerically the ergodic and outage capacity over single- and multiple-antenna channels for different quantizers. Finally, we show bit error rate simulations for BICM systems with LLR quantization using a rate 1/2 low-density parity-check code.
Two decoder structures for coded modulation over the Gaussian and flat fading channels are studied: the maximum likelihood symbol-wise decoder, and the (suboptimal) bit-wise decoder based on the bit-interleaved coded modulation paradigm. We consider a 16-ary quadrature amplitude constellation labeled by a Gray labeling. It is shown that the asymptotic loss in terms of pairwise error probability, for any two codewords caused by the bit-wise decoder, is bounded by 1.25 dB. The analysis also shows that for the Gaussian channel the asymptotic loss is zero for a wide range of linear codes, includin…
The wideband regime of bit-interleaved coded modulation (BICM) in Gaussian channels is studied. The Taylor expansion of the coded modulation capacity for generic signal constellations at low signal-to-noise ratio (SNR) is derived and used to determine the corresponding expansion for the BICM capacity. Simple formulas for the minimum energy per bit and the wideband slope are given. BICM is found to be suboptimal in the sense that its minimum energy per bit can be larger than the corresponding value for coded modulation schemes. The minimum energy per bit using standard Gray mapping on M-PAM or …
Bit-interleaved coded modulation (BICM) and multilevel coded modulation (MLC) are commonly used to combine polar codes with high order modulation. While BICM benefits from simple design and the separation of coding and modulation, MLC shows better performance under successive-cancellation decoding. In this paper we propose a hybrid polar coded modulation scheme that lies between BICM and MLC, wherein a fraction of bits are assigned to set-partition (SP) labeling and the remaining bits are assigned for Gray labeling. The SP labeled bits undergo sequential demodulation, using iterative demodulat…
Delayed bit-interleaved coded modulation (DBICM) generalizes bit-interleaved coded modulation (BICM) by modulating differently delayed sub-blocks of codewords onto the same signals. DBICM improves transmission reliability over BICM due to its capability of detecting undelayed sub-blocks with the extrinsic information of the decoded delayed sub-blocks. In this work, we propose a novel windowed decoding algorithm for DBICM, which uses the extrinsic information of both the decoded delayed and undelayed sub-blocks, to improve the detection on all sub-blocks. Numerical results show that the propose…
Linear precoded discrete multitone modulation (LP-DMT) system has been already proved advantageous with adaptive resource allocation algorithm in a power line communication (PLC) context. In this paper, we investigate the bit and energy allocation algorithm of an adaptive LP-DMT system taking into account the channel coding scheme. A coded adaptive LP-DMT system is presented in the PLC context with a loading algorithm which ccommodates the channel coding gains in bit and energy calculations. The performance of a concatenated channel coding scheme, consisting of an inner Wei's 4-dimensional 16-…
This paper investigates the design of spatially coupled low-density parity-check (SC-LDPC) codes constructed from connected-chain ensembles for bit-interleaved coded modulation (BICM) schemes. For short coupling lengths, connecting multiple SC-LDPC chains can improve decoding performance over single-chains and impose structured unequal error protection (UEP). A joint design of connected-chain ensembles and bit mapping to further exploit the UEP from codes and high-order modulations is proposed. Numerical results demonstrate the superiority of the proposed design over existing connected-chain e…
This paper develops a low-complexity near-optimal non-coherent receiver for a multi-level energy-based coded modulation system. Inspired by the turbo processing principle, we incorporate the fundamentals of bit-interleaved coded modulation with iterative decoding (BICM-ID) into the proposed receiver design. The resulting system is called bit-interleaved coded energy-based modulation with iterative decoding (BICEM-ID) and its error performance is analytically studied. Specifically, we derive upper bounds on the average pairwise error probability (PEP) of the non-coherent BICEM-ID system in the …
We consider the problem of using polar codes with higher order modulation over AWGN channels. Unlike prior work, we focus on using modulation independent polar codes. That is, the polar codes are not re-designed based on the modulation used. Instead, we propose bit-permuted coded modulation (BPCM): a technique for using the multilevel coding (MLC) approach for an arbitrary polar code. The BPCM technique exploits a natural connection between MLC and polar codes. It involves applying bit permutations prior to mapping the polar code to a higher order modulation. The bit permutations are designed,…
Caire, Taricco and Biglieri presented a detailed analysis of bit interleaved coded modulation, a simple and popular technique used to improve system performance, especially in the context of fading channels. They derived an upper bound to the probability of error, called the expurgated bound. In this correspondence, the proof of the expurgated bound is shown to be flawed. A new upper bound is also derived. It is not known whether the original expurgated bound is valid for the important special case of square QAM with Gray labeling, but the new bound is very close to, and slightly tighter than,…
In this paper, we present the diversity order analysis of bit-interleaved coded multiple beamforming (BICMB) combined with the constellation precoding scheme. Multiple beamforming is realized by singular value decomposition of the channel matrix which is assumed to be perfectly known to the transmitter as well as the receiver. Previously, BICMB is known to have a diversity order bound related with the product of the code rate and the number of parallel subchannels, losing the full diversity order in some cases. In this paper, we show that BICMB combined with the constellation precoder and maxi…
Multiple beamforming is realized by singular value decomposition of the channel matrix which is assumed to be known to both the transmitter and the receiver. Bit-Interleaved Coded Multiple Beamforming (BICMB) can achieve full diversity as long as the code rate Rc and the number of employed subchannels S satisfy the condition RcS<=1. Bit-Interleaved Coded Multiple Beamforming with Constellation Precoding (BICMB-CP), on the other hand, can achieve full diversity without the condition RcS<=1. However, the decoding complexity of BICMB-CP is much higher than BICMB. In this paper, a reduced complexi…
The performance of bit-interleaved coded modulation (BICM) with bit shaping (i.e., non-equiprobable bit probabilities in the underlying binary code) is studied. For the Gaussian channel, the rates achievable with BICM and bit shaping are practically identical to those of coded modulation or multilevel coding. This identity holds for the whole range of values of signal-to-noise ratio. Moreover, the random coding error exponent of BICM significantly exceeds that of multilevel coding and is very close to that of coded modulation.
When the Channel State Information (CSI) is known by both the transmitter and the receiver, beamforming techniques employing Singular Value Decomposition (SVD) are commonly used in Multiple-Input Multiple-Output (MIMO) systems. Without channel coding, there is a trade-off between full diversity and full multiplexing. When channel coding is added, both of them can be achieved as long as the code rate Rc and the number of employed subchannels S satisfy the condition RcS<=1. By adding a properly designed constellation precoder, both full diversity and full multiplexing can be achieved for both un…
The capacity of the AWGN broadcast channel is achieved by superposition coding, but superposition of individual coded modulations expands the modulation alphabet and distorts its configuration. Coded modulation over a broadcast channel subject to a specific channel-input modulation constraint remains an important open problem. Some progress has been made in the related area of unequal-error protection modulations which can be considered single-user broadcast transmission, but it does not approach all points on the boundary of the capacity region. This paper studies broadcast coded modulation u…
A new variant of bit interleaved coded modulation (BICM) is proposed. In the new scheme, called Parallel BICM, L identical binary codes are used in parallel using a mapper, a newly proposed finite-length interleaver and a binary dither signal. As opposed to previous approaches, the scheme does not rely on any assumptions of an ideal, infinite-length interleaver. Over a memoryless channel, the new scheme is proven to be equivalent to a binary memoryless channel. Therefore the scheme enables one to easily design coded modulation schemes using a simple binary code that was designed for that binar…
It is shown that while the mutual information curves for coded modulation (CM) and bit interleaved coded modulation (BICM) overlap in the case of a single input single output channel, the same is not true in multiple input multiple output (MIMO) channels. A method for mitigating fading in the presence of multiple transmit antennas, named coordinate interleaving (CI), is presented as a generalization of component interleaving for a single transmit antenna. The extent of any advantages of CI over BICM, relative to CM, is analyzed from a mutual information perspective; the analysis is based on an…
$2^m$-ary modulation creates $m$ bit channels which are neither independent nor identical, and this causes problems when applying polar coding because polar codes are designed for independent identical channels. Different from the existing multi-level coding (MLC) and bit-interleaved coded modulation (BICM) schemes, this paper provides a convolutional polar coded modulation (CPCM) method that preserves the low-complexity nature of BICM while offering improved spectral efficiency. Numerical results are given to show the good performance of the proposed method.
Multilevel coding (MLC) is a coded modulation technique which can achieve excellent performance over a range of communication channels. Polar codes have been shown to be quite compatible with communication systems using MLC, as the rate allocation of the component polar codes follows the natural polarization inherent in polar codes. MLC based techniques have not yet been studied in systems that use spatial modulation (SM). SM makes the polar code design difficult as the spatial bits actually select a channel index for transmission. To solve this problem, we propose a Monte Carlo based evaluati…
This paper addresses the performance of bit-interleaved coded multiple beamforming (BICMB) [1], [2] with imperfect knowledge of beamforming vectors. Most studies for limited-rate channel state information at the transmitter (CSIT) assume that the precoding matrix has an invariance property under an arbitrary unitary transform. In BICMB, this property does not hold. On the other hand, the optimum precoder and detector for BICMB are invariant under a diagonal unitary transform. In order to design a limited-rate CSIT system for BICMB, we propose a new distortion measure optimum under this invaria…
The space-time bit-interleaved coded modulation (ST-BICM) is an efficient technique to obtain high diversity and coding gain on a block-fading MIMO channel. Its maximum-likelihood (ML) performance is computed under ideal interleaving conditions, which enables a global optimization taking into account channel coding. Thanks to a diversity upperbound derived from the Singleton bound, an appropriate choice of the time dimension of the space-time coding is possible, which maximizes diversity while minimizing complexity. Based on the analysis, an optimized interleaver and a set of linear precoders,…
An energy-efficient approach is presented for shaping a bit-interleaved low-density parity-check (LDPC) coded amplitude phase-shift keying (APSK) system. A subset of the interleaved bits output by a binary LDPC encoder are passed through a nonlinear shaping encoder whose output is more likely to be a zero than a one. The "shaping" bits are used to select from among a plurality of subconstellations, while the unshaped bits are used to select the symbol within the subconstellation. Because the shaping bits are biased, symbols from lower-energy subconstellations are selected more frequently than …
A protograph-based low-density parity-check (LDPC) code design technique for bandwidth-efficient coded modulation is presented. The approach jointly optimizes the LDPC code node degrees and the mapping of the coded bits to the bit-interleaved coded modulation (BICM) bit-channels. For BICM with uniform input and for BICM with probabilistic shaping, binary-input symmetric-output surrogate channels are constructed and used for code design. The constructed codes perform as good as multi-edge type codes of Zhang and Kschischang (2013). For 64-ASK with probabilistic shaping, a blocklength 64800 code…
This letter investigates bit-interleaved coded multiple beamforming (BICMB) with perfect coding in millimeter-wave (mm-wave) massive multiple-input multiple-output (MIMO) systems to achieve both maximum diversity gain and multiplexing gain. Using perfect coding with BICMB enables us to do this. We show that by using BICMB and perfect coding, the diversity gain becomes independent from the number of transmitted data streams and the number of antennas in each remote antenna unit (RAU) at the transmitter and the receiver. The assumption is that the perfect channel state information (CSI) is known…
For broadband wireless communication systems, Orthogonal Frequency Division Multiplexing (OFDM) has been combined with Multi-Input Multi-Output (MIMO) techniques. Bit-Interleaved Coded Multiple Beamforming (BICMB) can achieve both spatial diversity and spatial multiplexing for flat fading MIMO channels. For frequency selective fading MIMO channels, BICMB with OFDM (BICMB-OFDM) can be applied to achieve both spatial diversity and multipath diversity, making it an important technique. However, analyzing the diversity of BICMB-OFDM is a challenging problem. In this paper, the diversity analysis o…
The problem of private information retrieval gets renewed attentions in recent years due to its information-theoretic reformulation and applications in distributed storage systems. PIR capacity is the maximal number of bits privately retrieved per one bit of downloaded bit. The capacity has been fully solved for some degenerating cases. For a general case where the database is both coded and colluded, the exact capacity remains unknown. We build a general private information retrieval scheme for MDS coded databases with colluding servers. Our scheme achieves the rate $(1+R+R^2+\cdots+R^{M-1})$…
Bit-interleaved coded modulation (BICM) has attracted considerable attention from the research community in the past three decades, because it can achieve desirable error performance with relatively low implementation complexity for a large number of communication and storage systems. By exploiting the iterative demapping and decoding (ID), the BICM is able to approach capacity limits of coded modulation over various channels. In recent years, protograph low-density parity-check (PLDPC) codes and their spatially-coupled (SC) variants have emerged to be a pragmatic forward-error-correction (FEC…
We revisit the information-theoretic analysis of bit-interleaved coded modulation (BICM) by modeling the BICM decoder as a mismatched decoder. The mismatched decoding model is well-defined for finite, yet arbitrary, block lengths, and naturally captures the channel memory among the bits belonging to the same symbol. We give two independent proofs of the achievability of the BICM capacity calculated by Caire et al. where BICM was modeled as a set of independent parallel binary-input channels whose output is the bitwise log-likelihood ratio. Our first achievability proof uses typical sequences, …
A hybrid ARQ (HARQ) scheme for polar code, which is called active-bit relocation under masks (ARUM), is proposed. In each transmission, the data bits are encoded and bit-wisely XOR-masked using a binary vector before being transmitted through the channel. The masking process combines multiple transmissions together which forms another step of inter-transmission channel transform. The reliabilities are updated after every transmission, and the less reliable bits in earlier ones are relocated to the more reliable positions at the latest transmitted block. ARUM is a very flexible HARQ scheme whic…
In this paper, diversity analysis of bit-interleaved coded multiple beamforming (BICMB) is extended to the case of general spatial interleavers, removing a condition on their previously known design criteria and quantifying the resulting diversity order. The diversity order is determined by a parameter Qmax which is inherited from the convolutional code and the spatial de-multiplexer used in BICMB. We introduce a method to find this parameter by employing a transfer function approach as in finding the weight spectrum of a convolutional code. By using this method, several Qmax values are shown …
A protograph-based low-density parity-check (LDPC) code design technique for bandwidth-efficient coded modulation is presented. The approach jointly optimizes the LDPC code node degrees and the mapping of the coded bits to the bit-interleaved coded modulation (BICM) bit-channels. For BICM with uniform input and for BICM with probabilistic shaping, binary-input symmetric-output surrogate channels for the code design are used. The constructed codes for uniform inputs perform as good as the multi-edge type codes of Zhang and Kschischang (2013). For 8-ASK and 64-ASK with probabilistic shaping, cod…
The Gauss--Newton Method for Phase Retrieval from Coded Diffraction Patterns
School of Mathematical Sciences, Beihang University（北京航空航天大学数学科学学院）；School of Artificial Intelligence, University of Science and Technology Beijing（北京科技大学人工智能学院）
Discrete multitone modulation (DMT) systems exploit the capabilities of orthogonal subcarriers to cope efficiently with narrowband interference, high frequency attenuations and multipath fadings with the help of simple equalization filters. Adaptive linear precoded discrete multitone (LP-DMT) system is based on classical DMT, combined with a linear precoding component. In this paper, we investigate the bit and energy allocation algorithm of an adaptive LP-DMT system taking into account the channel coding scheme. A coded adaptive LPDMT system is presented in the power line communication (PLC) c…
In this paper we carry out the asymptotic diversity analysis for millimeter-wave (mmWave) massive multiple-input multiple-output (MIMO) systems by using bit interleaved coded multiple beamforming (BICMB). First, a single-user mmWave system which employs $M_t$ antenna subarrays at the transmitter and $M_r$ antenna subarrays at the receiver is studied. Each antenna subarray in the transmitter and the receiver has $N_r$ and $N_t$ antennas, respectively. We establish a theorem for the diversity gain when the number of antennas in each remote antenna unit (RAU) goes to infinity. Based on the theore…
On the one hand, sign-bit shaping is a popular shaping scheme where the conditional probability of the sign bit is made non-equiprobable. On the other hand, probabilistic amplitude shaping (PAS) is a popular coding scheme, to combine shaping and a bit-interleaved coded modulation (BICM), where the sign bit should not be involved in the shaping. Indeed, with the PAS scheme the sign bit is the parity bit, i.e., the output of the systematic error-correcting code. As a result, sign-bit shaping has been used with multilevel coded modulations rather than BICM. In this paper, we show that with minor …
Multi-Input Multi-Output (MIMO) techniques have been incorporated with Orthogonal Frequency Division Multiplexing (OFDM) for broadband wireless communication systems. Bit-Interleaved Coded Multiple Beamforming (BICMB) can achieve both spatial diversity and spatial multiplexing for flat fading MIMO channels. For frequency selective fading MIMO channels, BICMB with OFDM (BICMB-OFDM) can be employed to provide both spatial diversity and multipath diversity, making it an important technique. In our previous work, the subcarrier grouping technique was applied to combat the negative effect of subcar…
We analyze the achievable information rates (AIRs) for coded modulation schemes with QAM constellations with both bit-wise and symbol-wise decoders, corresponding to the case where a binary code is used in combination with a higher-order modulation using the bit-interleaved coded modulation (BICM) paradigm and to the case where a nonbinary code over a field matched to the constellation size is used, respectively. In particular, we consider hard decision decoding, which is the preferable option for fiber-optic communication systems where decoding complexity is a concern. Recently, Liga \emph{et…
The minimum average signal-to-noise ratio (SNR) per bit required for error-free transmission over a fading channel is derived, and is shown to be equal to that of the additive white Gaussian noise (AWGN) channel, which is $-1.6$ dB. Discrete-time algorithms are presented for timing and carrier synchronization, as well as channel estimation, for turbo coded multiple input multiple output (MIMO) orthogonal frequency division multiplexed (OFDM) systems. Simulation results show that it is possible to achieve a bit error rate of $10^{-5}$ at an average SNR per bit of 5.5 dB, using two transmit and …
We propose a novel joint decoding technique for distributed source-channel (DSC) coded systems for transmission of correlated binary Markov sources over additive white Gaussian noise (AWGN) channels. In the proposed scheme, relatively short-length, low-density parity-check (LDPC) codes are independently used to encode the bit sequences of each source. To reconstruct the original bit sequence, a joint source-channel decoding (JSCD) technique is proposed which exploits the knowledge of both temporal and source correlations. The JSCD technique is composed of two stages, which are iteratively perf…
An analysis of the diversity gain for bit-interleaved coded multiple beamforming (BICMB) method in millimeter-wave (mm-Wave) massive multiple-input multiple-output (MIMO) systems is carried out for both the single-user and multi-user scenario. We show that the diversity gain is independent of the number of data streams and full spatial multiplexing order can be achieved in both scenarios. Also, we show that the diversity gain in the multi-user scenario is independent of the number of users in the system and only depends on the number of the remote antenna units (RAUs) at the transmitter side, …
A scheme is proposed that combines probabilistic signal shaping with bit-metric decoding. The transmitter generates symbols according to a distribution on the channel input alphabet. The symbols are labeled by bit strings. At the receiver, the channel output is decoded with respect to a bit-metric. An achievable rate is derived using random coding arguments. For the 8-ASK AWGN channel, numerical results show that at a spectral efficiency of 2 bits/s/Hz, the new scheme outperforms bit-interleaved coded modulation (BICM) without shaping and BICM with bit shaping (i Fabregas and Martinez, 2010) b…
Generalized product codes (GPCs) are extensions of product codes (PCs) where coded bits are protected by two component codes but not necessarily arranged in a rectangular array. It has recently been shown that there exists a large class of deterministic GPCs (including, e.g., irregular PCs, half-product codes, staircase codes, and certain braided codes) for which the asymptotic performance under iterative bounded-distance decoding over the binary erasure channel (BEC) can be rigorously characterized in terms of a density evolution analysis. In this paper, the analysis is extended to the case w…
3D tomographic phase retrieval under the Born approximation for discrete objects supported on a $n\times n\times n$ grid is analyzed. It is proved that $n$ projections are sufficient and necessary for unique determination by computed tomography (CT) with full projected field measurements and that $n+1$ coded projected diffraction patterns are sufficient for unique determination, up to a global phase factor, in tomographic phase retrieval. Hence $n+1$ is nearly, if not exactly, the minimum number of diffractions patterns needed for 3D tomographic phase retrieval under the Born approximation.
