Quantum error detection .I. Statement of the problem

This paper is devoted to the problem of error detection with quantum codes. We show that it is possible to give a consistent definition of the undetected error event. To prove this, we examine possible problem settings for quantum error detection. Our goal is to derive a functional that describes the probability of undetected error under natural physical assumptions concerning transmission with error detection with quantum codes. We discuss possible transmission protocols with stabilizer and unrestricted quantum codes. The set of results proved in the paper shows that in all the cases considered the average probability of undetected error for a given code is essentially given by one and the same function of its weight enumerators. We examine polynomial invariants of quantum codes and show that coefficients of Rains's (see ibid., vol44, p.1388-94, 1998) “unitary weight enumerators” are known for classical codes under the name of binomial moments of the distance distribution. As in the classical situation, these enumerators provide an alternative expression for the probability of undetected error     

Error probabilites for a.s.k./p.s.k. systems in an additive mixture of Gaussian and impulsive noise

A simple expression for the error probability for coherent a.s.k/p.s.k systems in an additive mixture of Gaussian and impulsive noise is obtained. It is shown that the impulsive noise considered may, in certain situations, be treated as conditionally Gaussian. This allows the above error probability to be expressed a weighted sum of the error probabilities due to Gaussian noises of different noise powers.     

Quantitative analysis of 2-f.s.k. traffic detection

Two different methods of detecting signal flow in a noncoherent 2-f.s.k. channel are analysed with respect to the necessary signal/noise ratio for a certain error probability. It is found that separate detection of each frequency requires s.n.r. ?18 dB compared to about 19.5 dB for sum detection when the error probability is 10?7.     

Beyond stabilizer codes .I. Nice error bases

Nice error bases have been introduced by Knill (1996) as a generalization of the Pauli basis. These bases are shown to be projective representations of finite groups. We classify all nice error bases of small degree, and all nice error bases with Abelian index groups. We show that, in general, an index group of a nice error basis is necessarily solvable.     

Asymptotically optimal quantizers for detection of i.i.d

The asymptotic probability of error for quantization in maximum-likelihood tests is analyzed. The authors assume quantizers with large numbers of levels generated from a companding function. A theorem that relates the companding function to the asymptotic probability of error is proved. The companding function is then optimized     

Bit error probability properties of Gray-coded m.p.s.k. signals

The bit error probability associated with the individual binary symbols is discussed for Gray-coded m.p.s.k. signals for large s.n.r.s. All Gray codes with uniform as well as with the most nonuniform bit error probability distribution are given for 16-phase signals; 8-phase signals and 32-phase signals are also considered.     

Multilevel coded modulation for unequal error protection andmultistage decoding .I. Symmetric constellations

In this paper, theoretical upper bounds and computer simulation results on the error performance of multilevel block coded modulations for unequal error protection (UEP) and multistage decoding are presented. It is shown that nonstandard signal set partitionings and multistage decoding provide excellent UEP capabilities beyond those achievable with conventional coded modulation. The coding scheme is designed in such a way that the most important information bits have a lower error rate than other information bits. The large effective error coefficients, normally associated with standard mapping by set partitioning, are reduced by considering nonstandard partitionings of the underlying signal set. The bits-to-signal mappings induced by these partitionings allow the use of soft-decision decoding of binary block codes. Moreover, parallel operation of some of the staged decoders is possible, to achieve high data rate transmission, so that there is no error propagation between these decoders. Hybrid partitionings are also considered that trade off increased intraset distances in the last partition levels with larger effective error coefficients in the middle partition levels. The error performance of specific examples of multilevel codes over 8-PSK and 64-QAM signal sets are simulated and compared with theoretical upper bounds on the error performance     

Performance analysis of both hybrid and frequency-hoppedphase-coherent spread-spectrum systems. I. A hybrid DS/FH system

The authors present and analyze a model for both hybrid and frequency-hopped spread-spectrum systems in which a fully digital coherent receiver is used to demodulate the data. A receiver for a hybrid DS/FH (direct-sequence/frequency-hopped) system using a digital delay-lock loop is considered. In the absence of frequency uncertainty, it is shown that the tracking error can be modeled as an ergodic Markov chain with a finite-state set, and the probability density function of the steady-state tracking error is evaluated. When there is a frequency uncertainty, the dynamics of the resulting nonstationary phase error can be obtained, and an expression to evaluate the probability distribution of the first time at which the phase error hits predetermined boundary values is derived. Bit error rate performance is determined in the presence of both additive white Gaussian noise and various types of interference, and the performance is compared to that of noncoherent FSK systems     

Error performance of frequency-hopped d.p.s.k. system

In order to accurately evaluate bit error rates for a frequency-hopped d.p.s.k. system, digital computer experiments were performed. The value of Eb/N0 at the 10?3 error rate without fading was 8.4 dB.     

Quantization of accumulated diffused errors in error diffusion.

Due to its high image quality and moderate computational complexity, error diffusion is a popular halftoning algorithm for use with inkjet printers. However, error diffusion is an inherently serial algorithm that requires buffering a full row of accumulated diffused error (ADE) samples. For the best performance when the algorithm is implemented in hardware, the ADE data should be stored on the chip on which the error diffusion algorithm is implemented. However, this may result in an unacceptable hardware cost. In this paper, we examine the use of quantization of the ADE to reduce the amount of data that must be stored. We consider both uniform and nonuniform quantizers. For the nonuniform quantizers, we build on the concept of tone-dependency in error diffusion, by proposing several novel feature-dependent quantizers that yield improved image quality at a given bit rate, compared to memoryless quantizers. The optimal design of these quantizers is coupled with the design of the tone-dependent parameters associated with error diffusion. This is done via a combination of the classical Lloyd-Max algorithm and the training framework for tone-dependent error diffusion. Our results show that 4-bit uniform quantization of the ADE yields the same halftone quality as error diffusion without quantization of the ADE. At rates that vary from 2 to 3 bits per pixel, depending on the selectivity of the feature on which the quantizer depends, the feature-dependent quantizers achieve essentially the same quality as 4-bit uniform quantization.     

Error probabilities due to atmospheric noise in practical m-c.p.s.k. and ideal f.s.k. receivers

An improved Cheby?shev approximative analytical expression for the probability distribution of the normalised envelope of atmospheric noise (a.n.) in narrowband receiver is presented. A new probabilistic interpretation of the derived expression is given, and the error probabilities for the m-c.p.s.k. receiver with finite-width decision threshold and an ideal f.s.k. receiver are calculated. The expressions obtained are explicit and they represent the weighted sum of the error probabilities due to Gaussian noises of different noise powers.     

Airborne imaging microwave radiometer. I. Radiometric analysis

The airborne imaging microwave radiometer (AIMR) was designed and built for regional scale sea ice mapping. It operates at 37 and 90 GHz (nominal), and collects radiance at two orthogonal polarizations from which one can compute horizontal and vertical polarizations. The sensitivity or precision (ΔT) of the radiometric data is on the order of 0.5-0.8 K for the 37 GHz channels and 0.8-1.5 for the 90 GHz channels. A detailed error analysis was conducted to assess the accuracy of the radiometric measurements. The error in the brightness temperatures of the original orthogonal polarizations channels was found to be on the order of 0.35-0.45 K for the 37 GHz channel and 0.55-0.65 K for the 90 GHz channel. The polarization conversion introduces additional errors and these are analyzed and computed for the LIMEX-89 data. The total error due to both calibration and polarization conversion for incidence angles greater than 20° is on the order of 0.65-0.70 K for 37 GHz and 0.75-0.85 K for 90 GHz. For incidence angles between 10° and 20° the error can be up to 1.5 K. As the incidence angle approaches zero the distinction between horizontal and vertical polarization breaks down and the error approaches infinity     

A direct geometrical method for bounding the error exponent for anyspecific family of channel codes. I. Cutoff rate lower bound for blockcodes

A direct, general, and conceptually simple geometrical method for determining lower and upper bounds on the error exponent of any specific family of channel block codes is presented. It is considered that a specific family of codes is characterized by a unique distance distribution exponent. The tight linear lower bound of slope -1 on the code family error exponent represents the code family cutoff rate bound. It is always a minimum of a sum of three functions. The intrinsic asymptotic properties of channel block codes are revealed by analyzing these functions and their relationships. It is shown that the random coding technique for lower-bounding the channel error exponent is a special case of this general method. The requirements that a code family should meet in order to have a positive error exponent and at best attain the channel error exponent are stated in a clear way using the (direct) distance distribution method presented     

Mean square error of s.s.b.a.m. class 4 partial response data signals due to slope, sag and ripple group delay components

An expression is derived for the minimum mean square error of an s.s.b.a.m. data signal using class 4 partial response coding when distorted by a channel with group delay slope. This result supplements previously published results which describe the effects of group delay sag and ripple. It is shown that the total minimum mean square error from a combination of all three group delay components can be determined by adding the distortion from each component independently.     

H.264视频误码掩盖综述

在无线网络和因特网上传输视频，误码掩盖的问题越来越重要。回顾了文献中介绍的误码掩盖的方法。误码掩盖后处理利用图像和视频信号的相关特性，在解码端对丢失区域进行掩盖，并总结了空域、时域，以及时空结合的误码掩盖方法。     

nd-convolutional codes. I. Performance analysis

A noncoherent coded system, which incorporates convolutional codes in conjunction of multiple symbol noncoherent detection, is presented in this two-part paper, where Part I focuses on the performance analysis of the system and Part II deals with the structural properties of the underlying convolutional codes. These convolutional codes are referred to as nd-convolutional codes. It is shown that nd-convolutional codes provide a general framework for various noncoherent coding systems, including differential systems. Two models of the carrier phase are examined and the relationships between them is established. For the first one, the carrier phase remains constant for L channels signals, whereas for the second one, it unvaries throughout the transmission period. The regular structure of nd-codes facilitates the evaluation of a simple upper bound on the pairwise and bit error probabilities, as well as a simple expression for the generalized cutoff rate. The exponential rate of the error probability, which is the single parameter governing the error performance at large signal-to-noise ratios, is identified via large deviations techniques. This parameter leads to the interesting conclusion that increasing L does not necessarily monotonically improve the error performance of the noncoherent system. The same conclusion is reached by examining upper bounds and computer simulation results of several interesting examples. These examples also reveal that optimal codes for coherent detection are not necessarily optimal for noncoherent detection and a search for good codes, some of which are tabulated in Part II of the paper, is required     

基于H.263视频码流的差错掩蔽算法

在H．263视频信号的传输过程中,由于采用了可变长编码,当发生比特错误时,很容易造成错误的扩散传播及图像质量下降。由于H．263为低速率传输模式,在进行错误恢复时不适于采用网络开销较大的前向纠错方式,所以采用了基于解码器端的差错掩蔽方法来消除错误的影响。使用的是自适应分类差错掩蔽方法,通过分析图像的运动剧烈程度,将图像分为三类进行掩蔽,既降低了运算复杂度,又取得了较好的掩蔽效果。     

The impact of G.826 [digital systems error performance]

The International Consultative Committee for Telephone and Telegraph (CCITT) recommendation G.826, the error performance recommendation for digital systems operating at or above the primary rate, is summarized. The relationship between block-based error performance parameters and the bit error rate (BER) is reviewed. The transition from the previous recommendation, G.821, to G.826 is discussed, including a brief statement of the main points found in G.826. The conversion of block-based error performance objectives (EPOs) to BER criteria is discussed in a nonmathematical way. The implications of G.826 and some conclusions are presented     

4-phase p.s.k. differential demodulator using a.s.w. delay lines

The letter describes a cheap and reliable solution, using acousticsurface-wave delay lines, for the differential demodulation of phase-shift-keyed digital signals at low digital rates (up to 20 Mbit/s). Results on 4-phase p.s.k. differential demodulators operating at 2.048 Mbit/s are reported in terms of bit error rate against energy per bit/noise spectral density ratio. In addition, using the same technology, a new multidifferential demodulator has been implemented. Owing to multiple differential demodulations carried out on the same signal, a significant improvement in both theoretical and experimental bit error rate is obtained.     

Quantum error detection .II. Bounds

For pt.I see ibid., vol.46, no.3, p.778-88 (2000). In Part I of this paper we formulated the problem of error detection with quantum codes on the depolarizing channel and gave an expression for the probability of undetected error via the weight enumerators of the code. In this part we show that there exist quantum codes whose probability of undetected error falls exponentially with the length of the code and derive bounds on this exponent. The lower (existence) bound is proved for stabilizer codes by a counting argument for classical self-orthogonal quaternary codes. Upper bounds are proved by linear programming. First we formulate two linear programming problems that are convenient for the analysis of specific short codes. Next we give a relaxed formulation of the problem in terms of optimization on the cone of polynomials in the Krawtchouk basis. We present two general solutions of the problem. Together they give an upper bound on the exponent of undetected error. The upper and lower asymptotic bounds coincide for a certain interval of code rates close to 1