Multistage recursive interleaver for turbo codes in DS-CDMA mobileradio

A multistage recursive block interleaver (MIL) is proposed for the turbo code internal interleaver. Unlike conventional block interleavers, the MIL repeats permutations of rows and columns in a recursive manner until reaching the final interleaving length. The bit error rate (BER) and frame error rate (FER) performance with turbo coding and MIL under frequency-selective Rayleigh fading are evaluated by computer simulation for direct-sequence code-division multiple-access mobile radio. The performance of rate-1/3 turbo codes with MIL is compared with pseudorandom and S-random interleavers assuming a spreading chip rate of 4.096 Mcps and an information bit rate of 32 kbps. When the interleaving length is 3068 bits, turbo coding with MIL outperforms the pseudorandom interleaver by 0.4 dB at an average BER of 10^-6 on a fading channel using the ITU-R defined Vehicular-B power-delay profile with the maximum Doppler frequency of f_D = 80 Hz. The results also show that turbo coding with MIL provides superior performance to convolutional and Reed-Solomon concatenated coding; the gain over concatenated coding is as much as 0.6 dB     

Probabilistic construction of large constraint length trellis codesfor sequential decoding

Probabilistic algorithms are given for constructing good large constraint length trellis codes for use with sequential decoding that can achieve the channel cutoff rate bound at a bit error rate (BER) of 10^-5-10^-6. The algorithms are motivated by the random coding principle that an arbitrary selection of code symbols will produce a good code with high probability. One algorithm begins by choosing a relatively small set of codes randomly. The error performance of each of these codes is evaluated using sequential decoding and the code with the best performance among the chosen set is retained. Another algorithm treats the code construction as a combinatorial optimization problem and uses simulated annealing to direct the code search. Trellis codes for 8 PSK and 16 QAM constellations with constraint lengths v up to 20 are obtained. Simulation results with sequential decoding show that these codes reach the channel cutoff rate bound at a BER of 10^-5-10^-6 and achieve 5.0-6.35 dB real coding gains over uncoded systems with the same spectral efficiency and up to 2.0 dB real coding gains over 64 state trellis codes using Viterbi decoding     

Approaching Shannon's capacity limit by 0.2 dB using simple Hammingcodes

The authors show that the Shannon capacity limit for the additive white Gaussian noise (AWGN) channel can be approached within 0.27 dB at a bit error rate (BER) of 10^-5 by applying long but simple Hamming codes as component codes to an iterative turbo-decoding scheme. In general, the complexity of soft-in/soft-out decoding of binary block codes is rather high. However, the application of a neurocomputer in combination with a parallelization of the decoding rule facilitates an implementation of the decoding algorithm in the logarithmic domain which requires only matrix additions and multiplications. But the storage requirement might still be quite high depending on the interleavers used     

Source-channel optimized trellis codes for bitonal imagetransmission over AWGN channels

We consider the design of trellis codes for transmission of binary images over additive white Gaussian noise (AWGN) channels. We first model the image as a binary asymmetric Markov source (BAMS) and then design source-channel optimized (SCO) trellis codes for the BAMS and AWGN channel. The SCO codes are shown to be superior to Ungerboeck's codes by approximately 1.1 dB (64-state code, 10^-5 bit error probability), We also show that a simple “mapping conversion” method can be used to improve the performance of Ungerboeck's codes by approximately 0.4 dB (also 64-state code and 10 ^-5 bit error probability). We compare the proposed SCO system with a traditional tandem system consisting of a Huffman code, a convolutional code, an interleaver, and an Ungerboeck trellis code. The SCO system significantly outperforms the tandem system. Finally, using a facsimile image, we compare the image quality of an SCO code, an Ungerboeck code, and the tandem code, The SCO code yields the best reconstructed image quality at 4-5 dB channel SNR     

Improved Performance Upper Bounds for Terminated Convolutional Codes

In this letter, we propose tight performance upper bounds for convolutional codes terminated with an input sequence of finite length. To obtain the upper bounds, a weight enumerator is defined to represent the relation between the Hamming distance of the coded output and the Hamming distance of the input bits of the code. The upper bounds on frame error rate (FER) and average bit error rate (BER) are obtained from the weight enumerator. A simple method is presented to compute the weight enumerator of a terminated convolutional code based on a modified trellis diagram.     

Iterative decoding for partial response (PR), equalized,magneto-optical (MO) data storage channels

Turbo codes and low-density parity check (LDPC) codes with iterative decoding have received significant research attention because of their remarkable near-capacity performance for additive white Gaussian noise (AWGN) channels. Previously, turbo code and LDPC code variants are being investigated as potential candidates for high-density magnetic recording channels suffering from low signal-to-noise ratios (SNR). We address the application of turbo codes and LDPC codes to magneto-optical (MO) recording channels. Our results focus on a variety of practical MO storage channel aspects, including storage density, partial response targets, the type of precoder used, and mark edge jitter. Instead of focusing just on bit error rates (BER), we also study the block error statistics. Our results for MO storage channels indicate that turbo codes of rate 16/17 can achieve coding gains of 3-5 dB over partial response maximum likelihood (PRML) methods for a 10^-4 target BER. Simulations also show that the performance of LDPC codes for MO channels is comparable to that of turbo codes, while requiring less computational complexity. Both LDPC codes and turbo codes with iterative decoding are seen to be robust to mark edge jitter     

Subband speech coding and matched convolutional channel coding formobile radio channels

The effects of digital transmission errors on a family of variable-rate embedded subband speech coders (SBC) are analyzed in detail. It is shown that there is a difference in error sensitivity of four orders of magnitude between the most and the least sensitive bits of the speech coder. As a result, a family of rate-compatible punctured convolutional codes with flexible unequal error protection capabilities have been matched to the speech coder. These codes are optimally decoded with the Viterbi algorithm. Among the results, analysis and informal listening tests show that with a 4-level unequal error protection scheme transmission of 12 kb/s speech is possible with very little degradation in quality over a 16 kb/s channel with an average bit error rate (BER) of 2×10^-2 at a vehicle speed of 60 m.p.h. and with interleaving over two 16 ms speech frames     

Multidimensional modulation used in BICM-ID

The performance of a BICM-ID system with multidimensional (MD) modulation is investigated. Design criteria of MD constellation labelling are proposed and BICM-ID schemes with QPSK of different dimensions are established. Simulation results show that when each data block contains 50000 information bits and the iteration number is 20, the BER of BICM-ID with six-dimensional QPSK and four-state rate 1/2 convolutional code is 10^-5 at SNR=0.9 dB over an AWGN channel which is only 0.2 dB away from Berrou's turbo code     

Interleaved concatenated coding for the turbulent atmosphericdirect detection optical communication channel

The performance of two concatenated coding systems using a K =3, R=1/2 convolutional inner code and a Reed-Solomon (RS) (15, 9) or (15, 7) outer code was measured over a 5.76-km-long atmospheric direct detection optical communication channel. Inner code interleaving of 100 μs combined with outer code interleaving of 240 bits (60 RS symbols) was found to be sufficient to obtain a decoded BER of less than 10^-6 under conditions of moderate channel turbulence and an average of 6-10 detected photons per channel bit     

On the design of low-density parity-check codes within 0.0045 dB ofthe Shannon limit

We develop improved algorithms to construct good low-density parity-check codes that approach the Shannon limit very closely. For rate 1/2, the best code found has a threshold within 0.0045 dB of the Shannon limit of the binary-input additive white Gaussian noise channel. Simulation results with a somewhat simpler code show that we can achieve within 0.04 dB of the Shannon limit at a bit error rate of 10^-6 using a block length of 10⁷     

Performance of RS coded M-ary modulation with and without symbol overlapping

In this paper, we present analytical bit error probability results for M-ary modulation concatenated with Reed Solomon (RS) codes. The analysis of bit error probability is nontrivial as the number of bits per symbol for the RS codes may not be an integer multiple of the number of bits per symbol for a modulation symbol. We propose a Markov chain technique which allows analytical evaluation of the bit error probability for such cases. The performance of RS coding with coherent biorthogonal, coherent/non-coherent orthogonal modulation over an additive white Gaussian noise (AWGN) channel is evaluated. Simulation of the bit error probability of RS code concatenated with a Nordstrom Robinson (NR) code as an inner code is performed and compared with the case of biorthogonal modulation. From the results, we notice that a stronger inner code gives better bit error probability. In addition, the throughput of the coded system with biorthogonal modulation over an AWGN channel is discussed. For a Rayleigh flat fading and block fading channel, we analyze the bit error probability of RS codes concatenated with biorthogonal modulation. From the result, we notice that a stronger outer code gives a better bit error probability for the case of Rayleigh flat fading channel.     

Adaptive channel error protection of subband encoded images

Protection of images that are encoded using subband coding from channel error is addressed. In this scheme the low-pass subband is encoded using DPCM (differential pulse-code modulation), and the other subbands are encoded using a scalar quantizer. The quantizers are all Lloyd-Max quantizers, from which the representation levels have fixed length codewords. First, considering only single errors in each codeword, a channel error distortion measure is derived for each quantizer, that is, for each subband. Codewords are assigned to the quantizer representation levels, yielding a low value of the distortion measure. Next, sets S_ij consisting of the jth bit from subband i are formed. Each set S _ij is assigned a particular BCH code C_ij. An algorithm that optimally assigns BCH codes C_ij to each set S_ij, based on a channel error distortion measure for the entire image, is derived. The protection scheme is adaptive, because each set of bits within each subband can be assigned a different error protection code. Examples show that this approach is preferable to assigning equal error protection codes to each set of bits. It is shown that in the case of a channel error probability of 10 ^-3, only 5% to 10% extra bits are needed for adequate channel error protection     

Iterative decoding of binary block and convolutional codes

Iterative decoding of two-dimensional systematic convolutional codes has been termed “turbo” (de)coding. Using log-likelihood algebra, we show that any decoder can be used which accepts soft inputs-including a priori values-and delivers soft outputs that can be split into three terms: the soft channel and a priori inputs, and the extrinsic value. The extrinsic value is used as an a priori value for the next iteration. Decoding algorithms in the log-likelihood domain are given not only for convolutional codes but also for any linear binary systematic block code. The iteration is controlled by a stop criterion derived from cross entropy, which results in a minimal number of iterations. Optimal and suboptimal decoders with reduced complexity are presented. Simulation results show that very simple component codes are sufficient, block codes are appropriate for high rates and convolutional codes for lower rates less than 2/3. Any combination of block and convolutional component codes is possible. Several interleaving techniques are described. At a bit error rate (BER) of 10^-4 the performance is slightly above or around the bounds given by the cutoff rate for reasonably simple block/convolutional component codes, interleaver sizes less than 1000 and for three to six iterations     

Design, analysis, and performance evaluation for BICM-ID withsquare QAM constellations in Rayleigh fading channels

We consider bit-interleaved coded modulation with iterative decoding (BICM-ID) for bandwidth-efficient transmission over Rayleigh fading channels. We propose the design criteria that utilize a large Hamming distance inherited in a low-rate code and a new labeling technique designed specifically for fading channels. This results in a large coding gain over noniterative coded modulation and performance close to that of “turbo” coded modulation with less complexity. We also show that BICM-ID designed for fading channels usually has a very good performance over the additive white Gaussian noise (AWGN) channel while the converse is difficult to achieve. When combined with signal space diversity, diversity order can be improved to twice the diversity order of conventional BICM-ID; therefore, the code complexity can further be reduced while maintaining the same level of performance. Specifically, with the bandwidth efficiency of 2 bits/s/Hz over Rayleigh fading channels, a bit error rate (BER) of 10^-6 can be achieved with 16-QAM, a four-state rate 1/2 code at E_b/N₀ of about seven dB. We also derive performance bounds for BICM-ID with and without signal space diversity over Rayleigh fading channels, which can be easily extended for other types of fading channels     

Transmission performance analysis of a new class of line codes for optical fiber systems

A family of mB(m+1)B binary, nonalphabetic, balanced line codes is presented that is suitable for high bit rate (>or=135 Mb/s) optical fiber transmission due to its relatively simple encoding and decoding rules. Here, B represents a block of m bits, where m is an odd number. The coding, decoding, and bit error rate (BER) performance of the codes are discussed. Statistical and spectral analysis for the specific case in which the number of bits, m, equals seven, is presented. This makes possible a detailed comparison of the proposed code with conventional 7B8B codes.<>     

Huffman code based error screening and channel code optimization for error concealment in perceptual audio coding (PAC) algorithms

The class of perceptual audio coding (PAC) algorithms yields efficient and high-quality stereo digital audio bitstreams at bit rates from 16 kb/sec to 128 kb/sec (and higher). To avoid "pops and clicks" in the decoded audio signals, channel error detection combined with source error concealment, or source error mitigation, techniques are preferred to pure channel error correction. One method of channel error detection is to use a high-rate block code, for example, a cyclic redundancy check (CRC) code. Several joint source-channel coding issues arise in this framework because PAC contains a fixed-to-variable source coding component in the form of Huffman codes, so that the output audio packets are of varying length. We explore two such issues. First, we develop methods for screening for undetected channel errors in the audio decoder by looking for inconsistencies between the number of bits decoded by the Huffman decoder and the number of bits in the packet as specified by control information in the bitstream. We evaluate this scheme by means of simulations of Bernoulli sources and real audio data encoded by PAC. Considerable reduction in undetected errors is obtained. Second, we consider several configurations for the channel error detection codes, in particular CRC codes. The preferred set of formats employs variable-block length, variable-rate outer codes matched to the individual audio packets, with one or more codewords used per audio packet. To maintain a constant bit rate into the channel, PAC and CRC encoding must be performed jointly, e.g., by incorporating the CRC into the bit allocation loop in the audio coder.     

Determination of best shortened linear codes

The quality of a block code is determined by its capability to protect data against undetectable errors and by the number of check bits that are required for that purpose. For a given number of check bits there are codes with optimum bit error detecting capability in shortened block lengths. These codes are determined and tabulated. The residual error characteristics of some of the tabulated codes are compared with those specified in ISO/CCITT or IEC standard data transmission protocols. For block lengths and bit error rates that are typical in process control applications, the residual error rate of the determined codes is more than six orders of magnitude smaller than that of codes specified by widely used standard transmission protocols     

Zigzag codes and concatenated zigzag codes

This paper introduces a family of error-correcting codes called zigzag codes. A zigzag code is described by a highly structured zigzag graph. Due to the structural properties of the graph, very low-complexity soft-in/soft-out decoding rules can be implemented. We present a decoding rule, based on the Max-Log-APP (MLA) formulation, which requires a total of only 20 addition-equivalent operations per information bit, per iteration. Simulation of a rate-1/2 concatenated zigzag code with four constituent encoders with interleaver length 65 536, yields a bit error rate (BER) of 10^-5 at 0.9 dB and 1.3 dB away from the Shannon limit by optimal (APP) and low-cost suboptimal (MLA) decoders, respectively. A union bound analysis of the bit error probability of the zigzag code is presented. It is shown that the union bounds for these codes can be generated very efficiently. It is also illustrated that, for a fixed interleaver size, the concatenated code has increased code potential as the number of constituent encoders increases. Finally, the analysis shows that zigzag codes with four or more constituent encoders have lower error floors than comparable turbo codes with two constituent encoders     

Test Method for Measuring Bit Error Rate of Pulsed Transceivers in Presence of Narrowband Interferers

The popularity of pulse-based transceivers can be attributed to the high information rates that can be achieved in such systems compared to traditional narrowband systems. However, such systems are usually low-power and transmission efficiency is severely affected by the interference signals from other moderate to high-power narrowband transmitters. Hence, during manufacturing, pulse-based systems must be characterized and tested for bit error rate (BER) performance in the presence of narrowband interferers. Usually, at large interference levels, the BER is moderate (10^-4) to high (10^-2). However, at low interference levels when very few bits are in error, the BER is low (10^-6 - 10^-10 ) and testing for the BER becomes time consuming. We propose a new measurement technique employing sinusoidal pulses such that the BER value obtained is significantly large (10^-3 - 10^-4 ) even at low interference levels. BER values obtained using sinusoidal pulses are highly correlated to the actual BER values. Hence, the actual BER can be accurately estimated in a much smaller time without actually performing the standard test. This method was implemented in hardware using an Altera field-programmable gate-array development board. From the measurements, BER estimation error was less than 3%. In addition, significant reduction (up to 100 x) in test time was obtained using the proposed method.     

Decoding performance of linear block codes using a trellis indigital mobile radio

Trellis decoding of linear block codes in a Rayleigh fading channel is discussed. Two methods for calculating metric values for each bit in a received block are considered: the values are calculated from the received signal envelope sample and from the demodulator output. Bit error rate (BER) performances of hard decision and trellis decoding are compared using Hamming (7, 4) and Golay (24, 12) codes in computer simulations and laboratory experiments. A simplified trellis decoding algorithm, in which the hard decision output of a bit with an envelope sample greater than the threshold value is accepted as correct, is presented. Laboratory experimental results for trellis decoding in combination with Gaussian minimum-shift-keying (GMSK) modulation and frequency detection are shown. The effect of n-bit A/D-conversion in signal envelope sampling is investigated experimentally. The results show that the trellis decoding algorithm improves BER performance