Low-power devices are massively deployed to facilitate real-time sensing and data transmission requested in low-power wide-area network. However, the dominating signaling, i.e., continuous phase modulation (CPM), barely gains attention in the context of supporting massive connectivity, not to mention ultra-reliable low-latency communications (URLLC), a primary concern in industrial network automation. To this end, autoconfigurable nonorthogonal multiple access (AC-NOMA) based on CPM signaling is proposed. The term autoconfiguration is used since each device selects a setup from a pool of configurations whenever connecting to the access point in a random and distributive way. It is proven, using ideal power allocation and phase shaping technique, that AC-NOMA offers drastically improved user load and near capacity performance even in finite blocklength regime. Moreover, to enable massive yet sporadic access, slotted ALOHA is combined with AC-NOMA. It is proven that the resultant scheme outperforms power-domain NOMA in terms of throughput even with reduced transmit power and simple forward error correction schemes, such as repetition and convolutional coding. The throughput is further improved using semi-AC-NOMA with slightly increased latency. It is demonstrated that both designs can support very high user load while enabling URLLC in finite blocklength regime, where the packet size is merely 256 bits while the error rate is$10^{-5}$, which are also desirable in a number of applications, including satellite communications, visible light communications, etc.

The acquisition of small-scale fading is essential but challenging in industrial Internet of Things (IIoT). This article concerns with the quality of service (QoS) provision for IIoT uplink networking, in terms of reliability, latency, and connectivity without resorting to small-scale fading. To this end, a scheme based on multiantenna nonorthogonal multiple access (mNOMA) technique is proposed. Different from existing setups, the proposed system is designed to work in overloaded regime where the number of devices$K$significantly outnumbers$N_{r}$, the antennas equipped by the access point, i.e.,$K\gg N_{r}$, such that concurrent massive connectivity is enabled using off-the-shelf equipment. The design starts with the development of asymptotic performance analysis in terms of signal to interference-plus-noise ratio, where a closed-form expression is obtained taking massive connectivity, finite blocklength, and code rate into consideration jointly. The analysis implies the possibility of QoS provision without estimating small-scale fading at transmitter side. Then the insight is leveraged to develop mNOMA with special focus on minimum shift keying type waveform, which is widely adopted in low power wide area IIoT standards including IEEE 802.15 series. Power allocation and code rate adaption are employed to offer QoS guaranteed performance as the blocklength is only 512 bits, whereas dozens of devices simultaneously transmit at block error rate down to$10^{-5}$or even lower, as required by industrial 5.0 and beyond.

This paper focuses on the design and performance analysis of single user detectable multiuser systems based upon continuous phase modulation. This problem is formulated as a special application of mismatched receiver theory, where a closed-form expression for achievable Euclidean distance can be derived. This expression inspires the constructions of two classes of spectrally efficient multiuser designs: nonorthogonal and orthogonal designs. They are both confirmed to offer strong robustness against nonlinear hardware impairment in downlink and uplink transmissions, but differ in spectral efficiency, complexity and achievable minimum Euclidean distance. Moreover, the proposed designs are able to reduce the convergence threshold in serially concatenated multiuser systems. As a general approach to constructing orthogonal and near-orthogonal signals, the proposed techniques find potential applications in multiantenna systems, and full-duplex communications. Compared with existing continuous phase modulated multiuser designs, the proposed techniques offer improved spectral efficiency and significantly reduced detection complexity.

This paper focuses on the design and performance analysis of continuous phase modulated orthogonal multiple access systems. The multiuser interference is firstly derived in the context of single user detection, which reveals that the interference is determined by the modulation format. Therefore, the interference can be successfully suppressed by carefully chosen modulation parameters. Based on this observation, a class of orthogonal multiple access schemes is proposed. It is shown that the proposed strategy not only facilitates simple detection at the receiver side but also offers improved sum rate.

The performance analysis of single user detection of continuous phase modulated multiuser systems is presented in this paper. The discussion is formulated in the context of a mismatched receiver and the performance of single user detection is analyzed in terms of Euclidean distance. Taking the modulation parameters into consideration, a closed-form expression for the performance evaluation is obtained. Principles of designing the transceiver for spectrally-efficient systems are proposed. Numerical and analytical results show that properly designed spectrally efficient multiuser systems can be successfully detected by the presented detector with low complexity.

This paper concerns with the design of spectrally-efficient Frequency-Division-Multiple-Access (FDMA) multiuser systems based on Continuous-Phase-Modulation (CPM). A class of FDMA-CPM is proposed to accommodate multiple users in a subchannel and therefore admits significantly improved spectral-efficiency. Phase spacing is incorporated into the system to improve the Euclidean distance property, especially for FDMA-CPM systems with small frequency spacing. The impacts of the phase spacing and frequency spacing are then evaluated in terms of the distance property, which shows that the proposed FDMA-CPM systems have a comparable error performance to the single user systems. The proposed strategy is exemplified by systems having spectral-efficiency varying from 1.67 bits/s/Hz to 12 bits/s/Hz, which shows significantly improved spectral efficiency is obtained comparing with existing systems given the same signal to noise ratio.

In this paper, we propose a distributed network coding (DNC) scheme based on the Raptor codes for wireless relay networks (WRNs), where a group of source nodes communicate with a single sink through a common relay network in a multi-hop fashion. At the sink, a graph-based Raptor code is formed on the fly. After receiving a sufficient number of encoded packets, the sink begins to decode. The main contributions of this paper are the derivations of upper and lower bit error rate (BER) bounds for the proposed Raptor-based DNC scheme.

This paper concerns developing a low complexity detector for spectrally-efficient Frequency-Division-Multiple-Access (FDMA) multiuser systems based on Continuous-Phase-Modulation (CPM). A class of tree-trellis-search based algorithms is proposed for joint/iterative detection and the complexity is further reduced employing a state-reduction technique based on mismatched filters. The proposed detector is evaluated in terms of minimum achievable Euclidean distance. Analytical and simulated results reveal that the proposed detector offers near optimum performance with significantly reduced complexity.

In this paper, we propose a distributed network coding (DNC) scheme based on the Raptor codes for wireless sensor networks (WSNs), where a group of sensor nodes, acting as source nodes, communicate with a single sink through some other sensor nodes, serving as relay nodes, in a multi-hop fashion. At the sink, a graph-based Raptor code is formed on the fly. After receiving a sufficient number of encoded packets, the sink begins to decode. The main contributions of this paper are the derivation of a bit error rate (BER) lower bound for the LT-based DNC scheme over Rayleigh fading channels under maximum-likelihood (ML) decoding, and the derivations of upper and lower BER bounds for the proposed Raptor-based DNC scheme on the basis of the derived BER bound of LT codes.

The performance of maximum likelihood detection of a coded modulation scheme over an additive white Gaussian noise channel can be predicted precisely by its Euclidean distance spectrum. In this paper, three algorithms are developed to generate the Euclidean distance spectra. The first algorithm is an exhaustive tree-search algorithm which is simplified by exploiting the correspondence between a code tree and the trellis. This algorithm is then modified to incorporate the A* algorithm and results in two heuristic algorithms. Moreover, the technique of trellis minimization is further developed for the multi user case and employed to reduce the complexity and memory requirements. Numerical results confirm that the proposed algorithms can generate the distance spectra of all systems with lower complexity relative to previous methods and are particularly suited for multi user CPM systems.

An Improved Maximum A Posteriori (IMAP) decoder for turbo-coded decode-and-forward (DF) relay channels which takes into account the decoding errors at the relay is analyzed. This decoder is implemented in an iterative manner similar to the traditional iterative decoder (TID) used for turbo codes. The performance of this IMAP decoder is compared with the performance of an iterative decoder normally used in the literature which does not take into account the decoding errors at the relay, by simulation. The comparison shows that although the proposed IMAP decoder provides a better performance, especially when many decoding errors occur at the relay, the improvement is not significant. Then, another heuristic modification for the iterative decoder is proposed. In spite of the lack of theoretical analysis, the numerical results show that the proposed heuristically modified iterative decoder (HMID) gives significantly better performance than the traditional one.

We study joint network and channel code design to optimize delay performance. Here the delay is the transmission time of information packets from a source to sinks without considering queuing effects. In our systems, network codes (network layer) are on top of channel codes (physical layer) which are disturbed by noise. Network codes run in a rateless random method, and thus have erasure-correction capability. For the constraint of finite transmission time, transmission errors are inevitable in the physical layer. A detection error in the physical layer means an erasure of network codewords. For the analysis, we model the delay of each information generation in the network layer as independent, identically distributed random variables. The calculation approaches for delay measures are investigated for coded erasure networks. We show how to evaluate the rate and erasure probability of a set of channels belonging to one cut. We also show that the min-cut determines the decoding error probability in the sinks if the number of information packets is large. We observe that for a given amount of source information, larger packet length leads to fewer packets to be transmitted but higher physical-layer detection error probabilities. Further, longer transmission time (delay) in the physical-layer causes smaller detection error probability at the physical layer. Thus, both parameters have opposite impacts on the physical and network layer, considering delay. We should find the optimal values of them in a cross-layer approach. We then formulate the problems of optimizing delay performance, and discuss solutions for them.

We use the sum-product algorithm (SPA) to propose a reduced-complexity detector for serially concatenated continuous phase modulated (SCCPM) signals with noisy channel state information (CSI) at the receiver. Two channel effects-phase noise (PN) and flat fading-are tackled using the statistical model and associated factor graph (FG) of the problem. Although optimal performance is not assured due to the loopy nature of the FG, as well as simplifying assumptions made in the algorithm development, simulations show that very significant performance gains are possible, and the gap between systems with noiseless and noisy CSI can be reduced to about 1 dB using the proposed methods. The increase in complexity compared to the conventional decoder, which treats CSI estimates as if they were equal to their true values, is small.

We investigate sink decoding approaches and performance analysis for a network with intermediate node encoding (coded network). The network consists of statistically independent noisy channels. The sink bit error probability (BEP) is the performance measure. First, we investigate soft-decision decoding without statistical information on the upstream channels (the channels not directly connected to the sink). Numerical results show that the decoder cannot significantly improve the performance from a hard-decision decoder. We develop union bounds for analysis. The bounds show the asymptotic (regarding SNR: signal-to-noise ratio) performance of the decoder. Using statistical information about the upstream channels, we can find the error patterns of final hop channels (channels directly connected to sinks).With the error patterns, maximum-likelihood (ML) decoding can be performed, and a significant improvement in the BEP is obtained. To evaluate the union bound for the ML decoder, we use an equivalent point procedure. It is reduced to the least-squares problem with a linear constraint in the medium-to-high SNR region. With deterministic knowledge of the errors in the upstream channels, a genie-aided decoder can further improve the performance. We give the union bound for the genie decoder, which is straightforward to evaluate. By analyzing these decoders, we find that knowledge about the upstream channels is essential for good sink decoding.

In a recent paper (Bai ,IEEETrans.Inf.Theory, 2007), an improved analysis concerning the analysis of List Decoding was presented. The event that the correct codeword is excluded from the list is central. For the additive white Gaussian noise (AWGN) channel an important quantity is the in (Bai ,IEEETrans.Inf.Theory, 2007) calledeffectiveEuclideandistance. This was earlier considered in (Aulin,IEEETrans.Commun., 1999) under the nameVectorEuclideanDistance(VED), where also a simple mathematical expression for this quantity was easily derived for any list size. In (Bai ,IEEETrans.Inf.Theory, 2007) , a geometrical analysis gives this when the list size is1,2,or3.

We investigate a systematic construction of binary deterministic rateless codes (BDRCs). The codes are for networks with erasure channels. With a maximum distance separable (MDS) property, non-systematic BDRCs were first proposed in [1] with encoding complexity O(K), and decoding complexity is O(K2). Here K is the length of information bits. To reduce complexity, we study systematic-BDRCs (SBDRCs). For SBDRCs, the source first transmits m - 1 uncoded blocks, where m is the number of source blocks. Then, the source produces and transmits coded blocks in a rateless way. These coded blocks are produced using only cyclic-shift and XOR (exclusive or). The SBDRCs can use a large number of information blocks (potentially infinite m). On receiving any m distinct blocks (uncoded or coded), a sink can rebuild the source. The SBDRCs have encoding complexity O(∈K), and decoding complexity O(∈2K2), where ∈ is the source-to-sink block erasure probability.

In this paper, a novel trellis source encoding scheme based on punctured ring convolutional codes is presented. Joint source and channel coding (JSCC) using trellis coded continuous phase modulation (CPM) with punctured convolutional codes over rings is investigated. The channels considered are the additive white gaussian noise (AWGN) channel and the Rayleigh fading channel. Optimal soft decoding for the proposed JSCC scheme is studied. The soft decoder is based on the a posteriori probability (APP) algorithm for trellis coded CPM with punctured ring convolutional codes. It is shown that these systems with soft decoding outperform the same systems with hard decoding especially when the systems operate at low to medium signal-to-noise ratio (SNR). Furthermore, adaptive JSCC approaches based on the proposed source coding scheme are investigated. Compared with JSCC schemes with fixed source coding rates, the proposed adaptive approaches can achieve much better performance in the high SNR region. The novelties of this work are the development of a trellis source encoding method based on punctured ring convolutional codes, the use of a soft decoder, the APP algorithm for the combined systems and the adaptive approaches to the JSCC problem.

We investigate the calculation approach of the sink bit error probability (BEP) for a network with intermediate node encoding. The network consists of statistically independent noisy channels. The main contributions are, for binary network codes, an error marking algorithm is given to collect the error weight (the number of erroneous bits). Thus, we can calculate the exact sink BEP from the channel BEPs. Then we generalize the approach to nonbinary codes. The coding scheme works on the Galois field 2m, wheremis a positive integer. To reduce computational complexity, a subgraph decomposition approach is proposed. In general, it can significantly reduce computational complexity, and the numerical result is also exact. For approximate results, we discuss the approach of only considering error events in a single channel. The results well approximate the exact results in low BEP regions with much lower complexity.

It has previously been shown that serially concatenated trellis coded modulation (SCTCM) with a non-recursive inner convolutional code can achieve convergence close to capacity. In this paper, we supplement the analysis by comparing recursive and non-recursive inner codes with respect to the bit error rate (BER) of the error floor region. We show that the high error floor, commonly attributed to serially concatenated codes using a non-recursive inner code, can be reduced significantly through careful code design. The considered system is iteratively decoded SCTCM with an outer single parity check (SPC) code and a two-state rate 3/4 inner convolutional code whose output is mapped to 8PSK. Simulation results are shown for the additive white Gaussian noise (AWGN) channel.

We investigate sink decoding methods and performance analysis approaches for a network with intermediate node encoding (coded network). The network consists of statistically independent noisy channels. The sink bit error probability (BEP) is the performance measure. We first discuss soft-decision decoding without statistical information on the upstream channels (the channels not directly connected to the sink). The example shows that the decoder cannot significantly improve the BEP from the hard-decision decoder. We develop the union bound to analyze the decoding approach. The bound can show the asymptotic (regarding SNR: signal-to-noise ratio) performance. Using statistical information of the upstream channels, we then show the method of maximum-likelihood (ML) decoding. With the decoder, a significant improvement in the BEP is obtained. To evaluate the union bound for the ML decoder, we use an equivalent signal point procedure. It can be reduced to a least-squares problem with linear constraints for medium-to-high SNR.

In this paper, we propose multi-labeled signal mapping as a means to improve the performance of bit interleaved coded modulation with iterative decoding (BICM-ID). By using an inner mapping that assigns multiple code symbols to each signal point we can improve the distance properties of the BICM- ID system. The use of an outer block code will compensate for the higher information loss that the multi-labeling inevitably entails. Hence, convergence at low signal-to-noise ratios (SNR's) will still be possible. The considered system is BICM-ID with 2 bits/symbol and 8PSK signaling. Simulation results are obtained for the additive white Gaussian noise (AWGN) channel.

We investigate a deterministic binary coding approach for combination networks. In the literature, network coding schemes with large alphabet sizes achieve the min-cut capacity. Here, we propose an approach using binary (GF(2)) sequences instead of going to a large alphabet size. In the encoding process, only cyclic-shifting and XOR operations are used. The encoding complexity is linear with the length of information bits. The transfer matrix is sparse, and the decoder can perfectly decode source information by a sparse- matrix processing approach. Our approach does not use any redundant bits, and achieves the min-cut capacity. Further, the code blocks can be produced in a rateless way. The sink can decode source information from any subset of code blocks, if the number of received distinct blocks is the same as that of the information blocks. Thus, we use the code for general networks with erasure channels. The proposed binary rateless codes have quite small overheads and can work with a small number of blocks. With high probability, the codes behave as maximum distance separable (MDS) codes.

Average performance bounds based on uniform interleaving are considered for a serial concatenation consisting of an outer code, an interleaver and an inner coded modulation system. The exact input–output weight spectrum is derived for the case when only the outer code has the uniform error property. Thereby, an upper bound to the bit error probability under maximum likelihood detection is obtained. However, numerous examples for serially concatenated continuous phase modulation show that neglecting the non-uniform error property of the inner system gives an accurate and less complex approximation. Other inner coded modulations may behave differently.

Joint source and channel coding (JSCC) using trellis coded quantization (TCQ) in conjunction with trellis coded continuous phase modulation (CPM) is studied. The channel is assumed to be the additive white gaussian noise (AWGN) channel. Analytical bounds on the channel distortion for the investigated systems with maximum-likelihood sequence detection (MLSD) are developed. The bounds are based on the transfer function technique, which was modified and generalized to include continuous-amplitude discrete-time signals. For a memoryless uniform source, the constructed bounds for the investigated systems are shown to be asymptotically tight for increasing channel signal-to-noise ratio (SNR) values. For a memoryless nonuniform source, the constructed bounds are not as tight as the one for the uniform source, however, it still can be used as an indication to how the system performs. It is concluded that the minimum Euclidean distance of the system alone is not enough to evaluate the performance of the considered systems. The number of error events having minimum Euclidean distance and the total distortion caused by those error events also affect the asymptotic performance. This work provides an analysis tool for the investigated systems. The analysis method is very general. It may be applied to any trellis based JSCC schemes.

We investigate the detection of space time codes (STC) modulated using continuous phase modulation (CPM), in quasi-static fading channels. A symbol-by-symbol iterative detector with an optimum front-end is derived, and its reduced-complexity implementation is considered. Also, we have introduced several full-rank STC-CPM systems, constructed by combining full-diversity STCs with simple, widely-used CPM schemes, and introducing a small frequency offset. According to the simulation results, the bit error rate (BER) performance of the detector for quasi-static fading is around 1 dB from the case without fading. Furthermore, a comparison of the performance of the proposed detector with results from previous work on STC-CPM showed an improvement in the range of 2-3 dB. Moreover, the detector is robust to the errors in estimating the channel state information (CSI), which is a desirable feature for practical implementation.

We investigate the analysis and design of low density generator matrix (LDGM) codes for continuous phase modulation (CPM). The system uses LDGM codes as an outer code for CPM. For additive white Gaussian noise channels, we derive the union bound to analyze the error floor performance. Design principles for lowering error floors are suggested from this analysis. We propose a design approach of jointly considering the LDGM code degree and the CPM modulation index. Then we consider the rate-adaptive system for slowly fading channels. By changing the rate of the LDGM codes, the information rate of the CPM signals is adapted according to channel variations. We use a low-rate LDGM code as the mother code. Higher rates are achieved by puncturing the output of these codes. To exploit the rate-flexible property of punctured LDGM codes, a rate function is proposed to calculate the rate of each transmitted block. Thus, we can have a quasi-continuous information rate. Numerical results show that this approach can improve the energy efficiency from a discrete-rate adaptation. Using the rate-adaptive approach, up to 11 dB transmitted energy gain can be achieved from the non-adaptive scheme in the low bit-error-rate region (smaller than 10-3) for minimum shift keying (MSK).

We investigate the physical layer bit error probability (BEP) of a network using network coding (coded network). The network consists of statistically independent noisy channels. Thus, transmitted bits are subject to the noise disturbance. An error marking algorithm is given to calculate the error weight (the number of erroneous bits) in the sinks. Then we can calculate the sink BEP of a coded network from the channel BEPs. We formulate the minimizing BEP problem. We show that coding schemes with the same flow may have different sink BEPs. Two approaches are suggested to reduce the complexity.

We use the sum-product algorithm to handle the problem of phase noise (PN) in the detection of a serially concatenated continuous phase modulated system. Although optimal performance is not assured due to the loopy nature of the factor graph describing the problem, as well as simplifications made in the algorithm development, simulations show that significant performance gains are possible, and the gap between systems with and without PN can be reduced to about 1 dB using the proposed method. The increase in complexity compared to ignoring PN is small, and no periodic pilot symbols are inserted within a code word, although we assume the availability of a phase-locking mechanism to set the initial phase offset to a small range around zero. We use a different Gaussian approximation from what has been previously proposed, and this results in a less complex algorithm

For serially concatenated codes, the general conception has been that one should only use recursive convolutional codes as inner codes. The reason for this is that non-recursive inner codes have error events of input distance one, that induces an irreducible error floor. However, as we show in this paper, this problem can largely be overcome by minimizing the role of input distance one error events for non-recursive inner convolutional codes. The system we consider is iteratively decoded serially concatenated trellis coded modulation (SCTCM) where the outer code is a single parity check code and the inner code is a two-state rate 3/4 non-recursive convolutional code whose output is mapped to 8PSK. Simulation results are obtained for the additive white Gaussian noise (AWGN) channel

We investigate energy-efficient network coding with a bit error probability (BEP) constraint. The network consists of statistically independent binary noisy channels. An error marking algorithm is used to calculate the number of erroneous bits in the sinks. Then we can accurately calculate the BEP in the sinks from the channel BEPs. For a given coding scheme, we formulate the optimal energy allocation problem. We show that the problem is convex for BPSK modulation. Then, we check the problem of the joint optimal network coding and energy allocation. A procedure is formulated to solve the problem. A pruning rule is suggested to reduce the search effort. Numerical results show the energy saving from the equal energy allocation and non-optimal coding

We investigate rate-adaptive continuous phase modulation (CPM) with punctured low-density generator matrix (LDGM) codes. We give an adaptive principal component analysis (PCA) CPM receiver for slow-fading channels. With the merit of the PCA receiver, the new receiver avoids the eigenvalue decomposition in every symbol interval, and uses a fixed filter-bank. By changing the rate of the LDGM codes, the information rate of CPM signals is adapted to the channel state information (CSI). We use a low rate LDGM code as the mother code, and higher rates are achieved by puncturing the output of the LDGM codes. To exploit the rate-flexible property of punctured LDGM codes, a rate function is proposed to calculate the rate of each transmitted block. Thus, we can have a quasi-continuous information rate. Numerical results for MSK show that up to 11 dB transmitted energy gain can be achieved in the low bit-error-rate region (smaller than 10-3), compared to the non-adaptive systems

We investigate the bit error probability (BEP) of a network using network coding (coded network). The network consists of independent binary noisy channels. An error marking algorithm is used to calculate the error weight (the number of erroneous bits) in the sinks. Hence, we can calculate the sink BEP of a coded network from the channel BEPs. Then we formulate the minimizing BEP problem. We show that coding schemes with the same flows may have different BEPs. Two approaches are given to reduce the complexity for the problem

In this paper, we investigate serially concatenated continuous phase modulation (SCCPM) with convolutional codes (CC) over rings. The transmitted signals are disturbed by additive white Gaussian noise. The properties for systems with both infinite and finite block lengths are investigated. For an infinite-length system, we check the convergence threshold using the extrinsic information transfer chart. For a finite-length system, we use union-bound techniques to estimate the error floors. In the union-bound analysis, we consider both the order and the position of nonzero permuted symbols. A simple method for determining a CPM error event through the sum of the input symbol sequence is shown. Thus, we can determine if the output symbol sequence of an error event in the ring CC can form an error event in CPM. Two properties concerning the interleaver gain (IG) are investigated. A recursive search algorithm for the maximal IG is shown. Compared with previous SCCPM with a binary CC, the proposed system shows an improvement concerning the convergence threshold or error floors

Joint source and channel (JSC) coding using combined trellis coded quantization (TCQ) and continuous phase modulation (CPM) is studied. The channel is assumed to be the additive white Gaussian noise (AWGN) channel. Optimal soft decoding for JSC coding using jointly designed TCQ/CPM is studied in this paper. The soft decoder is based on the a posteriori probability (APP) algorithm for trellis coded CPM. It is shown that the systems with soft decoding outperform the systems with hard decoding especially when the systems operate at low to medium signal-to-noise ratio (SNR). Furthermore, a TCQ design algorithm for the noisy channel is developed. It has been demonstrated that the combined TCQ/CPM systems are both power and bandwidth efficient compared with the combined TCQ/TCM/8PSK systems. The novelty of this work is the use of a soft decoder and the APP algorithm for combined TCQ/CPM systems.

We investigate the low density generator matrix (LDGM) codes for continuous phase modulation (CPM). The nonsystematic version of LDGM codes is used in the scheme. The overall system has linear encoding complexity due to the low complexity of the LDGM codes. A property of the check node degree of the LDGM code is shown. We use the EXIT chart/function to optimize the LDGM codes. The EXIT function of CPM with a fixed SNR (signal-to-noise-ratio) is shown. We derive the union bound to analyze the error floor performance. Design approaches for lowering error floors are suggested from the analysis process. Numerical results show that this scheme converges earlier (lower SNR) than best found serially concatenated CPM (SCCPM) for iterative decoding while maintaining comparable error floors.

We investigate the detection of space time codes (STC) modulated using continuous phase modulation (CPM) in quasi-static fading channels. An optimum symbol-by-symbol iterative detector is derived and its practical implementation using the concept of principal component analysis (PCA) is considered. To assess the performance, numerical examples for different STC-CPM schemes are presented. According to the simulation results, the proposed detector obtains very good power efficiencies. Moreover, the bit error rate (BER) performance of the detector for quasi-static fading is very close to the case without fading, highlighting its effectiveness to mitigate fading.

Joint source and channel (JSC) coding using combined trellis coded quantization (TCQ) and continuous phase modulation (CPM) is studied. The channel is assumed to be the additive white Gaussian noise (AWGN) channel. Analytical bounds on the channel distortion for jointly designed TCQ/CPM systems with maximum likelihood sequence detection (MLSD) are developed. For a memoryless uniform source, our upper bounds for the investigated TCQ/CPM systems are shown to be asymptotically tight for increasing channel signal-to-noise ratio (SNR) values. It is concluded that the minimum Euclidean distance of the CPM system alone is not enough to evaluate the performance of the combined TCQ/CPM systems. The number of the error events having the minimum Euclidean distance and the total distortion caused by those error events also affect the asymptotic performance of the investigated systems. This work provides an analysis tool to estimate the performance for a given combined TCQ/CPM system. The analysis method is very general, it may be applied to any trellis based JSC coding schemes.

Joint source and channel coding (JSCC) using combined trellis coded quantization (TCQ) and trellis coded continuous phase modulation (CPM) is studied. Optimal soft decoding based on the a posteriori probability (APP) algorithm for trellis coded CPM is developed for the considered systems. It has been demonstrated that the combined TCQ/CPM systems are both power and bandwidth efficient compared with the combined TCQ/TCM/8PSK systems. Furthermore, a TCQ design algorithm for the noisy channel is developed. The novelties of this work is the use of a soft decoder and the APP algorithm for combined TCQ/CPM systems.

An iterative soft decoding approach to joint source and channel coding using combined trellis coded quantization (TCQ) and continuous phase modulation (CPM) is proposed. This iterative procedure exploits the structure of the TCQ encoder and the continuous phase modulator. It is observed that the combined TCQ/CPM systems are both power and bandwidth efficient compared with the combined TCQ/TCM system. Based on extrinsic information density evolution, a convergence analysis of combined TCQ/CPM with iterative decoding is presented. The conventional convergence analysis techniques based on the uniform error property of geometrically uniform codes has been modified and generalized to the codes with a non-uniform error property.

In-line bit interleaving has over the last decade been used in a number of papers, where it has proven to he very effective. However, little or nothing has been said about its advantages and why it should be used in favor of conventional bit interleaving. In this paper, we offer an analysis of the inline bit interleaver and elucidate the advantages of using it. The mutual information (MI) constellation is introduced as a useful tool, both for the analysis and the code design. With its assistance, we show how some common code design principles rely on conventional bit interleaving. The system considered is serially concatenated trellis coded modulation (SCTCM), consisting of a serially concatenated convolutional code (SCCC) of rate 2/3 and an 8PSK mapper. Simulation results are given for the additive white Gaussian noise (AWGN) channel. The results show a clear improvement in the bit error rate (BER) performance for systems using in-line bit interleaving.

We investigate the performance of space time codes (STCs) modulated using continuous phase modulation (CPM) for different quasi-static Rayleigh fading channels. We propose few STC-CPM schemes that satisfy the full-rank condition, which is necessary to achieve maximum diversity, and also provide some guideline on how to find such STC-CPM schemes. To assess the performance, numerical examples are presented, and according to these results, the proposed STC-CPM systems perform similar to linearly modulated STCs

In this paper, a novel trellis coded quantization (TCQ) scheme based on ring convolutional codes is presented. Joint source and channel coding (JSCC) using trellis coded continuous phase modulation (CPM) with convolutional codes over rings is investigated. The channel is assumed to be an additive white Gaussian noise (AWGN) channel. Optimal soft decoding for the proposed JSCC scheme is studied. The soft decoder is based on the a posteriori probability (APP) algorithm for trellis coded CPM with ring convolutional codes. It is shown that these systems with soft decoding outperform the same systems with hard decoding especially when the systems operate at low to medium signal-to-noise ratio (SNR). It has been demonstrated that TCQ based on ring convolutional codes is superior to conventional TCQ of the same complexity. The novelties of this work are the development of a TCQ method based on ring convolutional codes, the use of a soft decoder and the APP algorithm for the combined systems

A new code design method for the inner code of bit interleaved serially concatenated trellis coded modulation (SCTCM) is proposed. The new design method is based upon the characteristics of bit interleaved coded modulation with iterative decoding (BICM-ID). We use the fact that the BICM encoder can be seen as an SCTCM encoder with a one-state inner code. New design rules for SCTCM are obtained by extending the design principles of BICM-ID to be valid for more than one state. The system considered is SCTCM of rate 2 bits/symbol with 8PSK signaling over the additive white Gaussian noise (AWGN) channel

We investigate the detection of space time coded (STC) signals modulated using continuous phase modulation (CPM) in fast Rayleigh fading channels. A symbol-by-symbol iterative detector with an optimum front-end is derived, and its practical implementation using the concept of principal component analysis (PCA) is considered. To assess the performance, numerical examples for different STC-CPM schemes are presented. According to the simulation results, the proposed detector obtains very good power efficiencies. Moreover, the bit error rate (BER) performance of the proposed detector is in the range of 0.5-1 dB from the case without fading, highlighting its effectiveness to mitigate fading.

In this paper, a new source encoding scheme is presented. It is a trellis coded quantization scheme which is based on ring convolutional codes. Joint Source and Channel Coding (JSCC) using trellis coded Continuous Phase Modulation (CPM) with convolutional codes over rings is investigated. The channel is assumed to be an Additive White Gaussian Noise (AWGN) channel. Optimal soft decoding for the proposed JSCC scheme is studied. The soft decoder is based on the A Posteriori Probability (APP) algorithm for trellis coded CPM with ring convolutional codes. It is shown that these systems with soft decoding outperform the same systems with hard decoding especially when the systems operate at low to medium Signal-to-Noise Ratio (SNR). It has been demonstrated that TCQ based on ring convolutional codes is superior to conventional TCQ of the same complexity. The novelties of this work are the development of a TCQ method based on ring convolutional codes, the use of a soft decoder and the APP algorithm for the combined systems.

In this paper, we will explain the usefulness of what we call the mutual information (MI) constellation. We will show how it can be used both in analysis and in design of codes. The MI constellation will be of special interest in case of in-line bit interleaving, since it is then very useful in the design of the outer code. The system considered is 2 bits/symbol serially concatenated trellis coded modulation (SCTCM) with XPSK signaling over the additive white Gaussian noise (AWGN) channel.

We propose a new scheme of serially concatenated continuous phase modulation (SCCPM) by using nonsystematic low density generator matrix (LDGM) codes as the outer code. A property of the LDGM code degree is investigated. We use the exit chart/function to optimize the LDGM codes. The exit function of CPM with a fixed SNR (signal-to-noise-ratio) is shown. We derive the union bound to analyze the error floor performance. Design principles are proposed from the analysis process. Numerical results show that this scheme converge earlier (lower SNR) than previous SCCPM for iterative decoding.

We investigate the union bound for symbol interleaved serially concatenated codes (SCC) with non-uniform error properties. We calculate the distance spectrum of the SCC using the input difference symbol sequence spectrum (DSSS) of the inner code. Then we modify the trellis search algorithm for general codes to calculate the DSSS. We also show the mapping approach between difference symbols with the different alphabet size. For the symbol interleaver, we use the multinomial coefficient to calculate the probability of the permuted sequence. Numerical results for symbol interleaved serially concatenated continuous phase modulation (SCCPM) show that the bound is tight in the medium to high signal-to-noise ratio (SNR) region.

An iterative soft decoding approach to joint source and channel coding using combined trellis coded quantization (TCQ) and continuous phase modulation (CPM) is proposed. The channel is assumed to be AWGN. The performance in terms of the signal-to-distortion ratio (SDR) is compared with that for a combined TCQ with trellis coded modulation (TCM) system. It is observed that the combined TCQ/CPM systems are both power and bandwidth efficient compared with the combined TCQ/TCM system. A convergence analysis, based on the uniform error property of geometrically uniform codes, has been generalized to codes with a nonuniform error property. For source encoding rate R=1 and R=2 bits/sample, the combined systems working at symbol level converge faster than those working at bit level. This can be explained by examining the error events of the combined system. The novelty of this work is the use of a soft decoder and an iterative decoding algorithm for TCQ based joint source channel coding systems. Also, combined TCQ/CPM systems operating at symbol level are considered for the first time.

Techniques for computing soft information in the presence of unknown intersymbol interference are presented, with a particular focus on iterative detection of serially concatenated continuous phase modulation. The techniques are centered around the recursive least-squares algorithm, thus enabling unsupervised detection. In particular, we employ bidirectional estimation.

An iterative decoding approach to joint source and channel coding (JSCC) using combined trellis-coded quantization (TCQ) and continuous phase modulation (CPM) is proposed. The channel is assumed to be the additive white Gaussian noise channel. This iterative procedure exploits the structure of the TCQ encoder and the continuous phase modulator. The performance in terms of the signal-to-distortion ratio (SDR) is compared with that of a combined TCQ/trellis-coded modulation (TCM) system. It is shown that the combined TCQ/CPM systems are both power- and bandwidth-efficient, compared with the combined TCQ/TCM system. For source encoding rate R=2 b/sample, it is observed that the combined TCQ/CPM systems with iterative decoding working at symbol level converge faster than the systems working at bit level. The novelty of this work is the use of a soft decoder and an iterative decoding algorithm for TCQ-based JSCC systems. The combined TCQ/CPM with iterative decoding is considered for the first time.

Serially concatenated continuous phase modulation (SCCPM) systems with symbol interleavers are investigated. The transmitted signals are disturbed by additive white Gaussian noise (AWGN). Iterative detection with extrinsic information is used at the receiver side. Compared to bit interleaved SCCPM systems, this scheme shows a substantial improvement in convergence threshold at the price of a higher error floor. In addition to showing this property, we also investigate the underlying reason by error event analysis. In order to estimate bit error rate performance, we generalize traditional union bounds for a bit interleaver to the non-binary interleaver. For the latter, both the order and the position of permuted non-zero symbols have to he considered. From the analysis, some principal properties are presented. Finally some design principles are proposed. The paper concentrates on SCCPM, but the proposed analysis methods and conclusions can he widely used in many other systems such as serially concatenated trellis coded modulation, et cetera.

We propose a new stopping criterion for iterative decoding of concatenated codes. The new stopping rule observes the a priori probabilities of different symbols and stops further iterations in the decoder whenever the average number of symbols in a trellis section below a certain a priori value, is above a predefined threshold. According to numerical examples for some practical transmission systems, the proposed criterion saves more decoding iterations than existing stopping rules. Also, it performs very close to the maximum possible limit of any stopping rule, with a very small degradation in bit error rate (BER) performance.

We propose a new reduced-complexity iterative decoding algorithm. The main concept is to approximate the aposteriori probabilities (APP) by a set of few sequences, and to reduce the size of this set with iterations. According to the numerical results on the bit error rate (BER) performance and the convergence behavior by extrinsic information transfer (EXIT) chart analysis, the algorithm could obtain a good balance between performance and complexity.

Serially concatenated continuous phase modulation (SCCPM) systems with ring convolutional codes (CC) are investigated. Both EXIT chart and union bound techniques are used to compare with the simulation results and are used as analysis tools. The latter is for the first time generalized to a nonbinary interleaver for a serially concatenated system, where both the order and position of the permuted nonzero symbols have to be considered.

A new algorithm to reduce the computational complexity of iterative decoding is presented. The algorithm reduces the complexity by removing certain branches from the corresponding trellis diagram. A threshold on a priori values is used as the elimination criterion. To assess the performance, numerical results for serially concatenated convolutional codes transmitted over an additive white Gaussian noise channel is presented. The results show that for the system considered, the algorithm can reduce the complexity by one third with a negligible degradation in error performance.

Symbol error probability bounds for maximum likelihood sequence detection (MLSD) of continuous phase modulation (CPM) signals are studied. The calculation of the upper bound is based on the transfer function technique, which has been generalized. A new method for constructing a lower bound for CPM systems is proposed. From numerical comparisons, it can be seen that the proposed algorithm can substantially improve the lower bound compared to previous approaches by considering the entire set of transmitted data sequences and not only the worst case. This generalized algorithm may be applied to any system which can be described as finite state machine.

We consider serially concatenated block codes in a hybrid ARQ scheme using iterative decoding. The extrinsic information generated in the iterative decoding process is saved and used when a retransmission is decoded. Two different strategies are examined; one using the extrinsic information only in the very first iteration, whereas the other uses it in all subsequent iterations until another retransmission arrives. The latter can be seen as turbo or concatenated code combining whereas the former, where the extrinsic information is used only once may be seen as code doping, providing an alternative perspective. The strategy of saving the extrinsic information is also compared to traditional type-III, equal gain diversity combining. Using the extrinsic information from previous retransmission is shown to improve performance not only in terms of bit error rate but also in terms of throughput and convergence speed and requires only negligible additional decoder complexity. The performance of this strategy is however not as good as simple equal gain combining. As a consequence, the investigated schemes are not competitive alternatives, however, the code doping procedure can be used in conjunction with traditional diversity combining schemes, improving further on convergence speed.

SA(B, C) is an algorithm penetrating a tree (or trellis) breadth first. It performs maximum-likelihood sequence detection (MLSD) under that structural constraint, and also under the complexity constraints determined by the parameters B and C. First, C is the number of partitions into which the states are distributed, and B denotes the number of paths in each partition. Recursively selecting paths which are closest to the received signal in the Euclidean distance (Hamming distance) sense guarantees constrained MLSD for the additive white Gaussian (binary symmetric) channel. The previously presented vector Euclidean distance (VED) is an important tool for analyzing SA(B, C) over the additive white Gaussian noise (AWGN) channel. A geometric interpretation of the signals involved clarifies the basic properties of this VED and other relevant general results (invariance, monotonicity). These results also form a basis for the construction of an algorithm for the efficient and fast calculation of minimum VEDs (of interest for large signal-to-noise ratio, SNR, detection performance). This, in turn, reveals the necessary complexity requirements to meet specified performance requirements for concrete trellis-coded systems. Here, the simple example of convolutionally coded (rate 1/2) with Gray-coded quaternary phase-shift keying over the AWGN channel is considered. When C=1 and choosing B/spl sim//spl radic/S (S being the number of states in the code trellis) gives the same asymptotic detection performance (large SNR) as unconstrained MLSD (e.g., implemented using the Viterbi algorithm).

An antenna array is proposed as a means of achieving a space-diversity effect that partly overcomes the severity of continuous-time Rayleigh fading channels. The investigated channel is assumed to be frequency-nonselective with correlated diversity links, where the correlation is related to the array geometry and the spatial and Doppler dispersions. Further, the error performance is improved by bit interleaving and channel coding, where the encoders/channel is viewed as a serially concatenated system: a convolutional code constitutes the outer code, whereas a differential encoder and the fading channel (having truncated memory) form a joint inner code. In order to obtain a practical detector structure it is desirable to perform iterative decoding by applying some a posteriori probability (APP) algorithms. For this purpose, we propose a novel generalization of the well-known Bahl-Cocke-Jelinek-Raviv (1974) algorithm that calculates the APPs over channels having memory. Numerical results indicate that iterative decoding becomes more powerful when the exploited channel memory depth is extended. Also, the error performance is significantly improved by introducing multiple antennas. The interleaver gain is, however, seen to be quite moderate, in contrast to additive white Gaussian noise channels.

The orthonormal basis for the space spanned by a given signal set can be chosen in many different ways. However, when the basis is truncated to fewer dimensions, the quality of the approximated signals differs, depending on the choice of the original basis. We study two energy-related quality measures and show that the optimal lower-dimensional approximation is given by the principal components (PC) method, which is also a simple and efficient alternative to Gram-Schmidt techniques. In addition, we derive and bound the average decrease in squared Euclidean distances over one symbol interval caused by the PC method. This measure is relevant in serially concatenated continuous phase modulation, where a manifold of signal pairs contributes to the bit-error rate for low-to-medium signal-to-noise ratios. By a numerical evaluation, we find that for this measure, the decrease is lower than that of a previous method by J. Huber and W. Liu (see IEEE J. Select. Areas Commun., vol.7, p.1437-49, 1989). Finally, we compare the minimum squared Euclidean distance for error events, which is relevant for uncoded CPM systems. Here, the loss with the PC method is generally larger than with Huber and Liu's method, although examples of the opposite exist.

Convergence analysis of iterative detectors for narrow-band multiple access is performed using extrinsic information transfer charts. The system has no bandwidth expansion, so K users use the same bandwidth as a single user. The load (the number of bits used per channel) of the system is therefore higher than the load in, for example, conventional CDMA systems. The analysis provides useful guidelines on how to combine outer codes with the chosen mapper for different loads of the system. Both memoryless mappers and differential mappers are evaluated. We present the limitations of the system measured with the maximum number of users and the required SNR for different scenarios.

Trellis-code multiple-access (TCMA) is a narrow-band multiple-access scheme based on trellis-coded modulation. There is no bandwidth expansion, so K users occupy the same bandwidth as one single user. The load of the system, in number of bits per channel use, is therefore much higher than the load in, for example, conventional code-division multiple-access systems. Interleavers are introduced as a new feature to separate the users. This implies that the maximum-likelihood sequence detector (MLSD) is now too complex to implement. Iterative detectors are therefore suggested as an alternative to the joint MLSD. The conventional interference cancellation (IC), detector has lower complexity than the MLSD, but its performance is shown to be far from acceptable. Even after a novel improvement of the IC detector, the performance is unsatisfactory. Instead of using IC, another iterative detector is suggested. This detector updates the branch metric for every iteration, and avoids the standard Gaussian approximation. Simulations show that the performance of this detector can be close to single-user performance, even when the interleaver and the phase offset are the only user-specific features in the TCMA system.

Trellis code multiple access (TCMA) is a narrowband multiple access scheme. There is no bandwidth expansion, so K users are using the same bandwidth as one single user. The load (the number of bits per channel use) of the system is therefore much higher than the load in, for example, conventional CDMA systems. It is shown here that the maximum number of users in a TCMA system is usually modest. This fact is based on constellation-constrained capacity, ie, the maximum capacity of a system with a specific constellation of the transmitted symbols. The minimum required signal-to-noise ratio for reliable transmission is also given for different code rates.

A convergence analysis of serially concatenated continuous phase modulation (SCCPM) with iterative decoding is presented, based on extrinsic information density evolution. Earlier conclusions from union bounds to maximum likelihood decoding that could not be verified by simulations can now be explained. This includes the effect of more complex outer codes and of inner CPM systems with input weight 1 error events. Due to their trivial feed-forward polynomial, all CPM systems provide good initial decoder convergence. Still the role of the CPM state complexity is not well explained. Finally, bit error rate performance per iteration is estimated from the diagrams, showing good coherence with simulation results.

Iterative detectors for trellis code multiple access (TCMA) are suggested as an alternative to the joint maximum likelihood sequence detector (MLSD). Interleavers are introduced in the TCMA system as a new feature to separate the users. This implies that MLSD is now too complex to implement. The conventional interference cancellation (IC) detector has lower complexity, but its performance is shown to be far from acceptable. Another iterative detector having close to single user performance is therefore suggested. Instead of using IC, this detector updates the branch metric for every iteration and avoids the standard Gaussian approximation. It is also shown that the users can be detected when the interleaver is the only user specific feature in the TCMA system.

The SA(B,C) algorithm is considered for continuous phase modulation (CPM). The SA(B,C) detector performs a breadth first maximum likelihood sequence detection in the trellis under structural and complexity constraints. Given a trellis, the SA(B,C) detector partitions the states into C classes and for every time interval selects B paths into each class. To find a partition that minimizes C for a given B and CPM system, such that the performance (error event probability) approaches unconstrained MLSD when the signal to noise ratio goes to infinity, is an NP-hard problem. Here a suboptimum method for finding the partitioning is considered. Substantial savings in complexity can be made by using the SA(B,C), while achieving almost optimal (unconstrained) error performance.

A multiple-antenna receiver is proposed for achieving a diversity effect that partly overcomes the severity of continuous-time frequency-flat Rayleigh fading channels, and fast fading channels in particular. The error performance is also improved by bit interleaving and channel coding, where the encoders/channel is viewed as a serially concatenated system: a convolutional code constitutes the outer code, whereas a differential encoder and the fading channel (having truncated memory) form a joint inner code. In order to obtain a feasible detector structure it is desirable to perform iterative decoding by applying some a posteriori probability (APP) algorithms. For this purpose, we derive a novel generalization of the well known BCJR (due to Bahl, Cocke, Jelinek, and Raviv 1974) algorithm that calculates the APPs for fading channels-or for channels having memory. Numerical results indicate that iterative decoding becomes more powerful when the exploited channel memory depth is extended, but the chief performance gain is achieved by introducing multiple antennas.

Trellis code multiple access (TCMA) is a narrow-band multiple access scheme. There is no bandwidth expansion, so K users are using the same bandwidth as one single user. The load (the number of bits per channel use) of the system is therefore much higher than the load in, for example, conventional CDMA systems. It is shown here that the maximum number of users in a TCMA system is usually modest. This fact is based on constellation-constrained capacity, i.e., the maximum capacity of a system with a specific constellation of the transmitted symbols. The minimum required signal-to-noise ratio for reliable transmission is also given for different code rates.

Serially concatenated and interleaved continuous phase modulation (CPM) with iterative decoding is investigated. An a posteriori probability (APP) algorithm for CPM is developed based on the classic APP algorithm for channel codes. The system is analyzed through upper bounds on the average bit error probability. For coded and interleaved minimum shift keying, the weight spectrum is computed, resulting in a transfer function bound. This is cumbersome for a general CPM system; instead, only the most significant error events contributing to the weight spectrum are identified. Simulations show that, firstly, these events give a satisfactory view of system performance when equal outer codes are used, and secondly, that remarkably good performance can be obtained for some simple systems. Finally, power spectral densities and bandwidths are computed, allowing for a bandwidth/performanee comparison of different combinations.

Serially concatenated continuous phase modulation (SCCPM) with iterative decoding is analyzed and simulated. Earlier results on outer rate 1/2 convolutional codes combined with inner binary and quaternary CPM systems are extended to rate 2/3 and 3/4 codes with low-complexity octal and hexadecimal CPM systems, having the 2RC and 3RC frequency pulses. Analysis and simulations show that-among the evaluated systems-the octal 2RC systems provide the highest power/bandwidth efficiency. As an example, rate 2/3-coded octal 2RC with modulation index 1/4 provides a 5.7 dB gain over uncoded minimum shift keying at a bit error rate of 10/sup -3/ and an input delay 4096 bits, without any bandwidth expansion.

Optimum (in the sense of minimum-error probability) single-symbol diversity detection for fading, noisy channels is too complex for practical implementation. A simplified, near-optimum array receiver is proposed, which is based on the statistics (i.e., the covariance-function matrix) of the fading channel. This detector is then analyzed by calculating the exact error probability. When there is a spread of the direction of arrival of the incident radio waves, the proposed detector significantly gains over an adaptive antenna array (which forms a weighted sum of the received antenna signals). Also, for this adaptive array, a fundamental difference between maximum-likelihood weights and least-mean-square weights is observed.

Traditional methods for multiple access (MA) are FDMA (frequency division MA), TDMA (time division MA) and CDMA (code division MA). The MA technique in this paper treats time, frequency and user signature sequences as a common resource to all users in the system. The only separation between users is accomplished by the use of a TCM (trellis coded modulation) scheme which is unique for each individual user. This access method is called trellis coded multiple access (TCMA). Assuming an additive, white and Gaussian (AWGN) channel, optimal (in the maximum likelihood sequence detection, MLSD, sense) multi-user detection is considered. The detection performance for each user is investigated using the union bound evaluated through a generalization of the transfer function (TF) approach and also by means of simulations. This paper outlines the analysis method and also gives some numerical results for TCMA systems having two or four users. The modulation is chosen to be M-ary phase shift keying (MPSK) for all users and it is possible that all users share the same modulation as well as having the same envelope magnitude. Thus, the only feature specific for each user is the choice of the convolutional encoder component in this case.

The problem of performing breadth-first maximum likelihood sequence detection (MLSD) under given structural and complexity constraints is solved and results in a family of optimal detectors. Given a trellis with S states, these are partitioned into C classes where B paths into each class are selected recursively in each symbol interval. The derived result is to retain only those paths which are closest to the received signal in the Euclidean (Hamming) distance sense. Each member in the SA(B, C) family of sequence detectors (SA denotes search algorithm) performs complexity constrained MLSD for the additive white Gaussian noise (AWGN) (BSC) channel. The unconstrained solution is the Viterbi algorithm (VA). Analysis tools are developed for each member of the SA(B, C) class and the asymptotic (SNR) probability of losing the correct path is associated with a new Euclidean distance measure for the AWGN case, the vector Euclidean distance (VED). The traditional Euclidean distance is a scalar special case of this, termed the scalar Euclidean distance (SED). The generality of this VED is pointed out. Some general complexity reductions exemplify those associated with the VA approach.

The optimal single symbol detector on the Rayleigh fading channel computes a functional quadratic form in the time continuous received signal. A drawback is that closed-form solutions of integral equations based on the channel statistics are required. This makes simplified discrete receivers attractive. A class of suboptimal receivers that transforms the received random process to a set of discrete observables is derived. The set of observables constitutes a random vector in a finite dimensional receiver signal space. Given this vector, the maximum likelihood detector computes a quadratic form in the received vector. The discretization implies a loss of information, therefore such a detector is not, in general, optimal given the received time continuous signal. The purpose is, however, to achieve close to optimal performance when the number of observables becomes large. This class of detectors is analyzed using exact error probability calculations, which reveal several interesting properties. The length of the observation interval and the number of discrete observables have significant influences on the error probability when the time variations of the fading process are rapid compared with the symbol duration. By increasing the number of observables, the error floor is lowered, and the implicit diversity order is increased. This implicit diversity arises as soon as more than one observable per symbol interval is used and is a consequence of the information bearing signal being a random process. Matched filter receivers use few discrete observables per symbol interval, and thus suffer from high error floors and low implicit diversity orders on fast fading channels. The error probability is highly dependent on the shapes and durations of the modulator waveforms. For instance, pulses of long duration give lower error probabilities than shorter pulses, and for a certain type of orthogonal waveforms there is no error floor.

In this letter, we derive a recursive, additive metric for complexity-constrained maximum likelihood detection for multiuser CDMA using breadth-first detection algorithms. The metric requires linear filtering of the matched-filtered received signal vector. It is shown that a class of filters fulfilling certain requirements lead to identical performance.

In this correspondence, we first correct a mistake in Best et al. (1995) and then present a method to compute the exact probability of information bit error as a function of truncation length K for a rate 1/2, 2 state code, Finally, we generalize the method to compute the exact bit-error probability of the above convolutional code when the information bits are transmitted in blocks. The simulation results are compared with the analytical results and good agreement is observed.

Concatenated encoders are found in many digital communication systems. A concatenated receiver for such a system consists of a chain of independently working estimators ended with an outer detector. Each stage of the receiver chain corresponds to a certain code in the system. Maximum likelihood sequence detection with a concatenated receiver requires soft information consisting of symbol aposteriori probabilities to be transferred between the receiver stages. In this paper the optimal inner estimator is derived and further simplified into a new asymptotically optimal one by imposing a tree structure on the inner code. By mapping several suboptimal inner estimators onto the tree structure and noting which approximations they make, a theoretical performance ranking is obtained. The asymptotical performance of the optimal concatenated detector at high signal-to-noise ratios (SNRs) is investigated. The decision variables underlying the pairwise error event probability are shown to be asymptotically Gaussian and consequently a free Euclidean distance can be found for the optimal concatenated detector.

In the above-named article [ibid., vol. 41, no.5, May 1993, pp. 649-654], code examples are given in the form of tables I-IV. Unfortunately, most of the highest order polynomials are incorrect. Two other polynomials are also incorrect. The correct values are given here.

The paper deals with detection of encoded discrete data transmitted over band-limited Gaussian channels with unknown parameters. The problem is to detect the data sequence and at the same time estimate the channel parameters such that the output from the decoder is as reliable as possible. Maximum likelihood (ML) detection fulfils this goal but, unfortunately, it leads to non-recursive and computationally complex algorithms. The approach taken in the present paper is to use the trellis corresponding to the combined states of the encoder and the channel. A modified Viterbi algorithm is then used. Simulation results show that when the channel is time-invariant and noisy, data can be detected with only 0.5 [dB] loss compared to ML-detection with the channel known to the receiver. The algorithm also performs well when the channel is time-varying. Compared to algorithms performing channel detection and decoding separately, the new algorithm is more complex but the performance loss is lower.>

An algorithmic approach is proposed whereby long convolutional codes of rate r/sub c/=k/n can easily be constructed for any chosen signal constellation in signal space. These algorithms are iterative, and in each step a number of candidate codes are found which locally maximize the distance (Hamming or Euclidean) between the codewords. The result is not necessarily a free-distance-optimizing code. However, since the construction complexity can be chosen, optimal codes are quite frequently found. The codes ae constructed such that a rapid growth of the column distance is achieved. A method of combining two codes into a single code of twice the constraint length is also presented.>

Partial response continuous phase modulation (CPM) gives constant envelope digital modulation schemes with excellent power spectra. Both narrow main lobe and low spectral tails can be achieved. When these signals are detected in an optimum coherent maximum likelihood sequence detector (Viterbi detector), power efficient schemes can also be designed, sometimes at the expense of receiver complexity. This paper describes a general class of simple Viterbi detectors with reduced complexity compared to the optimum case. The key idea is that the approximate receiver is based on a less complex CPM scheme than the transmitted scheme. The asymptotically optimum reduced-complexity receiver is found for a variety of transmitted schemes and various complexity reduction factors, for a specific class of receivers and modulation indexes. A new distance measure is introduced for the performance analysis. Smooth schemes based on raised cosine pulses are analyzed and simulated for the case of simplified reception. A graceful performance degradation occurs with the reduction of complexity.

Continuous phase frequency shift keying (CPFSK) is a constant amplitude modulation method with good spectral sidelobe properties. Good error probability properties can be obtained with coherent maximum-likelihood detection. In this paper we study the Euclidean distance properties of signals formed by a conventional rate1/2convolutional encoder followed by a binary or4-level CPFSK modulator. The minimum Euclidean distance is calculated for these signal sets as a function of the modulation index and the observation interval length. The optimum detector is discussed for rational modulation index values. The best obtainable codes are found for the case of short rate1/2codes with binary or4-level CPFSK modulation. Lists of the best codes are given. Among the results are that the noncatastrophic rate1/2convolutional codes with optimum free Hamming distance do not in general give the best Euclidean distance with CPFSK.

Recently bandwidth efficient constant envelope digital modulation schemes have been shown also to be power efficient, if detected coherently. In this paper the performance of such systems is analyzed for an optimum noncoherent or partially coherent detector, at large signal-to-noise ratios. The considered schemes are M-ary with arbitrary pulse shaping over one symbol interval (full response). The performance is analyzed by means of a parameter called equivalent minimum Euclidean distance, mathematically playing the same role as the minimum Euclidean distance used for coherent detectors. It is shown that noncoherent detectors perform as well as coherent, although the observation interval must be increased.

In sharp contrast to the task of calculating the power spectrum for a general digital FM scheme, the autocorrelation can easily be obtained. By considering a class of FM schemes having a time-limited autocorrelation function, truncation is avoided and, applying a Fourier transformation, power spectral densities are easily obtained. A method is applied taking these properties into account. Thus, a powerful tool is available for investigation of different parameters within the considered digital FM modulation format.

A class of smoothed phase modulation codes is analyzed in terms of power spectral density and error probability for large signal-to-noise ratios. The signals have constant envelope and the information-carrying phase function is of multi-hpartial response continuous phase type. A cyclic variation of modulation index from symbol time interval to symbol time interval is introduced. Coherent transmission and an additive white Gaussian channel are assumed. Both power- and bandwidth-efficient signals are to be found in the above class. Comparisons are made between fixed-hand multi-hsystems both for the binary and multilevel cases. Tradeoff is performed between bandwidth and error probability performance for large signal-to-noise ratios. It is concluded that quaternary and octal systems always seem preferable. Increasing the number of modulation indexes from one in the fixed-hcase to two in the simplest multi-hcase gives performance advantages at the expense of increased system complexity.

The problem of calculating accurate power spectral densities for an extended class of digital FM schemes is considered. This extended class includes partial response as well as multilevel schemes. A method suitable for numerical calculation using a computer is developed. This new method consists of two parts. First, the spectral main lobe is calculated. Then, the spectral tail is calculated by a different approach. The accuracy of the spectral tail is that of a correction term to an analytical asymptotic tail formula. This method yields accurate power spectra down to a level considerably below -100 dB. Numerical results obtained from calculations are given. The power spectra can be calculated for an arbitrary modulation index value.

The minimum Euclidean distance for a class of constant envelope phase modulation codes is studied. Bandwidth and power efficient signals with continuous phase are considered. The information carrying phase varies piecewise linearly and the slopes are cyclically changed for successive symbol time intervals, yielding the so-called multi-hsignals. It has previously been shown that this class of signals contains bandwidth and power efficient signals when coherent maximum likelihood sequence detection is used. Bounds on the achievable Euclidean distance for signals in the above class are given. Upper bounds are calculated as well as minimum distance results for specific multilevel multi-hsignals. It is concluded that quaternary and octal muiti-hschemes considerably outperform the binary schemes. Furthermore in the important small modulation index region,2-hcodes gain the maximum 3 dB. Larger gains are not available by increasing the number ofhvalues.

Constant envelope phase varying sinusoids of the form\sqrt{2E/T} \cos(\omega_{c}t + \phi(t))are studied, in which the phase function\phi(t)follows some coded pattern in response to data. Power and bandwidth performance are studied for such patterns. The patterns depend on a phase shaping function, a modulation index (h), and a sequence ofM-ary underlying changes in phase which are chosen at random. A cutoff rate-like parameter R0is computed, which guarantees existence of codes at all ratesR < R_{0}bits/T-interval whose error performance varies as exp[-N(R_{0} - R)], whereNis the code word length inT-intervals. Plots of R0are given as a function of interval energyE, the shaping functionhandM. Extensive spectral calculations give the spectra of these phased sinusoids, and their performance is plotted in the power-bandwidth plane. The results give strong evidence that phase codes can approximate any power-bandwidth combination consistent with Shannon's Gaussian channel capacity, and that linear channels are not required for narrow-band transmission.

Recently the minimum Euclidean distance and bandwidth properties of continuous phase modulated (CPM) signals have been considered. It has also been shown that, for rational modulation indexes, a state description of these signals is possible and that the Viterbi algorithm (VA) can be used for demodulation. In this paper the performance analysis of CPM systems is extended to obtain bounds on the symbol error probability when the VA is used. The calculation of these bounds is based on the transfer function technique, which has been generalized. From numerical comparisons of the upper and lower bounds, it is concluded that the minimum Euclidean distance is a good performance measure for a broad class of CPM signals, even when the symbol error probability is as large as 10-2.

An analysis of constant envelope digital partial response continuous Phase modulation (CPM) systems is reported. Coherent detection is assumed and the channel is Gaussian. The receiver observes the received signal over more than one symbol interval to make use of the correlative properties of the transmitted signal. The Systems areM-ary, and baseband pulse shaping over several symbol intervals is considered. An optimum receiver based on the Viterbi algorithm is presented. Constant envelope digital modulation schemes with excellent spectral tail properties are given. The spectra have extremely low sidelobes. It is concluded that partial response CPM systems have spectrum compaction properties. Furthermore, at equal or even smaller bandwidth than minimum shift keying (MSK), a considerable gain in transmitter power can be obtained. This gain increases withM. Receiver and transmitter configurations are presented.

The continuous phase modulation (CPM) signaling scheme has gained interest in recent years because of its attractive spectral properties. Data symbol pulse shaping has previously been studied with regard to spectra, for binary data and modulation index 0.5. In this paper these results have been extended to theM-ary case, where the pulse shaping is over a one symbol interval, the so-called full response systems. Results are given for modulation indexes of practical interest, concerning both performance and spectrum. Comparisons are made with minimum shift keying (MSK) and systems have been found which are significantly better inE_{b}/N_{0}for a large signal-to-noise ratio (SNR) without expanded bandwidth. Schemes with the same bit error probability as MSK but with considerably smaller bandwidth have also been found. Significant improvement in both power and bandwidth are obtained by increasing the number of levelsMfrom 2 to 4.