Optimization of signal sets for partial-response channels. I.Numerical techniques

Given a linear, time-invariant, discrete-time channel, the problem of constructing N input signals of finite length K that maximize minimum l₂ distance between pairs of outputs is considered. Two constraints on the input signals are considered: a power constraint on each of the N inputs (hard constraint) and an average power constraint over the entire set of inputs (soft constraint). The hard constraint, problem is equivalent to packing N points in an ellipsoid in min(K,N-1) dimensions to maximize the minimum Euclidean distance between pairs of points. Gradient-based numerical algorithms and a constructive technique based on dense lattices are used to find locally optimal solutions to the preceding signal design problems. Two numerical examples are shown for which the average spectrum of an optimized signal set resembles the water pouring spectrum that achieves Shannon capacity, assuming additive white Gaussian noise     

A convolutional single-parity-check concatenated coding scheme forhigh-data-rate applications

The coding scheme uses a set of n convolutional codes multiplexed into an inner code and a (n,n-1) single-parity-check code serving as the outer code. Each of the inner convolutional codes is decoded independently, with maximum-likelihood decoding being achieved using n parallel implementations of the Viterbi algorithm. The Viterbi decoding is followed by additional outer soft-decision single-parity-check decoding. Considering n=12 and the set of short constraint length K=3, rate 1/2 convolutional codes, it is shown that the performance of the concatenated scheme is comparable to the performance of the constraint length K=7, rate 1/2 convolutional code with standard soft-decision Viterbi decoding. Simulation results are presented for the K=3, rate 1/2 as well as for the punctured K=3, rate 2/3 and rate 3/4 inner convolutional codes. The performance of the proposed concatenated scheme using a set of K=7, rate 1/2 inner convolutional codes is given     

On the Hamming distance properties of group codes

Under certain mild conditions, the minimum Hamming distance D of an (N, K, D) group code C over a non-abelian group G is bounded by D⩽N -2K+2 if K⩽N/2, and is equal to 1 if K>N/2. Consequently, there exists no (N, K, N-K+1) group code C over an non-abelian group G if 1<K<N. Moreover, any normal code C with a non-abelian output space has minimum Hamming distance equal to D=1. These results follow from the fact that non-abelian groups have nontrivial commutator subgroups. Finally, if C is an (N, K, D) group code over an abelian group G that is not elementary abelian, then there exists an (N, K, D) group code over a smaller elementary abelian group G'. Thus, a group code over a general group G cannot have better parameters than a conventional linear code over a field of the same size as G     

Effect of conduction-band discontinuity on lasing characteristicsof 1.5 μm InGaAs/In(Ga)AlAs MQW-FP lasers

The characteristic temperature (T₀), relaxation frequency (f_r), differential gain (dg /dn) and nonlinear gain coefficient (ϵ) of 1.5-μm InGaAs/In(Ga)AlAs multiple-quantum-well (MQW) Fabry-Perot (FP) lasers grown by gas source molecular beam epitaxy (GSMBE) are reported. It is found that T₀ is little affected by the difference in the conduction band discontinuity. A maximum T₀ value of 86 K is obtained. The dg/dn and ϵ∈ were calculated from the slope of the f_r versus √ power plot and the damping K-factor. It is demonstrated that dg/dn and ϵ of InGaAs/In(Ga)AlAs MQW lasers increase with an increase in the conduction band discontinuity     

Comments on `GMSK with differential phase detection in thesatellite mobile channel'

The error probability results shown by I. Korn (see ibid., vol.38, no.11, p.1980-6, 1990) indicate that the error floor is higher for systems with decision feedback (DF). It was concluded that DF gives a lower error probability only for smaller values of the normalized bandwidth B_tT of the premodulation Gaussian filter, higher values of the ratio of powers in the direct and diffuse signal components K, and a lower range of signal-to-noise ratio. It is shown that this conclusion is not correct by theoretically analyzing the case of the land mobile channel where K=0 (or -∞ dB) and deriving a simple closed-form expression for the error probability for 1 bit differential detection with DF. It is shown that DF reduces the error probability for all values of B_tT and signal-to-noise ratios. The formula derived can be easily evaluated not only for Gaussian minimum shift keying (GMSK) but for all partial-response continuous-phase-modulation (PRCPM) signals     

The bandwidth performance of a two-element adaptive array withtapped delay-line processing

The bandwidth performance of a two-element adaptive array with a tapped delay line behind each element is examined. It is shown how the number of taps and the delay between taps affect the bandwidth performance of the array. An array with two weights and one delay behind each element is found to yield optimal performance (equal to that obtained with continuous-wave interference) for any value of intertap delay between zero and T₉₀/B, where T ₉₀ is a quarter-wavelength delay time and B is the fractional signal bandwidth. Delays less that T₉₀ yield optimal performance but result in large array weights. Delays larger than T₉₀/B yield suboptimal signal-to-interference-plus-noise ratio when each element has only two weights. For delays between T₉₀/B and 4T₉₀/B , the performance is suboptimal with only two taps but approaches the optimal if more taps are added to each element. Delays larger than T₉₀/B result in suboptimal performance regardless of the number of taps used     

A pragmatic approach to trellis-coded modulation

Since the early 1970s, for power-limited applications, the convolutional code constraint length K=7 and rate 1/2, optimum in the sense of maximum free distance and minimum number of bit errors caused by remerging paths at the free distance, has become the de facto standard for coded digital communication. This was reinforced when punctured versions of this code became the standard for rate 3/4 and 7/8 codes for moderately bandlimited channels. Methods are described for using the same K=7, rate 1/2 convolutional code with signal phase constellations of 8-PSK and 160PSK and quadrature amplitude constellations of 16-QASK, 64-QASK, and 256-QASK to achieve, respectively, 2 and 3, and 2, 4, and 6 b/s/Hz bandwidth efficiencies while providing power efficiency that in most cases is virtually equivalent to that of the best Ungerboeck codes for constraint length 7 or 64 states. This pragmatic approach to all coding applications permits the use of a single basic coder and decoder to achieve respectable coding (power) gains for bandwidth efficiencies from 1 b/s/Hz to 6 b/s/Hz     

Asymptotically optimal classification for multiple tests withempirically observed statistics

The decision problem of testing M hypotheses when the source is Kth-order Markov and there are M (or fewer) training sequences of length N and a single test sequence of length n is considered. K, M, n, N are all given. It is shown what the requirements are on M , n, N to achieve vanishing (exponential) error probabilities and how to determine or bound the exponent. A likelihood ratio test that is allowed to produce a no-match decision is shown to provide asymptotically optimal error probabilities and minimum no-match decisions. As an important serial case, the binary hypotheses problem without rejection is discussed. It is shown that, for this configuration, only one training sequence is needed to achieve an asymptotically optimal test     

Experiments on postdetection diversity for narrowband digital FMmobile radio

Postdetection diversity, in which the demodulator outputs are weighted in proportion to the vth power of each demodulator input signal envelope when they are added, is described for GMSK signal reception using a frequency demodulator and a one-bit decision feedback equaliser. Experiments on a 16 kbit/s GMSK with a premodulation filter bandwidth-bit duration product of B_bT=0.25, show that using v=2 provides a diversity gain about 1-1.5 dB larger than selection combining, at an average bit error rate of 10^-2 in a Rayleigh fading environment     

Trellis coding with multidimensional QAM signal sets

Trellis coding using multidimensional quadrature amplitude modulation (QAM) signal sets is investigated. Finite-size 2D signal sets are presented that have minimum average energy, are 90° rotationally symmetric, and have from 16 to 1024 points. The best trellis codes using the finite 16-QAM signal set with two, four, six, and eight dimensions are found by computer search (the multidimensional (multi-D) signal set is constructed from the 2-D signal set). The best moderate complexity trellis codes for infinite lattices with two, four six, and eight dimensions are also found. The minimum free squared Euclidean distance and number of nearest neighbors for these codes were used as the selection criteria. Many of the multi-D codes are fully rotationally invariant and give asymptotic coding gains up to 6.0 dB. From the infinite lattice codes, the best codes for transmitting J, J+1/4, J+1/3, J+1/2, J+2/3, and J+3/4 b/sym (J an integer) are presented     

Balancing sets of vectors

For n>0, d⩾0, n≡d (mod 2), let K(n, d) denote the minimal cardinality of a family V of ±1 vectors of dimension n, such that for any ±1 vector w of dimension n there is a v∈V such that |v- w|⩽d, where v-w is the usual scalar product of v and w. A generalization of a simple construction due to D.E. Knuth (1986) shows that K(n , d)⩽[n/(d+1)]. A linear algebra proof is given here that this construction is optimal, so that K(n, d)-[n/(d+1)] for all n≡d (mod 2). This construction and its extensions have applications to communication theory, especially to the construction of signal sets for optical data links     

A new extrapolation algorithm for band-limited signals using theregularization method

The ill-posedness of the extrapolation problem in the presence of noise is considered. A stable algorithm is constructed by solving a Fredholm equation based on a regularization method. The algorithm appears relatively robust, since the noise η_δ(t ) is taken as a function in L²[-T,T](T>0) such that the error energy ∫|η_δ(t)|² dt⩽δ², where integration is from - T to T, and the constructed extrapolation uniformly converges to the desired signal over (-∞, +∞) as δ→0. An estimate for the error energy of the constructed extrapolation over (-∞, +∞) and for the absolute error between the constructed extrapolation and the desired signal over (-∞, +∞) are presented     

Bounds for abnormal binary codes with covering radius one

The normality of binary codes is studied. The minimum cardinality of a binary code of length n with covering radius R is denoted by K(n,R). It is assumed that C is an (n,M)R code, that is, a binary code of length n with M codewords and covering radius R. It is shown that if C is an (n,M)1 code, then it is easy to find a normal (n ,M)1 code by changing C in a suitable way, and that all the optimal (n,M)1 codes (i.e. those for which M=K(n,1)) are normal and their every coordinate is acceptable. It is shown that if C is an abnormal (n,M) code, then n⩾9, and an abnormal (9118)1 code which is the smallest abnormal code known at present, is constructed. Lower bounds on the minimum cardinality of a binary abnormal code of length n with covering radius 1 are derived, and it is shown that if an (n,M)1 code is abnormal, then M⩾96     

New ternary line codes based on trellis structure

The trellis coding technique is applied to line-coded baseband digital transmission systems. For R=n/n+1(n=1,2,3) coding rates, a new codeword assignment model is proposed to accomplish basic requirements for line coding in which each length n binary data sequence is encoded into a length n+1 ternary (+,0,-) line codeword chosen among the code alphabet with 2ⁿ⁺² elements. Assuming Viterbi decoding, the system error performance is improved by increasing the free Euclidean distance between coded sequences. A new algorithm is given for the calculation of the free distance between line-coded sequences so obtained. For R=1/2 and R=3/4 rates, the analytical error performance upper bounds are derived. The power spectral densities of the new line codes are also calculated and compared with those of known line codes     

Nonequiprobable signaling on the Gaussian channel

Signaling schemes for the Gaussian channel based on finite-dimensional lattices are considered. The signal constellation consists of all lattice points within a region R, and the shape of this region determines the average signal power. Spherical signal constellations minimize average signal power, and in the limit as N →∞, the shape gain of the N-sphere over the N-cube approaches πe/6≈1.53 dB. A nonequiprobable signaling scheme is described that approaches this full asymptotic shape gain in any fixed dimension. A signal constellation, Ω is partitioned into T subconstellations Ω₀ , . . ., Ω_τ-1 of equal size by scaling a basic region R. Signal points in the same subconstellation are used equiprobably, and a shaping code selects the subconstellation Ω_i with frequency f_i. Shaping codes make it possible to achieve any desired fractional bit rate. The schemes presented are compared with equiprobable signaling schemes based on Voronoi regions of multidimensional lattices. For comparable shape gain and constellation expansion ratio, the peak to average power ratio of the schemes presented is superior. Furthermore, a simple table lookup is all that is required to address points in the constellations. It is also shown that it is possible to integrate coding and nonequiprobable signaling within a common multilevel framework     

Lower bounds for binary covering codes

G.D. Chen et al. (ibid., vol.IT-32, p.680-94, 1986) presented two new lower bounds for K(n,R), where K(n,R) denotes the minimum cardinality of a binary code of length n and covering radius R. The author shows that a slight modification gives further improvements and some examples are given to confirm the argument. Codes that have a certain partitioning property are considered     

Operation of poly bipolar transistors near liquid-heliumtemperatures (9 K)

Operation of bipolar transistors with poly emitters with current gain β>25 at 77 K and β>3 at 9 K has been demonstrated. In the temperature range of 300 K to 9 K, the plot of β vs. temperature (T) exhibits a minimum at T equal to 40 K: decreasing monotonically from 300 K to 40 K, remaining unchanged between 40 K to 30 K, and increasing slightly from 30 K to 9 K. This behavior is attributed to freezeout of acceptor impurities in the base for T<40 K     

Improved sphere bounds on the covering radius of codes

The sphere bound is a trivial lower bound on K(n,R), the minimal cardinality of any binary code of length n and with covering radius R. By simple arguments it is considerably improved, to K(n,1)⩾2 ⁿ/n for n even. A table of lower and upper bounds on K(n,R) for n⩽33, R ⩽10 is included     

The total PFA of the multiwindow harmonicdetector and its application to real data

The authors study the multiwindow spectral analysis method as it applies to the detection of sinusoidal signals. They examine the probability of false alarm P_FA. The total P _FA (sinusoidal frequency unknown) is shown analytically to be bounded below by the order statistics (minimum) of BM/K independent identically distributed (i.i.d.) beta variates, where M is the length of the data record used in the detection, K the number of windows, and B the width of the frequency band of interest. Simulation results indicate a much larger bound, the minimum of BM i.i.d. beta variates. It is shown that for real signals, the assumptions made in the derivation of the detector break down at frequencies close to zero and to half the sampling frequency