Trellis coding of quadrature frequency/phase modulated signals

The authors explore the trellis coding of continuous-phase quadrature frequency/phase modulated (CPQFPM) signal sets and continuous-phase FPM (CPFPM) signal sets, which are embodiments of the quadrature biorthogonal modulation (QBOM) technique. Conventional TCM and multiple TCM schemes with these modulation formats are examined using both the AWGN channel and the Rician fading channel design. Asymptotic coding gains in d²(free) are tabulated for trellis-coded rate 3/4, 5/6, and 6/7 QFPM schemes, in comparison with uncoded modulations (8AMPM, 32AMPM) and other trellis-coded modulations [TCM (2FSK/4PSK 16QAM, 64QAM), MTCM (2FSK/8PSK)] of equivalent throughput rate. Performance gains on the Rician fading channel are demonstrated by increased values of the design parameters for this channel, namely symbol diversity L_min and branch distance product P     

Performance analysis of nonblocking packet switch with input andoutput buffers

The performance of nonblocking packet switches such as the knockout switch and Batcher banyan switch for high-speed communication networks can be improved as the switching capacity L per output increases; the switching capacity per output refers to the maximum number of packets transferred to an output during a slot. The N×N switch with L=N was shown to attain the best possible performance by M.J. Karol et al. (1987). Here a N×N nonblocking packet switch with input and output buffers is analyzed for an arbitrary number of L such that 1⩽L⩽N. The maximum throughput and packet loss probability at input are obtained when N=∞     

Multifrequency trellis coding with low delay for fading channels

An L-frequency trellis coding scheme designed for fading channels that provides diversity up to order L is described. L-frequency coding is advantageous over channel coding schemes that achieve implicit time-diversity through trellis coding in conjunction with time-interleaving because it requires negligible end-to-end delay. The authors derive an upper-bound for the probability of symbol error of the L-frequency code. Based on this upper-bound, they then deduce the design criteria for optimum L-frequency codes. A numerical approximation to this upper-bound and the design of a two-state and two four-state, two-frequency codes are presented. Through simulations, it was found that relative to the traditional dual frequency diversity approach, the two-state and four-state codes achieve 3- and 4-dB coding gains, respectively. This energy efficiency gain can be translated into a spectral efficiency gain. A brief discussion on the difference in code design techniques for the Gaussian channel and the fading channel is also presented     

Error-correcting codes for list decoding

In the list-of-L decoding of a block code the receiver of a noisy sequence lists L possible transmitted messages, and is in error only if the correct message is not on the list. Consideration is given to (n,e,L) codes, which correct all sets of e or fewer errors in a block of n bits under list-of-L decoding. New geometric relations between the number of errors corrected under list-of-1 decoding and the (larger) number corrected under list-of-L decoding of the same code lead to new lower bounds on the maximum rate of (n,e,L) codes. They show that a jammer who can change a fixed fraction p<1/2 of the bits in an n-bit linear block code cannot prevent reliable communication at a positive rate using list-of- L decoding for sufficiently large n and an L⩽n. The new bounds are stronger for small n , but weaker for fixed e/n in the limit of large n and L than known random coding bounds     

Quadtree-structured recursive plane decomposition coding of images

The approximation of two-dimensional highly correlated grey value functions can be performed using a linear model of the type f( x, y)=a+bx+cy. The set of plane parameters (PPs) [a, b, c] can be determined in the least squares sense for a block of size N×N pixels, for example. Starting with a block size of 2×2 pixels, it is shown that the PPs obey a recursive law such that the PPs of a 2N×2N block can be computed recursively when only the PPs of the four adjacent subblocks of size N×N in the lower decomposition level are known. This concept of recursive plane decomposition (RPD) is embedded in a quadtree data structure to obtain a new variable block size image coding algorithm that offers a high performance at a low computational cost. Extensive comparisons to other state-of-the-art image coding algorithms are reported     

Bit error rate of binary and quaternary DPSK signals with multipledifferential feedback detection

A differentially coherent detection scheme with improved bit error rate (BER) performance is presented for differentially encoded binary and quaternary phase shift keying (PSK) modulation. The improvement is based on using L symbol detectors with delays of 1, 2, . . ., L symbol periods and on feeding back detected PSK symbols. Exact formulas for the bit error probability are derived for the case that correct symbols are fed back. The effect of symbol errors in the feedback path on the BER is determined by computer simulations     

Vector sets for exhaustive testing of logic circuits

(L, d)-universal sets are useful for exhaustively testing logic circuits with a large number of functional components, designed so that every functional component depends on at most d inputs. Randomized and deterministic constructions of ( L, d)-universal test sets are presented, and lower and upper bounds on the optimal sizes of such sets are proven. It is also proven that the design of an optimal exhaustive test set for an arbitrary logic circuit is an NP-complete problem     

The Zak transform and some counterexamples in time-frequencyanalysis

It is shown how the Zak transform can be used to find nontrivial examples of functions f, g∈L²(R) with f×g≡0≡F×G, where F, G are the Fourier transforms of f, g, respectively. This is then used to exhibit a nontrivial pair of functions h, k∈L²(R), h≠k, such that |h|=|k|, |H |=|K|. A similar construction is used to find an abundance of nontrivial pairs of functions h, k∈L² (R), h≠k, with |A_h |=|A_k| or with |W_h|=|W _k| where A_h, A_k and W_h, W_k are the ambiguity functions and Wigner distributions of h, k, respectively. One of the examples of a pair of h, k∈L²(R), h≠k , with |A_h|=|A_k| is F.A. Grunbaum's (1981) example. In addition, nontrivial examples of functions g and signals f₁≠f₂ such that f₁ and f₂ have the same spectrogram when using g as window have been found     

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     

On false lock in suppressed carrier MPSK tracking loops

The problem of false lock in suppressed-carrier minimum phase-shift keying (MPSK) tracking loops (M>4 in particular), such as Mth power phase-locked loop (PLL) and MPSK Costas loop carrier recovery subsystems, is investigated. It is demonstrated that such tracking loops false lock onto the received signal when the received carrier frequency and the reference signal frequency generated by MPSK tracking loops are mismatched by multiples of 1/M of the MPSK symbol rate. False lock margins (FLMs) for the suppressed carrier MPSK tracking loops are obtained for a noiseless system model as well as for a model corrupted by additive white Gaussian noise (AWGN). Numerical results are presented in order to explain the influence of system parameters, such as the time-bandwidth product of MPSK Costas loop arm filters (or the Mth power PLL prefilter), arm filters' output signal-to-noise ratio and the input signal-to-noise ratio, on the performance of MPSK tracking loops     

Optimization of signal sets for partial-response channels. II.Asymptotic coding gain

For Pt. I see ibid., vol.37, no.5, p.1327-141 (1991). For a linear, time-invariant, discrete-time channel with a given transfer function H(f), and information rate R bits/ T, where T is the symbol interval, an optimal signal set of length K is defined to be a set of 2^RK inputs of length K that maximizes the minimum l₂ distance between pairs of outputs. The author studies the minimum distance between outputs, or equivalently, the coding gain of optimal signal sets as K→∞. He shows how to estimate the coding gain, relative to single-step detection, of an optimal signal set length K when K is large     

On linear inductance- and capacitance-time conversions usingNIC-type configurations

The problem of linear inductance- and capacitance-time (L/T, C/T) conversion is approached through the systematic study of four approaches to building astable multivibrators using piecewise linear resistances obtained from one operational amplifier (OA) negative impedance converter (NIC) configuration. A new L/T converter with grounded inductance is found. Formulas for the time period taking into consideration the losses as well as the OA saturation output resistance are derived     

4-phase sequences with near-optimum correlation properties

Two families of four-phase sequences are constructed using irreducible polynomials over Z₄. Family A has period L =2^r-1. size L+2. and maximum nontrivial correlation magnitude C_max⩽1+√(L+1), where r is a positive integer. Family B has period L=2(2^r-1). size (L+2)/4. and C_max for complex-valued sequences. Of particular interest, family A has the same size and period as the family of binary Gold sequences. but its maximum nontrivial correlation is smaller by a factor of √2. Since the Gold family for r odd is optimal with respect to the Welch bound restricted to binary sequences, family A is thus superior to the best possible binary design of the same family size. Unlike the Gold design, families A and B are asymptotically optimal whether r is odd or even. Both families are suitable for achieving code-division multiple-access and are easily, implemented using shift registers. The exact distribution of correlation values is given for both families     

Performance analysis of the Josephson DC flip-flop

A linear analytical model of the Josephson DC flip-flop is proposed. The model describes both the Baechtold's and Hebard's flip-flops. The output signal line is treated as either a single inductance or a transmission line with a finite impedance. The former leads to the lumped model, while the latter leads to the distributed model. The lumped model gives the load condition for successful reset. This is given as a relationship between the CR and L/ R time constant, where C is the device capacitance, L is the load inductance, and R is the load resistance. The switching delay is also described as a linear function of the CR and L/R. With the distributed circuit model, the load condition for successful reset is Z₀⩾R. Minimum delay is obtained at Z₀=R. Grounding one end of the output signal line reduces the delay more than the nongrounded configuration. The scalar relationship of the switching delay and the power consumption to the design rule is discussed     

Maximum-rank array codes and their application to crisscross errorcorrection

A μ-[n×n,k] array code C over a field F is a k-dimensional linear space of n×n matrices over F such that every nonzero matrix in C has rank ⩾μ. It is first shown that the dimension of such array codes must satisfy the Singleton-like bound k⩽n(n-μ+1). A family of so-called maximum-rank μ-[n×n,k=n ( n-μ+1)] array codes is then constructed over every finite field F and for every n and μ, 1⩽μ⩽n . A decoding algorithm is presented for retrieving every Γ∈C, given a received array Γ+E, where rank (E)+1⩽(μ-1)/2. Maximum-rank array codes can be used for decoding crisscross errors in n×n bit arrays, where the erroneous bits are confined to a number t of rows or columns (or both). This construction proves to be optimal also for this model of errors. It is shown that the behavior of linear spaces of matrices is quite unique compared with the more general case of linear spaces of n×n. . .×n hyper-arrays     

Rotationally invariant nonlinear trellis codes for two-dimensionalmodulation

A general parity-check equation is presented that defines rotationally invariant trellis codes of rate k/(k+1) for two-dimensional signal sets. This parity-check equation is used to find rate k/(k+1) codes for 4PSK, 8PSK, 16PSK, and QAM signal sets by systematic code searches. The MPSK codes exhibit smaller free Euclidean distances than nonrotationally invariant linear codes with the same number of states. However, since the nonlinear codes have a smaller number of nearest neighbors, their performance at moderate signal to noise ratios is close to that of the best linear codes. The rotationally invariant QAM codes with 8, 32, 64, and 256 states achieve the same free Euclidean distance as the best linear codes. Transparency of user information under phase rotations is accomplished either by conventional differential encoding and decoding, or by integrating this function directly into the code trellis     

In the optimum design of the block adaptive FIR digital filter

A general optimum block adaptive (GOBA) algorithm for adaptive FIR (finite impulse response) filtering is presented. In this algorithm, the correction terms for the filter coefficients in each block, instead of the convergence factors, are optimized in a least squares sense. There are no constraints on the block length L and the filter tap number N. It is shown that the GOBA algorithm is reduced to the normalized LMS algorithm when L⩾N. The convergence of the GOBA algorithm can be assured if the correlation matrix of the input signal is positive definite. Computer simulations based on an efficient computing procedure confirm that the GOBA algorithm achieves faster convergence with slightly degraded convergence accuracy in stationary environments and better weight tracking capability in nonstationary environments as compared to existing block adaptive algorithms with no constraints on L and N     

Effect of parameter variations on the performance of GaAs/AlGaAsmultiple-quantum-well electroabsorption modulators

A theoretical investigation is presented of the dependence of electroabsorption in GaAs/Al_xGa_1-xAs multiple-quantum-well (MQW) structures on the MQW parameters (Al mole fraction x, well thickness L_z barrier thickness L_b and interface quality) and on the applied electric field studied. The on/off ratio of a modulator using MQWs with x=0.45, L_z=75 Å, and L _b=78 Å is predicted to increase by 20% compared to that of a modulator using MQWs with x=0.3, L_z =100 Å, and L_b=100 Å, when the MQW total active region thickness is 1 μm     

Light-to-input nonlinearities in an R-LED series network

The light-to-current (L-I) and light-to-voltage (L-V) differential nonlinearities in the simple network of a customary LED and an external resistor R in series are analyzed and calculated theoretically and compared with experimental data. Particular emphasis is placed on the influence of the log-arithmetic slope ν of the L-I characteristic and the bias current I upon the ratio of the corresponding nonlinearity parameters. It is thus deduced that, for a given optical power P, over superlinear portions of the L-I curve (ν>1) the L-I linearity is typically better than its corresponding L-V linearity. On the contrary, when the L-I dependence is sublinear (ν<1) the voltage driving scheme may ensure for the R-LED network, or the LED alone, a local L-V response much more linear than the L-I response, provided that appropriate (optimum) I and/or R values are chosen