Asymptotic properties of delay-time-weighted probability of error (Corresp.)

Asymptotic properties of expected distortion are studied for the delay-time-weighted probability of error distortion measured_n(x,tilde{x}) = n^{-1} sum_{t=0}^{n-1} f(t + n)[l - delta(x_t,tilde{x}_t)],, wherex = (x_0,x_1,cdots,x_{n-1})andtilde{x} = (tilde{x}_0,tilde{x}_1,cdots,tilde{x}_{n-1})are source and reproducing vectors, respectively, anddelta (cdot, cdot)is the Kronecker delta. With reasonable block coding and transmission constraintsx_tis reproduced astilde{x}_twith a delay oft + ntime units. It is shown that if the channel capacity is greater than the source entropyC > H(X), then there exists a sequence of block lengthncodes such thatE[d_n(X,tilde{X})] rigjhtarrow 0asn rightarrow inftyeven iff(t) rightarrow inftyat an exponential rate. However, iff(t)grows at too fast an exponential rate, thenE[d_n(X,tilde{X})] rightarrow inftyasn rightarrow infty. Also, ifC < H(X)andf(t) rightarrow inftythenE[d_n(X,tilde{X})] rightarrow inftyasn rightarrow inftyno matter how slowlyf(t)grows.     

On the capacity of certain additive non-Gaussian channels

A model of an additive non-Gaussian noise channel with generalized average input energy constraint is considered. The asymptotic channel capacityC_{zeta}(S), for large signal-to-noise ratioS, is found under certain conditions on the entropyH_{ tilde{ zeta}}( zeta)of the measure induced in function space by the noise processzeta, relative to the measure induced bytilde{zeta}, where is a Gaussian process with the same covariance as that ofzeta. IfH_{ tilde{zeta}}( zeta) < inftyand the channel input signal is of dimensionM< infty, thenC_{ zeta}(S)= frac{1}{2}M ln(1 + S/M) + Q_{zeta}( M ) + {o}(1), where0 leq Q_{ zeta}( M ) leq H_{ tilde{ zeta}}( zeta). If the channel input signal is of infinite dimension andH_{ tilde{ zeta}}( zeta) rightarrow 0forS rightarrow infty, thenC_{ zeta}(S) = frac{1}{2}S+{o}(1).     

On theepsilon-entropy and the rate-distortion function of certain non-Gaussian processes

Letxi = {xi(t), 0 leq t leq T}be a process with covariance functionK(s,t)andE int_0^T xi^2(t) dt < infty. It is proved that for everyvarepsilon > 0thevarepsilon-entropyH_{varepsilon}(xi)satisfies begin{equation} H_{varepsilon}(xi_g) - mathcal{H}_{xi_g} (xi) leq H_{varepsilon}(xi) leq H_{varepsilon}(xi_g) end{equation} wherexi_gis a Gaussian process with the covarianeeK(s,t)andmathcal{H}_{xi_g}(xi)is the entropy of the measure induced byxi(in function space) with respect to that induced byxi_g. It is also shown that ifmathcal{H}_{xi_g}(xi) < inftythen, asvarepsilon rightarrow 0begin{equation} H_{varepsilon}(xi) = H_{varepsilon}(xi_g) - mathcal{H}_{xi_g}(xi) + o(1). end{equation} Furthermore, ff there exists a Gaussian processg = { g(t); 0 leq t leq T }such thatmathcal{H}_g(xi) < infty, then the ratio betweenH_{varepsilon}(xi)andH_{varepsilon}(g)goes to one asvarepsilongoes to zero. Similar results are given for the rate-distortion function, and some particular examples are worked out in detail. Some cases for whichmathcal_{xi_g}(xi) = inftyare discussed, and asymptotic bounds onH_{varepsilon}(xi), expressed in terms ofH_{varepsilon}(xi_g), are derived.     

Optimum waveform signal sets with amplitude and energy constraints

It is desirable to choose the waveforms making up a signaling alphabet so that they are maximally separated one from another. This problem is considered, in the space of square-integrable functions, for signals which have finite duration, and are constrained in the ranges of their values as well as in energy. Corresponding to each of the following cases, we establish sharp bounds for the minimum distance and for the average distance between elements of a fixed size signal set, and construct sets of signals that attain both bounds simultaneously. begin{list} item {em Case A (Energy Constraint Only):} The average energy of the waveforms in the signal set is at mostsigma, where0 leq sigma < infty. item {em Case B (Energy and Peak Amplitude Constraints):} The average energy of the waveforms in the signal set isleq sigma (0 leq sigma < 1), and the absolute value of each waveform is at most1. item {em Case C (Energy and Value Constraints):} The average energy of the waveforms in the signal set is at mostb^{2}sigma + a^{2}(1 - sigma), and each waveform takes values in the set[a, b], where0 leq a < b < infty, and0 leq sigma leq 1. end{list} Cases A and B are applicable to signal design for communication in channels with additive noise (say Gaussian), and Case C is applicable to signal design for optical channels, where the signal represents the intensity of a photon stream. The general character of the results is that the minimum distance behaves likegamma sigmain Cases A and B, and likegamma sigma (1 - sigma)in Case C, withgammaa suitable constant.     

Hypothesis testing with multiterminal data compression

The multiterminal hypothesis testingH: XYagainstH̄: X̄Ȳis considered whereX^{n} (X̄^{n})andY^{n} (Ȳ^{n})are separately encoded at ratesR_{1}andR_{2}, respectively. The problem is to determine the minimumbeta_{n}of the second kind of error probability, under the condition that the first kind of error probabilityalpha_{n} leq epsilonfor a prescribed0 < epsilon < 1. A good lower boundtheta_{L}(R_{1}, R_{2})on the power exponenttheta (R_{1}, R_{2},epsilon)= lim inf_{n rightarrow infty}(-1/n log beta_{n})is given and several interesting properties are revealed. The lower bound is tighter than that of Ahlswede and Csiszár. Furthermore, in the special case of testing against independence, this bound turns out to coincide with that given by them. The main arguments are devoted to the special case withR_{2} = inftycorresponding to full side information forY^{n}(Ȳ^{n}). In particular, the compact solution is established to the complete data compression cases, which are useful in statistics from the practical point of view.     

Algorithms for the generation of full-length shift- register sequences

Two algorithms are presented for the generation of full-length shift-register cycles, also referred to as de Bruijn sequences. The first algorithm generates2^{k cdot g(n,k)full cycles of length2^{n}, using3n + k cdot g(n, k)bits of storage, wherekis a free parameter in the range1 leq k leq 2^{((n-4)/2)}, andg(n, k)is of the order ofn - 2 log k. The second algorithm generates about2^{n^{2}/4}full cycles of length2^{n}, using aboutn^{2}/2bits of storage. In both algorithms, the time required to produce the next bit from the lastnbits is close ton. A possible application to the construction of stream ciphers is indicated.     

Further classifications of codes meeting the Griesmer bound (Corresp.)

For any(n, k, d)binary linear code, the Griesmer bound says thatn geq sum_{i=0}^{k-1} lceil d/2^{i} rceil, wherelceil x rceildenotes the smallest integergeq x. We consider codes meeting the Griesmer bound with equality. These codes have parametersleft( s(2^{k} - 1) - sum_{i=1}^{p} (2^{u_{i}} - 1), k, s2^{k-1} - sum_{i=1}^{p} 2^{u_{i} -1} right), wherek > u_{1} > cdots > u_{p} geq 1. We characterize all such codes whenp = 2oru_{i-1}-u_{i} geq 2for2 leq i leq p.     

An upper bound for codes for the noisy two-access binary adder channel (Corresp.)

Using earlier methods a combinatorial upper bound is derived for|C|. cdot |D|, where(C,D)is adelta-decodable code pair for the noisy two-access binary adder channel. Asymptotically, this bound reduces toR_{1}=R_{2} leq frac{3}{2} + elog_{2} e - (frac{1}{2} + e) log_{2} (1 + 2e)= frac{1}{2} - e + H(frac{1}{2} - e) - frac{1}{2}H(2e),wheree = lfloor (delta - 1)/2 rfloor /n, n rightarrow inftyandR_{1}resp.R_{2}is the rate of the codeCresp.D.     

Effect of gap length on the input admittance of center fed coaxial waveguides and infinite dipoles

The input admittance of a coaxial waveguide fed by a gap of length2din the center conductor is evaluated using the dyadic Green's function of the guide and a band of equivalent magnetic surface current proportional to the gap's axial electric field via the equivalence principle. The axial electric field is expressed in terms of a rapidly convergent series of ultraspherical polynomials whose weighting function satisfies the edge conditions at each end of the gap. If the inner and outer radii of the coaxial guide areaandb, respectively, then the limiting case ofb rightarrow inftyis an infinite dipole in free space. Numerical results for the admittance are given as a function ofka (0.01 leq ka leq 0.50)with parameterb/a = 2, 5,10and 50 for the coaxial guide. For the infinite dipole the admittance is presented as a function ofd/a (10^{-3} leq d/a leq 10)withkaas a parameter (0.001 leq ka leq 0.1).     

When does the modular distance induce a metric in the binary case? (Corresp.)

The modular distance induces a metric if and only if the nonadjacent form of the modulusMhas one of the following forms:1) 2^{n}+2^{n-2} pm 2^{i}, wheren-igeq 4; 2) 2^{n} - 2^{j} pm 2^{i}, where2 leq n -j leq 5andj-igeq 2; 3) 2^{n} pm 2^{j}, wheren -j geq 2; 4) 2^{n}.     

A new class of codes meeting the Griesmer bound

An infinite sequence ofk-dimensional binary linear block codes is constructed with parametersn=2^{k}+2^{k-2}-15,d=2^{k-1}+2^{k-3}-8,k geq 7. Fork geq 8these codes are unique, while there are five nonisomorphic codes fork=7. By shortening these codes in an appropriate way, one finds codes meeting the Griesmer bound for2^{k-1}+2^{k-3}-15 leq d leq 2^{k-1}+2^{k-3}-8; k geq 7.     

Lower bounds for constant weight codes

LetA(n,2delta,w)denote the maximum number of codewords in any binary code of lengthn, constant weightw, and Hamming distance2deltaSeveral lower bounds forA(n,2delta,w)are given. Forwanddeltafixed,A(n,2delta,w) geq n^{W-delta+l}/w!andA(n,4,w)sim n^{w-l}/w!asn rightarrow infty. In most cases these are better than the "Gilbert bound." Revised tables ofA(n,2 delta,w)are given in the rangen leq 24anddelta leq 5.     

On certain projective geometry codes (Corresp.)

LetVbe an(n, k, d)binary projective geometry code withn = (q^{m}-1)/(q - 1), q = 2^{s}, andd geq [(q^{m-r}-1)/(q - 1)] + 1. This code isr-step majority-logic decodable. With reference to the GF(q^{m}) = {0, 1, alpha , alpha^{2} , cdots , alpha^{n(q-1)-1} }, the generator polynomialg(X), ofV, hasalpha^{nu}as a root if and only ifnuhas the formnu = i(q - 1)andmax_{0 leq l < s} W_{q}(2^{l} nu) leq (m - r - 1)(q - 1), whereW_{q}(x)indicates the weight of the radix-qrepresentation of the numberx. LetSbe the set of nonzero numbersnu, such thatalpha^{nu}is a root ofg(X). LetC_{1}, C_{2}, cdots, C_{nu}be the cyclotomic cosets such thatSis the union of these cosets. It is clear that the process of findingg(X)becomes simpler if we can find a representative from eachC_{i}, since we can then refer to a table, of irreducible factors, as given by, say, Peterson and Weldon. In this correspondence it was determined that the coset representatives for the cases ofm-r = 2, withs = 2, 3, andm-r=3, withs=2.     

On MDS extensions of generalized Reed- Solomon codes

An(n, k, d)linear code overF=GF(q)is said to be {em maximum distance separable} (MDS) ifd = n - k + 1. It is shown that an(n, k, n - k + 1)generalized Reed-Solomon code such that2leq k leq n - lfloor (q - 1)/2 rfloor (k neq 3 {rm if} qis even) can be extended by one digit while preserving the MDS property if and only if the resulting extended code is also a generalized Reed-Solomon code. It follows that a generalized Reed-Solomon code withkin the above range can be {em uniquely} extended to a maximal MDS code of lengthq + 1, and that generalized Reed-Solomon codes of lengthq + 1and dimension2leq k leq lfloor q/2 rfloor + 2 (k neq 3 {rm if} qis even) do not have MDS extensions. Hence, in cases where the(q + 1, k)MDS code is essentially unique,(n, k)MDS codes withn > q + 1do not exist.     

Convergence to the rate-distortion function for Gaussian sources

In this paper we derive an expression for the minimum-mean-square error achievable in encodingtsamples of a stationary correlated Gaussian source. It is assumed that the source output is not known exactly but is corrupted by correlated Gaussian noise. The expression is obtained in terms of the covariance matrices of the source and noise sequences. It is shown that ast rightarrow infty, the result agrees with a known asymptotic result, which is expressed in terms of the power spectra of the source and noise. The rate of convergence to the asymptotic results as a function of coding delay is investigated for the case where the source is first-order Markov and the noise is uncorrelated. WithDthe asymptotic minimum-mean-square error andD_tthe minimum-mean-square error achievable in transmittingtsamples, we findmid D_t - D mid leq O((t^{-1} log t) ^ {1/2})when we transmit the noisy source vectors over a noiseless channel andmid D_t - D mid leq O((t^{-1} log t)^ {1/3})when the channel is noisy.     

On mean-square aliasing error in the cardinal series expansion of random processes (Corresp.)

An upper bound is derived for the mean-square error involved when a non-band-limited, wide-sense stationary random processx(t)(possessing an integrable power spectral density) is approximated by a cardinal series expansion of the formsum^{infty}_{-infty}x(n/2W)sinc2W(t-n/2W), a sampling expansion based on the choice of some nominal bandwidthW > 0. It is proved thatlim_{N rightarrow infty} E {|x(t) - x_{N}(t)|^{2}} leq frac{2}{pi}int_{| omega | > 2 pi W}S_{x}( omega) d omega,wherex_{N}(t) = sum_{-N}^{N}x(n/2W)sinc2W(t-n/2W), andS_{x}(omega)is the power spectral density forx(t). Further, the constant2/ piis shown to be the best possible one if a bound of this type (involving the power contained in the frequency region lying outside the arbitrarily chosen band) is to hold uniformly int. Possible reductions of the multiplicative constant as a function oftare also discussed, and a formula is given for the optimal value of this constant.     

Some results on the minimum weight of primitive BCH codes (Corresp.)

For1 leq i leq m - s- 2and0 leq s leq m -2i, the intersection of the binary BCH code of designed distance2 ^{m-s-1} - 2 ^{m-s-t-1} - 1and length2^m - 1with the shortened(s + 2)th-order Reed-Muller code of length2^m -- 1has codewords of weight2^{m-s-1} - 2^{m-s-t-1} - 1.     

An approximation to the weight distribution of binary primitive BCH codes with designed distances 9 and 11 (Corresp.)

Recently Kasami {em et al.} presented a linear programming approach to the weight distribution of binary linear codes [2]. Their approach to compute upper and lower bounds on the weight distribution of binary primitive BCH codes of length2^{m} - 1withm geq 8and designed distance2t + 1with4 leq t leq 5is improved. From these results, the relative deviation of the number of codewords of weightjleq 2^{m-1}from the binomial distribution2^{-mt} left( stackrel{2^{m}-1}{j} right)is shown to be less than 1 percent for the following cases: (1)t = 4, j geq 2t + 1andm geq 16; (2)t = 4, j geq 2t + 3and10 leq m leq 15; (3)t=4, j geq 2t+5and8 leq m leq 9; (4)t=5,j geq 2t+ 1andm geq 20; (5)t=5, j geq 2t+ 3and12 leq m leq 19; (6)t=5, j geq 2t+ 5and10 leq m leq 11; (7)t=5, j geq 2t + 7andm=9; (8)t= 5, j geq 2t+ 9andm = 8.     

Mutual information of the white Gaussian channel with and without feedback

The following model for the white Gaussian channel with or without feedback is considered: begin{equation} Y(t) = int_o ^{t} phi (s, Y_o ^{s} ,m) ds + W(t) end{equation} wheremdenotes the message,Y(t)denotes the channel output at timet,Y_o ^ {t}denotes the sample pathY(theta), 0 leq theta leq t. W(t)is the Brownian motion representing noise, andphi(s, y_o ^ {s} ,m)is the channel input (modulator output). It is shown that, under some general assumptions, the amount of mutual informationI(Y_o ^{T} ,m)between the messagemand the output pathY_o ^ {T}is directly related to the mean-square causal filtering error of estimatingphi (t, Y_o ^{t} ,m)from the received dataY_o ^{T} , 0 leq t leq T. It follows, as a corollary to the result forI(Y_o ^ {T} ,m), that feedback can not increase the capacity of the nonband-limited additive white Gaussian noise channel.     

Linear feedback rate bounds for regressive channels (Corresp.)

This article presents new tighter upper bounds on the rate of Gaussian autoregressive channels with linear feedback. The separation between the upper and lower bounds is small. We havefrac{1}{2} ln left( 1 + rho left( 1+ sum_{k=1}^{m} alpha_{k} x^{- k} right)^{2} right) leq C_{L} leq frac{1}{2} ln left( 1+ rho left( 1+ sum_{k = 1}^{m} alpha_{k} / sqrt{1 + rho} right)^{2} right), mbox{all rho}, whererho = P/N_{0}W, alpha_{l}, cdots, alpha_{m}are regression coefficients,Pis power,Wis bandwidth,N_{0}is the one-sided innovation spectrum, andxis a root of the polynomial(X^{2} - 1)x^{2m} - rho left( x^{m} + sum^{m}_{k=1} alpha_{k} x^{m - k} right)^{2} = 0.It is conjectured that the lower bound is the feedback capacity.