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1.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1978,24(4):497-499
It is shown thatsqrt[8]{2} is an element of order2^{n+4} inGF(F_{n}) , whereF_{n}=2^{2^{n}}+1 is a Fermat prime forn=3,4 . Hence it can be used to define a fast Fourier transform (FFT) of as many as2^{n+4} symbols inGF(F_{n}) . Sincesqrt[8]{2} is a root of unity of order2^{n+4} inGF(F_{n}) , this transform requires fewer muitiplications than the conventional FFT algorithm. Moreover, as Justesen points out [1], such an FFT can be used to decode certain Reed-Solomon codes. An example of such a transform decoder for the casen=2 , wheresqrt{2} is inGF(F_{2})=GF(17) , is given. 相似文献
2.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1975,21(4):453-458
For a nondecreasing distortion characteristicphi(cdot) and a given signalx(cdot) , the "cross correlation" function defined byR_{phi} (tau) triangleq int_{-infty}^{infty} phi[x(t)]x(t - tau) dt is shown to satisfy the inequalityR_{phi}(tau) leq R_{phi}(0) , for alltau , generalizing an earlier result of Richardson that requiredphi(cdot) to be continuous and strictly increasing. The methods of the paper also show that, under weak conditions, begin{equation} R_{phi,psi}(tau) triangleq int_{-infty}^{infty} phi[x(t)]psi[x(t - tau)] dt leq R_{phi,psi}(0) end{equation} whenpsi is strictly increasing andphi is nondecreasing. In the case of hounded signals (e.g., periodic functions), the appropriate cross correlation function is begin{equation} mathcal{R}_{phi,psi}(tau} triangleq lim_{T rightarrow infty} (2T)^{-l} int_{-T}^T phi[x(t)]psi[x(t - tau)] dt. end{equation} For this case it is shown thatmathcal{R}_{phi,psi} (tau) leq mathcal{R}_{phi,psi}(0) for any nondecreasing (or nonincreasing) distortion functionsphi andpsi . The result is then applied to generalize an inequality on correlation functions for periodic signals due to Prosser. Noise signals are treated and inequalities of a similar nature are obtained for ensemble-average cross correlation functions under suitable hypotheses on the statistical properties of the noise. Inequalities of this type are the basis of a well-known method of estimating the unknown time delay of an observed signal. The extension to nondecreasing discontinuous distortion functions allows the use of hard limiting or quantization to facilitate the cross correlation calculation. 相似文献
3.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1984,30(2):385-388
LetV be 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 ifnu has 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-q representation of the numberx . LetS be 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 thatS is 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 . 相似文献
4.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1978,24(4):422-432
In 1964 the author proposed as an explication of {em a priori} probability the probability measure induced on output strings by a universal Turing machine with unidirectional output tape and a randomly coded unidirectional input tape. Levin has shown that iftilde{P}'_{M}(x) is an unnormalized form of this measure, andP(x) is any computable probability measure on strings,x , thentilde{P}'_{M}geqCP(x) whereC is a constant independent ofx . The corresponding result for the normalized form of this measure,P'_{M} , is directly derivable from Willis' probability measures on nonuniversal machines. If the conditional probabilities ofP'_{M} are used to approximate those ofP , then the expected value of the total squared error in these conditional probabilities is bounded by-(1/2) ln C . With this error criterion, and when used as the basis of a universal gambling scheme,P'_{M} is superior to Cover's measurebast . WhenHastequiv -log_{2} P'_{M} is used to define the entropy of a rmite sequence, the equationHast(x,y)= Hast(x)+H^{ast}_{x}(y) holds exactly, in contrast to Chaitin's entropy definition, which has a nonvanishing error term in this equation. 相似文献
5.
Maximum entropy and conditional probability 总被引:2,自引:0,他引:2
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1981,27(4):483-489
It is well-known that maximum entropy distributions, subject to appropriate moment constraints, arise in physics and mathematics. In an attempt to find a physical reason for the appearance of maximum entropy distributions, the following theorem is offered. The conditional distribution ofX_{l} given the empirical observation(1/n)sum^{n}_{i}=_{l}h(X_{i})=alpha , whereX_{1},X_{2}, cdots are independent identically distributed random variables with common densityg converges tof_{lambda}(x)=e^{lambda^{t}h(X)}g(x) (Suitably normalized), wherelambda is chosen to satisfyint f_{lambda}(x)h(x)dx= alpha . Thus the conditional distribution of a given random variableX is the (normalized) product of the maximum entropy distribution and the initial distribution. This distribution is the maximum entropy distribution wheng is uniform. The proof of this and related results relies heavily on the work of Zabell and Lanford. 相似文献
6.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1984,30(3):480-484
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, wherek is 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}/2 bits of storage. In both algorithms, the time required to produce the next bit from the lastn bits is close ton . A possible application to the construction of stream ciphers is indicated. 相似文献
7.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1972,18(5):659-661
Recently Mazo and Salz proved that if{ Y(t), t in T } is a stationary random process with mean-square derivative{ dot{Y}(t), t in T }, then the conditional expectation ofdot{Y} (t) givenY(t) is zero almost everywhere with respect to the distribution ofY(t) . We extend this property and obtain a characterization of stationary processes differentiable in mean square. 相似文献
8.
An algorithm for maximizing expected log investment return 总被引:3,自引:0,他引:3
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1984,30(2):369-373
Let the random (stock market) vectorX geq 0 be drawn according to a known distribution functionF(x), x in R^{m} . A log-optimal portfoliob^{ast} is any portfoliob achieving maximal expectedlog returnW^{ast}=sup_{b} E ln b^{t}X , where the supremum is over the simplexb geq 0, sum_{i=1}^{m} b_{i} = 1 . An algorithm is presented for findingb^{ast} . The algorithm consists of replacing the portfoliob by the expected portfoliob^{'}, b_{i}^{'} = E(b_{i}X_{i}/b^{t}X) , corresponding to the expected proportion of holdings in each stock after one market period. The improvement inW(b) after each iteration is lower-bounded by the Kullback-Leibler information numberD(b^{'}|b) between the current and updated portfolios. Thus the algorithm monotonically improves the returnW . An upper bound onW^{ast} is given in terms of the current portfolio and the gradient, and the convergence of the algorithm is established. 相似文献
9.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1986,32(1):106-108
LetC(B) denote the binary cyclicAN code with generatorA , whereAB=2^{n} - 1 . It is known thatC(B) is equidistant ifB is a prime powerp^{k} , where either2 or-2 is primitive moduloB providedpequiv 1 pmod{3}{rm if} k > 1 . It is conjectured that these are the onlyB such thatC(B) is equidistant. We have verified this forB < 100 000 . Several results are established that further limit the possibilities for counterexamples to the conjecture. 相似文献
10.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1972,18(1):20-27
The approach to Gaussianity of the outputy(t) of a narrow-band systemh(t) is investigated. It is assumed that the inputx(t) is ana -dependent process, in the sense that the random variablesx(t) andx(t + u) are independent foru > a . WithF(y) andG(y) the distribution functions ofy(t) and of a suitable normal process, a realistic boundB on the differenceF(y) -- G(y) is determined, and it is shown thatB rightarrow 0 as the bandwidthomega_o of the system tends to zero. In the special case of the shot noise process begin{equation} y(t) = sum_i h(t - t_i) end{equation} it is shown that begin{equation} mid F(y) - G(y) mid < (omega_o/lambda) frac{1}{2} end{equation} wherelambda_i is the average density of the Poisson pointst_i . 相似文献
11.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1986,32(6):846-847
A probabilistic polynomial-time algorithm for computing the square root of a numberx in {bf Z}/P{bf Z} , whereP = 2^{S}Q + 1(Q odd,s > 0) is a prime number, is described. In contrast to the Adleman, Manders, and Miller algorithm, this algorithm gets faster as s grows. As with the Berlekamp-Rabin algorithm, the expected running time of the algorithm is independent ofx . However, the algorithm presented here is considerably faster for values ofs greater than2 . 相似文献
12.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1967,13(3):356-359
The purpose of this paper is to show that decoding complexity need not grow exponentially with the code block length at rates close to channel capacity and also to show the expediency of the approach of imbedding codes in each other. It is demonstrated that it is possible to communicate over a memoryless channel of capacityC at any rateR < C with a probability of error of less than2^{-E(R)nu}, E(R) > 0 , per block of a length approximately proportional tonu^{2} and with a computational decoding complexity per digit which is asymptotically proportional tonu^{alpha} whennu is large,nu^{alpha} being finite forR < C .(alpha rightarrow mbox{as} R rightarrow C, alpha rightarrow 2 mbox{as} R rightarrow 0) . 相似文献
13.
In this communication we have analyzed the errors introduced by table lookup when an electromagnetic diffraction integral is performed numerically. We have shown that in order to set the3sigma error of both the real and imaginary components equal tovarepsilon , the required table should have at leastK entries uniformly distributed within a2pi range, whereK geq 3.85/varepsilonsqrt{N} andN is the number of terms used for numerical integration. The maximum deviation of the amplitude (bar{R} ) of this integral in this case is also less thanvarepsilon and the maximum phase deviation istan^{-1} (varepsilon/bar{R}) . A numerical example is presented which confirms this prediction. 相似文献
14.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1972,18(2):310-312
Given a binary data streamA = {a_i}_{i=o}^infty and a filterF whose output at timen isf_n = sum_{i=0}^{n} a_i beta^{n-i} for some complexbeta neq 0 , there are at most2^{n +1) distinct values off_n . These values are the sums of the subsets of{1,beta,beta^2,cdots,beta^n} . It is shown that all2^{n+1} sums are distinct unlessbeta is a unit in the ring of algebraic integers that satisfies a polynomial equation with coefficients restricted to +1, -1, and 0. Thus the size of the state space{f_n} is2^{n+1} ifbeta is transcendental, ifbeta neq pm 1 is rational, and ifbeta is irrational algebraic but not a unit of the type mentioned. For the exceptional values ofbeta , it appears that the size of the state space{f_n} grows only as a polynomial inn ifmidbetamid = 1 , but as an exponentialalpha^n with1 < alpha < 2 ifmidbetamid neq 1 . 相似文献
15.
Exact formulas are derived for the quality factorQ of strip and line source antennas. Contrary to popular opinion, none of them is equal to Taylor's superdirectivity ratiogamma or togamma - 1 . But in the case ofE -plane strip sources (the complement of the type of strip source treated by Woodward and Lawson) the value ofQ is precisely equal togamma_{alpha}^{-1/2}- 1 , wheregamma_{alpha}^{beta} is a generalized superdirectivity ratio that reduces to Taylor'sgamma when the edge exponentalpha and the pattern weighting exponentbeta are both zero. In the case ofH -plane strip sources the value ofQ is approximately equal togamma_{alpha}^{1/2} - 1 , and forH -plane line sources of vanishing widtha it is approximately equal to[(2/pi) ln (2.516lambda/pi a)]gamma_{alpha}^{1} . 相似文献
16.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1974,20(5):598-601
In comparing a signalf(t) with its amplitude-distorted formg(f(t)) , whereg(cdot) is a monotonically increasing function of its argument, one is led to consider the correlation function begin{equation} R(s) triangleq int_{-infty}^{infty} dtg (f(t))f(t-s). end{equation} A rigorous proof is given of the inequalityR(s) leq R(O) . Generalizations are presented for the cases of finite domains and of signals defined in two-dimensional space. 相似文献
17.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1976,22(2):138-147
An extensive study of binary triple-error-correcting codes of primitive lengthn = 2^{m} - 1 is reported that results in a complete decoding algorithm whenever the maximum coset weightW_{max} is five. In this regard it is shown thatW_{max} = 5 when four dividesm , and strong support is provided for the validity of the conjecture thatW_{max} = 5 for allm . The coset weight distribution is determined exactly in some cases and bounded in others. 相似文献
18.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1983,29(3):345-348
Ifm is odd andsigma /in Aut GF(2^{m}) is such thatx rightarrow x^{sigma^{2}-1} is1-1 , there is a[2^{m+1}-1,2^{m+l}-2m-2] nonlinear binary codeP(sigma) having minimum distance 5. All the codesP(sigma) have the same distance and weight enumerators as the usual Preparata codes (which rise asP(sigma) whenx^{sigma}=x^{2}) . It is shown thatP(sigma) andP(tau) are equivalent if and only iftau=sigma^{pm 1} , andAut P(sigma) is determined. 相似文献
19.
20.
On the distribution of de Bruijn sequences of given complexity 总被引:1,自引:0,他引:1
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1984,30(4):611-614
The distributiongamma (c, n) of de Bruijn sequences of ordern and linear complexityc is investigated. It is shown that forn geq 4, gamma (2^{n} - 1, n) equiv 0 pmod{8} , and fork geq 3, gamma (2^{2k} - 1,2k) equiv 0 pmod{l6} . It is also shown thatgamma (c, n) equiv 0 pmod{4} for allc , andn geq 3 such thatcn is even. 相似文献