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1.
《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
2.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1978,24(2):254-256
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/ pi is 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 oft are also discussed, and a formula is given for the optimal value of this constant. 相似文献
3.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1974,20(4):517-524
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 > 0 thevarepsilon -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_g is 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) < infty then, asvarepsilon rightarrow 0 begin{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 asvarepsilon goes 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) = infty are discussed, and asymptotic bounds onH_{varepsilon}(xi) , expressed in terms ofH_{varepsilon}(xi_g) , are derived. 相似文献
4.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1967,13(3):414-422
This paper considers the performance of a communication system which transmits forT seconds the real part of a sample function of one ofM stationary complex Gaussian processes whose spectral densities are all frequency translations of the functionS_{xi (f) . At the receiver white Gaussian noise of one-sided densityN_{0} is added. The center frequencies of the processes are assumed to be sufficiently separated that theM covariance functions are orthogonal overT . Exponently tight bounds are obtained for the error probability of the maximum likelihood receiver. It is shown that the error probability approaches zero exponentially withT for all ratesR = (ln M)/T up toC= int_{-infty}^{infty} [S_{xi (f)/N_{0}] df - int_{- infty}^{infty} ln [1 + S_{xi}(f)/N_{0}] df which is shown to be the channel capacity. Similar results are obtained for the case of stochastic signals with specular components. 相似文献
5.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1987,33(6):759-772
The multiterminal hypothesis testingH: XY againstH̄: X̄Ȳ is 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 epsilon for 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} = infty corresponding 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. 相似文献
6.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1986,32(3):410-412
For a complex-valued deterministic signal of finite energy band-limited to the normalized frequency band|w| leq pi explicit coefficients{a_{kn}} are found such that for anyT satisfying0 < T leq 1/2 ,left| f(t)-sum^{2n}_{k=1}a_{kn}f(t - kT)right| leq E_{f}cdot beta^{n} whereE_{f} is the signal energy andbeta doteq 0.6863 . Thus the estimate off(t) in terms of2n past samples taken at a rate equal to or in excess of twice the Nyquist rate converges uniformly at a geometric rate tof(t) on(- infty , infty) . The suboptimal coefficients{a_{kn}} have the desirable property of being pure numbers independent of both the particular band-limited signal and of the selected sampling rate1/T . It is also shown that these same coefficients can be used to estimate the value ofx(t) of a wide-sense stationary random process in terms of past samples. 相似文献
7.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1981,27(1):132-136
A randomized decision rule is derived and proved to be the saddlepoint solution of the robust detection problem for known signals in independent unknown-mean amplitude-bounded noise. The saddlepoint solutionphi^{0} uses an equaUy likely mixed strategy to chose one ofN Bayesian single-threshold decision rulesphi_{i}^{0}, i = 1,cdots , N having been obtained previously by the author. These decision rules are also all optimal against the maximin (least-favorable) nonrandomized noise probability densityf_{0} , wheref_{0} is a picket fence function withN pickets on its domain. Thee pair(phi^{0}, f_{0}) is shown to satisfy the saddlepoint condition for probability of error, i.e.,P_{e}(phi^{0} , f) leq P_{e}(phi^{0} , f_{0}) leq P_{e}(phi, f_{0}) holds for allf andphi . The decision rulephi^{0} is also shown to be an eqoaliir rule, i.e.,P_{e}(phi^{0}, f ) = P_{e}(phi^{0},f_{0}) , for allf , with4^{-1} leq P_{e}(phi^{0},f_{0})=2^{-1}(1-N^{-1})leq2^{-1} , N geq 2 . Thus nature can force the communicator to use an {em optimal} randomized decision rule that generates a large probability of error and does not improve when less pernicious conditions prevail. 相似文献
8.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1986,32(3):436-440
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 infty andR_{1} resp.R_{2} is the rate of the codeC resp.D . 相似文献
9.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1971,17(4):368-371
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} wherem denotes 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 messagem and 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. 相似文献
10.
《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
11.
Ifr(t) is the linear scattering response of an object to an excitation waveforme(t) , thenr(t) = (e ast h) (t) . One would like to deconvolve and solve forh(t) , the impulse response. It is well-known that this is often an ill-conditioned problem. Two methods are discussed. The first method replaces the discretized matrix formE cdot H = R by the following problem. Minimize|h_{1}|+ ldots + |h_{n}| subject toR - lambda leq E cdot H leq R + lambda wherelambda is a column vector chosen sufficiently small to yield acceptable residuals, yet large enough to make the problem well-conditioned. This problem is converted to a linear programming problem so that the simplex algorithm can be used. The second method is to minimizeparallel E cdot H - R parallel^{2} +lambda parallel H parallel^{2} where againlambda is chosen small enough to yield acceptable residuals and large enough to make the problem well-conditioned. The method will be demonstrated with a Hilbert matrix inversion problem, and also by the deconvolution of the impulse response of a simple target from measured data. 相似文献
12.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1974,20(2):278-279
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_t is reproduced astilde{x}_t with a delay oft + n time units. It is shown that if the channel capacity is greater than the source entropyC > H(X) , then there exists a sequence of block lengthn codes such thatE[d_n(X,tilde{X})] rigjhtarrow 0 asn rightarrow infty even iff(t) rightarrow infty at an exponential rate. However, iff(t) grows at too fast an exponential rate, thenE[d_n(X,tilde{X})] rightarrow infty asn rightarrow infty . Also, ifC < H(X) andf(t) rightarrow infty thenE[d_n(X,tilde{X})] rightarrow infty asn rightarrow infty no matter how slowlyf(t) grows. 相似文献
13.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1969,15(4):440-444
Forf(t) a real-valued signal band-limited to- pi r leq omega leq pi r (0 < r < 1) and represented by its Fourier integral, upper bounds are established for the magnitude of the truncation error whenf(t) is approximated at a generic timet by an appropriate selection ofN_{1} + N_{2} + 1 terms from its Shannon sampling series expansion, the latter expansion being associated with the full band[-pi, pi] and thus involving samples off taken at the integer points. Results are presented for two cases: 1) the Fourier transformF(omega) is such that|F(omega)|^{2} is integrable on[-pi, pi r] (finite energy case), and 2)|F(omega)| is integrable on[-pi r, pi r] . In case 1) it is shown that the truncation error magnitude is bounded above byg(r, t) cdot sqrt{E} cdot left( frac{1}{N_{1}} + frac{1}{N_{2}} right) whereE denotes the signal energy andg is independent ofN_{1}, N_{2} and the particular band-limited signal being approximated. Correspondingly, in case 2) the error is bounded above byh(r, t) cdot M cdot left( frac{1}{N_{1}} + frac{1}{N_{2}} right) whereM is the maximum signal amplitude andh is independent ofN_{1}, N_{2} and the signal. These estimates possess the same asymptotic behavior as those exhibited earlier by Yao and Thomas [2], but are derived here using only real variable methods in conjunction with the signal representation. In case 1), the estimate obtained represents a sharpening of the Yao-Thomas bound for values ofr dose to unity. 相似文献
14.
《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
15.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1972,18(5):662-664
Forx(t) either a deterministic or stochastic signal band-limited to the normalized frequency intervalmidomegamid leq pi , explicit coefficients{ a_{kn} } are exhibited that have the property that begin{equation} lim_{n rightarrow infty} parallel x(t) - sum_{1}^n a_{kn} x(t - kT) parallel = 0 end{equation} in an appropriate norm and for any constant intersample spacingT satisfying0 < T < fac{1}{2} ; that is,x(t) may be approximated arbitrarily well by a linear combination of past samples taken at any constant rate that exceeds twice the associated Nyquist rate. Moreover, the approximation ofx(t) is uniform in the sense that the coefficients{ a_{kn} } do not depend on the detailed structure ofx(t) but are absolute constants for any choice ofT . The coefficients that are obtained provide a sharpening of a previous result by Wainstein and Zubakov where a rate in excess of three times the Nyquist rate was required. 相似文献
16.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1975,21(1):95-96
Some integrals are presented that can be expressed in terms of theQ_M function, which is defined as begin{equation} Q_M(a,b) = int_b^{infty} dx x(x/a)^{M-1} exp (- frac{x^2 + a^2}{2}) I_{M-1}(ax), end{equation} whereI_{M-1} is the modified Bessel function of orderM-1 . Some integrals of theQ_M function are also evaluated. 相似文献
17.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1984,30(6):859-865
The variance of the output of a cross-correlation detector, which is called a quadrature cross-correlation detector, is estimated. In this type of detector two zero-mean Gaussian quadrature processesalpha(t) andbeta(t) of a complex process(alpha(t) -j beta(t)) are cross correlated. This cross-correlation functionR_{A} (tau) is estimated when neither of the two processes is distorted (the analog method), when both processes are distorted by a signum function before being cross correlated (the polarity coincidence method), and when one of the two processes is distorted by either a signum function or by a "comparator logic" function (the relay method). These quadrature cross-correlation detectors then are compared on the basis of output signal-to-noise ratio (s/n) and the clipping and relay losses are computed for two test quadrature processes of an Edgeworth-expansion-approximated power spectrum. SinceR_{A} (0) is zero, the four corresponding differential estimators, such as(R_{A} (tau) - R_{A} (0)) are also estimated and are compared on the basis of s/n. For these differential estimators, the clipping and relay losses are computed for the two test processes. In all cases the exact expressions for the s/n are derived as a function oftau . Some applications of these correlation detectors are outlined. The mathematical techniques employed here are thought to have potential usefulness for related problems in statistical communication theory and signal processing. 相似文献
18.
《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
19.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1965,11(3):330-335
The optimum test statistic for the detection of binary sure signals in stationary Gaussian noise takes a particularly simple form, that of a correlation integral, when the solution, denoted byq(t) , of a given integral equation is well behaved(L_{2}) . For the case of a rational noise spectrum, a solution of the integral equation can always be obtained if delta functions are admitted. However, it cannot be argued that the test statistic obtained by formally correlating the receiver input with aq(t) which is notL_{2} is optimum. In this paper, a rigorous derivation of the optimum test statistic for the case of exponentially correlated Gaussian noiseR(tau) = sigma^{2} e^{-alpha|tau|} is obtained. It is proved that for the correlation integral solution to yield the optimum test statistic whenq(t) is notL_{2} , it is sufficient that the binary signals have continuous third derivatives. Consideration is then given to the case where a, the bandwidth parameter of the exponentially correlated noise, is described statistically. The test statistic which is optimum in the Neyman-Pearson sense is formulated. Except for the fact that the receiver employsalpha_{infty} (which in general depends on the observed sample function) in place ofalpha , the operations of the optimum detector are unchanged by the uncertainty inalpha . It is then shown that almost all sample functions can be used to yield a perfect estimate ofalpha . Using this estimate ofalpha , a test statistic equivalent to the Neyman-Pearson statistic is obtained. 相似文献
20.
Anderson J. Sundberg C.-E. Aulin T. Rydbeck N. 《Communications, IEEE Transactions on》1981,29(3):187-195
Constant envelope phase varying sinusoids of the formsqrt{2E/T} cos(omega_{c}t + phi(t)) are studied, in which the phase functionphi(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 R0 is computed, which guarantees existence of codes at all ratesR < R_{0} bits/T -interval whose error performance varies as exp[-N(R_{0} - R)] , whereN is the code word length inT -intervals. Plots of R0 are given as a function of interval energyE , the shaping functionh andM . 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. 相似文献