There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Consider a Gaussian multiple-access channel shared by K users who transmit asynchronously independent data streams by modulating a set of assigned signal waveforms. The uncoded probability of error achievable by optimum multiuser detectors is investigated. It is shown that the K -user maximum-likelihood sequence detector consists of a bank of single-user matched filters followed by a Viterbi algorithm whose complexity per binary decision is O(2^{K}) . The upper bound analysis of this detector follows an approach based on the decomposition of error sequences. The issues of convergence and tightness of the bounds are examined, and it is shown that the minimum multiuser error probability is equivalent in the Iow-noise region to that of a single-user system with reduced power. These results show that the proposed multiuser detectors afford important performance gains over conventional single-user systems, in which the signal constellation carries the entire burden of complexity required to achieve a given performance level.

Minimum probability of error for asynchronous Gaussian multiple-access channels

The Global Positioning System (GPS) is a satellite-based navigation and time transfer system developed by the U.S. Department of Defense. It serves marine, airborne, and terrestrial users, both military and civilian. Specifically, GPS includes the Standard Positioning Service (SPS) which provides civilian users with 100 meter accuracy, and it serves military users with the Precise Positioning Service (PPS) which provides 20-m accuracy. Both of these services are available worldwide with no requirement for a local reference station. In contrast, differential operation of GPS provides 2- to 10-m accuracy to users within 1000 km of a fixed GPS reference receiver. Finally, carrier phase comparisons can be used to provide centimeter accuracy to users within 10 km and potentially within 100 km of a reference receiver. This advanced tutorial will describe the GPS signals, the various measurements made by the GPS receivers, and estimate the achievable accuracies. It will not dwell on those aspects of GPS which are well known to those skilled in the radio communications art, such as spread-spectrum or code division multiple access. Rather, it will focus on topics which are more unique to radio navigation or GPS. These include code-carrier divergence, codeless tracking, carrier aiding, and narrow correlator spacing.

Global Positioning System: Signals, Measurements, and Performance

Binary maximal-length linear feedback shift register sequences (m-sequences) have been successfully employed in communications, navigation, and related systems over the past several years. For the early applications, m-sequences were used primarily because of their excellent periodic autocorrelation properties. For many of the recent systems applications, however, the crosscorrelation properties of such sequences are at least as important as the autocorrelation properties, and the system performance depends upon the aperiodic correlation in addition to the periodic correlation. This paper presents a survey of recent results and provides several new results on the periodic and aperiodic crosscorrelation functions for pairs of m-sequences and for pairs of related (but not maximal-length) binary shift register sequences. Also included are several recent results on correlation for complex-valued sequences as well as identities relating the crosscorrelation functions to autocorrelation functions. Examples of problems in spread-spectrum communications are employed to motivate the choice of correlation parameters that are considered in the paper.

Crosscorrelation properties of pseudorandom and related sequences

Under the assumption that each pair of participants can communicate secretly, we show that any reasonable multiparty protocol can be achieved if at least 2n/3 of the participants are honest. The secrecy achieved is unconditional. It does not rely on any assumption about computational intractability.

/pdf/multiparty-unconditionally-secure-protocols-14nd15ffz7.pdf

Multiparty unconditionally secure protocols

Identities are established which relate certain crosscorrelation parameters for periodic real-valued sequences to their autocorrelation functions. Applications to multiple-access communication systems and to the design of uncorrelated sequences are discussed.

New correlation identities for periodic sequences

The extended Euclidean algorithm (EEA) for polynomial greatest common divisors is commonly used in solving the key equation in the decoding of Reed-Solomon (RS) codes, and more generally in BCH decoding. For this particular application, the iterations in the EEA are stopped when the degree of the remainder polynomial falls below a threshold. While determining the degree of a polynomial is a simple task for human beings, hardware implementation of this stopping rule is more complicated. This paper describes a modified version of the EEA that is specifically adapted to the RS decoding problem. This modified algorithm requires no degree computation or comparison to a threshold, and it uses a fixed number of iterations. Another advantage of this modified version is in its application to the errors-and-erasures decoding problem for RS codes where significant hardware savings can be achieved via seamless computation.

Modified Euclidean algorithms for decoding Reed-Solomon codes

Most versions of the Peterson-Gorenstein-Zierler (PGZ) decoding algorithm are not true bounded distance decoding algorithms in the sense that when a received vector is not in the decoding sphere of any codeword, the algorithm does not always declare a decoding failure. For a t-error-correcting BCH code, if the received vector is at distance i, i/spl les/t from a codeword in a supercode with BCH distance t+i+1, the decoder will output that codeword from the supercede. If that codeword is not a member of the t-error-correcting code, then decoder malfunction is said to have occurred. We describe the necessary and sufficient conditions for decoder malfunction, and show that malfunction can be avoided in the PGZ decoder by checking t-/spl nu/ equations, where /spl nu/ is the number of errors hypothesized by the decoder. A formula for the probability of decoder malfunction is also given, and the significance of decoder malfunction is considered for PGZ decoders and high-speed Berlekamp-Massey decoders. >

Malfunction in the Peterson-Gorenstein-Zierler decoder

A set of complementary sequences is a set of sequences such that, for each nonzero offset, the out-of-phase aperiodic autocorrelations of all the sequences sum to zero. It is shown that the sequences from any coset of an (N, m) nondegenerate minimal linear cyclic code over GF(p) form a set of complex-valued complementary sequences of length N.

Sets of complementary sequences

From (15) and (II), it follows that because N(0) = IV(O). The resulting estimate of X(w) is given by m bee (2211 c J(Wi-w/&(w) = J(w-Wk), k = 1,. .. ,m. (16) i=l With Aik the elements of the inverse of a matrix whose elements X(w) = X(w) + &) [N(w)-iv(O) +$q equal d(wi-wk), the system (16) yields and the mean-square error by [see (21)] m @i(w) = C A;J(w-wk). k=l (17) From (16), it follows that Without the correction, the mean-square value of the error is @'.i(Wk) = 1, k=i o > k # i. (18) Inserting (17) into (9), we conclude that the optimum estimator N(w) is given by m N(W) = kgl BkJ(w-Wk) (19) where the random variables & are linearly dependent on the data Wi. Setting w = wi in (19), we obtain the system fJ BkJ(wi-Wk) = A(Ui) = Wi, k=l i = 1, fee ,m. (20) The last equality follows from (9) and (18). Thus N(w) is a linear combination of the functions J(w-Wk) and it interpolates N(W) at the zero-crossings wi of Y(w). The coefficients & can be found by solving the system (20). Clearly [see (9) and (ll)], the difference N(w)-R(w) is orthogonal to N(w), hence the mean-square error e is given by e = E(I N(w)-AVOW) = E([iV(w)-N(w)]iV*(w)]. The above yields e = J(0)-f %pi(W)J(Wi-W). i=l (21) The error is zero at w = wi [see (18)]. With N(w) so corrected, the best estimate of X(w) takes the form S(w) = j& [W(w)-mJd1 = X(w) + & [iv(w)-AYW,]. The correction has the following time-domain interpretation: Since the inverse of J(w) equals Z(t), it follows from (19) that the inverse a(t) of the estimator R(w) is given by m fi(t) = z(t) c &ej \" 'kt. k=l (2% Thus, to improve the estimation of x (t) with the above method, we deconvolve not the data w(t), but the difference w(t)-a(t). Example: We wish to deconvolve the function w(t) = x(t) * y(t) + n(t), where y(t) is a known function whose transform Y(w) has a single zero at w = 0. We assume that w(t) vanishes for ] t ] > T and that n(t) is stationary white noise with R(t1,tz) = LWl-t'd, It11 < T, ltzl < 7'. Clearly, Z(t) equals Z, for (t ] < T and is 0 otherwise. Hence, J(w) = Z,s_TT e-jut & = 2z0 'r wT. …

Dilip V. Sarwate

Papers

New correlation identities for periodic sequences

Modified Euclidean algorithms for decoding Reed-Solomon codes

Malfunction in the Peterson-Gorenstein-Zierler decoder

Sets of complementary sequences

Comments on "A class of balanced binary sequences with optimal autocorrelation properties" by Lempel, A., et al.