Hadamard transform

About: Hadamard transform is a research topic. Over the lifetime, 7262 publications have been published within this topic receiving 94328 citations.

Performance of orthogonal CDMA codes for quasi-synchronous communication systems

Abstract: In this paper, the performance of quasi-synchronous direct-sequence CDMA communication systems based on different sets of orthogonal codes is investigated. The authors search for sets of sequences that minimize the probability of bit detection error, given that there is imperfect synchronization among the signals. Orthogonal sequences based on the Sylvester-type Hadamard matrices (Walsh functions) are shown to provide a large improvement over the case where an orthogonal set is chosen at random. For these sequences closed form expressions are derived for the average bit error rate. Computer searches indicate that this set of codes has special properties with respect to minimizing the average cross-correlations between the different signals for small errors in chip synchronization. It appears that these codes are optimal. A multi-carrier signalling scheme designed to help synchronize the CDMA signals at the chip level is also discussed.

Progressive transmission of grey-scale and binary pictures by simple, efficient, and lossless encoding schemes

Abstract: Efficient, simple, and lossless methods for encoding and transmitting grey-scale and binary (B/W) pictures are described. The methods are such that an initial segment of the transmission permits the receiving device to construct an approximate picture; as transmission proceeds, the receiver progressively constructs better approximations. Some of the low-bandwidth channel usages include telebrowsing -where one wants to abort transmission of unwanted pictures as soon as they are recognized, and teleconferencing-where one wants to receive a gestalt first, followed by the details. In the method for grey-scale pictures, the transmitter first defines a hierarchical structure of picture subdivisions, and assigns, simply by table lookup, approximate average light values to all pairs of cells from the smallest size to that encompassing the entire picture. During tranmission the latter "primary" value is sent, followed by numbers which permit successive cell splitting, thus producing progressively finer picture representations in approximate values, culminating in an exact reconstruction at the pel level. A picture of p pels quantized into b bits each is perfectly reconstructed upon receipt of exactly b × p bits. If the recipient deems an intermediate version satisfactory, then "picture compression" is effectively achieved. This scheme thus provides some of the benefits of transform encoding (Fourier, Hadamard, Hotelling, singular-value decomposition) but with no truncation error and no need for handling coefficients of various sizes. Furthermore, computation varies only linearly with number of pels. The method for binary (B/W) pictures also involves a hierarchical structuring of the picture but with attention paid to picture compression-by declaring subdivisions, at as high a level as possible, to be solid black or solid white (and in need of no further description). Compressions thus achieved are 8:1 in cases where the "blob" coding method of Frank gives 11½:1

Computing Medians and Means in Hadamard Spaces

Abstract: The geometric median as well as the Frechet mean of points in a Hadamard space are important in both theory and applications. Surprisingly, no algorithms for their computation are hitherto known. To address this issue, we use a splitting version of the proximal point algorithm for minimizing a sum of convex functions and prove that this algorithm produces a sequence converging to a minimizer of the objective function, which extends a recent result of Bertsekas [Math. Program., 129 (2011), pp. 163--195] into Hadamard spaces. The method is quite robust, and not only does it yield algorithms for the median and the mean, but also it applies to various other optimization problems. We, moreover, show that another algorithm for computing the Frechet mean can be derived from the law of large numbers due to Sturm [Ann. Probab., 30 (2002), pp. 1195--1222]. In applications, computing medians and means is probably most needed in tree space, which is an instance of a Hadamard space, invented by Billera, Holmes, and Vo...

Unified Matrix Treatment of the Fast Walsh-Hadamard Transform

Abstract: A unified matrix treatment is presented for the various orderings of the Walsh-Hadamard (WH) functions using a general framework. This approach clarifies the different definitions of the WH matrix, the various fast algorithms and the reorderings of the WH functions.