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Published in Philip A. Laplante, Comprehensive Dictionary of Electrical Engineering, 2018
The scaling factor allows the same matrix to be used for the inverse transform. The 4 × 4 Haar transform can be interpreted as follows: first apply the 2×2 transform to two independent pairs of samples; then apply the 2 × 2 transform to the two average coefficients just computed. Larger Haar transforms are constructed by continuing this process recursively. The Haar transform yields coefficients equal to the subband values generated by dyadic decomposition with the Haar wavelet. This transform has achieved rather less use than the other transforms in this family, such as the discrete cosine, Fourier, and Hadamard transforms. Haar wavelet the orthonormal wavelet pair 1 1 1 -1 ( , ), ( , ). Analysis and synthesis pairs 2 2 2 2 are identical. This is the most compact wavelet pair. Dyadic subband decomposition with the Haar wavelets yields coefficients equal to those from the Haar transform. hacker a person who explores computer and communication systems, usually for intellectual challenge, commonly applied to those who try to circumvent security barriers (crackers). Hadamard matrix an n × n matrix H with elements ±1 is a Hadamard matrix of order n if H H T = n I , i.e., the rows are all mutually orthogonal, as are the columns. Hadamard matrices can only exist for n = 1, 2 or n an integer multiple of 4. Hadamard matrices of order 2i can be constructed by the recursion H1 = (1) H2n = Hn Hn Hn -Hn
Wireless Communication Systems
Published in Keshab K. Parhi, Takao Nishitani, Digital Signal Processing for Multimedia Systems, 2018
There are many different strategies to assign PN codes to the different users in a CDMA network. The key issue it to choose PN codes in such a manner that the cross-correlation between any two codes is as low as possible. Perhaps the first issue to consider here is whether or not it is possible to synchronize the different transmissions at the chip level. If this is possible then we can actually design a family of orthogonal spreading codes for the different users. In such a case there will be no interference between users and we can support a number of users equal to the processing gain N. Such a set of spreading codes may be represented as the rows of a matrix. The required matrix for orthogonal codes is a Hadamard matrix. A special case of a Hadamard matrix is the matrix generated recursively as follows: H0 = 1, and () Hn+1=[HnHnHn−Hn].
Modelling and analysis of skin pigmentation
Published in Ahmad Fadzil Mohamad Hani, Dileep Kumar, Optical Imaging for Biomedical and Clinical Applications, 2017
Ahmad Fadzil Mohamad Hani, Hermawan Nugroho, Norashikin Shamsudin, Suraiya H. Hussein
Similar to a single-slit system, the spectrometer of a multi-slit system also consists of a disperse element (a prism) and a detector. The difference is that it employs a special mask known as Hadamard mask. Hadamard masks are made up of a pattern of reflective and transmissive slits of various widths constructed from a Hadamard matrix. A Hadamard matrix is a square matrix whose entries are either +1 or −1 and whose rows are mutually orthogonal. The spectral information is then generated from Hadamard transform [109]. As illustrated in Figure 4.27, the object is illuminated and spectral information is obtained using a multiple slit (Hadamard mask) spectrophotometer with a disperse element and a linear array of CCD detector. A complete image (a spectral cube) is obtained by varying the mask and performing Hadamard transform.
A family of orthogonal main effects screening designs for mixed-level factors
Published in Journal of Quality Technology, 2023
Bradley Jones, Ryan Lekivetz, Christopher Nachtsheim
In this paper, we introduce and explore a new family of mixed, two- and three-level designs that are orthogonal for main effects and often have generally low levels of absolute correlation between main effects columns and two-factor interaction columns. Our designs are obtained by concatenating two replicates of a (three-level) definitive screening design (DSD) with a folded-over (two-level) Hadamard matrix design (Hedayat and Wallis 1978). When the number of three-level factors is k, where is a multiple of four, the number of runs is and the designs can accommodate up to k three-level factors and up to 2k two-level factors, the latter of which can be either continuous or categorical. We show that when k or fewer two-level factors are to be employed and n is a multiple of 16, the two-level columns can be chosen in such a way that main effects are completely independent of two-factor interactions. We note that the three-level factors must be continuous—the new designs are not appropriate for three-level categorical factors.
Chaos-Based transmitted-reference ultra-wideband communications
Published in International Journal of Electronics, 2019
Marijan Herceg, Denis Vranješ, Ratko Grbić, Josip Job
The CM-TR scheme exploits a similar scheme as the one proposed in (Goeckel & Zhang, 2007). The difference is that the orthogonality between reference and data pulses is obtained by amplitude modulating of reference and data pulse trains by two different orthogonal codes. The set of orthogonal codes is obtained from the Hadamard matrix rows. In (D’Amico & Mengali, 2010; D’Amico & Mengali, 2008) is shown that the CM-TR scheme outperforms both, the standard and the SFS-TR scheme for higher data rates and shows the best performance in a multi-user scenario, while still keeping the low hardware complexity. A similar method based on Walsh codes is proposed in (Nie & Chen, 2009), while a generalized CM-TR scheme is given in (Zhou, Ma, & Lottici, 2013).
A note on balanced incomplete block designs and projective geometry
Published in International Journal of Mathematical Education in Science and Technology, 2021
Ömür Deveci, Anthony G. Shannon
A Hadamard 2-design is based on Hadamard matrices (Cooper & Wallis, 1972). A Hadamard matrix is a symmetric matrix of size m, with mutually orthogonal rows and with cell entries ±1 such that HHT = mIm; for example, which has an incidence matrix (a translation of the identity matrix) which has the structure of a Hadamard 2-design. Every such design is extendable to a Hadamard 3-design. The incidence matrix of block designs provides a natural source of interesting block codes that can be used as error-correcting codes (McWilliams & Sloane, 1977).