Orthogonal signals with jointly balanced spectra: Application to cdma transmissions
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RESEARCH
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Orthogonal signals with jointly balanced spectra: Application to cdma transmissions Thierry Chonavel
Abstract This paper presents a technique for generating orthogonal bases of signals with jointly optimized spectra, in the sense that they are made as close as possible. To this end, we propose a new criterion, the minimization of which leads to signals with close energy inside a set of prescribed subbands. Starting with the case of a single subband, we illustrate it by building orthogonal signals with maximum energy concentration in time and in frequency, with the same energy rate outside a fixed frequency interval or a fixed time interval, by resorting to Slepian sequences or Slepian functions, respectively. Then, we present spectrum balancing in a set of frequency intervals. We apply this method to Slepian sequences and Slepian functions, as well as to Walsh-Hadamard codes. On these examples, we point out a number of nice properties of the so-built orthogonal families that are of interest for signaling applications. PACS: signal processing techniques and tools; modulation techniques Keywords: orthogonal signaling bases, spectrum balancing, Slepian sequences, Slepian functions, Walsh-Hadamard, scrambling, CDMA, UWB
1 Introduction A few studies have been carried out to build orthogonal signals with flat spectrum. Several of these studies are based on invariance property of Hadamard matrices w.r. t. orthogonal transforms. More specifically, approaches presented in [1] and [2] account for the fact that when collecting orthogonal codes represented by column vectors in a matrix, then any permutation of the lines of the matrix yields columns that represent a new family of orthogonal codes. In [1], this principle is applied to Walsh codes and authors mention the fact that new codes spectra may be more flat than initial Walsh codes. However, permutations are performed randomly, and no criterion is supplied to optimize spectrum flatness. In fact, flatness will occur randomly in generated codes. In [2], the same approach is considered, but spectrum flatness is achieved by changing codes at each data transmission by considering a new random permutation at each time. Thus, flatness is not achieved by each code but only as a mean spectrum property among codes. Correspondence: [email protected] Télécom Bretagne, UEB, Lab-STICC UMR CNRS 3192, Technopôle Brest-Iroise, Institut Télécom, CS 83818, 29238 Brest Cedex 3, France
Alternatively, for controlling the spectra of the codes, one can generate white noise vectors and then apply amplitude distortion in the Fourier domain to achieve desired spectra. Finally, orthonormality of the codes is achieved by means of a singular value decomposition [3]. Another technique that enables better control of spectral shape consists in splitting code sequences spectra in a set of subbands of interest. In each subband, the Fourier transforms of the sequences are chosen as orthogonal Walsh codes with fixed amplitudes [4]. Proceeding so in each subband yi
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