zbMATH — the first resource for mathematics

Unit time-phase signal sets: bounds and constructions. (English) Zbl 1335.94014
Summary: Digital signals are complex-valued functions on \(\mathbb Z_n\). Signal sets with certain properties are required in various communication systems. Traditional signal sets consider only the time distortion during transmission. Recently, signal sets taking care of both the time and phase distortion have been studied, and are called time-phase signal sets. Several constructions of time-phase signal sets are available in the literature. There are a number of bounds on time signal sets (also called codebooks). They are automatically bounds on time-phase signal sets, but are bad bounds. The first objective of this paper is to develop better bounds on time-phase signal sets from known bounds on time signal sets. The second objective of this paper is to construct four series of time-phase signal sets, one of which is optimal.

94A12 Signal theory (characterization, reconstruction, filtering, etc.)
Full Text: DOI arXiv
[1] Allop, W.O.: Complex sequences with low periodic correlations. IEEE Trans. Inf. Theory 26(3), 350–354 (1980) · Zbl 0432.94011 · doi:10.1109/TIT.1980.1056185
[2] Bajwa, W.U., Calderbank, R., Mixon, D.G.: Two are better than one: fundamental parameters of frame coherence. Appl. Comput. Harmon. Anal. 33(1), 58–76 (2012) Preprint. arXiv:1103.0435v2 (2011) · Zbl 1246.42026
[3] Berndt, B.C., Evans, R.J., Williams, K.S.: Gauss and Jacobi Sums. Wiley-Interscience (1998) · Zbl 0906.11001
[4] Calderbank, A.R., Cameron, P.J., Kantor, W.M., Seidel, J.J.: Z 4-Kerdock codes, orthogonal spreads, and extremal Euclidean line-sets. Proc. Lond. Math. Soc. 75(3), 436–480 (1997) · Zbl 0916.94014 · doi:10.1112/S0024611597000403
[5] Conway, J.H., Harding R.H., Sloane, N.J.A.: Packing lines, planes, etc.: packings in Grassmannian spaces. Exp. Math. 5(2), 139–159 (1996) · Zbl 0864.51012 · doi:10.1080/10586458.1996.10504585
[6] Ding, C.: Codebooks from combinatorial designs. IEEE Trans. Inf. Theory 52(9), 4229–4235 (2006) · Zbl 1237.94001 · doi:10.1109/TIT.2006.880058
[7] Ding, C., Feng, T.: A generic construction of complex codebooks meeting the Welch bound. IEEE Trans. Inf. Theory 53(11), 4245–4250 (2007) · Zbl 1237.94002 · doi:10.1109/TIT.2007.907343
[8] Ding, C., Yin, J.: Signal sets from functions with optimum nonlinearity. IEEE Trans. Commun. 55(5), 936–940 (2007) · doi:10.1109/TCOMM.2007.894113
[9] Grassl, M.: Computing equiangular lines in complex space. In: Calmet, J., Geiselmann, W., Muller-Quade, J. (eds.) Beth Festschrift, LNCS 5393, pp. 89–104 (2008) · Zbl 1178.68690
[10] Gurevich, S., Hadani, R., Sochen, N.: The finite harmonic oscillator and its applications to sequences, communication and radar. IEEE Trans. Inf. Theory 54(9), 4239–4253 (2008) · Zbl 1205.94034 · doi:10.1109/TIT.2008.926440
[11] Heath, R.W. Jr., Strohmer,T., Paulraj, A.J.: On quasi-orthogonal signature for CDMA systems. IEEE Trans. Inf. Theory 52(3), 1217–1226 (2006) · Zbl 1316.94101 · doi:10.1109/TIT.2005.864469
[12] Helleseth, T., Kumar, V.P.: Sequences with low correlation. In: Pless, V.S., Huffman, W.C. (eds.) Handbook of Coding Theory, vol. II, pp. 1765–1853. Elsevier, Amsterdam (1998) · Zbl 0924.94027
[13] Herman, M.A., Strohmer, T.: High-resolution radar via compressed sensing. IEEE Trans. Signal Process. 57, 2275–2284 (2009) · Zbl 1391.94236 · doi:10.1109/TSP.2009.2014277
[14] Howard, S.D., Calderbank, A.R., Moran, M.: The finite Heisenberg-Weyl groups in radar and communications. EURASIP J. Appl. Signal Process. 2006 1–12 (2006) · Zbl 1122.94015 · doi:10.1155/ASP/2006/85685
[15] Karystinos, G.N.: Optimum binary signature set design and short-data-record adaptive receivers for CDMA communications. Ph.D Thesis, University of New York at Buffalo (2003)
[16] Kretschmer, F.F., Lewwis, B.L., Jr.: Doppler properties of polyphase coded pulse compression waveforms. IEEE Trans. Aero. Elec. Syst. 19(4), 521–531 (1983) · doi:10.1109/TAES.1983.309340
[17] Levenstein, V.I.: Bounds on the maximal cardinality of a code with bounded modules of the inner product. Soviet Math. Dokl. 26, 526–531 (1982) · Zbl 0528.94017
[18] Li, W.C.W.: Number theory with applications. In: Series on University Mathematics, vol. 7 (1995) · Zbl 0849.11006
[19] Love, D.J., Heath, R.W., Strohmer, T: Grassmannian meamingforming for multiple-input multiple-output wireless systems. IEEE Trans. Inf. Theory 49(10), 2735–2747 (2003) · Zbl 1301.94080 · doi:10.1109/TIT.2003.817466
[20] Nelson, J.L., Temlyakov, V.N.: On the size of incoherent systems. J. Approx. Theory 163, 1238–1245 (2011) · Zbl 1242.41037 · doi:10.1016/j.jat.2011.04.001
[21] Renes, J.M., Blume-Kohout, R., Scott, A.J., Caves, C.M.: Symmetric informationally complete quantum measurements. J. Math. Phys. 45(6), 2171–2180 (2007) · Zbl 1071.81015 · doi:10.1063/1.1737053
[22] Sarwate, D.: Meeting the Welch bound with equality. In: Proc. of the Sequences and their Applications, SETA’ 98, pp. 79–102. Springer, London (1999) · Zbl 1013.94510
[23] Scott, A.J., Grassl, M.: SIC-POVMs: a new computer study. J. Math. Phys. 51, 042203 (2010) · Zbl 1310.81022 · doi:10.1063/1.3374022
[24] Sidelnikov, V.M.: On mutual correlation of sequences. Probl. Kibern. 24, 15–42 (1971)
[25] Strohmer, T., Heath, R.W. Jr.: Grassmannian frames with applications to coding and communication. Appl. Comput. Harmon. Anal. 14(3), 257–275 (2003) · Zbl 1028.42020 · doi:10.1016/S1063-5203(03)00023-X
[26] Welch, L.: Lower bounds on the maximum cross correlation of signals. IEEE Trans. Inf. Theory 20(3), 397–399 (1974) · Zbl 0298.94006 · doi:10.1109/TIT.1974.1055219
[27] Xia, P., Zhou, S., Giannakis, G.B.: Achieving the Welch bound with difference sets. IEEE Trans. Inf. Theory 51(5), 1900–1907 (2005) · Zbl 1237.94007 · doi:10.1109/TIT.2005.846411
[28] Zauner, G.: Quantendesigns: Grundzuege einer nichtkommutativen Designtheorie. Dissertation, Universitaet Wien (1999)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.