Physical Communication 7 (2013) 92–104 Contents lists available at SciVerse ScienceDirect Physical Communication journal homepage: www.elsevier.com/locate/phycom Full length article Optimal design of perfect DFT sequences João S. Pereira a,b,∗ , Henrique J.A. da Silva a,c a Instituto de Telecomunicações, Universidade de Coimbra, Portugal b DEI/ESTG do Instituto Politécnico de Leiria, 2401-951 Leiria, Portugal c DEEC, Faculdade de Ciências e Tecnologia, Universidade de Coimbra, Portugal article info Article history: Received 30 November 2011 Received in revised form 26 June 2012 Accepted 13 October 2012 Available online 24 October 2012 Keywords: CDMA GDFT OFDM Perfect sequences abstract In this paper, we propose a novel scheme to construct a large set with N(N + 1) perfect sequences of length N, derived from an Inverse Discrete Fourier Transform (IDFT) of Chu and maximum-length sequences. This optimum set of perfect periodic autocorrelation sequences has a maximum absolute value of periodic cross-correlation strictly lower than N and close to the well known lower bound SQRT(N). Moreover, we present a method to transform these perfect sequences into orthogonal sequences. A similar method is also proposed to obtain optimum bipolar codes derived from an alternative set of N 2 perfect sequences. In the design of perfect sequences, the difficulty is to achieve both low cross-correlation and low peak-to-average power ratio (PAPR). Many of the proposed perfect DFT sequences should have low PAPR and thus can be applied in an OFDM–CDMA (Orthogonal Frequency- Division Multiplexing–Code Division Multiple Access) communication system or in a simple CDMA communication system. Alternatively, the proposed perfect DFT sequences may be useful for pre-coded mapping in OFDM communication systems or for the design of radar waveform diversity sets. © 2012 Elsevier B.V. All rights reserved. 1. Introduction The selection of appropriate coding sequences is a complex choice in the design of various communica- tion systems. One of them is the Code Division Multiple Access (CDMA) communication system where the se- quences should have a perfect periodic autocorrelation (with null side lobes) [1], excellent cross-correlation prop- erties (equal or near the null value for most of the se- quences selected) and should exist, if possible, in large enough numbers for the planned application. Some well known orthogonal codes that can be found in CDMA communication systems are orthogonal Gold codes [2,3], Walsh–Hadamard codes [4], and many other codes such ∗ Corresponding author at: Instituto de Telecomunicações, Universi- dade de Coimbra, Portugal. Tel.: +351 244 820300; fax: +351 244 820310. E-mail addresses: joao.pereira@ipleiria.pt (J.S. Pereira), hjas@ci.uc.pt (H.J.A. da Silva). as perfect sequences [5]. Benefits and drawbacks of other CDMA spreading sequences can be found in [6,7]. It is well known that perfect sequences are complex sequences such that all out-of-phase periodic autocor- relation values are zero. Unfortunately, perfect bipolar sequences of length N > 4 and perfect quadri-phase sequences of length N > 16 are unknown. Perfect se- quences with a low maximum absolute value of periodic cross-correlation (MaxCC) have been proposed in the lit- erature. One of the largest sets of perfect sequences with the lowest MaxCC ( √ N ) is the Chu polyphase set [5], which has N − 1 perfect sequences. Other well known sets, with a number of perfect sequences smaller than N , are the Generalized Chirp-Like Polyphase Sequences [8], Frank Perfect Sequences [5], Frank–Zadoff–Chu (or FZC) Perfect Sequences [9], and Generalized Chu polyphase se- quences [10]. Larger sets of perfect sequences have also been reported. For example, the union of Frank and Chu sequences [5] is an alternative solution to obtain a larger 1874-4907/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.phycom.2012.10.001