On the PAPR of Binary Reed-Muller OFDM Codes Kaustuvmani Manji Senior Research Engineer, Center for Development of Telematics, Telecom Technology Center of Government of India, 71/1 Miller Rd, Bangalore: 560052, India. e-mail:
[email protected] Abstract — We present a lower bound on the Peak to Average Power Ratio (PAPR) for the second order cosets of binary Reed-Muller code by classifying the codewords using their Walsh Hadamard Transform (WHT) vectors.
I. Introduction A major drawback of the Orthogonal Frequency Division Multiplexing (OFDM) system is that the OFDM signal exhibits a very high Peak to Average Power Ratio (PAPR). It has been shown by Boyd [1] and Popovi´c [2] that the use of Golay Complementary Sequences as codewords restricts the PAPR of the OFDM signal to at most 3dB. Subsequently OFDM code has been characterized by Davis, Jedwab and Paterson [4],[3] in terms of generalized boolean function to arrange them as second order cosets of first order Reed-Muller code. In this paper we present a lower bound on PAPR for the second order cosets of binary Reed-Muller code by classifying the WHT spectrum of the codewords. This result categorizes the second order cosets of binary Reed-Muller code as per their PAPR by observing their WHT spectrum.
II. Second order Cosets of Reed-Muller code as OFDM codes An n-carrier OFDM signal is composed by adding together n equally spaced, phase-shifted sinusoidal carriers. Information is carried in the phase shift applied to each carrier. When the sinusoidal signals of the n carriers of a OFDM signal add constructively the peak envelope power can be as high as n times the average power. It has been established in [4],[3] that the PAPR of an OFDM signal modulated by a codeword from a Golay complementary set of size N is at most N . For q ≥ 2, a length n linear code, C, over Zq is defined to be a set of Zq valued vectors, called codewords, of length n that is closed under the operation of taking Zq -linear combination of codewords. A coset of C implies a set of the form a + C where a is some fixed vector over Zq . The Vector a is called the coset representative for the coset a + C. Definition 1 For q = 2, the r-th order Reed-Muller code, RM2 (r, m), is defined to be the binary code whose codewords are the vectors identified with the Boolean functions of degree at most r in x0 , x1 , . . . , xm−1 . Let Q : {0, 1}m → Z2 be a quadratic form in variables x0 , x1 , · · · , xm−1 . Then Q + RM2 (1, m) is a second order coset of the first order Reed-Muller code. Results in [4],[3] show the 1 This work was partly supported by the DRDO-IISc program on Advanced Research in Mathematical Engineering through a grant to B.S.Rajan
Dr. B. Sundar Rajan
1
Associate Professor, Dept. of Electrical Communication Engineering, Indian Institute of Science, Sir C.V.Raman Avenue, Bangalore: 560012, India. e-mail:
[email protected] availability of large number of OFDM codewords with low PAPR as second order cosets of the generalized first order Reed-Muller code (RM2 (1, m)).
III. Walsh Hadamard Transform (WHT) To transform the codewords of RM2 (1, m) and its second order cosets Walsh Hadamard Transform (WHT) has been employed. Definition 2 (WHT) If f = (fo , f1 , . . . , fn−1 ) denotes a vector of real numbers, then the WHT, F = (Fo , F1 , . . . , Fn−1 ) of f is given by Pm−1 P i j k=0 k k fi ; Fj = n−1 0 ≤ j ≤ n − 1, n = 2m . i=0 (−1)
IV. WHT Characterization of Binary Reed-Muller OFDM codes As RM2 (1, m) is defined over the binary alphabet {0, 1} and the domain of WHT is real arithmetic the mapping 0 7→ 1 and 1 7→ −1 is used in order to take the WHT of the codewords. We now present our main result of this paper which gives a lower bound on the PAPR of second order Binary Reed-Muller OFDM codes in Walsh Hadamard Transform domain. Theorem 1 If the WHT spectrum of the codewords of a second order coset of RM2 (1, m) has 22j nonzero components then the PAPR for that coset ≥ 2m−2j ; j = 0 to b m c. 2 Example 1 The following table describes the maximum PAPR of all possible 8 cosets of RM2 (1, 3) in RM2 (2, 3) along with their WHT spectrum. Coset Representative 0 0 0 1 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 1 0 0 0 1 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0
WHT Spectrum 4 4 0 0 0 0 4 -4 4 0 4 0 0 4 0 -4 4 0 0 4 4 0 0 -4 4 0 4 0 4 0 -4 0 0 4 4 0 4 0 0 -4 4 4 4 -4 0 0 0 0 4 4 0 0 4 -4 0 0 8 0 0 0 0 0 0 0
Max PAPR 2.000000 2.000000 2.000000 3.350368 3.420438 3.442696 3.450439 8.000000
References [1] S. Boyd, Multitone Signals With Low Crest Factor, IEEE Trans. Circuit and Systems, Vol: CAS-33, No.10, pp 1018-1022, Oct 1986. [2] B.M. Popovi´c Synthesis of Power Efficient Multitone Signals With Flat Amplitude Spectrum, IEEE Trans. Comm, Vol: 39, No.7, pp 1031-1033, July 1991. [3] James A. Davis, Jonathan Jedwab and Kenneth G. Paterson, Codes Correlation and Power Control in OFDM, Technical Report HPL-98-199, Hewlett-Packard Labs, Bristol, Dec 1998. [4] Kenneth G. Paterson, Generalized Reed-Muller Codes and Power Control in OFDM Modulation, IEEE Trans. Info. Th. Vol 46, No.1, pp 104-120, Jan 2000.
