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Peak-to-Average-Power-Ratio (PAPR) Reduction of OFDM Systems Using Generalized Rudin-Shapiro Construction Fu-Hsuan Chiu University of Southern California, Los Angeles, CA 90089 Email: [email protected]

I. I NTRODUCTION Large Peak-to-Average-Power-Ratio (PAPR) is a critical issue in OFDM systems and many techniques has been proposed to overcome it [1](and the references therein); among them, the most well-known algebra approach is Golay sequence [1]. This article summarizes recently results on PAPR reduction techniques of OFDM systems using the generalized Rudin-Shapiro construction. This article is inspired by Zhao’s extended Rudin-Shapiro construction [2]. However, there are some errors in that paper and we propose the generalized Rudin-Shapiro construction to correct them. Before discussing the approach, we overiew the basic idea of the extended Rudin-Shapiro construction. II. RUDIN -S HAPIRO C ONSTRUCTION The Rudin-Shapiro construction proceeds as follows: Let P0 (z) = z. For any nonnegative integer i, we define Pi+1 (z) ≡ Pi (z 2 ) + z −1 Pi (−z 2 )

(1)

It can be shown that Pk (z) satisfies |Pi+1 (z)|2 + |Pi+1 (−z)|2 = 2i+2 Oct. 29th 2004

∀ |z| = 1

(2) DRAFT

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III. G ENERALIZED RUDIN -S HAPIRO C ONSTRUCTION We are interested in obtaining low PMPR m-PSK (m = 2h ) symbol sequences with length n = 2k . Let Q0 (z) = z The generalized Rudin-Shapiro construction is defined as Qi+1 (z) ≡ where αi , βi , γi ∈ {ej

2πp m

1 γi+1

{αi+1 Qi (γi+1z 2 ) + βi+1 z −1 Q(−γi+1 z 2 )}|αl =1,γl =1,∀

l≤i ,

(3)

: p = 0, 1, · · · , m − 1}.

Then it is straightforward to show that |Qi+1 (z)|2 + |Qi+1 (−z)|2 = 2i+2

∀ |z| = 1.

(4)

Therefore, we claim that the modulation symbol sequences obtained from the coefficients of Qk (z) have PMPR less or equal to 2. IV. D ISCUSSION A. Insight of the Construction The reason we restrict the parameters within the desired constellation is to fix the problem of signal constellation shift caused by the arbitrary phase-shift {α i , βi , γi }. By this way, we can easily find the maximal number of sequences corresponding to each n = 2 k (length) and size of m-PSK signal constellation (m = 2h ). However, the uncountable property does not hold. The following example explains how the construction works. B. Simple Example Consider k = 2, h = 2, i.e., we have four subcarriers (n = 4) with QPSK modulation. By direct calculation, (β1 β2 , −α2 β1 , β2 γ2 , α2 γ2 ) is the corresponding coefficients of Q2 (z). The remaining question is how to get the corresponding Z 4 sequences. We know that each Z4 symbol a is mapped to a QPSK symbol j a = ej

2πa 4

. Let a2 , b1 , b2 ,and r2 be the corresponding Z4 symbols of

α2 , β1 , β2 , and γ2 . Then the coefficients sequence of the polynomial Q2 (z) is the mapping from Z4 sequence {b1 + b2 , a2 + b1 + 2, b2 + r2 , a2 + r2 }. The remaining problem is how to find the

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actual size of those sequences. It can be determined by the degree of freedom of the variable. ⎡ ⎤ ⎤ a2 ⎡ ⎡ ⎤ ⎥ 0 1 1 0 0 ⎢ ⎥ ⎢x1 ⎥ ⎥⎢ ⎢ b 1⎥ ⎢ ⎢1 1 0 0 1⎥ ⎢ ⎥ ⎢x ⎥ ⎥ ⎢ ⎥ ⎢ 2⎥ ⎢ (5) ⎥ b2 ⎥ = ⎢ ⎥ ⎢ ⎢0 0 1 1 0⎥ ⎢ ⎢x3 ⎥ ⎢ ⎥ ⎦⎢ ⎥ ⎣ ⎦ ⎣ ⎣ r2 ⎦ 1 0 0 1 0 x4 2 By the row operator elimination, we find that the four entries of the Z 4 sequence are not linear independent, i.e., one entry is fully described by the other three. Thus, the size of the sequence obtained by extended Rudin-Shapiro construction in this case is 4 3 = 64. C. Golay Sequences Construction The number of polynomials satisfied the extended Rudin-Shapiro construction with m-PSK should be bounded by the size of the Golay sequences proposed by Davis and Jedwab [1] with the same modulation size. For the simple example case, the size predicted by the Golay sequences is

k! h k+1 (2 ) 2

=

2! 2 2+1 (2 ) 2

= 43 = 64. A theoretic connection between the extended Rudin-

Shapiro construction and the Goaly sequences should be carefully studied further. However, unlike the QPSK Golay sequences introduced by Davis and Jedwab, the generalized RudinShapiro construction provides a more explicit way to construct the same sequences. R EFERENCES [1] J. A. Davis and J. Jedwab, “Peak-to-mean power control in OFDM, Golay complementary sequences and Reed-Muller codes,” Trans. Inform. Theory, Vol. 45, No. 7, Nov. 1997, pp. 2397-2417. [2] J. Zhao, “A Peak-to-mean power control scheme: the extended Rudin-Shapiro construction,” in Proceeding 60th IEEE VTC, Los Anegels, CA, Sep. 2004.

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