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IEEE COMMUNICATIONS LETTERS, VOL. 16, NO. 12, DECEMBER 2012
PAPR Reduction in SC-FDMA by Pulse Shaping Using Parametric Linear Combination Pulses Cesar A. Azurdia-Meza, Kyujin Lee, and Kyesan Lee
Abstract—A linear combination between two intersymbol interference (ISI) free parametric linear pulses was proposed in order to obtain a new family of Nyquist pulses. The new family of pulses is utilized for pulse shaping to reduce peak-to-average power ratio (PAPR). The proposed pulse contains a new design parameter, μ, giving an additional degree of freedom to minimize PAPR for a given roll-off factor, α, and transmission scheme. While keeping the same bandwidth, the frequency responses of the proposed pulses differ with different values of the parameter μ for a fixed roll-off factor. Simulations showed that PAPR reduction is achieved when compared to that of other existing filters for the interleaved subcarrier mode of single carrier frequency division multiple access (SC-FDMA). Index Terms—Intersymbol interference (ISI), Nyquist’s first criterion, peak-to-average power ratio (PAPR), single carrier frequency division multiple access (SC-FDMA).
Both of these families of filters hold a new design parameter, giving an additional degree of freedom to minimize PAPR for a given roll-off factor, but are characterized by complex impulse response expressions. In this letter, a linear combination of two ISI-free parametric linear pulses [4] was proposed. The proposed family of filters has an additional parameter μ, giving an extra degree of freedom to minimize PAPR for a given roll-off factor, α, and transmission scheme. The proposed pulse shaping filter has a much simpler impulse response expression compared to those of the existing filters in [7] and [8]. Our proposed filter can be applied in any single carrier transmission scheme, but this letter only focuses on SC-FDMA. II. OVERVIEW OF THE SC-FDMA T RANSMITTER
I. I NTRODUCTION
D
URING the last few years, demands for media-rich wireless services have brought much attention to new high speed broadband wireless technologies. In the 3GPP LTE standard, the technologies being implemented for downlink and uplink consist of orthogonal frequency division multiple access (OFDMA) and SC-FDMA, respectively. Both of these technologies are based on the OFDM scheme; however, SCFDMA is being implemented as the uplink access technology due to its lower peak-to-average power ratio (PAPR) [1]–[3]. Nowadays, PAPR plays an important role in the design of wireless communication systems. A bandlimited signal with high PAPR requires a large back-off to ensure that the power amplifier operates inside its linear region in order not to distort signals [1], [2]. Additionally, a high PAPR requires higher dynamic ranges of digital-to-analog (D/A) converters. The total PAPR at the transmitter side is determined by the combination of the modulation scheme and the pulse shaping filter implemented. It has been shown in [4] that the Nyquist pulses have different decay rates, and according to [5] and [6], the decay rate of the filter has a big impact on PAPR. In order to reduce the PAPR of the modulated signal, we should design a pulse with a reduced tail size because the relative magnitudes of the filter’s two largest sidelobes are the ones that have the biggest effect on PAPR [5], [6]. In [7], a new family of generalized raised cosine (RC) filters was derived to reduce PAPR. Recently, a new pulse shaping filter, known as the K-Exponential Filter [8], has been derived to reduce PAPR. Manuscript received August 22, 2012. The associate editor coordinating the review of this letter and approving it for publication was Y. Li. The authors are with the Electronics and Radio Engineering Department, Kyung Hee University, Republic of Korea (e-mail:
[email protected]). Digital Object Identifier 10.1109/LCOMM.2012.111612.121891
SC-FDMA is considered a discrete Fourier transform spread OFDMA scheme because time domain symbols are transformed into the frequency domain via discrete Fourier transform (DFT) before OFDMA modulation takes place. At the transmitter side, a baseband modulator transforms the binary sequence into a multilevel sequence of complex numbers using one of several digital modulation techniques. Next, the transmitter groups the modulated symbols into blocks each containing N symbols. To generate a frequency domain version of the input signal, an N-point DFT is done. Then, subcarrier mapping is performed, and each of the DFT outputs is mapped to one of the transmittable M (>N) orthogonal subcarriers. In SC-FDMA, subcarrier mapping is achieved by implementing either the localized or distributed subcarrier mode [1]–[3]. In the distributed subcarrier mode, the DFT outputs are spread over the entire bandwidth and zeros are introduced in the unused subcarriers. A special case of distributed SC-FDMA is called interleaved SC-FDMA. According to [1] and [3], the interleaved mode of SC-FDMA is more desirable than the localized mode in terms of PAPR and power efficiency; therefore, we implemented the interleaved mode of SC-FDMA in our analysis in order to further reduce its PAPR. In the interleaved SC-FDMA mode, the occupied subcarriers are equally spaced over the entire bandwidth. After subcarrier mapping, an M-point inverse DFT (IDFT) transforms the subcarrier amplitudes into a complex time domain signal. The transmitter performs two operations prior to transmitting the symbols sequentially. The IDFT output is followed by a cyclic prefix (CP) insertion. The CP is inserted to implement a guard time between consecutive blocks to prevent ISI. The transmitter also executes a linear filtering operation known as pulse shaping, which is used to reduce the out-of-band signal energy [1]–[3]. A convolution between the modulated subcarriers and the filter’s impulse response takes place.
c 2012 IEEE 1089-7798/12$31.00
AZURDIA-MEZA et al.: PAPR REDUCTION IN SC-FDMA BY PULSE SHAPING USING PARAMETRIC LINEAR COMBINATION PULSES
III. PARAMETRIC L INEAR C OMBINATION P ULSES Nyquist’s ISI-free criterion for distortionless transmissions within a bandlimited channel is defined as follows [10], [11] 1, n = 0 h (nT ) = (1) 0, n = ±1, ±2, . . . where h(t) is the impulse response, and T is the symbol period. Whereas in the frequency domain, the Fourier transform of (1) is given as follows m=∞ 1 m = 1, (2) H f+ T m=−∞ T where H(f) is the Fourier transform of h(t). The excess bandwidth of a Nyquist ISI-free pulse is determined by the roll-off factor, α, 0 ≤ α ≤ 1, and T = 1/2B is the symbol repetition rate for a bandwidth B > 0. Moreover, we considered the linear combination of two parametric linear pulses (PLP) [4]. The prior Nyquist pulses were chosen because they possess an explicit time-domain expression, and according to [12], the linear combination of two Nyquist pulses ensures that the resulting pulse will be ISIfree. The proposed linear combination is defined as follows h (t) = μh(t)P LPn=1 + (1 − μ) h(t)P LPn=2 ,
(3)
where n is a parameter that defines different parametric linear pulses with different decay rates according to [4, eq. (5)]. The impulse responses of P LPn=1 and P LPn=2 are given in [4], and decay as 1/t2 and 1/t3 , respectively. Whereas μ is the constant that corresponds to the linear combination and it is defined for all real numbers [13]. The introduced linear combination constant adds an additional degree of freedom to minimize PAPR for a given roll-off factor, α, and single carrier transmission scheme. Furthermore, the impulse response of the proposed ISI-free parametric linear combination pulse (PLCP) is given as follows h(t)PLCP = μ [sinc (πτ ) sinc (απτ )] + (1 − μ) sinc (πτ ) sinc2 (0.5απτ ) ,
(4)
where τ is the normalized time (τ = t/T ). In the rest of this paper we will express the time-domain functions in terms of the normalized time for simplicity reasons. After some algebraic manipulations, the ISI-free PLCP is given as follows sin (πτ ) h(t)PLCP = πτ 4 (1 − μ) sin2 (πατ /2) + παμτ sin (πατ ) . × π 2 α2 τ 2 (5)
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SC-FDMA is a single carrier modulation scheme, and according to [1] and [9], pulse shaping is required to bandlimit the transmitted signal. Nevertheless, pulse shaping to limit the frequency bandwidth enlarges the PAPR of the SC-FDMA transmitted signals. As a result, there is a trade-off between PAPR and bandwidth reduction in conventional single carrier systems. Research has been ongoing over the last years trying to design a pulse shaping filter that limits PAPR without degrading the system’s performance, based on the knowledge that, and compared to other PAPR reduction techniques, pulse shaping is an effective and simple method [2], [5], [7], [8].
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Fig. 1. Impulse response of the RC filter and the ISI-Free PLCP for different values of μ and α = 0.35.
The pulse defined in (5), evaluated for t = 0, and for any value of μ, is always equal to one. Additionally, the proposed pulse, evaluated for n = ±1, ±2, . . ., and for any value of μ, is always equal to zero. Therefore, the proposed pulses in (5) fulfill Nyquist’s ISI-free criterion, equation (1). The frequency response of the proposed ISI-free PLCP given in (5) is described by the even-frequency spectrum in [4, eq. (5)] as follows. H(f )PLCP ⎧ T, ⎪ ⎪ ⎪ ⎪ T ⎪ ⎪ μ{1 − f − B(1 − α)}+ ⎪ ⎪ ⎪ 2αB ⎪ ⎪ ⎪ ⎪ {1 − f − B(1 − α)}2 ⎪ ⎨ (1 − μ) , αB μ{1 − f + B(α = − 1)} ⎪ ⎪ + (1 − μ) T ⎪ ⎪ 2αB ⎪ ⎪ ⎪ ⎪ {f − 1 + B(1 − α)}2 ⎪ ⎪ , 1 − ⎪ ⎪ 2α2 B 2 ⎪ ⎪ ⎩ 0,
0 ≤ f < B(1 − α) B(1 − α) ≤ f ≤ B
B < f ≤ B(1 + α) B(1 + α) < f (6)
The time and frequency responses of the proposed filter, as well as for the raised cosine (RC) filter, are shown in Figs. 1 and 2, respectively, for a roll-off factor, α, and different values of μ. Regarding the impulse response of the PLCP for different values of μ, the RC filter has larger sidelobes than those of the proposed filter with μ = 1, μ = 1.6, μ = 2, and μ = 2.5 as examples. From examination of Fig. 1, it can be inferred that by selecting the proper μ, a reduction in the relative amplitudes of the two largest sidelobes can be accomplished. As a result, a filter with smaller sidelobes would lead to PAPR reduction [5], [6], [8]. The proposed filter maintains the same bandwidth compared to that of the RC pulse for a roll-off factor, α, and different values of μ; therefore, it does not introduce additional out-of band radiation into the transmitted signal, which is a major concern in the design of pulse shaping filters. A. Optimality of μ It is desired to determine the optimum value of μ to minimize PAPR according to the transmission scheme and
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Fig. 2. Frequency response of the RC filter and the ISI-Free PLCP for different values of μ and α = 0.35.
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Fig. 3. CCDF of PAPR for a SC-FDMA system with 16QAM and α = 0.35.
TABLE I SIMULATION PARAMETERS −1
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Parameter Modulation Number of Subcarriers Input Data Block Size Spreading Factor Transmission Bandwidth Block Oversampling Uniform Random Data Points Subcarrier Allocation Roll-off Factor
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roll-off factor. We refer to this value of μ as μopt , and the pulse obtained by using μopt in (5) is known as the optimum PLCP. Let us consider the case for |μ| 1 and t = 0 in (3) as follows h(t)PLCP
= μ h(t)PLPn=1 − h(t)PLPn=2 .
RC Pulse, [8] Convex(d=5), [8] K−Exponential Pulse (k=10), [8] Concave (d=1), [8] PLCP (u=1.6)
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As previously mentioned, P LPn=1 and P LPn=2 decay as 1/t2 and 1/t3 , respectively. As the tails of P LPn=2 decay faster than those of P LPn=1 , it can be inferred the larger the value of μ, the larger will be the sidelobes in (7). These large sidelobes will make the proposed pulse more sensitive to PAPR. Therefore, it is unlikely that PAPR, as a function of μ, and for any roll-off factor and single carrier transmission scheme, has a minimum for |μ| 1. This expected behavior was corroborated during the numerical evaluation shown in the next section. In general, there is an optimum μopt for every roll-off factor and single carrier transmission scheme, although it might not be unique. In this paper, and for illustration purposes, the interleaved mode of SC-FDMA was evaluated for α = 0.35, which is a roll-off factor commonly used in literature. IV. P ERFORMANCE E VALUATION To evaluate the PAPR of individual system configurations, we simulated the transmission of 106 system blocks. After calculating the PAPR of each block, the data was presented as an empirical complementary cumulative distribution function
Fig. 4. CCDF of PAPR for a SC-FDMA system with 16QAM and α = 0.35.
(CCDF). As it was done in [8], PAPR was measured and analyzed on the spectrum of the proposed optimum ISI-free PLCP and the other filters implemented for comparison. Table I illustrates the parameters implemented in the simulation for the SC-FDMA system using interleaved allocation. The parameters applied in our simulations agree with the ones in [8] for comparison purposes. The optimal value of the linear combination constant μ for PAPR reduction was selected through extensive computer simulations because an analytical solution for the optimal μ seems unrealizable, as it was done in [8] for selecting the optimal K, and the parameter d proposed in [7] for the convex and concave filters. Fig. 3 illustrates the PAPR performance of the SC-IFDMA system by using different values of μ. For the sake of clarity and lack of space, only PAPR curves evaluated with positive values of μ were plotted, but the trend is the same for negative values of μ. The PAPR of the proposed system increases with a larger value of |μ|. The proposed system did not achieve a minimum PAPR
AZURDIA-MEZA et al.: PAPR REDUCTION IN SC-FDMA BY PULSE SHAPING USING PARAMETRIC LINEAR COMBINATION PULSES 0.1
V. C ONCLUSION In this letter, a novel Nyquist-I pulse has been derived by implementing a linear combination between two ISI-free pulses. The proposed filter keeps the same bandwidth for different values of μ and a fixed roll-off factor. The new design parameter μ gives an additional degree of freedom to minimize PAPR for a given roll-off factor, α, and single carrier transmission scheme. The optimum value of μ to minimize the PAPR of the interleaved SC-FDMA scheme was found to be 1.60 for α = 0.35. Simulations showed that the proposed optimum filter provides better performance regarding PAPR reduction compared to that of the existing filters described in [7] and [8]. R EFERENCES
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PLCP (u = 1.6) K−Exponential Pulse (k = 10) [8] Concave (d = 1) [7] Convex (d = 5) [7] 1
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Fig. 5. Impulse response of the PLCP and other existing pulses for α = 0.35.
for the interval 0 ≤ |μ| ≤ 1.6. According to our numerical analysis, the optimum μopt for the proposed system is equal to 1.60. In Fig. 4, we demonstrate the effectiveness of the proposed pulse shaping filter for the interleaved mode of the SC-FDMA system with 16QAM. The proposed optimum filter has the lowest PAPR among the evaluated pulses for comparison. The reasons in PAPR reduction are explained as follows. The better performance of the optimum PLCP can be attributed to the fact that its first two sidelobes are smaller than those of the pulses implemented for comparison, as depicted in Fig. 5. Consequently, a filter with smaller sidelobes would lead to a smaller PAPR according to [5], [6], [8]. Despite that the filters proposed in [7] and [8] also have an additional design parameter, the optimum PLCP has superior performance in PAPR reduction. Additionally, our novel filter has a simpler impulse response expression in comparison to the prior pulses. The proposed pulses in [7] require two complex mathematical expressions, while the pulse described in [8] has a very complex impulse response compared to that of our proposed pulse.
[1] H. G. Myung, J. Lim, and D. J. Goodman, “Single carrier FDMA for uplink wireless transmission,” IEEE Veh. Technol. Mag., vol. 1, no. 3, pp. 30–38, Sep. 2006. [2] H. G. Myung, J. Lim, and D. Goodman, “Peak-to-average power ratio of single carrier FDMA with pulse shaping,” in Proc. 2006 IEEE International Symposium on PIMRC, pp. 1–5. [3] H. G. Myung, “Introduction to single carrier FDMA,” in Proc. 2007 European Signal Processing Conference, pp. 2144–2148. [4] N. C. Beaulieu and M. O. Damen, “Parametric construction of Nyquist-I pulses,” IEEE Trans. Commun., vol. 52, no. 12, pp. 2134–2142, Dec. 2004. [5] B. Farhang-Boroujeny, “A square-root Nyquist (M) filter design for digital communication systems,” IEEE Trans. Signal Process., vol. 56, no. 5, pp. 2127–2132, May 2008. [6] B. Chatelain and F. Gagnon, “Peak-to-average power ratio and intersymbol interference reduction by Nyquist optimization,” in Proc. 2004 IEEE Vehicular Technology Conference, vol. 2, pp. 954–958. [7] P. S. Rha and S. Hsu, “Peak-to-average ratio (PAR) by pulse shaping using a new family of generalized raised cosine filters,” in Proc. 2003 IEEE Vehicular Technology Conference, vol. 1, pp. 706–710. [8] Y. D. Wei and Y. F. Chen, “Peak-to-average power ratio (PAPR) reduction by pulse shaping using the K-exponential filter,” IEICE Trans. Commun., vol. E93-B, no. 11, pp. 3180–3183, Nov. 2010. [9] G. Huang, A. Nix, and S. Armour, “Impact of radio resource allocation and pulse shaping on PAPR of SC-FDMA signals,” in Proc. 2007 IEEE International Symposium on PIMRC, vol. 1, pp. 1–5. [10] H. Nyquist, “Certain topics in telegraph transmission theory,” AIEE Trans., vol. 47, no. 1, pp. 617–644, 1928. [11] J. O. Scanlan, “Pulses satisfying the Nyquist criterion,” Electron Lett., vol. 28, no. 1, pp. 50–52, Jan. 1992. [12] P. Sandeep, S. Chandan, and A. K. Chaturvedi, “ISI-free pulses with reduction sensitivity to timing errors,” IEEE Commun. Lett., vol. 9, no. 4, pp. 292–294, Apr. 2005. [13] G. Strang, Introduction to Linear Algebra. Wellesley-Cambridge Press, 1998.
