JOURNAL OF TELECOMMUNICATIONS, VOLUME 30, ISSUE 1, APRIL 2015

8

Broadband Pulse shapes for PAPR

Mitigation Technique Ashwini Saykar and Debashis Adhikari

Abstract—The desire for higher data rates of transmission has resulted in the evolution of many new communication

technologies over the years. Orthogonal Frequency Division Multiplexing (OFDM) is one such technology which is capable of

achieving a high rate system. Although OFDM has many advantages of being a bandwidth efficient and high rate system, the

problem of peak to average power ratio (PAPR) has resulted in a considerable disadvantage to this technology. In this paper we

discuss a pulse shaping method to reduce the PAPR. Few broadband pulse shapes are analysed that can be used to modulate

the sub-carrier frequencies. It is established that PSWF based pulses could lead to minimize the problem of PAPR.

Index Terms— PAPR, OFDM, PSWF.

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1 INTRODUCTION

IRELESS multimedia applications has emerged as the fastest growing voice and data communication technology. This rapid development of wide are-

amultimedia networks has built up general greed for higher data rates. However such a goal is bound by opti-mal spectrum utilization and major channel impairments like delay spread, noise, interference and channel varia-tions as well as battery life of portable devices.

High data rates are primarily governed by the symbol rate and transmit constellation complexity. Also a disper-sive multipath channel results in severe intersymbol in-terferernce (ISI), thereby limiting the signaling rate. This would call for complex equalizers as the data rates are higher. For a single carrier system, the increase in signal bandwidth due to increase in symbol rate results in fre-quency selective fading as the coherence bandwidth be-comes less than the signal bandwidth.

Orthogonal frequency division multiplexing (OFDM) is one such technology that is best suited to achieve this goal of high data rate. It can be viewed as a kind of multi-carrier system with a narrow-band transmission scheme. OFDM has the advantage of combating multipath fading by virtue of multiplexing serial data into large number of subcarrier frequencies, each of narrow bandwidth. Thus, with a large number of subcarriers (N), the subcarrier bandwidth (B/N, with B as the total bandwidth) becomes much less than the coherent bandwidth thereby satisfying the conditions of a frequency flat fading channel. In con-trast, a multi-carrier transmission scheme uses fewer sub-carrier frequencies resulting in broader subcarrier band-widths as in the case of HSPA. A proper choice of symbol duration and carrier spacing makes the OFDM technolo-gy an efficient one. The system complexity largely reduc-es by IDFT – DFT process. OFDM is bandwidth efficient

signaling system as the spectra of orthogonal subcarriers are overlapped.

However there are few limitations of OFDM inspite of the avantages of being a high rate system. One of the ma-jor limitations is the high peak-to-average power ratio (PAPR) and its sensitivity against carrier frequency offset (CFO). An outcome of CFO is loss of orthogonality among the carriers resulting in inter carrier interference (ICI).

In this paper PAPR and its mitigation techniques using different pulse shapes is presented. In Section 2 the prob-lem of PAPR and many proposed reduction techniques in literature is discussed. Section 3 discusses about the sys-tem model of PAPR reduction scheme with different pulse shapes for OFDM symbols. In Section 4 we discuss few broadband waveforms that can be used as pulse shapes and analyze these pulse shapes with simulation results in Section 5.

2 PEAK TO AVERAGE POWER RATIO IN OFDM

2.1 PAPR mathematical foundation

A major drawback in OFDM because of multiple carriers is the non-constant envelope with high peaks leading to high peak – to – average power ratio (PAPR). PAPR is defined as the ratio between the maximum power and the average power of a complex bandpass signal s(t)

𝑃𝐴𝑃𝑅{�̌�(𝑡)} =𝑚𝑎𝑥|𝑅𝑒(�̌�(𝑡)𝑒𝑗2𝜋𝑓𝑐𝑡)|2

𝐸{|𝑅𝑒(�̌�(𝑡)𝑒𝑗2𝜋𝑓𝑐𝑡)|2}=𝑚𝑎𝑥|𝑠(𝑡)|2

𝐸{|𝑠(𝑡)|2} (1)

For a single carrier system with BPSK modulated symbols x(0), x(1) .. x(N-1) the average power in each symbol is given as

𝐴𝑣. 𝑃𝑜𝑤𝑒𝑟 = 𝐸{|𝑥(𝑘)|2} = 𝑎2 (2) where the power in each symbol is 𝑎2 which is also the peak power. The PAPR for a single carrier system is

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Ashwini Saykar is with Symbiosis Institute of Technology, Symbiosis International University, Lavle, Pune, India.

D. Adhikari is a Faculty at Synbiosis Institute of Technology, Symbiosis International University, Lavle, Pune.

W

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therefore

𝑃𝐴𝑃𝑅|𝑆𝑖𝑛𝑔𝑙𝑒 𝑐𝑎𝑟𝑟𝑖𝑒𝑟 =𝑃𝑒𝑎𝑘 𝑝𝑜𝑤𝑒𝑟

𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑝𝑜𝑤𝑒𝑟= 1 = 0 𝑑𝐵 (3)

This indicates that there is no significant deviation from the mean power level for a single carrier system. For N sub-carrier OFDM system, the kth sample of IFFT is given as

𝑥(𝑘) =1

𝑁∑𝑋(𝑖)

𝑁−1

𝑖=0

𝑒𝑗2𝜋𝑘𝑖𝑁 (4)

where 𝑋(𝑖) is the ith information symbol. The average power is given as 𝐴𝑣. 𝑃𝑜𝑤𝑒𝑟 = 𝐸{|𝑥(𝑘)|2}

= 1

𝑁2∑𝐸{|𝑋(𝑖)|2}

𝑁−1

𝑖=0

𝐸 {|𝑒𝑗2𝜋𝑘𝑖𝑁|2

}

=1

𝑁2∑𝐸{|𝑋(𝑖)|2}

𝑁−1

𝑖=0

=1

𝑁2∑𝑎2𝑁−1

𝑖=0

=𝑎2

𝑁 (5)

To find the peak power for the zeroth sample

𝑥(0) =1

𝑁∑𝑋(𝑖)

𝑁−1

𝑖=0

𝑒𝑗2𝜋𝑘𝑖𝑁 =

1

𝑁∑𝑋(𝑖)

𝑁−1

𝑖=0

(6)

If 𝑋(0) = 𝑋(1) = ⋯ = 𝑋(𝑁 − 1) = 𝑎 then

𝑥(0) =1

𝑁∑𝑋(𝑖) =

1

𝑁∑𝑎 =

𝑁−1

𝑖=0

𝑁−1

𝑖=0

𝑎𝑁

𝑁= 𝑎 (7)

Therefore the peak power is 𝑎2. The PAPR for the OFDM system with N sub-carriers is given as

𝑃𝐴𝑃𝑅|𝑂𝐹𝐷𝑀 =𝑃𝑒𝑎𝑘 𝑝𝑜𝑤𝑒𝑟

𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑝𝑜𝑤𝑒𝑟= 𝑁 (8)

In general for MPSK modulated symbols in an OFDM system with N subcarriers, the maximum power occurs when all of the N subcarrier components add up with identical phases. This results in a high peak value signal causing different types of non-linearities. The sources of non-linearity can be [1] in the FFT and IFFT blocks due to limited binary word lengths, signal clipping and quantization errors and non-linearity of power amplifiers (PA). Of these, the non-linearity in the PA is most dominant in a multicarrier modulations sys-tem, because of high PAPR. As long as the signal swing is within the dynamic range of the PA, no problem is encountered. However due to high peaks encountered in OFDM, the peak signals re-sulting in high PAPR is likely to drive the PA into satura-tion. An input back-off (IBO) is required to shift the oper-ating point to the left [2], where,

𝐼𝐵𝑂(𝑑𝐵) = 10 log10 (𝑃𝑠𝑎𝑡

𝑃𝑎𝑣) =10 log10 (

𝑥𝑠𝑎𝑡2

𝐸{|𝑥(𝑡)|2}) (9)

To ensure that the amplified peaks of the OFDM signal do not exceed the saturation level, IBO should be atleast equal to PAPR. 2.1 PAPR reduction techniques

The PAPR reduction techniques is classified as clipping techniques, coding technique, probabilistic (scrambling) technique, adaptive predistortion technique and DFT spreading technique. In the first technique the peak of the resultant summed carrier output is clipped by block-scaling, filtering, peak cancellation, Fourier projection and decision aided recon-struction techniques. The coding technique reduces the PAPR without causing any distortion and out-of band noise, but suffers from bandwidth efficiency [1]. The probabilistic technique is to scramble an input data block of OFDM symbols and transmit the one with minimum PAPR. This reduces the probability of incurring high PAPR. This technique includes selective mapping (SLM), partial transmit sequence (PTS), tone reservation (TR) and tone injection (TI). In the adaptive pre-distortion tech-nique the non-linear effects of the HPA is compensated by automatically modifying the input constellation with the least hardware requirement. In the DFT spreading technique, the input signal is spreaded with DFT which can be subsequently taken as IFFT. This reduces the PAPR of OFDM signal to the level of single carrier trans-mission. This technique, also known as Single Carrier–FDMA (SC-FDMA) is adopted for uplink transmission in 3GPP LTE standard.

A modified SLM technique for PAPR reduction of cod-ed OFDM signal is proposed in [3]. In this technique the phase sequence is embedded in check sequence of coded data blocks. Based on SLM approach, [4] proposed a post IFFT PAPR reduction technique for determining a unique set of time-domain sequences per OFDM block that can minimize PAPR. In [5] a semi-blind SLM technique is proposed where the same PAPR as that of a classical SLM is maintained while increasing the overall throughput. A non-linear companding technique to reduce high PAPR of OFDM signals is discussed in [6]. A non-linear com-panding transform (NLT) technique is proposed in [7] for further reducing the PAPR of SC-FDMA. Modification of dummy subcarriers to reduce PAPR is proposed in [8].

Attempts have also been made to reduce the problem of PAPR by adopting suitable pulse shaping techniques. An efficient technique based on proper selection of time waveforms of the different subcarriers of an OFDM mod-ulation scheme to reduce PAPR is proposed in [9]. It was shown that with broadband pulse shaping the PAPR of OFDM modulated signals can be made very close to that of single carrier signals. The complimentary CDF (CCDF) of PAPR to ascertain the reduction in performance was derived in [10] with pulse shaping filters.

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3 SYSTEM MODEL FOR PAPR MITIGATION BY

PULSE SHAPING TECHNIQUES

The system model is shown in Fig. 1. For a N – subcarriers system with pulse-shaping. The incoming data is fed to the constellation mapper to obtain the baseband modu-lated output. The modulation technique selected should be bandwidth efficient. The baseband modulated stream is then split into N parallel sreams. Each stream is shaped by a pulse shape and transmitted over a given carrier. The OFDM transmitted signal is expressed as [11]

𝑥(𝑡) = ∑𝑋𝑛(𝑘)𝑝𝑘

𝑁−1

𝑖=0

(𝑡)𝑒𝑗2𝜋𝑘𝑡𝑇, 𝑛𝑇 ≤ 𝑡 ≤ (𝑛 + 1)𝑇 (10)

where 𝑋𝑛(𝑘) is the modulated data symbol for sub-carrier k and T is the duration of the OFDM block. The pulse shape 𝑝𝑘(𝑡) is of duration T having a bandwidth equal to or less than the OFDM signal x(t) used with subcarrier k with

∫|𝑝𝑘(𝑡)|2

𝑇

0

𝑑𝑡 = 𝑇 (11)

From the definition of PAPR in (1) we have

𝑃𝐴𝑃𝑅 = 𝑚𝑎𝑥|𝑥(𝑡)|2

𝐸|𝑥(𝑡)|2

Using phase shift keyed modulated symbols in the con-stellation mapper and considering (1) we obtain from above

𝑃𝐴𝑃𝑅 ≤ 𝑃𝐴𝑃𝑅𝑚𝑎𝑥 =1

𝑁max0≤𝑡≤𝑇

(∑|𝑝𝑘(𝑡)|

𝑁−1

𝑛=0

)

2

(12)

The above equation signifies that the PAPR depends on the number of sub-carriers N as well as the pulse shape 𝑝𝑘(𝑡) used for each sub-carrier. As shown in [slimanne] the above expression reduces to the following if the same pulse shape is used for each sub-carrier,

𝑃𝐴𝑃𝑅𝑚𝑎𝑥 =1

𝑁max0≤𝑡≤𝑇

(∑|𝑝𝑘(𝑡)|

𝑁−1

𝑛=0

)

2

= 𝑁 max0≤𝑡≤𝑇

|𝑝𝑘(𝑡)|2 (13)

The expression in (12) can therefore be minimized if the sub-carrier waveforms have the following properties [2002] (i) Broadband pulse shapes are desirable. (ii) All selected pulse shapes should be different and satis-

fy the orthogonality criterion

∫𝑝𝑘(𝑡)𝑝𝑙∗

𝑇

0

𝑒𝑗2𝜋(𝑓𝑘−𝑓𝑙)𝑡𝑑𝑡 = {𝑇, 𝑘 = 𝑙0, 𝑘 ≠ 𝑙

(14)

Based on the above we propose few broadband pulse

shapes that mitigate the problem of PAPR.

4 BROADBAND PULSE SHAPES FOR PAPR

MITIGATION

4.1 Gaussian Pulse shapes

Gaussian pulse shapes are derived from the Gausssian function

𝐺(𝑥) =1

√2𝜋𝜎2𝑒𝑥𝑝 (−

𝑥2

2𝜎2) (15)

In general the nth order derivative of Gaussian pulse is given recursively as

𝑥(𝑛)(𝑡) = −(𝑛 − 1)

𝜎2𝑥(𝑛−2)(𝑡) −

𝑡

𝜎2𝑥(𝑛−1)(𝑡) (16)

where 𝜎 is the scaling factor −∞ < 𝑡 < ∞ . Higher or-derwaveforms are formed by highpass filtering of the Gaussian pulses. The time domain plots for derivatives of Gaussian pulses for order n = 0 to 4 is shown in Fig.2. The Gaussian monocycle (n = 1) has a single zero-crossing and each of the further derivatives adds one more zero crossing. The spectrum of nth order derivative of Gaussian

pulse is obtained by using the transform properties of derivative of functions.

Fig. 1 System model for OFDM with different waveforms

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𝑋𝑛(𝑓) = (𝑗2𝜋𝑓)𝑛𝑋(𝑓) = (𝑗2𝜋𝑓)𝑛𝑒𝑥𝑝 (−

(2𝜋𝑓𝜎)2

2) (17)

where 𝑋(𝑓) is the Fourier spectrum of the first order de-rivative of the Gaussian pulse. The power spectral densi-ty on nth order derivative of Gaussian pulse is 𝜑𝑛𝐺(𝑓) = |𝑋𝑛(𝑓)|

2 = (2𝜋𝑓)2𝑛𝑒𝑥𝑝(−(2𝜋𝑓𝜎)2) (18)

The autocorrelation function function of 𝑥(𝑛)(𝑡) is found by computing the inverse Fourier transform of 𝜑𝑛𝐺(𝑓) to obtain

𝑅𝑛(𝜏) = (2𝜋)2𝑛(−𝑗)2𝑛

1

√2𝜎2

𝑑2𝑛

𝑑𝜏2𝑛𝑒𝑥𝑝 (−

𝜏2

4𝜎2) (19)

4.2 Modified Hermite Pulse shapes

Hermite pulses of nth order are obtained from derivatives of Gaussian pulse as

ℎ𝑛(𝑡) = (−𝜏)𝑛𝑒𝑥𝑝 (

𝑡2

4𝜎2)𝑑𝑛

𝑑𝑡𝑛𝑒𝑥𝑝 (−

𝑡2

2𝜎2) (20)

The time domain pulse shapes are shown in Fig. 3 and 4 for even and odd orders respectively. The ACF of the MHP is given as

𝑅𝑛(𝜏) = √2𝜋𝑒𝑥𝑝 (−𝜂2

2)∑(

𝑛! 𝑛!

𝑘! 𝑘! (𝑛 − 𝑘)!)

𝑛

𝑘=0

(−1)𝑘𝜂2𝑘 (21)

where the scaling factor 𝜎 is assumed to be 1. Taking the Fourier transform of 𝑅𝑛(𝜏) we obtain the power spectral density of MHP as ∅𝑛𝐻(𝑓)

= √2𝜋∑𝑛! 𝑛!

𝑘! (𝑛 − 𝑘)! 𝑘!

𝑛

𝑘=0

(−1)𝑘1

2(𝑗)2𝑘

𝑑2𝑘

𝑑𝜔2𝑘𝑒𝑥𝑝 (−

𝜔2

2) (22)

4.3 Raised cosine Pulse shapes

The mathematical equation for raised cosine pulse is given as,

𝑧(𝑡) =cos (𝜋𝛼

𝑡𝑇𝑠)

1 − (2𝛼𝑡𝑇𝑠)2 .sin (𝜋

𝑡𝑇𝑠)

𝜋𝑡𝑇𝑠

(23)

where 𝛼 is a roll-off factor and it’s value lies between zero and one. As the value of 𝛼 increase from zero to one, the pulse shape becomes sharper and the side-lobe reduces. Fig. 6 shows the raised cosine pulse for different roll-off factors. As the roll factor increases, side lobe level is sup-pressed significantly and the energy content is bound in the main lobe only. Autocorrelation function becomes sharper as we go on increasing the roll-off factor. Fig. 5 shows the time domain autocorrelation of raised cosine pulse shapes of different roll-off factors.

4.4 Prolate Spheroidal Wave Function (PSWF) based pulse shape

PSWF of the form 𝜓 n(c,t) arevreal, continuous functions of time t for 𝑐 ≥ 0 having the property of orthogonality over the time and frequency intervals. Here n represents the order of the pulse and c represents the time-bandwidth product. The PSWF are solution of the Helm-holtz’s differential equation

(1 − 𝑡2)𝑑2𝜓𝑛

𝑑𝑡2− 2𝑡

𝑑𝜓𝑛

𝑑𝑡+ (𝜒𝑛 − 𝑐

2𝑡2)𝜓𝑛 = 0 (24)

and the integral equation

𝜆𝑛𝜓𝑛(𝑡) = ∫sin 𝑐(𝑡 − 𝑠)

𝜋(𝑡 − 𝑠)

1

−1

𝜓𝑛(𝑠)𝑑𝑠 (25)

where 𝜒𝑛 and 𝜆𝑛 are the corresponding eigenvalues. The above set of equations are assumed to have a solution of the form

𝑆0𝑛1 (𝑐, 𝑡) =

{

∑ 𝑑𝑘

𝑛(𝑐)𝑃𝑘(𝑡)

∞

𝑘=0,2,..

∑ 𝑑𝑘𝑛(𝑐)𝑃𝑘(𝑡)

∞

𝑘=1,23,..

(26)

For odd and even values of order k. Here 𝑃𝑘(𝑡) represents the Legendre polynomials. The final expression for the PSWF pulse shape is given as [12]

𝜓𝑛(𝑐, 𝑡) = (𝜆𝑛(𝑐)

2𝑇𝑝)

1/2

(2𝑛 + 1

2) ∑ 𝑑𝑘

𝑛(𝑐)𝑃𝑘(𝑡)

∞

𝑘=0,2,..

(27)

The time domain PSWF pulse shapes are shown in Fig.7 and 8. The pulse shapes are characterized by exactly n zero crossings in the interval 𝑇𝑝. The zero-crossings shift towards the origin with larger values of c. This signifies time compression of PSWF pulses and consequently a higher change in the random process within the same time duration.

5 EVALUATION OF PAPR FOR BROADBAND

PULSE-SHAPES

The PAPR evaluated for the broadband pulse shapes dis-cussed in the previous section is shown in in Table. 1. (i) It is seen that the the PAPR is lowest for PSWF based

pulse shapes and highest for raised cosine pulse shapes.

(ii) The PAPR for raised cosine pulse shapes increases as we increase the roll-off factor 𝛽 .

(iii) The variation of PAPR for PSWF based pulses is due to the time-bandwidth product c which is governing factor in the pulse shape design. The higher value of PAPR for n = 7 is contributed to the asymptotic be-haviour of pulse shape as the value of c and n in-creases.

(iv) The higher value of PAPR in case of Gaussian pulses compared to MHP for orders more than 2 is pri-marily due to smaller decorrelation time resulting in

12

higher autocorrelation value.

6 CONCLUSION

In this paper the effect on PAPR due to application of different broadband pulse shapes in an OFDM system is studied. It is analysed that PSWF based pulses due to their unique property of double orthogonality both in time and frequency domain, is best suited for a low PAPR. Also the advantage of the time bandwidth product as being an additional degree of freedom in the design procedure would result in larger number of orthogonal pulse shapes.

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Table 1. PAPR for different pulse shapes of different orders

Fig.3 MHP of even orders, n = 2, 4 and 6

Fig. 4 MHP of odd orders, n = 1, 3 and 5

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Fig. 5 Autocorrelation of raised cosine pulse shapes

Fig. 6 Raised cosine pulse shapes for 𝛽 =0.25. 0.5, 0.75 and 1.0

Fig. 7 PSWF pulses for n = 5, 6, 7 and 8 with c = 2

Fig. 8 PSWF pulses for n = 5, 6, 7 and 8 with c = 8