Peak-to-average power Reduction (PAPR) by pulse shaping using a modified raised cosine filters

Srabani Mohapatra, Susmita Das IEEE member

Abstract- Orthogonal Frequency Division (OFDM)

technologies allow data transmission over band limited channels requires pulse shaping to eliminate or control inter-symbol interference (ISI). Nyquist filters provide ISI-free transmission. Here we introduce a new family of modified raised cosine filters is formulated in an effort to reduce the peak-to-average power ratio (PAPR) of the baseband signal over that obtained using the conventional raised cosine filter, while maintaining the same excess bandwidth and the zero inter-symbol interference condition. The modified filters contain a new design parameter d, giving an additional degree of freedom to minimize PAPR for a given roll-off factorα . It is shown that PAPR can be reduced by more than .5 dB over most of the α range. Closed form expressions for the frequency and impulse responses of the modified raised cosine filters are presented, along with the optimum value of d that yields minimum PAPR as a function ofα .

I. INTRODUCTION

Orthogonal frequency division multiplexing (OFDM) has become a popular technique adopted in various high-speed wireless communication standards such as digital video broadcasting (DVB-T) [1], digital audio broadcasting (DAB) [2], IEEE 802.11 and IEEE 802.16. The rebirth of this old technique is due to its ability against multi-path fading, immunity to impulse interference and high spectral efficiency. However, one of the main disadvantages is the high peak to average power ratio (PAPR) of the OFDM transmitting signals. As a result, linear behavior of the system over a large dynamic range is needed in the OFDM transmitter and the efficiency of the RF power amplifier is reduced. Several PAPR attenuation schemes such as magnitude clipping [3], coding [4] and partial transmit sequence (PTS) [5] have been proposed to solve this problem. Pulse shaping is a PAPR reduction technique based on a proper selection of the time-limited waveforms for different subcarriers. Its implementation complexity is low in comparison with the previous methods. The effectiveness of PAPR reduction by using pulse shaping in OFDM has been investigated in [6], [7]. However, no numerical comparison or analytical framework exists to evaluate the capacity of PAPR reduction after applying pulse shaping. In this paper our main goal is to reduce the PAPR of the baseband signal by improving over the conventional raised cosine filter, the most widely used pulse shaping technique. To our knowledge, there is no analytical or systematic methodology for doing this. The approach we take is to first modify the conventional raised cosine filter by introducing a

new design parameter called the shaping parameter d . This parameter, which can be varied independently of the roll-off factorα , is then used to shape the impulse response in minimizing PAPR, while maintaining the zero ISI condition. The value of d yielding the minimum PAPR is then obtained as a functionα . This paper is organized as follows: Background on the Nyquist zero ISI condition and the conventional raised cosine solution is first reviewed. Next, the approach taken in the modification and the resulting solutions of both the frequency and impulse responses of modified raised cosine filters are presented. Finally, the performance of the generalized filters is discussed.

II. BACKGROUND

A pulse )(th is set to have zero ICI if it meets the Nyquist criterion.

⎩⎨⎧

±±==

=,...2,1,0

0,1)(

kk

kTh (1)

Where k is the bit (symbol) time index and T is the bit (Symbol) period. The Fourier transform of )(kTh in Eq. 1 can be obtained to be

∑∞

−∞=

=+m T

mfHT

1)(1 (2)

Where )( fH is the Fourier transform of the continuous-time pulse )( th . A widely known solution that satisfies Eq.2 is the conventional raised cosine filter. Its spectral magnitude is given by [1].

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

+>

+≤≤−

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −−+

−≤≤

=

Tf

Tf

TTfTT

TfT

fH

210

21

21

21cos1

2

210

)(

α

αααα

π

α(3)

And its phase is assumed to be linear; here α is the roll-off factor that determines the filter bandwidth

TB 2/)1( α+= .The corresponding impulse response, obtained by taking the inverse Fourier Transform of Eq. 3, and is given by [1].

978-1-4244-4859-3/09/$25.00 ©2009

2241)cos()(sin)(

ταπατττ

−= ch (4)

Where Tt /=τ is the normalized time.

III. MODIFIED RAISED COSINE FILTER

There have been studies to improve or to modify the conventional raised cosine filter over the years as well as recently. Comparing only to those of continuous-time solutions, it will be shown later that the improved filter given in [4] can be considered to be a special case of the modified solution given in this paper. The complicated modified given in [5] is restricted to solutions that give higher asymptotic decay rates than the conventional raised cosine filter, which actually result in increased PAPRs. The modified solution given in this paper is simple and probably represents the most natural extension of the conventional raised cosine filter.

A. Frequency Domain Solution The conventional raised cosine solution, given in Eq. 3, is obtained by fitting a (raised) half-cycle of cosine, thus the name, in the transition region

Tf

T 21

21 αα +≤≤− . Our

modified is based on simply allowing any multiple or fraction of cosine cycles to be fitted in the transition region; this is done by introducing a multiplicative factor d that scales the period (in the frequency domain) of the raised cosine function. As d increases over 1, the cosine period increases and Thus results in a fraction of half-cycle cosine fitted in the transition region. In other words, d is inversely related to the length of cosine cycle fitted in the transition region, measured relative to one half-cycle

0.0 0.2 0.4 0.6 0.8 1.0-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

P(f)

f

Conventional RC Concave d=0.8 Convex d=0.8 Convex d=0.9 Concave d=0.9

Figure 1 modified raised cosine spectra for d=1

Figure 1 shows the range of spectral shaping possible for d varying 0.8 to 0.9 for a given α . As shown, there are two types of modified solutions: denoted as “convex” and “concave” - the names denoting the curvature of the response in the first-half portion of the transition region. Using the modification method just described, the frequency responses of the convex and concave filters in the transition region are obtained to be, respectively,

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −−+=

Tdf

dTqTfH CV 2

).1(.

cos12

)( ααπ (5)

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −−−−+=

Tdf

dTqfTTfH CC 2

).1(.

cos)21(212

)( ααπ

α

(6) Where ⎟

⎠⎞

⎜⎝⎛ −= −

ddq )1(

2cos 1 π represents an amplitude

normalization factor needed to make the frequency responses to be continuous across the borders between different regions. The subscripts CV and CC denote the convex and concave solutions, respectively. The spectra in the other two regions remain the same as those given in Eq. 3. It can be seen that q = 1 if d = 1 and Eq. 5 for the convex filter reduces to the conventional raised cosine filter. Note that q becomes singular for the ratio (d-1)/d equal to an odd integer: thus solutions do not exist for certain values of d less than 1 - this is not of concern for us as it lies outside the range of d yielding optimum PAR performance.

B. Time Domain Solution The corresponding impulse responses for the convex and concave filters are obtained, by taking the inverse Fourier Transform of HCV (f) and HCC (f), respectively, to be

)()()( 21 τττ hhhCV += (7)

)()()()( 321 ττττ hhhhCC +−= (8) Where )cos()(sin)(1 πατττ ch =

⎟⎠⎞

⎜⎝⎛ +−−= πτατπτατατ sin)

21(sinsin)

21(sin

2.)(2 d

ctd

cqh

(9)

( ))cos()(sin)(sin.2)(3 πατατττ −= cch (10) To illustrate how d affects the impulse responses, Figs. 3 and 4 Show the pulses of convex and concave filters, respectively, for d = .8, 1 and 1.5, with α = 0.35. Again, d = 1 for the convex filter corresponds to the conventional filter. As can be seen, d affects the side lobe responses in a significant way, whose effect on the PAPR Performance is discussed in the next section.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

tau

hcc

concave filter for d=.8,1,and 1.5 (alp=.35)

d=1.5

d=1d=.8

Figure 2 concave filter for d=.8, 1, and 1.5for (alp=.35)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

tau

hcv

convex filter for d=.8,1,and 1.5 (alp=.35)

d=1.5

d=1d=.8

Figure 3 convex filter for d=.8, 1, and 1.5 for (alp=.35)

IV. PAPR PERFORMANCE

PAPR is defined as

)(.20)(.20 prms

p VLogvv

LogPAPR == (11)

Where Vp, the normalized peak voltage, does not necessarily Represent the absolute peak voltage, but rather can be more Generally defined in a probabilistic way [2,3], [ ] pp xVtV =≥)(Pr (12)

Where probability xp can be whatever value deemed most suitable for the problem at hand: xp of 10-6 is used, for example, in [2] based on simulation and load pull measurements of RF amplifiers used for IS-95 CDMA. In this paper, to make the PAPR evaluation easier over wide ranges of α and d, mainly for relative performance comparison purposes, we use the absolute peak voltage for Vp. It will be shown later that this is justified.

For an FIR filter with tab coefficients given by h(t-Ttab)where n = -N,…,0,…N, PAPR can be approximated using the following expression

[ ]

∑

∑

−=

−

=

−

⎟⎟⎠

⎞⎜⎜⎝

⎛+−

≈= N

Nntab

MN

ktab

bb

p

MnTth

TMkMth

TEPAPR v

/)(

).2/.(.2

2

21

0

2

(13)

where tabb TTM /= is the number of tabs per bit period bT

and bE is the energy per bit. This approximation arises from the assumption that the peak occurs at the midpoint between ideal sampling points when the values of 2N+1consecutive data bits are such that the contributions from all the corresponding pulses add coherently to produce the peak either on the positive or negative side. The probability of this occurrence is, thus, 2-

2N. Equation 13 simplifies the computation for PAPR as a function of d and α, which otherwise would require time-consuming FIR simulation with random bit generations.

1.0 1.5 2.0 2.5 3.03.5

4.0

4.5

5.0

5.5

6.0

6.5

7.0

7.5

PAR

,dB

Shaping parameter d

Concave alp=.45 Convex alp=.45 Concave alp=.35 Convex alp=.35 Concave alp=.25 Convex alp=.25 Concave alp=.15 Convex alp=.15

Figure 4 PAPR vs. d for concave and convex filters Figure 4 shows how PAPR varies as a function of d for both convex and concave filters for the indicated values of α. It can be seen that the concave filter yields minimum PAPRs, although at different values of d, for α = .15, .25 and .35, whereas the convex filter yields the minimum for α = .45. It can be seen that PAPR reduction of More than .5 dB is possible with an appropriate choice of d using the modified filters over the conventional raised cosine Filter. For example, for α = .35, PAPR is about 4.5 dB for the concave filter at d ≅ 1.5, as compared to about 5.1 dB for the conventional raised cosine filter.

To show that the usage of absolute peak to determine PAPR using Eq. 12 is justified, Fig. 5 shows the complimentary CDF distribution of instantaneous signal power at the output of conventional and modified filters.

0 2 4 6 8 10 12 141E-4

1E-3

0.01

0.1

1C

CD

F (P

APR

>PAP

R0)

PAPR0

Modified Pulse Conventional Pulse

Figure 5 Complimentary CDF Distribution of power for 35.0=α

0.2 0.4 0.6 0.8 1.01

2

3

4

5

6

7

8

PAR

,dB

Roll-off factor alp

Conventional RC Beaulieu [4] Concave Convex

Figure 6 Minimum PAPR vs. alp

Figure 6 shows the minimum PAPR possible as a function of α with the modified filters, along with those of the conventional raised cosine and Beaulieu [4] filters for comparison purposes. It can be seen that the modified filters Can give a PAPR reduction of .5 to 1 dB up to about α = 0.9. The PAPR performance of the improved filter given by Beaulieu is similar to that of the optimized modified filter, although not quite as good over some α ranges, up to about α = 0.6. As one would expect, at α where the PAPR performance is similar between the modified and Beaulieu filters, the two filters are almost identical in their frequency and impulse responses – this is good indication that the modified filters presented in this paper are indeed general.

V. CONCLUSIONS

A new family of modified raised cosine filters, containing a new shaping parameter d, is introduced. It was shown that this new parameter can be varied independently of the roll-off factor α, while maintaining the same bandwidth and the zero ISI condition. In this paper, the effect of d was considered mainly in terms of the PAPR performance. The optimal value of d that produces the minimum PAPR was presented as a function of α. It was shown that PAPR can be reduced between .5 to 1 dB using the modified filter, with an appropriate choice of d, over the conventional raised cosine filter for most of α range. The pulse shaping design for superior PAPR performance over the conventional raised cosine filter is made simple.

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Average Ratio for Estimating the Spectral Regrowth of an RF/Microwave Power Amplifier,’ IEEE Transactions on Microwave Theory and Techniques, VOL. 48, NO.6, JUNE 2000, pp. 1068-1071.

[2] V. K. N. Lau, ‘On the Analysis of Peak-to-Average Ratio (PAR) for IS95 and CDMA2000 Systems,’ IEEE Transactions on Vehicular Technology, VOL. 49, NO. 6, Nov. 2000, pp. 2174-2188.

[3] N. C. BEAULIEU, C. C. TAN, and M. O. Damen, ‘A “Better Than” Nyquist Pulse,’ IEEE Communications Letters, VOL. 5, NO. 9, pp. 367- 338, SEPTEBER 2001

[4] N. S. ALAGHA and P. KABAL, ‘Generalized Raised-Cosine Filters,’ IEEE Transactions on Communications, VOL. 47, NO. 7, pp. 989-997, JULY 1999.

[5] B. Farhang-Boroujeny and G. Mathew, “Nyquist Filters with Robust Performance Against Timing Jitter,” IEEE Transactions on Signal Processing, VOL. 46, NO. 12, Dec. 1998, pp. 3427-3431.

[6] Yuwen Jian, Zhuquan Zang, Wei-Yong Tan, “ PAPR Distribution Analysis of OFDM signals with pulse shaping ” Asia –pacific conference on Communications, perth, western Australia, 3-5 october-2005. Pp. 473-477.

[7] J. G. PROAKIS, Digital Communications, 4th edition, New-work: McGrak-Hill, 2001.

[8] Slimane, S.B., “Peak-to-Average Power Ratio Reduction of OFDM Signals using Broadband Pulse shaping,” Vehicular Technology Conference, 2002. Proceedings, Vol.2, pp. 889-893, 24-28 Sept. 2002.

[9] Peter S. Rha and Sage Hsu, “Peak-to-Average Ratio (PAR) Reduction by Pulse Shaping Using New Family of Generalized Raised Cosine Filters,” IEEE Vehicular Technology Conference,2003. Vol.1, pp.706-710, 6-9 Oct..2003.