Complex-Weighted OFDM Transmission with Low PAPR

Homayoun Nikookar and Djoko Soehartono

International Research Centre for Telecommunications and Radar (IRCTR) Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology

Mekelweg 4, 2628 CD, Delft, The Netherlands h.nikookar@tudelft.nl

d.soehartono@student.tudelft.nl Abstract— In this paper the novel method of complex

weighting for peak-to-average power (PAPR) reduction of OFDM signal is addressed. Combination of different amplitude weighting factors (including Rectangular, Bartlett, Gaussian, Raised cosine, Half-sin, Shannon, and subcarrier masking) with phasing of each OFDM subcarrier using random phase updating algorithm is studied. The impact of complex weighting of OFDM signal on the PAPR reduction is investigated by means of simulation and is compared for the above mentioned weighting factors. Results show that by either amplitude weighting or random phase updating the PAPR can be reduced. Applying both techniques together will further reduce the PAPR. For an OFDM system with 32 subcarriers and by Gaussian weighting combined with random phase updating, a PAPR reduction gain of 3.2 dB can be achieved. In order to reduce the complexity, grouping of amplitude weighting and/or phasing is applied. Results show that grouping of amplitudes weighting and phases reduces the hardware complexity while not much impacting the PAPR reduction gain of the method. For even further reduction of the PAPR, complex weighting method with dynamic threshold is investigated. Results show that the PAPR can be further reduced by factor of 4.8 dB.

I. INTRODUCTION Orthogonal frequency-division multiplexing (OFDM) is a parallel

transmission method where the input data is divided into several parallel information sequences, and each sequence modulates a subcarrier. OFDM signal has a non-constant envelope characteristic since modulated signals from orthogonal subcarriers are summed. The PAPR problem is occurred when these signals are added up coherently, resulting in a high peak. The high PAPR of OFDM signal is not favorable for the power amplifiers working in non-linear region.

Different methods have been proposed to mitigate the PAPR problem of OFDM. These techniques are mainly divided into two categories: signal scrambling and signal distortion techniques. Signal scrambling techniques are all variations on how to modify the phases of OFDM subcarriers to decrease the PAPR. The signal distortion technique is developed to reduce the amplitude of samples whose power exceeds a certain threshold [1]. The simplest method of this kind is clipping. However, clipping introduces both in-band and out-of-band radiations [2]. Windowing with a number of window functions gives better spectral properties than clipping, and peak cancellation is similar to clipping followed by filtering [2].

This paper addresses the PAPR reduction of OFDM by combination of both signal scrambling and signal distortion techniques. The PAPR is to be reduced by both amplitude weighting and random phase updating, mainly the combination of methods

described in [3] and [4]. The paper is organized as follows: In Section II the complex-weighted OFDM modulation is discussed and its PAPR is addressed in Section III. Different weighting factors and phasing for OFDM signal are studied in Section IV along with their effects on the PAPR reduction. The simulation results of the method and the PAPR reduction gains are depicted in Section V. Concluding remarks appear in Section VI.

II. COMPLEX WEIGHTING FOR MULTICARRIER MODULATION The OFDM signal for one symbol interval 0 ≤ t ≤ T is written as

1 2

0

( )mM j tT

m mm

s t b w eπ−

=

= ∑ (1)

where M is number of subcarriers, bm is modulation data of the m-th subcarrier, T is the OFDM symbol period, and wm is a complex factor defined as 0,1, 2, , 1mj

m mw e m Mϕα= = −… (2) where αm is a positive real value and φm is the phase of m-th subcarrier. The block diagram of an OFDM modulator with complex weighting factors is shown in Fig. 1.

02πj f te0

0 0jw e ϕα=

12πj f te1

1 1jw e ϕα=

12π Mj f te −1

1 1Mj

M Mw e ϕα −− −=

Fig. 1 Block diagram of complex-weighted OFDM modulator.

III. PAPR OF COMPLEX-WEIGHTED OFDM SIGNAL For the calculation of PAPR, first we obtain the instantaneous

power of OFDM signal from (1) as

2 ( )1 1

2 *

0 0

( ) ( ) ( )m nM M j tT

m m n nm n

P t s t b w b w eπ −− −

= =

= = ∑ ∑ (3)

which can be written as

2 ( )1 1 1

2 *

0 0 0,

( ) ( )m nM M M j tT

m m m m n nm m n n m

P t b w b w b w eπ −− − −

= = = ≠

= +∑ ∑ ∑ (4)

978-2-87487-003-3 © 2007 EuMA October 2007, Munich Germany

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Averaging the power P(t) over a symbol period of T, we obtain

[ ]2π( )1 1 1

2 2 * *

0 0 0,

( )m- nM M M j tT

m m m n m nm m n n m

E P t b E b b w w eα− − −

= = = ≠

= + ∑ ∑ ∑ (5)

The symbols on different carriers are assumed to be independent [3] i.e., E[bmbn

*] = δ(m–n), therefore, the second term in (5) is zero and the average power becomes

1

2

0

[ ( )]M

mm

E P t w−

=

= ∑ (6)

The variation of the instantaneous power of OFDM signal from the average is

[ ]2π( )1 1

*

0 0,

( ) ( ) ( ) ( )m- nM M j tT

m m n nm n n m

P t P t E P t b w b w e− −

= = ≠

∆ = − = ∑ ∑ (7)

The variance is obtained by averaging of (∆P(t))2 over a symbol period of T which yields [3]

1

22

01

1 ( ( )) ( )MT

cci

P t dt R iT

ρ−

=

= ∆ = ∑∫ (8)

where Rcc(i) is the autocorrelation function of the complex sequence cm = bm . wm.

1

*

0

( )M i

cc m m im

R i c c− −

+=

= ∑ (9)

The parameter ρ is the power variance of the OFDM signal and it has been shown that is a good measure of the PAPR [4]. As will be shown in the following sections, we consider different weighting functions in this work. In order to be able to compare them with each other we assume the power of all weighting factors be constant, i.e.,

1

2

0

1M

mm

w−

=

=∑ (10)

Using (6) and (10) the PAPR of OFDM signal can be written as

{ }{ } { } max

( )( )

( )Max P t

PAPR Max P t PMean P t

= = = (11)

Referring to (4) and by considering a large number of terms in the summations, and by using the Central Limit Theorem, P(t) can be approximated by a Gaussian random process with mean 1 and variance ρ. Thus PAPR can be related to the power variance ρ. Let β denote the probability that P(t) be less than or equal to Pmax, i.e.,

[ ] max2

max 0

1 ( 1)Pr exp22

P PP P dPβρπρ

−= ≤ = −

∫ (12)

Using (11) and (12) the relationship between PAPR and power variance is [3]:

1 1PAPRQ Q βρ ρ

− + =

(13)

where

( ) 2 212

u

xQ x e du

π∞ −= ∫ (14)

Using (14) and knowing ρ and β, the exact value of PAPR can be calculated. For a fixed β the multicarrier signal with high PAPR has a high value of power variance ρ. In the simulations power variance has been obtained because of its computational simplicity expressed by (8) and (9). However, the corresponding PAPR can be calculated from power variance as is obvious from (13).

IV. COMPLEX WEIGHTING AND ITS EFFECT ON THE PAPR REDUCTION

Each complex weighting factor has two components: amplitude (αm) and phase (φm) as defined in (2). In this section we will discuss different amplitude weighting factors as well as the phasing

algorithm and their combination applied to the OFDM subcarriers to reduce the PAPR.

A. Different Amplitude Weighting Factors In this paper, different amplitude weighting factors are considered.

They are: Rectangular, Bartlett (Triangular), Gaussian (with two standard deviations, std = M/8 and std = M/16), Raised Cosine, Half-Sin, and Shannon window functions [3]. The Rectangular window has a flat shape; therefore throughout this paper we use this amplitude weighting as the reference for the comparison of PAPR reduction results. For the Gaussian shape, the parameter std is the standard deviation of the function around its peak (index M/2 of the subcarriers). As mentioned before, the amplitudes in the weighting factors are selected in such a way that the power of all weighting factors be constant as expressed in (10).

To reduce the complexity of implementation, we can also group the subcarriers in OFDM signal by assigning a single amplitude value for a group of subcarriers. By doing so, we obtain the piecewise version of seven weighting factors mentioned above.

Another way to reduce PAPR with amplitude weighting factors is by applying subcarrier masking to OFDM signal. This is usually done in some applications, such as adaptive OFDM or non-contiguous OFDM for cognitive radio applications [6]. By masking, some of the subcarriers are deactivated (amplitudes of weighting factors are set to zero) while the others are maintained with the same amplitudes.

B. Phasing of OFDM Subcarriers For the phasing of OFDM subcarriers, we consider the random

phase updating algorithm [4]. In random phase updating, the phase of each subcarrier in each OFDM signal is updated by a random increment as ( ) ( ) ( )1 0,1, , 1m m mn n n m Mϕ ϕ ϕ−= + ∆ = −… (15) where n is the iteration index and (∆φm)n is the phase increment of the mth subcarrier at nth iteration. The initial phase, i.e., (φm)0 can be considered zero without loss of generality.

Different distributions for the random phase increments have been considered, they are Uniform (∆φm = 2π·Unif[0,x]), Gaussian (∆φm = 2π·N(0,x)) and von Misses distribution with parameter κ, as described in detail in [5]. We consider values of x∈ {0.01, 0.05, 0.1, 0.25, 0.5, 0.75, 1.0} for Uniform and Gaussian distribution and κ∈ {1, 3, 5, 8, 10, 50, 100} for von Misses distribution.

Several random phase updating algorithms are discussed in detail and the flowcharts of the algorithms are presented in detail in [4] and will be briefly mentioned in this paper. The first algorithm is random phase updating with a certain power variance threshold. In this method, the phase updating with increments are performed until the resulting power variance of the OFDM signal is less than the threshold. The other algorithm is random phase updating with a limited number of iterations. In this method the iteration numbers are restricted not to exceed a certain limit for a given power variance threshold. Grouping, as described above, can also be done to reduce the complexity of implementation.

The random phase updating for OFDM subcarriers can also be carried out with dynamic power variance threshold for further reduction of the PAPR as discussed in detail in [4]. In this paper we combine this method with amplitude weighting factors applied to the OFDM subcarriers.

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C. Combination of Amplitude Weighting and Random Phase Updating for OFDM

This section conveys the main idea of this paper. As presented in [3] and [4], either amplitude weighting factors or random phase updating can be used to reduce the PAPR of OFDM signal. We can combine both methods to further reduce the PAPR. Several combination scenarios are considered and their effects to the PAPR reduction are investigated. The combinations of amplitude weighting and phasing are:

• Amplitude weighting and random phase updating with a certain power variance (or PAPR) threshold.

• Amplitude weighting and grouping of subcarriers phases in random phase updating with a certain power variance (or PAPR) threshold.

• Subcarriers grouping in both amplitude weighting and phasing with a certain power variance (or PAPR) threshold.

• Subcarriers masking and random phase updating with a certain power variance (or PAPR) threshold.

• Amplitude weighting and random phase updating with dynamic thresholds.

V. SIMULATION RESULTS In this section the simulation results of PAPR reduction of

complex-weighted OFDM signal are presented. We generate 5000 OFDM signals with each subcarrier is modulated by a BPSK signal. In the simulations, the number of subcarriers is M = 32. The Uniform, Gaussian, and von Misses distributions are considered for the random phase updating. In this section only results with Uniform random phase updating are presented since no noticeable PAPR differences among these distributions were found.

The CDF of PAPR of OFDM signal with several scenarios of weighting and phasing are depicted in Fig. 1 for M = 32. In Fig. 1, phasing is applied by random phase updating with Uniform distribution (x = 1.0) and power variance threshold ρTh = -4 dB.

0 2 4 6 8 10 120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

PAPR (dB)

CD

F

c1

c2c3

c4

Fig. 2 CDF plots of PAPR for different weightings with and without phasing.

c1: Rect. weighting, no phasing, c2: Rect. weighting with phasing, c3: Gaussian weighting (std = M/16), no phasing, c4: Gaussian weighting (std =

M/16) with phasing.

We compute the PAPR value at the 90% CDF level for the simulated OFDM signals. We also define the PAPR reduction gain in dB as the difference between PAPR of standard OFDM signals (OFDM with Rectangular weighting) and PAPR of OFDM signals with other weighting factors, at the 90% CDF level. The PAPR reduction gain by phasing is defined as the difference of PAPR value in dB between phasing and without phasing at the 90% CDF level for a specific weighting factor. The total gain in PAPR reduction shown in Tables I–IV, is defined as

Total gain (dB) =

PAPR reduction gain by amplitude weighting (dB) +

PAPR reduction gain by phasing (dB) (16)

A. Application of Amplitude Weighting and Random Phase Updating to OFDM signal

Table I shows the total PAPR reduction gain of OFDM signal with both amplitude weighting and random phase updating. In this table, the PAPR reduction gain by only amplitude weighting for Rectangular weighting is 0 dB and for Gaussian (std = M/16) is 2 dB. Using (16), the PAPR reduction gain by phasing can be obtained by subtracting PAPR reduction gain due to amplitude weighting from the total gain.

Simulations with different power variance threshold values are done, i.e. ρTh = -3 dB and ρTh = -4 dB. The average iteration numbers to reach the threshold are also presented in Table I. We can see that for a small threshold value we obtain more PAPR reduction by phasing and subsequently more total gain. However, for a lower threshold value more iteration in random phase updating process is needed. For a narrower weighting shape, i.e. Gaussian window with std = M/16, we need less iteration to reach the threshold since weighting alone has reduced the power variance and more likely PAPR goes below the threshold before random phase updating is applied. Furthermore, the larger dynamic range in random phase updating, i.e. x = 1.0, will result in a larger PAPR reduction gain by phasing (and also total gain) and a lower average number of iterations.

TABLE I PAPR REDUCTION GAIN FOR OFDM SIGNAL WITH WEIGHTING AND RANDOM

PHASE UPDATING (M = 32) Rectangular Gaussian (std = M/16)

ρTh=-3 dB ρTh=-4 dB ρTh=-3 dB ρTh=-4 dB x Avg.

Iteration

Total Gain (dB)

Avg. Iterati

on

Total Gain (dB)

Avg. Iterati

on

Total Gain (dB)

Avg. Iterati

on

Total Gain (dB)

0.01 7.3 0.5 20.0 1.1 3.0 2.5 6.7 2.70.05 2.5 0.7 7.1 1.2 1.0 2.7 2.3 2.9

0.1 1.6 0.8 4.6 1.3 0.7 2.8 1.5 3.00.25 0.9 0.9 3.0 1.3 0.4 3.0 0.8 3.1

0.5 0.6 0.8 2.4 1.3 0.3 3.0 0.6 3.20.75 0.6 1.0 2.3 1.3 0.2 3.0 0.6 3.2

1 0.6 0.9 2.3 1.3 0.2 3.0 0.6 3.2

B. Application of Amplitude Weighting and Grouping of Subcarriers Phases

The PAPR gain for OFDM signal with amplitude weighting and grouping of subcarriers phases in random phase updating is shown in Table II. For M = 32, by grouping of phases into 16 groups of 2 subcarriers, the average number of iterations is a bit increased as compared to that in Table I. In this case, we obtain a trade-off between PAPR reduction performance and complexity reduction.

C. Application of Subcarriers Grouping in Both Amplitude Weighting and Phasing

If we apply grouping in both weighting amplitudes and phases of subcarriers, a lower total PAPR reduction gain is obtained when compared to Table II, and a slight increase in the average number of iterations (due to space limitation, simulation results are not shown here).

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TABLE II PAPR GAIN FOR OFDM SIGNAL WITH WEIGHTING AND GROUPING OF

SUBCARRIERS PHASES (M = 32, 16 GROUPS OF 2 SUBCARRIERS) Rectangular Gaussian (std=M/16)

ρTh=-3 dB ρTh=-4 dB ρTh=-3 dB ρTh=-4 dB x Avg.

Iteration

Total Gain (dB)

Avg. Iterati

on

Total Gain (dB)

Avg. Iterati

on

Total Gain (dB)

Avg. Iterati

on

Total Gain (dB)

0.01 8.0 0.5 22.7 1.1 4.2 2.5 9.2 2.80.05 2.7 0.7 7.9 1.2 1.4 2.7 3.2 3.0

0.1 1.7 0.7 5.4 1.2 0.9 2.7 2.1 3.10.25 1.0 0.8 3.7 1.3 0.5 2.8 1.3 3.1

0.5 0.7 1.0 3.1 1.3 0.4 2.8 1.0 3.20.75 0.7 1.0 3.0 1.3 0.3 3.0 0.9 3.2

1 0.7 1.0 2.8 1.3 0.3 2.8 0.9 3.2

D. Application of Subcarrier Masking and Random Phase Updating

In a non-contiguous OFDM system, some of the subcarriers are deactivated due to interference in that frequency region with the licensed users [6]. Activation and deactivation of subcarriers can be performed as a special form of the weighting function, i.e. for the m-th subcarrier activation means αm = constant and deactivation means αm = 0.

In this paper we consider two masking shapes. Masking #1 corresponds to deactivation of 3M/4 subcarriers and Masking #2 corresponds to deactivation of M/2 subcarriers of total M subcarriers. The PAPR reduction gains by amplitude weighting are 1.5 dB and 0.8 dB for Masking #1 and Masking #2, respectively. It is obvious that the more subcarriers are deactivated, the more PAPR reduction gain is obtained.

Table III illustrates the total PAPR reduction gain by amplitude weighting and phasing for those two maskings. The more subcarriers are deactivated, the more total PAPR reduction gain (due to amplitude weighting and phasing) is achieved. For lower power variance threshold (ρTh ≤ -4 dB) and larger dynamic range x values, in general deactivation of more subcarriers requires a less number of iteration to achieve the threshold.

Simulation results (not shown in this paper because of space limit) also indicate that the positions of deactivated subcarriers in an OFDM signal have only a negligible effect on the PAPR reduction gain. PAPR reduction gain by weighting is mostly affected by the number of deactivated subcarriers.

TABLE III PAPR GAIN FOR COMBINED SUBCARRIER MASKING AND RANDOM PHASE

UPDATING (M = 32) Masking #1 Masking #2

ρTh=-3 dB ρTh=-4 dB ρTh=-3 dB ρTh=-4 dB x Avg.

Iteration

Total Gain (dB)

Avg. Iterati

on

Total Gain (dB)

Avg. Iterati

on

Total Gain (dB)

Avg. Iterati

on

Total Gain (dB)

0.01 6.1 1.7 9.4 2.3 5.8 1.4 12.9 1.80.05 2.1 1.9 3.1 2.5 2.0 1.5 4.6 1.9

0.1 1.3 2.0 2.1 2.6 1.3 1.5 3.0 1.90.25 0.7 2.0 1.2 2.7 0.7 1.8 1.7 1.9

0.5 0.5 2.1 0.8 2.8 0.5 1.8 1.4 2.00.75 0.5 2.1 0.8 2.8 0.5 1.8 1.3 1.9

1.0 0.5 2.1 0.8 2.8 0.5 1.8 1.3 1.9

E. Application of Amplitude Weighting and Random Phase Updating with Dynamic Thresholding

Table IV shows the OFDM PAPR reduction by phasing as well as total gain obtained by applying both amplitude weighting and

random phase updating with dynamic thresholds. The number of iterations of random phase updating is I = 5 and I = 10. In each iteration, the threshold is reduced if the power variance is below it. If the limit in number of iterations is reached, the dynamic range of phases (i.e. x) is changed to obtain a more proper phasing for the PAPR reduction. In Table IV, the PAPR reduction gain for all seven amplitude weighting factors is shown.

TABLE IV PAPR GAIN FOR COMBINED WEIGHTING FUNCTION AND RANDOM PHASE

UPDATING WITH DYNAMIC THRESHOLD (M = 32) Uniform phases

I=5 I=10 Weighting FunctionPhasing

Gain (dB)Total

Gain (dB) Phasing

Gain (dB) Total

Gain (dB)Rectangular 2.0 2.0 2.2 2.2Bartlett 2.2 2.8 2.4 3.0Gaussian (std=M/8) 2.5 3.7 2.8 4.0Gaussian (std=M/16) 2.5 4.5 2.8 4.8Raised Cosine 2.3 3.0 2.5 3.2Half-Sin 2.0 2.6 2.3 2.9Shannon 2.3 2.8 2.4 2.9 It is seen from Table IV that amplitude weighting with random

phase updating with dynamic thresholds will provide larger PAPR reduction gains compared to the previous results with fixed thresholds. In the algorithm, by allowing more iteration in lowering the thresholds, larger PAPR reductions (by phasing) are achieved. Results show that dynamic thresholding has a significant effect on the PAPR reduction of OFDM with both weighting and phasing.

VI. CONCLUSION In this paper we have addressed the novel method of PAPR

reduction for OFDM signal by applying both amplitude weighting and phasing of OFDM subcarriers. This joint application gives more PAPR reduction gain than only weighting or phasing.

Employing both weighting and phasing to subcarriers implies more complex implementation. However, the complexity can be reduced by grouping of the subcarriers when weighting or phasing is applied. A good trade-off can be obtained as a small degradation in PAPR reduction performance was noticed by grouping.

Furthermore, the complex weighting with dynamic threshold was studied. Combining amplitude weighting, phasing and dynamic thresholding will result in a larger PAPR reduction gain of the proposed algorithm.

REFERENCES [1] K. S. Lidsheim, “Peak-to-Average Power Reduction by Phase Shift

Optimization for Multicarrier Transmission,” MSc. thesis, Delft University of Technology, Delft, The Netherlands, April 2001.

[2] R. Prasad and R. D. J. van Nee, OFDM for Wireless Multimedia Communications. Boston: Artech House, 1999.

[3] H. Nikookar and R. Prasad, “Weighted OFDM for wireless multipath channels,” IEICE Commun. Trans., vol. E83-B, no. 8, pp. 1864-1872, Aug. 2000.

[4] H. Nikookar and K. S. Lidsheim, “Random phase updating algorithm for OFDM transmission with low PAPR,” IEEE Broadcasting Trans., vol. 48, no. 2, pp. 123-128, Jun. 2002.

[5] K. V. Mardia, Statistics of Directional Data, pp. 57-68. London, U.K.: Academic Press, 1972.

[6] R. Rajbanshi, A. M. Wyglinski, and G. J. Minden, “An efficient implementation of NC-OFDM transceivers for cognitive radios,” in Proc. CROWNCOM’06, 2006.

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