Application of the Method of Conditional Expectations for Reduction of PAPR and Cubic Metric of OFDM Signals

The OFDM waveform exhibits high fluctuation in the signal envelope, which causes the nonlinear power amplifier of the transmitter to produce distortion. Peak-to-Average Power Ratio (PAPR) and Cubic Metric (CM) are the most commonly used metrics to quantify the phenomenon. Originally proposed in the literature for PAPR reduction, the Sign Selection problem is an approach for minimizing the metric of interest by altering the signs of the data symbols, which implies an exponential complexity. In this paper, the Method of Conditional Expectations (CE Method) is proposed to obtain a competing suboptimal solution to the Sign Selection problem. For PAPR reduction, a surrogate metric is introduced which allows for a more efficient application of the CE Method compared to a direct application to the PAPR metric itself without considerable performance degradation. For CM reduction, the tractability of the definition of CM is exploited to efficiently apply the CE Method. The reduction performance is analyzed to obtain a constant upper bound on the reduced metric value for every realization of the data symbols. Simulations show a persistent reduction of the effective PAPR - the value at which the distribution function of PAPR equals 0.999 - to about 6.5dB for a wide range of 64 to 1024 subcarriers. The steady performance is observed for CM reduction as well with a reduction of roughly 3dB. A pruned version of the sign selection approach is made possible by the CE Method, such that it reduces the rate loss from log_2 M to 1/2log_2 M bits per symbol for M-ary modulation order with insignificant loss in performance

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I Introduction

Orthogonal Frequency Division Multiplexing (OFDM) is a well-known multicarrier waveform which has been used in the major wireless communication systems. A main drawback of OFDM scheme is the high dynamic range of its signal envelope, which causes nonlinear distortion at the output of the power amplifier [16]. In order to avoid the distortion, the so-called power back-off needs to be applied in the power amplifier. Consequently, the power amplifier operates with a low energy efficiency. Especially for mobile equipments where battery life is limited and power amplifiers cannot have a large linear range due to cost constraints, the problem is more pressing [8].

It is therefore critical to reduce the required power back-off. The problem is commonly formulated as the minimization of a metric which captures the physical phenomenon and determines the power back-off. The classical metric is the ratio of the peak instantaneous signal power to the average power over consecutive signal segments referred to as Peak-to-Average-Power-Ratio (PAPR) [16]. An alternative metric called Cubic Metric (CM), which is based on the energy in the nonlinear distortion, was more recently proposed and reported to predict the required back-off more accurately [14].

The PAPR reduction problem has been tackled by several approaches, which can be broadly categorized into two groups. Methods based on deliberately introduced distortion constitute one category, with Clipping and Filtering [2] as a well-known example. The second category consists of the distortionless methods which typically provide PAPR reduction at the expense of some reserved resources which incurs rate loss, such as Selected Mapping (SLM) [3], Tone Reservation (TR) and Tone Injection (TI) [21]. The methods differ significantly at least in terms of reduction gain, rate loss, transmission power and complexity. A comparison of the pros and cons requires a separate study as provided, for instance, in [10]. A refreshed and fundamental review of the problem is as well provided in [26]. The CM reduction problem, on the other hand, has received limited attention compared to PAPR. In particular, very few of the already known methods from PAPR reduction research are examined for CM reduction, such as in [7], [27] and [20] for TR, Clipping and Filtering and SLM, respectively. It will be emphasized in this paper that CM has a more amenable mathematical structure, which indicates that there is room to improve on the performance and complexity of the back-off reduction problem by considering CM instead of PAPR, besides its reportedly higher accuracy.

Sign Selection is a promising distortionless approach based on altering the signs of the data symbols to reduce the PAPR, which has shown potentials for considerable reduction performance at the price of a rate loss equivalent to one bit per complex data symbol for each utilized sign variable [18, 1, 23, 19, 22]. Considering subcarriers, there are

possible sign combinations, which implies an exponential complexity order for the optimal sign selection. This has motivated research for competing suboptimal solutions. Some proposals with noticeable performance include the application of the method of Conditional Probabilities in

[18, 1], a sign selection method guided by clipping noise in [23], a greedy algorithm in [19] and a cross-entropy-based algorithm in [22]. In this work, the method of Conditional Expectations (CE Method), originally proposed in fields of discrete mathematics and graph theory [13], is used to treat the Sign Selection problem to develop a simple algorithm with a competitive performance for both PAPR and CM reduction requiring only sign bits.

The core idea of the CE method is to treat the optimization variables, i.e. the signs of the complex data symbols, as random variables. This artificial randomness is then employed to optimize the signs using conditional expectations. In addition to a direct application of the method to PAPR, a new metric, or a surrogate function for PAPR, referred to as Sum-Exp (SE) is proposed to gain indirect PAPR reduction. Unlike the other metrics, SE has no physical interpretation and is not directly related to power back-off. However, it will be shown that its reduction results in the reduction of the PAPR with lower complexity. The CE method is also applied to CM reduction, where the benefit of the mathematical tractability of CM in deriving low complexity closed-form expressions for the required calculations is demonstrated. As a rather uncommon characteristic among the solutions of the Sign Selection problem in the literature, an increasing reduction gain in PAPR and CM for increasing number of subcarriers is shown by simulations, which implies a roughly constant back-off for a large range of

. Furthermore, the CE method allows the analysis of the reduction performance by providing upper-bounds on reduced PAPR and CM values for any combination of the data symbols.

Notation

A random variable is distinguished from a realization

by using upper and lower case letters, respectively. Vectors are shown by bold-face letters. For a vector

, the notation is the compact form for . The expected value of with respect to the random variable is denoted by , where the subscript may be omitted if clear from the context. Cardinality of a set is denoted by .

Ii Preliminaries

In this section, the OFDM signal model as well as the definitions of the metrics PAPR, SE and CM are first presented. Then the Sign Selection problem is formalized and discussed.

Ii-a Signal Model

Consider an OFDM scheme with subcarriers. Let be the set of the complex-valued constellation points from which the data symbols that modulate the subcarriers are equiprobably and independently generated with zero mean, which implies that . Accordingly, the random vector denotes the vector of data symbols in an OFDM symbol. Denoting the frequency separation of the first and the last subcarriers as , the baseband continuous-time signal model for an OFDM symbol is

(1)

where and the signal power is normalized by . With the sampling frequency , where is the oversampling factor, the discrete-time signal model for an OFDM symbol is

(2)

The oversampling is necessary for reliable measurement of PAPR and CM from the discrete-time signal [25, 11].

Ii-B Peak to Average Power Ratio (PAPR)

Definition 1.

The PAPR metric is a function of the random data vector and is defined as

(3)

where is given in (2) and is the oversampling factor.

It will be seen that the maximum operator in the definition of the PAPR makes the required derivations of the CE Method difficult. Here we propose the Sum-Exp (SE) metric, which will be shown to be a suitable objective function to replace PAPR such that a desirable indirect PAPR reduction is gained by SE reduction.

Definition 2.

The SE metric is a function of the random data vector and is defined as

(4)

where is given in (2), is an adjustable parameter and is the oversampling factor.

The SE metric is obtained from the log-sum-exp function of the squared magnitude of the signal samples, i.e. , which is a well-known approximation of the maximum function [6] since

The first inequality is strict unless and approaches an equality as the maximum becomes larger relative to the rest of the samples, while the second inequality holds when all values are equal. That is, the approximation improves when the spread of the amplitudes of the signal samples is larger. Therefore, high ratio of the peak power to the average power of the OFDM signal implies that log-sum-exp is likely to be an acceptable approximation for PAPR. Furthermore, it motivates the introduction of the scaling factor to modify the log-sum-exp function as to increase the spread. The SE metric is obtained from the modified log-sum-exp function by omitting the monotonically increasing function as well as the constant .

Ii-C Cubic Metric (CM)

CM [14] is based on the assumption of a third-order (cubic) polynomial model for the input-output relation of the power amplifier. That is, the output signal for a passband input signal is assumed to be

where the linear gain and the non-linear gain are constant and related to the amplifier design. While PAPR is based only on the peaks of the instantaneous power, CM directly captures the energy in the distortion term and is calculated as

where the subscript refers to the value in logarithmic scale and the Raw Cubic Metric (RCM) of a signal is defined as

(5)

The reference signal , the slope factor and the bandwidth scaling factor [15] are independent of and are not discussed here. The Root Mean Square (RMS) of a signal over a large enough interval is .

Consider that reduction of CM for is essentially equivalent to reduction of its RCM. In addition, CM and RCM are constants calculated for the whole continuous-time passband signal, whereas practical reduction algorithms operate over individual discrete-time baseband OFDM symbols. Therefore, the discrete-time baseband version of the RCM of an OFDM symbol is actually used for CM reduction, as done in [7, 27, 20], which is referred to as Symbol RCM (SRCM) in this paper.

Definition 3.

SRCM is a function of the random data vector and is defined as

(6)

where is given in (2) and is the oversampling factor.

In order to show the relation of RCM and SRCM, we shall first briefly discuss the baseband representation of . Let the baseband equivalent representation of be as a function of complex data symbols pertaining to consecutive OFDM symbols. By a suitable choice of the normalization factor, it follows from the standard procedure of passband to baseband conversion that [4]. Ignoring the scaling factors, it can as well be shown that is the baseband representation of the frequency component of at the carrier frequency [4], where is the complex conjugate of . Consequently, for some scalar gives the RCM in terms of the baseband continuous signal. Next, the discrete-time version of is . Replacing the summation with an integral in calculation of the RMS of a discrete-time signal, we have given adequate oversampling. Finally, RCM can be written as

(7)

Therefore, RCM of the OFDM signal is the average of the SRCM values of the underlying OFDM symbols.

Ii-D The Sign Selection Problem

Altering the signs of the data symbols in an OFDM symbol in order to reduce one of the metrics defined before implies that one bit per transmitted symbol is consumed for this purpose. Consequently, for a constellation , bits of each transmitted symbol actually carry information. Initially consider taking a random sign bit to complete a -bit block which can then be mapped to a point in . For an OFDM symbol with a symbols vector , the Sign Selection approach seeks a solution for the problem

(8)

where can be any of the metrics defined before and denotes element-wise multiplication of vectors. Accordingly, will be the actually transmitted symbols. Considering that (8) is a NP-hard combinatorial problem, the proposal of the CE Method for deriving an efficient algorithm to obtain a desirable suboptimal solution is the objective of this paper.

Now we shall justify that the random sign bits which initially complete the -bit blocks do not alter the minimization problem. Assume that the constellation is symmetric such that for each point , the negated value is in the set. Let be a non-unique choice of points of such that if , then . A sample choice of for 16-QAM is shown in Fig. 1. For every , let . The whole space of the data symbols can be partitioned into the sets such that and when . Therefore, every in (8) belongs to a partition such that and for some . Having all possible sign vectors as the solution space, it is clear that the Sign Selection problem always seeks the minimum of the partition which contains . Formally, for every . Notice that although the starting vector does not affect the solution of (8) for the partition , it may change the suboptimal solution provided by a proposed algorithm.

The (bit-to-)symbol mapping in the transmitter and the decoding in the receiver are based on a predetermined . Specifically, the data symbols are obtained by mapping bits to a point in . On the other side, the decoding of the symbol of each subcarrier is performed by choosing when one of is detected and reversing the symbol mapping accordingly. Notice that the decoding adds no complexity to the receiver. It must be mentioned that the choice of plays a role only in the symbol mapping and decoding and is otherwise immaterial to the Sign Selection problem. Particularly, it can be shown that the partitioning described before is independent of .

Fig. 1: A non-unique choice of from for the 16-QAM constellation.

As the final comment, sign selection clearly incurs rate loss. Consider signs to be reserved for the sign selection. Consequently, the remaining data symbols are mapped from bits to . The incurred amount of rate loss, i.e. the ratio of the bits used for Sign Selection to the total number of bits in an OFDM symbol, is

(9)

Clearly, the rate loss decreases for a larger constellation size.

Iii Method of Conditional Expectations

The CE Method [13] is represented here for obtaining a suboptimal solution to the Sign Selection problem for reduction of an arbitrary metric . For a given data vector , a random vector of sign variables is initially assumed with equiprobable and independent elements, which are then sequentially decided and fixed. Consider the iteration where the random signs are fixed to . The expected values of conditioned on with and are compared and the sign that yields the smaller expectation is chosen as . Formally, a sub-optimal solution to the minimization problem stated in (8) can be obtained by sequentially choosing the sign variables as

(10)

for .

The decision rule given in (10) is based on introducing random sign variables and then reduction of conditional expectation of the original objective function. The justification that (10) leads to a desirable suboptimal solution of (8) is explained partly here for the general metric and will be finalized in Section V for PAPR and SRCM. For the -th sign decision, let

(11)

Following the decision criterion in (10), we have

whereas for the -th step with , it holds that

Therefore,

for . This shows that for a given , the non-increasing sequence of the conditional expectations begins with the initial expectation and ends with where no randomness is left. That is, the last conditional expectation coincides with a metric value such that

(12)

This justifies that the decision criterion given in (10) leads to a value of the original metric with the property stated above. Proving the reduction and the upper-bound on the reduced values is not known for the general case of the arbitrary metric and will be treated in Section V specifically for PAPR and CM. Calculation of the conditional expectations required at each step is indeed the main part of the algorithm and will be discussed in Section IV.

Pruned Sign Selection It has been observed through simulations that the impact of a sign decision increases for the sign variables with higher indices. That is, the reduction steps in the trajectory of the conditional expectations, as the algorithm performs sign decisions for to , become statistically larger. This motivates pruning the sign bits whose contribution is insignificant. Formally, in the pruned Sign Selection, the first symbols fully carry data and the sign bits of last symbols are determined by (10).

Iv Calculation of CEs

For a given vector of data symbols , the decision on requires calculation of in (11) which is rewritten here as

(13)

where

The obvious way of calculating the conditional expectations for practically any metric is to use the empirical average

to estimate

, which is

(14)

where is the number of realizations of the random sign vector used for the estimation and

(15)

where the random variables are independent and equiprobable.

Deriving more efficient ways of calculation of the conditional expectations is a pivotal part of the proposed method. The PAPR metric does not lend itself well to mathematical manipulations which could allow for closed-form expressions. Consequently, the conditional expectations are estimated by a sample average, which will be further discussed in the next part. On the contrary, the definitions of SRCM and SE together with the statistical properties of the signal samples make it possible to derive elegant closed-form expressions for . These results depend on convergence of the signal samples in distribution to a Gaussian random variable, proof of which is not trivial due to the specific signal model imposed by the Sign Selection problem. This will be clarified in the second part of this section before treating the calculation of SE and SRCM.

Iv-a PAPR metric

In order to calculate in (13) for , a sample average with realizations of the sign vector is used which gives the estimate

(16)

where and the random vectors were defined in (15). Consequently, the sign decision rule for PAPR is

(17)

It is clear that . Consequently,

as the variance of

is finite, although not known for finite . In order to obtain a relation between the reliability of the estimation and , concentration inequalities are used to bound the probability of deviation of the estimate from its true value as stated in the following theorem. The proof is given in Appendix A.

Theorem 1.

Consider the estimate of . For any and

where .

A lowerbound on the required can be deduced from the theorem as

which guarantees the probability of deviation by from the true value to be smaller than . In particular, it indicates that does not depend on as . The independence of the estimation accuracy promised by the concentration inequality of Theorem 1 stems from the fact that the bound on the difference in PAPR value due to a single sign change in , as shown in the proof, is normalized by . Establishing a connection between the probability of error in sign decision and , however, needs further research.

Iv-B Distribution of

The following discussion begins with the more general subject of characterizing the distribution of the continuous-time OFDM symbol in Theorem 2, which is required in performance analysis of Section V. Subsequently, the distribution of the discrete-time version follows automatically and is stated in Corollary 1, which is used in the derivation of the conditional expectations of SRCM and SE. Consider the centered random variables

(18)

where subscripts and denote the real and imaginary parts respectively. The following Lemma gives the covariance functions of these random variables as

, which is a necessary step to obtain their joint distribution in Theorem 

2.

Lemma 1.

Consider where is a rational number. For randomly distributed in , let the variances and covariances of and with respect to as and at any time instances with be denoted as

Then

with probability one, hence omitting the argument from the notation. Clearly, .

The proof is given in Appendix B. The following theorem characterizes the distribution of the OFDM signal.

Theorem 2.

For randomly distributed in and as specified in Lemma 1, consider as defined in (18) at any set of time instances . Omitting to save space, the vector

(19)

converges in distribution as to the vector

(20)

of jointly Gaussian random variables with , and as given in Lemma 1.

Proof.

The proof follows a standard procedure and is only outlined here. It essentially consists of the application of the Cramer-Wold device [5] to the vector in (19) which requires that every linear combination of the elements of the vector in (19) converges in distribution to the same linear combination of the elements of the vector in (20). This can be verified by the Lindeberg condition. In this procedure, the existence of the covariances of the linear combination is shown in Lemma 1. ∎

From Theorem 2, the following result is immediate.

Corollary 1.

For any given , and as defined in Lemma 1, it holds that

where

Remark 1.

A pivotal result which enables the analytical derivations in the remainder of this paper is that at every iteration of the algorithm, the distribution of in the limit is independent of . In addition, the distribution of , i.e. prior to any sign decision, is identical to that of as .

Remark 2.

In the following sections, the asymptotically Gaussian distribution shown in Corollary 

1 is used to approximate the distribution of for a finite but large enough number of random sign variables . Therefore, the approximation can be used to derive closed-form expressions of the sign decision criterion (10) only for . The number of the excluded final signs , for which the approximation is unacceptable, will be determined based on simulations in Section VI.

Iv-C SE Metric

By substituting for in (13), we have

(21)

It was shown in Corollary 1 that the real and imaginary components of are Gaussian and independent in the limit with equal variances. For , let as obtained in Lemma 1. Then the real and imaginary parts of

(22)

have approximately unit variances with accordingly scaled expected values. Therefore, for large enough is approximately a non-central

-distributed random variable with two degrees of freedom. Consider the moment generating function of

which is

where the non-centrality parameter is

and and were given in Corollary 1. It can be seen that the terms in (21) are identical to the definition of . Consequently,

(23)

where . Finally, a closed-form decision rule can be obtained as

(24)

The number of the last sign decisions which do not follow the closed-form expression in (23), i.e. , will be determined in Section VI. A sample average must be inevitably used instead for signs as in (14).

0:  : data symbols
1:  
2:  
3:  
4:   {Element-wise operations on arrays are assumed in this pseudocode.}
5:  for  to  do
6:     
7:     
8:      {element-wise absolute values by }
9:     if  then
10:        
11:     else
12:        
13:     end if
14:  end for
15:  return  
Algorithm 1 Implementation of the CE Method for CM reduction by Sign Selection.

Iv-D Cubic Metric

Replacing with in (13), we have

(25)

The expected values are the third moments of , which can be obtained from the third derivative of the moment generating function of the random variable as defined in (22). That is,

Obtaining the derivative and substituting it in (25), we have

and the decision rule in (10) can be written in closed form as

(26)

for . For the sign variables , consider using sample averages as in (14) with a high , which is the number of realizations of the random sign variables to calculate the conditional expectations. Simulations have shown that the CE Method delivers the same performance for several nonzero values of as for . That is, using accurate sample averages for the final sign variables does not improve the performance.

The application of the CE Method to the Sign Selection problem essentially leads to the explicit sign decision criteria derived in this section for PAPR and its substitute SE as well as for the SRCM. For better readability, the pseudocode for SRCM reduction is shown in Algorithm 1, where the expected values required for obtaining are constructed by adding the contribution of one subcarrier at each iteration (see lines 6 and 7).

V Performance analysis

The CE Method guarantees (12), which is rewritten here for convenience:

for a given . In order to characterize , one approach can be to establish a relation between the distribution of the initial expectation and that of the uncoded metric values . The analysis will be done for PAPR and SRCM with the help of some useful results from the literature. Concerning the SE metric, a relevant analysis would include a relation between SE reduction and the resulting indirect PAPR reduction, which requires further research.

V-a PAPR metric

Theorem 3.

For any , the reduced PAPR value obtained by the CE Method is bounded in the limit as

where , and is the Euler constant.

Proof.

Consider the PAPR of the continuous-time OFDM symbols given in (1) which is defined as

Clearly, for any finite oversampling factor ,

Therefore, It directly follows from (12) that

for any . Therefore [17],

(27)

In order to obtain the right hand side limit, recall that the covariance functions of , as emphasized in Remark 1, was shown to be identical to that of as . In addition, Extreme Value Theory [12] has been employed in [24] to obtain the asymptotic distribution of as