Abstract’ In the current scenario, the increasing demand of high speed data transfer in wireless communication has been raised the requirement and optimization of orthogonal frequency division multiplexing (OFDM) based system like Advanced Long Term Evolution system. But, the main problem with OFDM based system is the high value of Peak-to-Average Power Ratio (PAPR) of the transmitted signals. In this work, an improved nonlinear companding technique has been proposed that can transform the Gaussian distributed OFDM signals into the set of threshold based uniform distributed signals while maintaining the same average power level. The simulation results indicate that the proposed companding technique performs better than the -law, exponential and nonlinear companding techniques in the terms of PAPR reduction as well as reduced numerical value of bit error rate.
Keywords’ Complementary cumulative distribution function, peak-to-average power ratio, orthogonal frequency division multiplexing, bit error rate, high power amplifier.
I. INTRODUCTION
In 1966 Chang has introduced the concept of multicarrier communication [1]. In his suggestion multicarrier technique utilize the parallel data transmission using FDM with overlapping subcarriers. This scheme has been proven good technique for the efficient utilization of bandwidth and to combat the impact of multipath propagation. Moreover need of high speed equalization techniques has been also removed [2].
Before introduction of FFT, OFDM based system with large number of sub-carriers was not efficient due to high complexity and its cost because it needs the N number of sinusoidal generators and coherent demodulators to perform its parallel operations. To make it free from the crosstalk between the subcarriers, system should not have frequency and timing offsets.
An OFDM based standard 802.11a get approval from IEEE in 1999, to support a maximum data rate 54 Mbps. During this period European Telecommunications Standards Institute (ETSI) has also standardized the High Performance Radio Local Area Networks (HIPERLAN/2) standard, which adopted OFDM for their PHY standards. Later the Federal Communications Commission came with some new guidelines for modulation techniques functioning in the 2.4 GHz range, resulting that IEEE allowed to extend 802.11b to 802.11g standard [3]. However, a major drawback of OFDM systems is the high value of PAPR of the transmitted signals, resulting that low power efficiency, distorted signal and radiation outside the allowed band when the HPA is used. Several companding schemes have been analyzed by researchers in the past for reducing the PAPR but their complexity is very high. For example, ??-law companding provides better performance in the terms of PAPR reduction as compared to clipping technique [7]. But ??-law companding increases the average power level of the signals and provides non-uniform distribution [8].
The proposed ‘novel companding technique’ transforms the Rayleigh distributed OFDM signals magnitude into non-uniform distribution like exponential companding but in much better way. This technique obtained the benefit of maintaining average power level constant in the nonlinear region.
The remainder of this paper is organized as follows: The typical OFDM system and formulation of high PAPR problem has been addressed in section II. The proposed companding and decompanding functions have been presented in section III. We have discussed simulation results of proposed scheme and compared it with original OFDM, ??-law companding, exponential companding and NCT scheme in section IV. Finally, the conclusion is given in section V.
II. OFDM
The general block diagram of an OFDM based system using companding technique for reducing the PAPR of OFDM signals is presented in [12]. Let the number of subcarriers is represented by and these sub-carriers are used for transmitting data in parallel form. Let be the kth complex modulated data symbol out of N data symbols. The time domain OFDM signal sampled at Nyquist rate over one symbol interval is denoted by and it is generated by taking the N-point Inverse Fast Fourier transform (IFFT) of as
(1)
where denotes the time index. According to central limit theorem, the amplitude has a Rayleigh distribution while its CDF has chi-square distribution as follows.
(2)
The OFDM signal power can be computed as
(3)
For the one symbol period, the PAPR of OFDM signal is evaluated as
(4)
where E [.] is the expectation operator has its usual meaning. Therefore, PAPR is the ratio of peak power to average power of the OFDM signals. Peak power occurs in OFDM system when N modulated symbols get added while having same phase at the same time instant [14].
By applying proposed companding scheme, the OFDM signals get companded before Digital to Analog (D/A) conversion and amplified through HPAs. At the transmitter, the companded signal is expressed as
= h ( ), n=0, 1 ‘.N-1 (5)
where companding function alter only the amplitudes of the applied signals at the input. Now, OFDM signal after D/A conversion can be transmitted over the radio channel. Considering Additive White Gaussian Noise (AWGN) channel and after Analog to Digital conversion signal at the receiving side can be expressed as
= +
= h ( ) + , n=0, 1 ‘.N-1 (6)
where is the sampled AWGN signal.
After performing decompanding operation we get
= h-1 ( )
= (7)
where decompanding function is the inverse of .
III. PROPOSED COMPANDING TECHNIQUE
In this section, an improved nonlinear companding technique has been derived with h(x) is a companding function. It transforms the original probability distribution function of OFDM signals magnitude into a uniform distribution as shown in Fig. 1. Unlike the ??-law companding, main advantage of this technique is that it will maintain the average power level constant and provides less companding distortion as compared to referred companding techniques [13]-[15].
Fig. 1. Distribution for proposed technique
The notation of symbols used throughout in this paper is listed in Table 1 for convenience.
Table 1: List of symbols used in paper
kth modulated data symbol
nth sample of discrete time domain OFDM Signal
PDF of original OFDM signal (without companding)
CDF of original OFDM signal (without companding)
PDF of OFDM signal after companding
CDF of OFDM signal after companding
Upper-bound of the peak value of OFDM signal
Proposed Companding function
Proposed Decompanding function
Exponential companding function
Exponential decompanding function
Exponential companding function
Exponential decompanding function
Let power of exponentially companded OFDM signal amplitude | | has uniform distribution throughout the range [0, ??]. Where d denotes the degree of companding technique.
The CDF of | |d is given by
(8)
The CDF of exponentially companded signal amplitude | | is evaluated as
(9)
The inverse of the CDF of exponential companded signals amplitude is evaluated as
(10)
As companding function is a strictly monotonic increasing function, therefore, we have
(11)
If we consider the phase of the input signals then companding function can be expressed as
(12)
Where ?? is the positive constant, which is used to calculate the average power of the output signals. To keep the average power level constant at the input and output, ?? can be evaluated as
(13)
The inverse function of i.e., the decompanding function is expressed as
(14)
In the similar manner, companding and decompanding functions has been derived for the NCT by author as given below.
(15)
The decompanding function for the NCT scheme is given by
(16)
Let now an amplitude of the companded signals denoted by which has to be designed in such a way to have uniform distribution in the range of [0, ] for . Consequently, the CDF of is evaluated as
(17)
This is also strictly monotonic increasing function, therefore, the inverse function for this can be expressed as
(18)
Since the CDF’s of both companded and original signals are monotonically increasing functions therefore the proposed companding technique should have monotonically increasing function and its corresponding inverse function. When these all conditions get fulfilled then the relationships between CDF’s can be expressed as
(19)
Now according to equation (19) here, we have derived the companding function for the proposed technique while maintaining the average power constant and thus the value has been obtained.
Therefore, the companding function can be expressed as (20)
The decompanding function for the proposed scheme is given by
(21)
The CCDF for the PAPR of companded OFDM signals can be derived by
where is the given threshold
(22)
The average power for the companded signals is remains almost same as average power for original OFDM signals i.e. . While the maximum power of the signals after companding is given by
(23)
From (20) and (21) we get
(24)
Where is given by
(25)
IV. PERFORMANCE ANALYSIS
The results obtained after simulation of the proposed method are presented in this section. This method has considered an OFDM system with N = 1024 subcarriers, the oversampling factor l = 4 and 16-QAM modulation respectively. The three more companding schemes i.e. the ?? ‘law, exponential and NCT companding has also been considered in this paper for the purpose of performance comparison with the proposed companding scheme.
The effectiveness of PAPR-reduction technique is generally computed by the CCDF. Fig. 2 represents the pdf of original OFDM signals.
Fig. 2. Distribution of original OFDM signals
As observed from Fig. 1, the Gaussian-distributed signals (as shown in Fig. 2) are transformed into an almost uniform distribution based on a companding function by the proposed method.
This method adjusts both small and large signals and does not increase the distribution of large signals like ??-law companding scheme. Besides, by properly changing the companding parameters, a better tradeoff between the PAPR and BER performance can be achieved by this method. The waveforms (envelopes) of the original OFDM signals with referred schemes and proposed companding transform are depicted in Fig. 3, from that it can be observed that the proposed method may compress large signals as well as enlarging small signals simultaneously.
Fig. 3. Original, referred and companded signals waveforms
In Fig. 4, plots of CCDFs of proposed companding transform with original OFDM, ??-law companding, exponential companding and NCT scheme has been given. As this method improves the PAPR reduction by almost 2 dB with reference to the same parameter with NCT companded OFDM signal at CCDF=10-8. The performance comparison of proposed companding transform with original OFDM, ??-law companding, exponential companding and NCT scheme has been carried out on AWGN channel.
Fig. 4. PAPR performance comparision of original, referred
and companded signal
V. CONCLUSION
PAPR reduction schemes which are based on companding having very low complexity, higher efficiency, no bar on modulation type and the total number of sub-carriers used. In this work, an improved nonlinear companding technique has been proposed and analyzed. It adjusts the amplitudes of small as well as large signals, while maintains the constant average power level by selecting proper companding parameters. It also provides less companding distortion as compared to NCT and EC. After simulation, the obtained results have proven that the proposed method has offered better system performance in terms of PAPR reduction over the ??-law, exponential companding and NCT scheme.
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