Nonlinear Generalized Companding Transform
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Published: Tue, 17 Jul 2018
Nonlinear Generalized Companding Transform for Peak-to-Average Power Ratio Reduction in OFDM
- Eashendra Singh
Abstract — One of the main drawback of Orthogonal Frequency Division Multiplexing (OFDM) system is its high Peak-to-Average Power Ratio (PAPR) of the OFDM signal. In this paper a novel non-linear generalized companding scheme called “Quadrilateral Companding Transform (QCT)” has been proposed to reduce the PAPR of OFDM signal. The proposed method provides additional degrees of freedom in comparison to existing trapezoidal companding, exponential companding and trapezium distribution based companding schemes. This allows more flexibility in designing the companding function, which is useful for the overall OFDM system to achieve low BER with good PAPR reduction capability.
Keywords – Complementary cumulative distribution function (CCDF), peak-to-average power ratio (PAPR), orthogonal frequency division multiplexing (OFDM), bit error rate (BER).
The modern day phenomenon of increased thirst for more information and the explosive growth of new multimedia wireless applications have resulted in an increased demand for technologies that support very high speed transmission rates, mobility and efficiently utilize the available spectrum and network resources. OFDM is one of the best solutions to achieve this goal and it offers a promising choice for future high speed data rate systems , . OFDM has been standardized as part of the IEEE 802.11a and IEEE 802.11g for high bit rate data transmission over wireless LANs . It is incorporated in other applications and standards such as digital audio broadcasting (DAB), digital video broadcasting (DVB), the European HIPERLAN/2 and the Japanese multimedia mobile access communications (MMAC) , . However, a major drawback of FDM systems is the high peak-to-average power ratio (PAPR) of the transmitted signals, resulting in the lower power efficiency, serious signal distortion and out-of-band radiation when the high power amplifier (HPA) is utilized.
Many companding schemes - have been proposed in the literature to reduce the PAPR of OFDM signal. The conventional μ – law and A-law companding schemes can be used for PAPR reduction, by choosing the suitable value of the parameters μ or A, controlling the nonlinearity of the μ -law  or A -law companding function respectively. But the error performance of both the schemes degrades as both of them introduce high companding distortion in OFDM signal at higher values of μ or A. A nonlinear companding transform  has been proposed by Jiang et al. to effectively reduce the PAPR of the OFDM signal. In this scheme , the Gaussian distributed in-phase (I) and quadrature-phase (Q) components of discrete time complex OFDM signal are transformed into a quasi-uniform distribution. In this scheme, the companding function is separately applied to I and Q components of the OFDM signal. The large values of I or Q components of the OFDM signal are compressed, whereas those with small I and Q components are enlarged. The PAPR reduction capability and BER performance of this scheme , can be optimized by properly choosing the parameters of the companding function. Jiang et al. proposed “Exponential Companding (EC)” scheme  to transform Rayleigh distributed OFDM signal magnitude into uniform distribution. Exponential companding has the advantage of maintaining the constant average power level in the nonlinear companding operation. However, the distribution of large signals is increased by the uniform companding, which makes the PAPR reduction was very limited under the bit error rate (BER) performance degradation. In this paper proposed technique transform the Rayleigh distributed OFDM signal magnitude into Quadrilateral distribution function as shown in figure 2 to achieve an additional degree of freedom over TC . The parameters of quadrilateral distribution are chosen in such a way that it produces least possible companding distortion to achieve low BER for a given PAPR.
The remainder of this paper is organized as follows: In section II, the OFDM system model with quadrilateral companding. The proposed quadrilateral companding and decompanding functions are derived in section III. Mathematical analysis of the PAPR performance of proposed scheme is presented in section IV, simulation results for PAPR performances of the proposed scheme are presented and discussed in the same section and conclusion is summarized in section V.
The block diagram of an OFDM system using companding scheme for PAPR reduction is shown in Fig. 1. Here, I have considered an OFDM system with N subcarriers, in which each of the subcarrier is each of the subcarrier is modulated by M-PSK or M-QAM. As shown in Figure 1.The input binary data sequence is first converted into N parallel data substreams and then these are mapped to the constellation points of M- PSK or M-QAM to achieve desired modulation on each of the subcarriers. After this, subcarrier modulation is performed using IFFT block to obtain the discrete time domain OFDM signal. Let be the N complex modulated data symbols to be transmitted over N subcarriers. The discrete time domain OFDM signal generated after taking IFFT of a block of N modulated data symbols. Discrete time domain OFDM signal is passed through the parallel to serial (S/P) converter and then applied to the compander for reducing the dynamic range or PAPR of the OFDM signal. The companded OFDM signal is applied to digital to analog (D/A) converter to get analog signal and then finally amplified using HPA. At the receiver, the received signal is first converted into digital signal using A/D converter.
Figure 1. Block diagram of OFDM with companding
The digital signal is then expanded by inverse companding function known as decomapnding function. After that subcarrier demodulation is performed by taking the FFT of OFDM signal obtained from expander. Finally, M-PSK or M-QAM decoder is used to decode the received data signal.
PROPOSED COMPANDING TECHNIQUE
The quadrilateral companding function h(x) is a nonlinear companding function. It transforms the original probability distribution function of OFDM signal magnitude into a quadrilateral distribution as shown in Figure 2, and hence the name “Quadrilateral Companding Transform”.This may also be called nonlinear generalized companding transform.
Figure 2. Quadrilateral distribution for proposed QCT
The symbols notation used throughout this paper are listed in Table 1 for convenience.
Table 1: List of symbols used in QCT
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
Quadrilateral Companding function
Quadrilateral Decompanding function
The pdf of quadrilateral trapezium distribution can be read from Figure 2 as
where h1 , h2, l, a and b are the parameters of quadrilateral distribution as shown in the Figure 2.These parameters (h1 , h2, l, a and b) control the nonlinearity of the companding functions. The cumulative distribution function (CDF) of quadrilateral distribution function can be calculated using the following relationship
Using (1) and (2) we have
Quadrilateral distribution function is bounded in the interval [0,l]. Like EC, TC and TDBC, in this scheme also average power of the OFDM signal before and after companding is kept same, therefore we have
As shown in Figure 2, the PDF of quadrilateral trapezium companded OFDM signal lies in the interval [0,l] , therefore, we have,
For given values of l, a and b, the parameters ( h1 , h2 ) of the companding function h(x) can be easily calculated using (3) and (4). Therefore, three parameters (l, a and b ) can be chosen independently to control the nonlinearity of companding function h(x) . Hence the proposed QCT has three degree of freedoms. The values of l, a and b should be chosen independently to provide low PAPR and BER.
The expression of QCT function h(x) can be derived after equating the CDF of original and companded OFDM signal. Therefore, we have
Where is the CDF of original OFDM signal given by following:
Therefore we have
The output of the N-point Inverse Fast Fourier Transform (IFFT) of are the OFDM signal sample over one symbol interval, or mathematically,
Where E [.] denotes the expectation operator.
In , the PAPR and BER performance of TC has been evaluated for (a = 0.4,b = 0.1 and l = 1.633) , (a = 0.2,b = 0.7 and l = 2.164) , (a = 0,b = 0 and l = 1.732) , (a = 0.9, b = 0.1 and l = 1.488) and (a = 0,b = 1 and l = 2.449) , here we refer to them as ‘TC-1’, ‘TC-2’, ‘EC’, ‘TC-3’ and ‘TC-4’ respectively. In , it has been shown that TC-3 provides the best PAPR reduction capability among all the cases under consideration, but its BER performance is very poor, on the other extreme TC-4 provides very less PAPR reduction. Therefore, we ignore these two cases (TC-3 and TC-4) and the remaining three cases i.e. (TC-1, TC-2 and EC), which offer reasonable PAPR are considered in my simulations for comparison with the proposed scheme.
To show the outperformance of the proposed scheme (QCT), the PAPR and BER performances are evaluated for two sets of companding function parameters i.e. (a = 0.2,b = 0.7,l = 2.174, h1 = 0.8596 and h2 = 0.8275) and (a = 0.4,b = 0.1,l = 1.643, h1 = 0.8276 and h2 = 0.7874) . Here, we call them as ‘QCT-1’ and ‘QCT-2’.
Figure 3. PAPR performance comparision of original and companded signal
Figure 4. BER performance comparison of various companding schemes
The QCT provides extra degrees of freedom to design the companding function and hence by choosing the suitable values of design parameters of the proposed companding function, a good trade-off between the PAPR reduction and the BER can be achieved. The proposed QCT provides better PAPR reduction and BER performance in comparison to TC, EC and TDBC. QCT can achieve a minimum PAPR of 0dB, whereas TC and EC can achieve a minimum PAPR of 3dB and 4.771dB respectively. QCT-2 has superior PAPR performance in comparison to QCT-1 but its BER performance is inferior in comparison to QCT-1.
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