Two-Stage Companding Scheme to Reduce Peak-to-Average Power Ratio for Safety Transportation Scenarios

Reliable wireless data transmission in intelligent transportation systems (ITS) has drawn a wide concern in recent years. As an efficient wireless transmission technology, multi-carrier communication (MCM) finds its way in many applications. However, the high peak-to-average power ratio (PAPR) of the MCM signal degrades the system performance. To remedy this, this paper introduces a novel two-stage distribution-based companding technique for the PAPR reduction of MCM transmission in transportation. The main principle of this technique is to transform the original MCM signal distribution into a desirable two-stage distribution, that is, compressing the large signals while maintaining the small signals constantly. In addition, a new variable parameter is also proposed to give the system more design flexibility. The simulation evaluation demonstrates the validity of the proposed companding technique.

SECTION I. Introduction

Intelligent transportation systems (ITS) have gained significant attentions in recent years. Particularly, reliable wireless communication among vehicles, road side units and base stations enable safety of life in future ITS. As an extraordinarily influential and reliable transmission technology in transportation, multi-carrier communication (MCM) has been extensively utilized in wireless standards, such as the IEEE 802.11 a/g/n standard, the terrestrial digital TV (DVB-T), 3GPP Long Term Evolution (3GPP LTE) and the IEEE 802.16 standard (WiMAX), [1], [2]. However, one major disadvantage is its high transmission peak-to-average power ratio (PAPR). A high PAPR leads to signal distortion and dramatically decreases the amplifier efficiency. Therefore, numerous techniques have been introduced to relieve the high PAPR problem [3], among which, clipping is the simplest method for PAPR reduction. Nevertheless, additional distortion noise caused by clipping deteriorates the MCM performance [4].

As an alternative method, a $\mu$ -law based companding scheme ($\mu \text{C}$ ) [5] is introduced to achieve a better PAPR performance than the clipping method. However, the $\mu \text{C}$ scheme reduces the PAPR at the cost of increasing transmission power. In [6] and [7], two novel companding schemes, e.g., exponential companding (EC) scheme and error companding scheme (ErC), have been proposed to remedy the power increasing problem. By transforming the transmitted signal into uniform distribution or error distribution, these two schemes can offer a more substantial performance than the $\mu \text{C}$ scheme. Nevertheless, these two schemes increase the probability of large signals. Therefore, when a heavy nonlinearity amplifier is applied, the bit error rate (BER) will be degraded dramatically. Literature [8] and [9] proposed two nonlinear companding schemes for further PAPR reduction, where BER performance are still unsatisfactory. Several piecewise companding transform (PCT) schemes with low computational complexity were also proposed in [10]–​[12] for further PAPR reduction. However, an inappropriate transform parameter value may further degrade the system performance. Root-based nonlinear companding (RNC) and enhanced nonlinear companding (ENC) are introduced in [13] and [14], respectively, while these methods reduce the PAPR performance mainly relies on large power spectrum sacrifice or BER deterioration.

Considering that, further motivated by the system improvement and tradeoffs among PAPR, power spectrum and BER performance, a novel two-stage distribution companding scheme is proposed in this paper. By transforming the distribution of transmitted MCM signal into the desirable distribution, this scheme can achieve an effectively PAPR reduction. In addition, a new variable parameter is also proposed in order to maintain the transmit power constant and increase the flexibility of system design. Simulation results demonstrate that the proposed two-stage distribution-based companding scheme outperforms other companding schemes in terms of PAPR, distortion noise and BER performance.

This paper is formulated as follows: the high PAPR problem in MCM transmission is described in Section II. Section III introduces the proposed two-stage distribution-based companding scheme, the PAPR expression in terms of the novel proposed transform parameter is also present. The performance comparison through Monte Carlo simulation are present in Section IV. Section V gives conclusions.

SECTION II. PAPR Problem Formulation

As shown in Fig. 1, after quadrature amplitude modulation (QAM) or phase shift keying (PSK) modulation, the MCM data symbols are independently sent on the subcarriers. Let $X=[X_{1}, X_{2}, \ldots, X_{N-1}]^{T}$ , where $N$ denotes the subcarrier index, respectively. The functions of ‘CT($\cdot$ )’ and ‘CT$^{-1}(\cdot)$ ’ in Fig. 1 represent the typical companding transform and the corresponding de-companding transform operation, respectively. Therefore, the complex representation of an MCM signal is defined as

$$\begin{equation*} x\left ({t }\right) = \frac {1}{\sqrt {N} }\sum \limits _{k = 0}^{N - 1} {X_{k}} {e^{j2\pi n\Delta ft}},\quad 0 \le t{\mathrm{ < }}NT, \tag{1}\end{equation*}$$ where $j =\sqrt {-1}$ , $\Delta f$ and $T$ represent the subcarrier spacing and the input data block period, respectively. The PAPR definition of the transmit MCM signal $x(t)$ can be expressed as $$\begin{equation*} PAPR = \frac {{\max \limits _{0 \le t < NT} {{\left |{ {x\left ({t }\right)} }\right |}^{2}}}}{{{1 \mathord {\left /{ {\vphantom {1 {NT}}} }\right. } (NT)} \cdot \int _{0}^{NT} {{{\left |{ {x\left ({t }\right)} }\right |}^{2}}dt} }}. \tag{2}\end{equation*}$$

FIGURE 1.

Diagram of traditional MCM system.

In practical application, $x(t)$ is oversampled in order to achieve a discrete time signal. Therefore, an $L$ -time oversampled MCM signal can be expressed as

$$\begin{equation*} {x_{n}} \!=\! \frac {1}{\sqrt {N} }\sum \limits _{k = 0}^{N - 1} {X_{k}}{e^{{j2\pi nk\Delta fT \mathord {\left /{ {\vphantom {j2\pi nk\Delta fT L}} }\right. } L}}},\quad n \!=\! 0,1, \ldots,NL - 1.\qquad \tag{3}\end{equation*}$$ Then, the PAPR calculated in Eq. (3) with $L$ times oversampled signal is re-defined as $$\begin{equation*} PAPR = \frac {{\max \limits _{0 \le n < NL - 1} \left \{{ {{{\left |{ {x_{n}} }\right |}^{2}}} }\right \}}}{{E\left [{ {{{\left |{ {x_{n}} }\right |}^{2}}} }\right]}}, \tag{4}\end{equation*}$$ where $E[\cdot]$ denotes the expectation operation.

SECTION III. Proposed Companding Scheme and Analysis

In section III, a new companding scheme is introduced to transform the amplitude distribution of original signals into two-stage distribution. In addition, a novel transform parameter is also proposed in order to increase the flexibility of the system design.

A. Proposed Companding Scheme Without Power Restriction

Let the input MCM symbols follows statistically independent and identically distribution (e.g., i.i.d distribution). When the subcarriers number is larger than sixty-four, according to the central limit theory, $x_{n}$ can approximate a complex Gaussian process. Assume the mean and variance of the original signal $x_{n}$ are assume to be zero and $\sigma ^{2}$ (note that $\sigma ^{2}=E[|X_{k}|^{2}]/2$ ), respectively. Therefore, the probability distribution function (PDF) of $|x_{n}|$ can be expressed as

$$\begin{equation*} {f_{\left |{ {x_{n}} }\right |}}\left ({x }\right) = \frac {2x}{\sigma ^{2}}\exp \left ({{ - \frac {x^{2}}{\sigma ^{2}}} }\right),\quad x \ge 0. \tag{5}\end{equation*}$$ Accordingly, the cumulative distribution function (CDF) of $|x_{n}|$ can be computed as $$\begin{align*} {F_{x}}\left ({x }\right)=&Prob\left \{{ {\left |{ {x_{n}} }\right | \le x} }\right \} \\=&\int _{0}^{x} {\frac {2y}{\sigma ^{2}}} \exp \left ({{ - \frac {y^{2}}{\sigma ^{2}}} }\right)dy \\=&1 - \exp \left ({{ - \frac {x^{2}}{\sigma ^{2}}} }\right),\quad x \ge 0. \tag{6}\end{align*}$$

Assuming that the proposed companding function is defined as

$$\begin{equation*} {t_{n}} = C\left ({{x_{n}} }\right), \tag{7}\end{equation*}$$ where $t_{n}$ denotes the proposed companded signal. For an additive white Gaussian noise (AWGN) channel by using the proposed scheme, the received signal can be written as $$\begin{equation*} {r_{n}} = {h_{n}} * {t_{n}} + {u_{n}} = {h_{n}} * C\left ({{x_{n}} }\right) + {w_{n}}, \tag{8}\end{equation*}$$ where $h_{n}$ and $w_{n}$ represent the channel response and the AWGN noise.

Generally, the proposed scheme only compressing the large signals follows the linear distribution. Specifically, during the interval $[0, Y]$ the proposed companded signal $|t_{n}|$ maintains the original Rayleigh distribution, while in the interval of $[Y, Z]$ the signal $|t_{n}|$ follows the linear distribution, where $Y$ and $Z$ are the companding distribution transform parameter and the blockage parameter in the PDF of the proposed companded signal $|t_{n}|$ , respectively. Therefore, the PDF function of the proposed companded signal $|t_{n}|$ can be written as

$$\begin{equation*} {f_{\left |{ {t_{n}} }\right |}}\left ({x }\right) = \begin{cases} {\dfrac {2x}{\sigma ^{2}}\exp \left ({{ - \dfrac {x^{2}}{\sigma ^{2}}} }\right)}&{x < Y}\\ {\dfrac {{2\alpha \exp \left ({{ - {\alpha ^{2}}} }\right)}}{{\alpha {\sigma ^{2}} - Z\sigma }}\left ({{x - Z} }\right)}&{Y < x \le Z}, \end{cases} \tag{9}\end{equation*}$$

Accordingly, the corresponding CDF function of $|t_{n}|$ is shown in Eq. (10), as shown at the bottom of this page,

where the parameter $\alpha$ is defined as the ratio between the companding distribution transform parameter $Y$ and the root mean square (rms) level of the MCM signal. The blockage parameter $Z$ can be obtained by the definition of the PDF function $\int _{ - \infty }^\infty {f\left ({x }\right)} dx = 1$ , e.g., $$\begin{equation*} Z = \alpha \sigma + \frac {\sigma }{\alpha }, \tag{11}\end{equation*}$$

Considering that the proposed two-stage distribution companding function $C(x)$ follows a strictly monotonic increasing function

$$\begin{align*} {F_{\left |{ {x_{n}} }\right |}}\left ({x }\right)=&Prob\left \{{ {\left |{ {x_{n}} }\right | \le x} }\right \} \\=&Prob\left \{{ {C\left ({{\left |{ {x_{n}} }\right |} }\right) \le C\left ({x }\right)} }\right \} \\=&{F_{\left |{ {t_{n}} }\right |}}\{ {C\left ({x }\right)} \}. \tag{12}\end{align*}$$

With the phase information injection of the signal $x_{n}$ , the proposed companding function $C(x)$ can be written as

$$\begin{equation*} C\left ({x }\right) = sgn\left ({x }\right) \cdot {F_{\left |{ {t_{n}} }\right |}}^{ - 1}\{{F_{\left |{ {x_{n}} }\right |}}\left ({x }\right)\}, \tag{13}\end{equation*}$$ where the $sgn(x)$ represents the standard sign function. Furthermore, with the help of Eqs. (6), (10), and (13), the proposed companding and de-companding function can now be computed, the results are shown in Eqs. (14) and (15), as shown at the bottom of this page.

B. Proposed Scheme Under Strictly Constant Average Power Restriction

The original proposed companding can reduce the PAPR effectively. However, the derivation and calculation operation of the proposed companding scheme neglect the impact of power increase during the transmission, which, in turn, degrades the whole system performance. To remedy this and to further maintain a strictly constant average power, an optimized structure of the proposed scheme with a new companding distribution transform parameter is also introduced in this section. Following the similar derivation process in Section III-A, the novel two-stage distribution-based nonlinear companding transform function can be obtained.

Assume the amplitude of the new companded symbol $t'_{n}$ follows the original Rayleigh distribution and linear distribution in the interval of $[0, \alpha '\sigma]$ and $[\alpha `\sigma, Z']$ , where $\alpha '\sigma$ and $Z'$ are the new companding distribution transform parameter and blockage parameter in the PDF function of the signal $|t'_{n}|$ , respectively. Note that the new companding distribution transform parameter $h=C'(\alpha '\sigma)$ is discontinuous. By properly choosing the value of parameter $\alpha '$ , the new proposed scheme can achieve variable PAPR performance while maintain the transmission power strictly constant. Therefore, the PDF and CDF of $|t'_{n}|$ can be achieved, the corresponding expressions are shown in Eqs. (16) and (17), as shown at the bottom of this page.

Fig. 2 depicts the probability distribution function of proposed scheme under strictly constant average power restriction. $$\begin{align*} {f_{\left |{ {t_{n}'} }\right |}}\left ({x }\right) = \begin{cases} {\dfrac {2x}{\sigma ^{2}}\exp \left ({{ - \dfrac {x^{2}}{\sigma ^{2}}} }\right)}&{0 \le x < \alpha '\sigma }\\[4pt] {\dfrac {h}{\alpha '\sigma - Z'}\left ({{x - Z'} }\right)}&{\alpha '\sigma \le x \le Z'} \end{cases}, \tag{16}\end{align*}$$

FIGURE 2.

probability distribution function of the proposed power constant companding scheme.

The new blockage parameter $Z'$ can also be achieved by the definition of the PDF function $\int _{ - \infty }^\infty {f _{| {t_{n}'}|} \left ({x }\right)} dx = 1$ , e.g.,

$$\begin{equation*} Z' = \frac {2}{h}\exp \left ({{ - \alpha {'^{2}}} }\right) + \alpha '\sigma. \tag{18}\end{equation*}$$

On one hand, the power of the signal after the proposed two-stage distribution companding can be represented as

$$\begin{align*}&\hspace{-1.9pc}\bar P = \int _{0}^\infty {x^{2}{f_{|t'_{n}|}}\left ({x }\right)dx} = \int _{0}^{\alpha '\sigma } {\frac {x^{2} \cdot 2x}{\sigma ^{2}}\exp \left ({{ - \frac {x^{2}}{\sigma ^{2}}} }\right)dx} \\&\qquad \quad \qquad \qquad + \int _{\alpha '\sigma }^{Z'} {\frac {x^{2} \cdot h}{\alpha '\sigma - Z'}\left ({{x - Z'} }\right)dx}. \tag{19}\end{align*}$$ On the other hand, the power of original MCM signal $x_{n}$ can be written as $$\begin{equation*} \int _{0}^\infty {x^{2}{f_{x}}\left ({x }\right)dx} = \int _{0}^\infty {\frac {x^{2} \cdot 2x}{\sigma ^{2}}\exp \left ({{ - \frac {x^{2}}{\sigma ^{2}}} }\right)dx}. \tag{20}\end{equation*}$$ Considering the average power level must constant during the new proposed two-stage distribution companding operation, e.g., $\bar P = {\sigma ^{2}}$ , it follows that $$\begin{equation*} h = \frac {1}{3\sigma }\exp \left ({{ - \alpha {'^{2}}} }\right) \cdot \left ({{2\alpha ' + \sqrt {4\alpha {'^{2}} + 3} } }\right). \tag{21}\end{equation*}$$ Considering (13), the proposed companding function $C'(x)$ can be written as $$\begin{equation*} C'\left ({x }\right) = sgn\left ({x }\right) \cdot {F_{\left |{ {t'_{n}} }\right |}}^{ - 1}({F_{\left |{ {x_{n}} }\right |}}\left ({x }\right)), \tag{22}\end{equation*}$$ With the help of Eq. (17), the final proposed companding and the decompanding function can be derived, as shown in Eqs. (23) and (24), as shown at the bottom of this page.

Fig. 3 shows the PAPR performance versus value of $\alpha$ or $\alpha '$ in proposed companding scheme. As shown, the proposed scheme under strictly constant average power restriction can increase the flexibility of the system design by adjusting the transform parameter ($\alpha$ or $\alpha '$ ).

FIGURE 3.

PAPR performance versus value of $\alpha$ or $\alpha '$ in proposed companding scheme.

C. Theoretical Analysis

Based on the Bussgang theorem [12], complex Gaussian inputs can be formed as a rescale input signal duplication plus an irrelevant nonlinear distortion noise. Therefore, the new signal $t_{n}'$ after the proposed companding operation can be defined as the sum of the rescale MCM signal and the companding distortion noise,

$$\begin{equation*} {t_{n}} = \eta {x_{n}} + {b_{n}}, \tag{25}\end{equation*}$$ where $\eta$ is the attenuation factor and $b_{n}$ is the companding noise, respectively. Literature [13] has proven that the parameter$\eta$ is independent on the stationarity of the original MCM signal and, therefore, time invariant. Under this condition, the corresponding expression are represented as follows: $$\begin{equation*} \eta = \frac {{E\left \{{ {t'_{n}{\bar x_{n}}} }\right \}}}{{E\left \{{ {x_{n}{\bar x_{n}}} }\right \}}} = \frac {1}{\sigma ^{2}}\int _{0}^\infty {x \cdot C'(x)} {f_{\left |{ {x_{n}} }\right |}}(x)dx, \tag{26}\end{equation*}$$ where $C'(x)$ is the proposed companding function. From this equation, we can calculate the attenuation factor parameter $\eta$ of the proposed companding scheme. Correspondingly, the complementary cumulative distribution function (CCDF) expression of proposed power changed companded signal is given by $$\begin{align*} CCDF(t_{n})=&CCDF\left ({{\frac {Z^{2}}{\sigma ^{2}}} }\right) \\[5pt]=&CCDF\left [{ {\frac {{{{\left ({{\alpha \sigma + {\sigma \mathord {\left /{ {\vphantom {\sigma \alpha }} }\right. } \alpha }} }\right)}^{2}}}}{\sigma ^{2}}} }\right] \\[5pt]=&CCDF{\left ({{\frac {{1{\mathrm{ + }}{\alpha ^{2}}}}{\alpha }} }\right)^{2}}. \tag{27}\end{align*}$$ In addition, the CCDF expression of proposed companded signals with constant power during the process can be represented as $$\begin{align*} CCDF(t_{n}')=&CCDF\left ({{\frac {Z'^{2}}{\sigma ^{2}}} }\right) \\[3pt]=&CCDF\left \{{ {\frac {{{{\left [{ {\frac {2}{h}\exp \left ({{ - \alpha {'^{2}}} }\right) + \alpha '\sigma } }\right]}^{2}}}}{\sigma ^{2}}} }\right \} \\[3pt]=&CCDF{\left ({{\sqrt {4\alpha {'^{2}} + 6} - \alpha '} }\right)^{2}}. \tag{28}\end{align*}$$

Accordingly, with reference to Eq. (4), the PAPR value of the new companded signal under power change and constant can be calculated as

$$\begin{align*} PAPR(t_{n})=&10 \cdot {\log _{10}}\frac {{\text {Max}\left ({{{{\left |{ {C\left ({{x_{n}} }\right)} }\right |}^{2}}} }\right)}}{{E\left [{ {{{\left |{ {x_{n}} }\right |}^{2}}} }\right]}} \\[3pt]=&20 \cdot {\log _{10}}\left ({{\frac {{1{\mathrm{ + }}{\alpha ^{2}}}}{\alpha }} }\right) [dB], \tag{29}\\[3pt] PAPR(t_{n}')=&10 \cdot {\log _{10}}\frac {{\text {Max}\left ({{{{\left |{ {C'\left ({{x_{n}} }\right)} }\right |}^{2}}} }\right)}}{{E\left [{ {{{\left |{ {x_{n}} }\right |}^{2}}} }\right]}} \\[3pt]=&20 \cdot {\log _{10}}{\left ({{\sqrt {4\alpha {'^{2}} + 6} - \alpha '} }\right)^{2}} [dB]. \tag{30}\end{align*}$$

SECTION IV. Performance Evaluation

In this section, simulation results are present based on an 802.11a/g WLAN MCM transmission of transportation. Random quadrature amplitude modulation symbols are input to the MCM system and then through to the wireless channel. Here, the oversampling factor and the subcarriers number are set to 4 and 64, receptively. Furthermore, a wireless local area network high power amplifier (WLAN-HPA No. AP-1093) produced by Radio Frequency Integrated Corp. [13] is applied for the purpose of sufficient transmit power.

According to Eq. (26) in Section III, we can calculate the attenuation factor $\eta$ in the proposed companding scheme with $\alpha '=0.7$ , $\alpha '=0.9$ and the EC scheme are 0.996, 0.995 and 0.992, respectively. Considering that the proposed companding scheme under the strictly constant average power restriction, it follows that

$$\begin{align*} {P_{x_{n}}}=&{P_{t_{n}}} = {P_{\eta {x_{n}}}} + {P_{b_{n}}} = {\eta ^{2}}{P_{x_{n}}} + {P_{b_{n}}}, \tag{31}\\ {P_{b_{n}}}=&(1 - {\eta ^{2}}){P_{x_{n}}}. \tag{32}\end{align*}$$

It also can be obtained that the power of the distortion noise of the proposed companding scheme with the parameter $\alpha '=0.7$ , $\alpha '=0.9$ and EC scheme are $0.004^{2}P_{x_{n}}$ , $0.005^{2}P_{x_{n}}$ and $0.008^{2}P_{x_{n}}$ , respectively. Based on this, it can be concluded that compared with EC scheme, the nonlinear distortion noise power caused by the proposed companding scheme ($\alpha '=0.7$ ) decrease 75%, with only about 0.8 dB increase in PAPR, which can be clearly seen in Fig. 4. The cubic metric (CM) can effectively predict the power capability reduction in the power amplifier. Fig. 5 shows the raw CM (RCM) performance under various PAPR reduction schemes.

FIGURE 4.

PAPR performance for various SLM techniques in various scenarios.

FIGURE 5.

The RCM performance under various PAPR reduction schemes.

Fig. 6 represents the influence of various PAPR reduction schemes on power spectrum. As seen, with high PAPR, the original MCM power spectrum maintaining a rectangular-liked characteristic. However, this characteristic is significantly influenced by the companding operation, e.g., higher adjacent channel leakage power ratio, more spectrum side-lobes and slower spectrum roll-off. As shown in Fig. 5, the EC and RMC schemes involve heavy side-lobes re-generation, while the proposed companding scheme bring less spectrum side-lobes. In addition, compared with the EC and RMC schemes, the proposed scheme with $\alpha '=0.7$ causes less effect on the original MCM power spectrum. Since little spectral regrowth during the proposed companding operation, it follows that the proposed companded signals can maintain a good spectrum characteristic, which, in turn, can enhance the system anti-interference capability from out-of-band noise. Note that the power spectrums of the proposed scheme with other parameter values (not shown) are found similar to that of $\alpha '=0.7$ .

FIGURE 6.

power spectrum caused by various PAPR reduction schemes.

Fig. 7 describes the performance of BER versus $E_{b}/N_{0}$ with various schemes in the case of 16QAM and 256 subcarriers. The curve ‘Original’ in the figure is achieved by ignoring the power amplifier affection and sending signals to the AWGN channel directly. Specifically, the proposed scheme with parameter $\alpha '=0.7$ and 0.9 are all have a better BER performance than other companding schemes. Moreover, by adjusting the proposed parameter $\alpha '$ , it can obtain different BER performance. Compared with the ideal BER curve, it also can be concluded that there is only 0.35 dB BER degradation results in the proposed scheme with $\alpha '=0.7$ . In addition, from Fig. 3, the optimal BER performance in MCM systems is the proposed scheme with the parameter $\alpha '$ at around 0.7. Taking all into account, the proposed scheme has high application prospect.

FIGURE 7.

BER comparison of the various schemes under 16QAM and 256 subcarriers.

Fig. 8 provides the BER performance of various schemes under the 64QAM with 256 subcarriers. Due to the nonlinear effect of the PA, the BER for all schemes suffer large degradation at the high SNRs. As shown in Fig. 7, it is remarkable that the proposed scheme can maintain a preferable performance compared with other companding schemes. As a summary, it can be concluded that with a proper value of the $\alpha '$ , the proposed companding scheme can obtain an excellent tradeoff between the BER, PAPR and the distortion noise performance. Therefore, the proposed two-stage companding scheme is an effective method to solve PAPR problem in applications.

FIGURE 8.

BER comparison of the various schemes under 64QAM and 256 subcarriers.

SECTION V. Conclusion

In order to ensure reliable wireless transmission of intelligent transportation systems (ITS), a novel two-stage distribution companding scheme have been proposed. By transforming the distribution of original MCM signal into a desirable distribution, this proposed scheme can increase the flexibility of the system design. Monte Carlo simulation results have verified the validity of the proposed two-stage scheme in the respect of PAPR, power spectrum, BER and distortion noise performance.