The sensitivity of modulation fidelity on PA envelope variation in OFDM transmitter systems

Abstract

Intermodulation distortion (IMD) caused by nonlinear power amplifier (PA) is a major weakness in orthogonal frequency-division multiplexing (OFDM) transmitter systems, and most behavioral studies of IMD are based on specific example PAs. In recent papers, PA percentage linearization (PL), a general measurement for PA envelope variation, is investigated for its influence on system performance. Meanwhile, PL still somehow depends on specific example PA, and it mostly considers the PA saturation point (SP) variation. In this paper, a comparison analysis is presented to investigate the system performance when both SP and nonlinearity onset point (NOP) variations are considered. The simulations show that how the PA envelope variation can be represented by the signal impairment over IBO and OBO domains, respectively. It is interesting to find that although SP attracts much concern in current studies, NOP actually has a clear influence on modulation fidelity of nonlinear OFDM transmitter systems. Moreover, it is observed that when input power gets close to saturation range, the PAs having relatively higher nonlinearity show relatively lower distortion level over the input power domain. These interesting simulation observations above are analytically explained based on a mixed time domain and statistical analysis of IMD mechanism. An example IEEE 802.11a OFDM signal is driving through five example solid state PAs, and the system performance is measured by error vector magnitude (EVM) and adjacent channel power ratio (ACPR).

1 Introduction

Orthogonal frequency-division multiplexing (OFDM)-modulated signals are widely used in wireless communications systems for their advantage of high spectrum efficiency [1, 2]. However, OFDM signals have a nature of high peak to average power ration (PAPR), since they consist of lots of independent input subcarriers. This high PAPR nature makes OFDM transmitter systems very sensitive to the nonlinearity of RF power amplifiers (PAs) integrated within transmitters, and significant nonlinear distortion generated in the amplification process is a major weakness of OFDM signal systems [3, 4]. The nonlinear distortion can be evaluated by the modulation fidelity over wanted subcarriers and the out-of-band spectral regrowth, and they can be quantitatively expressed by error vector magnitude (EVM) and adjacent channel power ration (ACPR), respectively [5].

For the designers of OFDM transmitter systems, the influence of PA envelope character on system modulation fidelity is a fundamental concern, and it attracts lots of studies [6–9]. However, most relevant studies are based on a specific example PA, and the influence of PA envelope variation on system performance is not well explored yet. In recent papers [10, 11], the PA envelopes are characterized by percentage linearization (PL) determined by PA saturation point (SP), and the influence of PL on system modulation fidelity is investigated. Meanwhile, the influence of PA envelope is not investigated yet as both SP and nonlinear onset point (NOP) variations are considered. Thus, a couple of questions can be asked here. What is the influence of PA envelope on modulation fidelity performance of nonlinear OFDM transmitter systems when both NOP and SP variations are considered? Can the simulation observations be explained analytically? This paper presents our efforts to answer these questions.

In this paper, the characteristic of PAs is considered as some kind of degradations from that of the ideal PA, i.e., a fully linearized hard limiter. This degradation is characterized by both NOP and SP defined in [10, 11]. Totally, five solid-state PA (SSPAs) envelopes with various settings of NOP and SP are designed, where the AM/PM conversion of these PA envelopes are neglected as in [10, 12], and these example PAs are modeled by Bessel-Fourier (BF) behavioral model [13, 14]. A 16QAM-modulated IEEE 802.11a OFDM signal is driven through these designed PAs. The intermodulation distortion (IMD) of output signal of example PAs is measured by EVM and ACPR, and the obtained EVM and ACPR results are displayed over input backoff (IBO) and output backoff (OBO) domains, respectively. This allows us to see how the variations of PA envelopes are represented through their associated EVM/ACPR results. An interesting outcome is that when these example PAs are driven close to their saturation ranges, PAs having relatively higher nonlinearity show relatively lower distortion level over IBO domain. On the contrary, PAs having relatively higher nonlinearity show relatively higher distortion level over the whole OBO domain. Another interesting outcome observed from the comparative experiment is that NOP is not much concerned in the characterization of PA envelope in current studies, but the system performance actually is more sensitive to the variation of NOP than to that of SP. These simulation observations are analytically discussed by using a recent mixed time-domain and statistical analysis approach of IMD [15, 16]. It can be seen that when input power increases, the weights of high-order terms in PA Bessel-Fourier model become comparable with those of dominated low-order terms. Therefore, the IMD of a PA envelope becomes more sensitive to the input power as it has more high-order model terms. This finding explains the simulation observations.

Section 2 presents the designed example PA envelopes. Section 3 presents the BF PA envelope model and the mixed time-domain and statistical analysis approach of IMD. The simulation observations and associated discussions are presented in Section 4, and this is followed by the conclusion in Section 5.

2 PA envelopes having various NOPs and SPs

In simulation-based analysis of nonlinear OFDM transmitter systems, the single SSPA transfer characteristic used can vary from device to device. Without comparative analysis of multiple-PA-based simulation experiment, valuable outcomes can be missed. In this paper, several PA envelopes having various envelope characters are designed, and the design scheme is presented below.

As in [13], generally, the input-output map of an amplification transfer system can be represented by the following nonlinear differential equation:

$$F\left[s_{\mathrm{o}}\left(t\right),\frac{d{s}_{\mathrm{o}}\left(t\right)}{\mathit{\text{dt}}},\dots ,\frac{d^p{s}_{\mathrm{o}}\left(t\right)}{d{t}^p},s_{\mathrm{i}}\left(t\right),\frac{d{s}_{\mathrm{i}}\left(t\right)}{\mathit{\text{dt}}},\dots ,\frac{d^r{s}_{\mathrm{i}}\left(t\right)}{d{t}^r}\right]=0$$

(1)

where s_o(t) and s_i(t) are the output and input signals, respectively. In PA transmitters, the fading memory effects of transfer character is related to the input signal only. As in [9, 10, 13, 17, 18], memoryless nonlinearity dominates PA transfer character, and it can be characterized by memoryless PA envelope character, i.e.,

$$s_{\mathrm{o}}\left(t\right)=g\left[A_{\mathrm{e}}\right(t\left)\right]·e^{j\left[\Phi \right(A_{\mathrm{e}}\left(t\right)\left)\right]}·\frac{s_{\mathrm{i}}\left(t\right)}{A_{\mathrm{e}}\left(t\right)}$$

(2)

where g[.] and Φ[.] denote PA AM/AM and AM/PM conversion, respectively; and A_e(t) is the envelope amplitude of input signal. Since AM/PM conversion of SSPAs does not have strong variation [13], Φ[.] of example PAs is set as zero.

As in [10, 11], PL is developed for the characterization of PA envelope, which depends on the SP variation. In this paper, five PA envelopes are presented with independent SP and NOP variation applied. The concepts of NOP, SP, and the characteristic’s ideal saturation point (CISP) are shown in Figure 1. The five example PA envelopes, i.e., L₀ to L₄, are designed to have various NOP and SP degradation referring to L₀, i.e., their NOPs and SPs depart from CISP. A smooth envelope characteristic is designed to connect NOP and SP when they depart from CISP, and the derivative function of designed PA envelope smoothly decreases from 1 to 0 when the input power sweeps from NOP to SP. In this paper, the distance of NOP and SP to CISP, written as D_NOP-CISP and D_SP-CISP, is measured along the asymptotic ideal linearity character (AILC) line and asymptotic output saturation power (AOSP) line, respectively. Table 1 shows the D_NOP-CISP and D_SP-CISP of all five example PAs, and the percentage linearization area (PL_A), defined in [10, 11], of all designed PAs, is presented in Table 1 as well. The five example PA envelopes are presented in Figure 2, and their input and output are normalized to those of CISP.

Figure 1

The characters of a typical SSPA and related concepts. The definitions of NOP, SP, and CISP are marked.

Table 1 Characters of five example PA envelopes

Figure 2

The envelope characters of five example SSPAs. These five example PAs have various combinations of NOP and SP.

3 IMD analysis in nonlinear OFDM transmitter systems

The input OFDM signal including N subcarriers, denoted as s_i(t) here, can be written as

$$s_{\mathrm{i}}\left(t\right)=\sum _{l=1}^N\left\{A_l\left(t\right)·e^{j[2\pi f_{l}t+{\phi }_l(t\left)\right]}\right\}$$

(3)

where A _l (t)φ _l (t) and f _l denote the amplitude, phase, and frequency of the l th subcarrier. In this paper, Bessel-Fourier behavioral model is selected for the example PA envelopes for its unique advantage when dealing with OFDM-like multi-carrier signals [13–15]. Using Bessel-Fourier model, the zonal band PA output signal, s_o(t), can be written as

$$\begin{array}{c}{s}_{\mathrm{o}}\left(t\right)=\sum _{k=1}^L{b}_k\sum _{\begin{array}{c}{n}_1,\dots n_N=-\infty \\ \sum n_l=1\end{array}}^{\infty }\hfill \\ \times \left\{\underset{l=1}{\overset{N}{\Pi }}\left[J_{n_l}\left(\frac{2\pi }{D_{\text{mod}}}k{A}_l\left(t\right)\right)\right]·e^{j\sum _{l=1}^N{n}_l[2\pi f_{l}t+{\phi }_l(t\left)\right]}\right\}\hfill \end{array}$$

(4)

where D_mod is the model dynamic range [18]; $J_{n_l}$ is the $n_l^{\text{th}}$ order Bessel function of the first kind; b _k is the k th coefficient of an L th order Bessel-Fourier model. As in [13, 14], s_o(t) includes fundamental carriers and zonal band intermodulation products (IMPs), and each of them maps to a unique realization of the parameter set [ n₁, n₂,…, n _N ]. The fundamental carriers IMPs can be distinguished by the condition of$\sum \left|n_l\right|=\lambda$, where λ=1 indicates fundamental carriers and other values of λ(λ≥3) indicates the λ th order IMPs.

Equation 4 requires an unacceptable computation cost in IMD simulation, and it can be simplified. As derived in [15, 16], the power level of individual fundamental carrier or IMP mapping to [ n₁,…, n _N ], denoted as $P_{[n_1,\dots n_N]}$, can be written as

$$P_{[n_1,\dots n_N]}\left({\sigma }^2\right)={\left|\sum _{k=1}^K\left\{b_k·\underset{l=1}{\overset{N}{\Pi }}{J}_{n_l}\left(\frac{2\mathrm{\pi k}}{D_{\text{mod}}}\sqrt{\frac{{\sigma }^2}{N}}\right)\right\}\right|}^2$$

(5)

where σ² denotes the power of input OFDM signal. Thus, as in [15], the power spectra of fundamental carriers and IMD, denoted as PS_fund(f,σ²) and PS_IMD(f,σ²), can be written as

$$\begin{array}{c}{\text{PS}}_{\text{fund}}(f,{\sigma }^2)=\left(\sum _{l=1}^N\delta \left(f-f_l\right)\right)·\left|\sum _{k=1}^L{b}_k·J_1\left(\frac{2\mathrm{\pi k}}{D_{\text{mod}}}\sqrt{\frac{{\sigma }^2}{N}}\right)\right.\hfill \\ {\left.\times {\left[J_0\left(\frac{2\mathrm{\pi k}}{D_{\text{mod}}}\sqrt{\frac{{\sigma }^2}{N}}\right)\right]}^{N-1}\right|}^2\hfill \end{array}$$

(6)

and

$$\begin{array}{c}{\text{PS}}_{\text{IMD}}\left(f,{\sigma }^2\right)={\text{DDF}}_3\left(f\right)·\left|\sum _{k=1}^L{b}_k·{\left[J_1\left(\frac{2\mathrm{\pi k}}{D_{\text{mod}}}\sqrt{\frac{{\sigma }^2}{N}}\right)\right]}^3\right.\hfill \\ \times {\left.{\left[J_0\left(\frac{2\mathrm{\pi k}}{D_{\text{mod}}}\sqrt{\frac{{\sigma }^2}{N}}\right)\right]}^{N-3}\right|}^2\hfill \end{array}$$

(7)

where f denotes the frequency; δ[.] is the Dirac function; DDF₃(f) denotes the distribution density function of third-order IMPs as defined in [15], and it can be easily counted according to the number of subcarriers. Here, IMD is represented by the dominated third-order IMPs, and higher-order IMPs are omitted.

The relation between D_mod and σ² is discussed in [18]. As derived in [16], we can have that

$$\begin{array}{c}\frac{2\pi }{D_{\text{mod}}}\sqrt{\frac{{\sigma }^2}{N}}<0.05\text{and}1\approx J_0\left(\frac{2\mathrm{\pi k}}{D_{\text{mod}}}\sqrt{\frac{{\sigma }^2}{N}}\right)\hfill \\ >J_1\left(\frac{2\mathrm{\pi k}}{D_{\text{mod}}}\sqrt{\frac{{\sigma }^2}{N}}\right)>0\hfill \end{array}$$

(8)

Applying (8), (6) and (7) can be re-written as

$$\begin{array}{c}{\text{PS}}_{\text{fund}}\left(f,{\sigma }^2\right)=\left(\sum _{l=1}^N\delta \left(f-f_l\right)\right)·{\left|\sum _{k=1}^L{b}_k·J_1\left(\frac{2\mathrm{\pi k}}{D_{\text{mod}}}\sqrt{\frac{{\sigma }^2}{N}}\right)\right|}^2\end{array}$$

(9)

and

$$\begin{array}{c}{\text{PS}}_{\text{IMD}}\left(f,{\sigma }^2\right)={\text{DDF}}_3\left(f\right)·{\left|\sum _{k=1}^L{b}_k·{\left[J_1\left(\frac{2\mathrm{\pi k}}{D_{\text{mod}}}\sqrt{\frac{{\sigma }^2}{N}}\right)\right]}^3\right|}^2\end{array}$$

(10)

Thus, the signal-to-noise ratio (SNR) of nonlinearly amplified OFDM signal, R_non(σ²), can be written as

$$\begin{array}{c}{R}_{\text{non}}\left({\sigma }^2\right)=\frac{\underset{\text{ch}}{\int }{\text{PS}}_{\text{fund}}\left(f,{\sigma }^2\right)·\mathit{\text{df}}}{\underset{\text{ch}}{\int }{\text{PS}}_{\text{IMD}}\left(f,{\sigma }^2\right)·\mathit{\text{df}}}\hfill \\ =\frac{N·{\left|\sum _{k=1}^L{b}_k·J_1\left(\frac{2\mathrm{\pi k}}{D_{\text{mod}}}\sqrt{\frac{{\sigma }^2}{N}}\right)\right|}^2}{\left[\underset{\mathit{\text{ch}}}{\int }{\text{DDF}}_3\left(f\right)·\mathit{\text{df}}\right]\times {\left|\sum _{k=1}^L{b}_k·{\left[J_1\left(\frac{2\mathrm{\pi k}}{D_{\text{mod}}}\sqrt{\frac{{\sigma }^2}{N}}\right)\right]}^3\right|}^2}\hfill \end{array}$$

(11)

where ch denotes the signal band of OFDM fundamental carriers. Based on the range estimation of Bessel series kernel in (7), i.e., $\pi \sqrt{{\sigma }^2}/\left(D_{\text{mod}}\sqrt{N}\right)$, it can be derived that

$$\partial \left\{{\left[J_1\left(\frac{2\mathrm{\pi k}}{D_{\text{mod}}}\sqrt{\frac{{\sigma }^2}{N}}\right)\right]}^3\right\}/\left(\partial {\sigma }^2\mathrm{\partial k}\right)>0$$

(12)

This indicates that the power of IMD increased as input power as σ² increases, and the power of high-order IMD components are more sensitive to the variation of σ² than that of the low-order ones.

4 Simulation observations and discussions

In this section, the modulation fidelity of an IEEE 802.11a OFDM signal driving through five example SSPAs are presented. The EVM and ACPR are two specific measures of system SNR, which focuses on the in-band distortion and out-band spectrum regrowth, respectively. The EVM and ACPR results shown in this section are simulated based on the classical statistical (Stat) approach in [14]. The simulation results reveal how the PA nonlinearity is represented by the signal impairment over IBO and OBO domains, respectively, and the associated simulation observations are discussed based on above analysis.

In Figure 3, the simulated EVM and ACPR results of five example PAs are presented. It can be seen that when the input power is around linear or quasi-linear ranges (e.g., around 5 dB), the order of nonlinear impairment of all five example PAs (from high to low) is L₀, L₁, L₂, L₃, and L₄. However, when the input power is close to saturation range (e.g., around -4 dB), the former order changes to L₄, L₃, L₂, L₁, and L₀. Referring to Table 1, here we can notice that NOP has a very clear influence on system modulation fidelity and that the PAs have better linearity generate higher IMD as they are driven close to saturation ranges.

Figure 3

Simulation results of five example PAs. A 16QAM IEEE 802.11a signal is applied, and the ACPR graphs at 11 MHz frequency offset from center of the signal band of fundamental carriers.

In Figure 4, the constellation diagrams of amplified OFDM signals driven through L₀ and L₄ are presented, which are driven at operating points of IBO 5 dB and -3 dB, respectively. It can be seen that, from the constellation diagrams, L₄ generates higher IMD than L₀ does within low input power range, while L₀ generates higher IMD than L₄ does within high input power range. This observation is consistent with that of Figure 3.

Figure 4

The constellation diagrams of L ₀ and L ₄ . An 16QAM IEEE 802.11a OFDM signal is applied. (a,b) Output diagrams of L₀ at operating points of IBO 6 and -2 dB, respectively. (c,d) L₄ as the same configuration applied.

In Figure 5, the power spectrum graphs of all example PAs at IBO 6 and -2 dB are presented. It can be seen that L₀ generates the lowest spectra regrowth than others do within low-input power range, while it generates the highest spectra regrowth within high-input power range. This observation is consistent with Figures 3 and 4, and it can be explained as follows. Since NOP has the clear influence on the convergence of coefficient spectrum of Bessel-Fourier PA model, and as shown in Figure 6 below, the weight of high-order model terms clearly increases as D_NOP-CISP decreases. For example, the Bessel-Fourier PA model of L₀ has more high-order terms than that of L₄ does, since L₀ has a smaller D_NOP-CISP than L₄ does. Furthermore, As analyzed around (12), the power of high-order IMD components is more sensitive to the variation of σ² than that of the low-order ones. That is, as σ² increases, the high-order model components show clear weights in total IMD. Therefore, IMD generated by L₀ is higher than that generated by L₄ as operating point is close to saturation points.

Figure 5

The power spectra of amplified OFDM signals. All example PAs are investigated as a 16QAM IEEE 802.11a signal applied. (a,b) Operation point of IBO 6 and -2 dB, respectively.

Figure 6

The weights of Bessel-Fourier model terms of example PA envelopes. The weights of model terms are normalized, and the high-order terms of L₀ is more significant than those of L₄.

In Figure 6, the weights of Bessel-Fourier model terms of L₀ and L₄ example PAs are presented. It can be seen that as analyzed above, the weights of high-order terms of L₀ PA are higher than those of L₄. This is because L₀ has a relatively sharp variation at CISP. Thus, as observed in above figures, the modulation fidelity of L₀ shows more significant degradation than L₄ does when input power increased.

In Figure 7, the EVM and ACPR results are displayed over OBO domain. It can be seen here that the PAs having relatively higher linearity, such as L₀, show better modulation fidelity over the whole OBO range. This is because that OBO is not sensitive to IBO when PAs are driven close to saturation range.

Figure 7

EVMs and ACPR results of all example PAs. The results are displayed over OBO domain, and a 16QAM IEEE 802.11a signal is applied.

5 Conclusions

In this paper, a comparison analysis is presented to investigate the system performance when both SP and nonlinearity onset point (NOP) variations are considered. The simulations show that how the PA envelope variation can be represented by the signal impairment over IBO and OBO domains, respectively. It is interesting to find that although SP attracts much concern in current studies, NOP actually has the clear influence on modulation fidelity of nonlinear OFDM transmitter systems. Moreover, it is observed when input power gets close to saturation range, the PAs having relatively higher nonlinearity show relatively lower distortion level over the input power domain. These interesting simulation observations above are analytically explained based on a mixed time domain and statistical analysis of the IMD mechanism.

Authors’ information

YL, MD, and YJ are from State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Electronic Engineering and Computer Science, Peking University, Haidian District, Beijing, China.