Complexity-reduced digital predistortion for subcarrier multiplexed radio over fiber systems transmitting sparse multi-band RF signals

Yinqing Pei; Kun Xu; Jianqiang Li; Anxu Zhang; Yitang Dai; Yuefeng Ji; Jintong Lin

doi:10.1364/OE.21.003708

1. Introduction

The wireless services have become increasingly diverse over the past decade. As the 2nd Generation network is still providing qualified mobile voice services to the low-end users and the 3rd generation (3G) network is under massive deployment all over the world, the 4th generation (4G) network, which is able to support much higher speed wireless access, has already been expected. In parallel, wireless local area network (WLAN) carries the major portions of internet traffic and has rapidly ingrained in our daily lives [1,2]. Deploying these multiple wireless services by separated network infrastructures will cause a great amount of capital expenditure (CAPEX) and operation expenditure (OPEX). To address this issue, subcarrier multiplexed radio-over-fiber (SCM-RoF) system is the most attractive solution. The SCM technique feeds the multiple wireless services with only one analog optical fiber link by reusing the subcarriers of one optical carrier, which enables full utilization of the network infrastructure as well as the broad bandwidth of the optical fiber [3,4].

One of the main limitations of the SCM-RoF system is the intermodulation distortion (IMD) introduced by the inherent nonlinearity of electrical to optical (E/O) conversion. Both photonic and electronic methods have been employed to improve the linearity of the RoF systems [4–6]. Of all the techniques, digital linearization, i.e. digital predistortion, is among the most cost effective because of its high flexibility, low cost, and high precision [7–9]. However, the maximum processing capacity of bandwidth of the existing DPD techniques is less than 100MHz limited by the sample rate of the digital to analog convertors (DAC). This bandwidth is not sufficient to compensate for the nonlinear distortions of these wireless services in multiple different RF bands which may span from 400MHz to up to 6GHz [10,11]. Consequently, the traditional DPD technique meets severe challenges. In fact, the combined multi-band RF signal can be considered as a sparse signal with much blank spectra between the constituent RF bands. This provides the possibility of simplification by disregarding the blank spectra.

In this paper, we proposed a novel simplified multi-band DPD technique for SCM-RoF systems. Instead of processing the combined multi-band RF signal as a single entity, our proposed DPD is performed on the baseband signal of each individual RF band before frequency up-conversion and RF combination. The compensation function is synthesized for each RF band by involving the nonlinear impacts and memory effects from all existing RF bands. Therefore, the digital signal processing speed and the DAC bandwidth are only determined by the baseband signal bandwidth of each individual RF band, rather than the entire bandwidth of the combined multi-band RF signal. As a result, the requirement of hardware bandwidth is greatly reduced. In this paper, experimental demonstration was performed in a two-band directly modulated SCM-RoF system transmitting two channels of 64QAM modulated OFDM signals on the WLAN band (2.4GHz) and the LTE band (3.6GHz). The results show that the adjacent channel powers (ACP) of the signals on the two bands are suppressed by up to 15dB and their EVM performances are significantly improved.

2. Operation principle of the multi-band DPD technique for SCM-RoF system

The block diagram of the multi-band DPD technique is shown in Fig. 1 . The DPD acts on the complex baseband signal of each band, X_i(n) (i = 1...I), before up-conversion and RF combination. The baseband expressions of the outputs of the RoF system are denoted by Y_i(n). The objective of the proposed technique is to obtain a inversed model of the RoF system at the multi-band DPD block to yield an overall linear output, i.e. Y_i(n) = g_iX_i(n), where g_i is the small signal gain of each multiplexed band applied to the RoF system. Instead of identifying the nonlinear characteristics of the RoF system, we directly train the predistorter in the RoF feedback path, which has Y_i(n)/g_i as its inputs and ${\hat{Z}}_{i} (n)$ as outputs. The coefficients of the predistorter training block are converged when ${| e_{i} (n) |}^{2} = {| {\hat{Z}}_{i} (n) - Z_{i} (n) |}^{2}$ is minimized, and the coefficients of the multi-band DPD block are an exact copy of the training block.

Fig. 1 Block diagram of the multi-band predistortion technique for SCM-RoF system.

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The compensation function of the predistorter in the conventional DPD technique is based on the memory polynomial [12],

z (n) = \sum_{q} \sum_{p} K_{p q} x (n - q) {| x (n - q) |}^{p - 1}

Where K_pq is the normalized coefficient of the p^th order nonlinearity and q^th order of memory length, x(n) and z(n) are the complex baseband input and output of the predistorter. In linearizing the SCM-RoF system transmitting sparse multi-band RF signal, we disregard the blank spectra between the multiplexed RF bands and pre-distort the signal on each band separately. Therefore, the compensation function synthesized for each band should consider IMDs from all frequency components, which is relatived with the complex amplitudes and carrier frequencies of signals on all the multiplexed bands. The compensation model of IMDs can be derived from the inversed continuous-time model of the RoF system written by

v_{o u t} = \sum_{p} K_{p} {(v_{i n} / g)}^{p}

v_in and v_out are the instantaneous voltages of the input and output signals of the DPD module, g is the linear gain and K_p is the normalized coefficient of the p-th order nonlinearity. Note that, different from Eq. (1), which describes the sampled baseband signal characteristics, Eq. (2) describes the continuous-time signals, such as the multi-band signal with multiple channels carried by different RF carriers. In this case, the input multi-band signal with I channels between the (n-1)^th and n^th sample period can be written by

v_{i n} (n) {= \sum_{i = 1}^{I} X_{i} (n) \cos (2 π f_{i} t + φ_{i}) |}_{(n - 1) Δ t < t < n Δ t}

Where

Δ t

is the sampling interval,

f_{i}

and

φ_{i}

are the carrier frequency and initial phase of the i^th band. Applying Eqs. (1) and (3) to (2), v_out(n) can be derived as

v_{o u t} (n) = \sum_{i = 1}^{I} {[\sum_{q} \sum_{p} K_{p q} \frac{X_{i} (n - q)}{| X_{i} (n - q) |} \frac{1}{2^{p - 1}} \sum_{f_{i} = \sum_{j = 1}^{p} \pm f_{j}} \prod_{j = 1}^{p} | X_{j} (n - q) |] \cos (2 π f_{i} t + φ_{i})}

where K_pq in Eq. (1) replaces K_p in Eq. (2) to involve the memory effects. The IMD products appeared in Eq. (4) by the items with p>1 and non-identical

f_{j}

(j = 1...p). Then, the inversed baseband model of the RoF system for the i^th (i = 1...I) channel is obtained by

Z_{i} (n) = \sum_{q} \sum_{p} K_{p q} \frac{X_{i} (n - q)}{| X_{i} (n - q) |} \frac{1}{2^{p - 1}} \sum_{f_{i} = \sum_{j = 1}^{p} \pm f_{j}} \prod_{j = 1}^{p} | X_{j} (n - q) |

which is exactly the compensation function for the i^th channel in the multi-band DPD block

3. Coefficients extraction

It is found from Eq. (5) that $Z_{i} (n)$ is linear in the parameters K_pq, thus the latter can be estimated by a simple least mean squares (LMS) method. Since the output Y_i = g_iX_i, is expected, a normalized matrix, $U_{i} = [u_{i}^{10 (1)}, ..., u_{i}^{1 Q (1)}, \frac{1}{2^{2}} u_{i}^{30 (1)}, ..., \frac{1}{2^{2}} u_{i}^{30 (M)}, \frac{1}{2^{2}} u_{i}^{3 Q (1)}, ..., \frac{1}{2^{2}} u_{i}^{3 Q (M)}, ......., \frac{1}{2^{P - 1}} u_{i}^{P Q (L)}],$ is defined to train the predistorter, where $u_{i}^{pq(m)} = {[u_{i}^{p q (m)} (0), ..., u_{i}^{p q (m)} (N - 1)]}^{T},$ and

u_{i}^{p q (m)} (n) = \frac{Y_{i} (n - q)}{| Y_{i} (n - q) |} \prod_{\begin{matrix} j = 1 \\ f_{i} = \sum_{j = 1}^{p} \pm f_{_{j}} \end{matrix}}^{p} | \frac{1}{g_{i}} Y_{j} (n - q) |

the superscript m denotes the m^th permutation of

f_{i} = \sum_{j}^{p} \pm f_{j}

. At convergence, we should have

z_{i} {=U}_{i} k_{i}

where

z_{i} = {[Z_{i} (0), Z_{i} (1), ..., Z_{i} (N - 1)]}^{T},

k_{i} = {[K_{i}^{10 (1)}, .., K_{i}^{1 q (1)}, K_{i}^{30 (1)}, .., K_{i}^{30 (M)}, K_{i}^{3 Q (1)}, .., K_{i}^{3 Q (M)}, ...., ...., K_{i}^{P Q (L)}]}^{T} .

Then the converged solution of

k_{i}

can be obtained by the LMS method,

{\hat{k}}_{i} {=(U}_{i}^{H} U_{i})^{-1} U_{i}^{H} z_{i}

Where (.)^H denotes complex conjugate transpose.

For example, to predistort a two-band multiplexed signals on carriers f₁ and f₂ by considering up to 3rd order nonlinear distortion and 2nd order memory effects, two times of Eq. (8) should be computed to solve ${\hat{k}}_{1}$ and ${\hat{k}}_{2}$ , respectively. For channel-1, i.e. i = 1, when p = 1, there is one permutation of $f_{j}$ (j = 1) $f_{i} = \sum_{j = 1}^{1} \pm f_{j},$ i.e. $f_{j} = f_{1},$ leading to $u_{1}^{1 q (1)} (n) = \frac{Y_{1} (n - q)}{g_{1}};$ when p = 3, there are nine permutations of $f_{j}$ (j = 1..3) of $f_{i} = \sum_{j = 1}^{3} \pm f_{j},$ i.e.① $f_{1}, f_{1}, - f_{1},$ ② $f_{1}, - f_{1}, f_{1},$ ③ $- f_{1}, f_{1}, f_{1},$ ④ $f_{1}, f_{2}, - f_{2},$ ⑤ $f_{1}, - f_{2}, f_{2},$ ⑥ $f_{2}, f_{1}, - f_{2},$ ⑦ $- f_{2}, f_{1}, f_{2},$ ⑧ $- f_{2}, f_{2}, f_{1},$ ⑨ $f_{2}, - f_{2}, f_{1},$ the first three permutations lead to $u_{1}^{3 q (1)} (n), u_{1}^{3 q (2)} (n), u_{1}^{3 q (3)} (n) = \frac{1}{4} \frac{Y_{1} (n - q)}{g_{1}} {[\frac{| Y_{1} (n - q) |}{g_{1}}]}^{2}$ and the rest six lead to $u_{1}^{3 q (4)} (n), ..., u_{1}^{3 q (9)} (n) = \frac{1}{4} \frac{Y_{1} (n - q)}{g_{1}} [\frac{{| Y_{2} (n - q) |}^{2}}{g_{2}^{2}}] .$ Moreover, as the values of the 1st to 3rd and 4th to 9th items are identical, they can be combined into 2 items as $u_{1}^{3 q (1)} (n) = \frac{3}{4} \frac{Y_{1} (n - q)}{g_{1}} {[\frac{| Y_{1} (n - q) |}{g_{1}}]}^{2}$ and $u_{1}^{3 q (2)} (n) = \frac{3}{2} \frac{Y_{1} (n - q)}{g_{1}} {[\frac{| Y_{2} (n - q) |}{g_{2}^{}}]}^{2} .$ Then, $U_{1}$ and ${\hat{k}}_{1}$ are given by

U_{1} = [u_{1}^{10 (1)} {,u}_{1}^{11 (1)} {,u}_{1}^{12 (1)}, u_{1}^{30 (1)}, u_{1}^{31 (1)}, u_{1}^{32 (1)}, u_{1}^{30 (2)}, u_{1}^{31 (2)}, u_{1}^{32 (2)}]

{\hat{k}}_{1} = {[K_{1}^{10 (1)}, K_{1}^{11 (1)}, K_{1}^{12 (1)}, K_{1}^{30 (1)}, K_{1}^{31 (1)}, K_{1}^{32 (1)}, K_{1}^{30 (2)}, K_{1}^{31 (2)}, K_{1}^{32 (2)}]}^{T}

Similarly, $U_{2}$ and ${\hat{k}}_{2}$ can be obtained by

U_{2} = [u_{2}^{10 (1)} {,u}_{2}^{11 (1)} {,u}_{2}^{12 (1)}, u_{2}^{30 (1)}, u_{2}^{31 (1)}, u_{2}^{32 (1)}, u_{2}^{30 (2)}, u_{2}^{31 (2)}, u_{2}^{32 (2)}]

{\hat{k}}_{2} = {[K_{2}^{10 (1)}, K_{2}^{11 (1)}, K_{2}^{12 (1)}, K_{2}^{30 (1)}, K_{2}^{31 (1)}, K_{2}^{32 (1)}, K_{2}^{30 (2)}, K_{2}^{31 (2)}, K_{2}^{32 (2)}]}^{T}

where $u_{2}^{3 q (1)} (n) = \frac{3}{4} \frac{Y_{2} (n - q)}{g_{2}} {[\frac{| Y_{2} (n - q) |}{g_{2}}]}^{2}$ and $u_{2}^{3 q (2)} (n) = \frac{3}{2} \frac{Y_{2} (n - q)}{g_{2}} {[\frac{| Y_{1} (n - q) |}{g_{1}^{}}]}^{2}$ .

4. Experimental setup and results

An experiment was performed to demonstrate the proposed multi-band DPD technique in a two-band SCM-RoF system based on direct modulation, as it is shown in Fig. 2 . Two 20MHz bandwidth, 64QAM modulated OFDM signals complied with the physical definition of IEEE 802.11g 54Mbps mode were used as the signal sources. Note that the proposed multi-band DPD technique is not sensitive to the signal modulation format and can be applied to any kinds of signals. The sampled baseband signals of the two channels, X₁ and X₂, were processed by the multi-band DPD processing block in Matlab, which yielded the predistorted signals, Z₁ and Z₂. Then, the two baseband signals were fed into two vector signal generators (VSG, Agilent E4438C, N5182A) with sampling rates of 100MSamples/sec for DAC and up-conversion to 2.412GHz and 3.6GHz. The two VSGs were connected by a 10MHz reference signal and a trigger signal from E4438C to N5182A to obtain the exact sample synchronization. After combined by an electrical combiner, the predistorted multi-band signal was applied to the RoF link. In the RoF link, a commercial directly modulated laser diode (MITEQ) with a wavelength of 1550nm and optical output power of 5mW was used to perform the E/O conversion. Then, a photodetector (u²t) with a responsivity of 0.6A/W was used to convert the optical signal to electrical domain. The small signal gains of the RoF link at 2.412GHz and 3.6GHz, g₁ and g₂, were measured to be 0.19 and 0.17, respectively, by a vector network analyzer (VNA, Agilent N5224A). Demultiplexing, downconversion, and ADC of the multi-band RF signals after RoF link were performed by a vector signal analyzer (VSA, Agilent N9030A). Note that as the two VSGs generated certain baseband signal sequences cyclically, the outputs of the RoF link were also cyclical. This enabled us to use one VSA to record the baseband signals of the two bands in different cycles, but regard them as the parallel outputs.

Fig. 2 Experimental setup.

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Then, the predistorter training was performed. Two channels of 20MHz 64QAM-OFDM signals on 2.412GHz and 3.6GHz without predistortion were directly injected to the SCM-RoF system. The coefficients of the predistorter are obtained by applying the inputs, Z₁, Z₂, (which are equals to X₁, X₂ in this step) and outputs, Y₁, Y₂, of the SCM-RoF system to Eq. (8). The LMS algorithm was implemented by 16384 samples (8192 samples for each channel) with considering up to 3rd odd order nonlinearity, and memory length of 2 as it was mentioned in Section 3. According to Eqs. (9)-(12), ${\hat{k}}_{1}$ and ${\hat{k}}_{2}$ have 9 elements respectively and the values were obtained by Eq. (8) as shown in Table 1 .

Table 1. Extracted Coefficients of the Predistorter

View Table

The obtained coefficients were re-used to predistort another two 20MHz, 64QAM modulated OFDM signals on 2.412GHz and 3.6GHz. Figures 3(a) –3(c) show the RF spectrum of the received signal at input power of −5dBm/channel. The RF spectrum of the entire multi-band RF signal, occupying over 1.2GHz bandwidth, is illustrated in Fig. 3(a). To predistort such signal with the traditional DPD technique, the DSP devices with several GHz bandwidth are required, which are not practical in the commercial wireless telecommunications systems. However, using our proposed multi-band DPD technique with DAC/ADC bandwidth at 100MHz, the nonlinearity was well compensated, as it can be seen from Figs. 3(b) and 3(c). The ACP was substantially suppressed by 14dB and 15dB of the signals on the two bands.

Fig. 3 (a) RF spectrum of the entire SCM signal; (b) RF spectrum of the signal on 2.412GHz, (c) RF spectrum of the signal on 3.6GHz. (blue dash line for without DPD, red solid line for with DPD).

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The corresponding EVMs and constellations of the received signals at input powers of −5dBm/channel are analyzed by the VSA. For channel-1 on 2.412GHz, the EVMs are obtained as 6.13% and 2.12% without/with DPD (Figs. 4(a) and 4(b)); and for channel-2 on 3.6GHz, the EVMs are 6.34% and 2.33% without/with DPD (Figs. 4(c) and 4(d)). It can be seen that the constellations of the received signals are much clearer and better separated with the proposed multi-band DPD technique.

Fig. 4 (a) and (b) Constellations of the received signal on 2.412GHz without and with the proposed DPD technique; (c) and (b) constellations of the received signal on 3.6GHz without and with the proposed DPD technique.

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5. Conclusion

We proposed a novel multi-band DPD technique for nonlinearity compensation of the SCM-RoF system transmitting sparse multi-band signal. This technique can greatly reduce the digital processing bandwidth and enables the employment of low-cost narrow-band DSP devices and DACs to linearize the RoF system with multiplexed subcarriers spanning over a wide frequency range. In the experimental demonstration, the proposed multi-band DPD technique successfully compensated the nonlinearity of a two-band directly modulated SCM-RoF system regardless of its entire operation bandwidth over 1.2GHz.

Acknowledgments

This work was supported in part by 863 Program (2011AA010306), National 973 Program (2012CB315705),NSFC Program (61271042, 61107058and 61120106001), Beijing Excellent Doctoral Thesis Project under Grant YB20101001301, the Cooperation Project between Province and Ministries under Grant 2011A090200025, the 111 Project under Grant B07005, and the Fundamental Research Funds for the Central Universities. The authors would like to thank Agilent Beijing open lab for providing the experimental instruments.

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	${\hat{k}}_{1}$			${\hat{k}}_{2}$
$K_{1}^{10 (1)}$ = 0.912 + 0.072i	$K_{1}^{11 (1)}$ = 0.033- 0.024i	$K_{1}^{12 (1)}$ = −0.005 + 0.01i	$K_{2}^{10 (1)}$ = 0.946 + 0.015i	$K_{2}^{11 (1)}$ = 0.052- 0.029i	$K_{2}^{12 (1)}$ = −0.004 + 0.008i
$K_{1}^{30 (1)}$ = 11.12 + 2.12i	$K_{1}^{31 (1)}$ = −1.26 + 0.337i	$K_{1}^{32 (1)}$ = −0.923- 0.148i	$K_{2}^{30 (1)}$ = 7.87 + 1.53i	$K_{2}^{31 (1)}$ = −1.01 + 0.267i	$K_{2}^{32 (1)}$ = −0.738 - 0.112i
$K_{1}^{30 (2)}$ = −9.91-6.04i	$K_{1}^{31 (2)}$ = 1.06 + 0.191i	$K_{1}^{32 (2)}$ = −0.846- 0.262i	$K_{2}^{30 (2)}$ = −11.076 - 6.78i	$K_{2}^{31 (2)}$ = 1.181 + 0.210i	$K_{2}^{32 (2)}$ = −1.02 - 0.328i

Complexity-reduced digital predistortion for subcarrier multiplexed radio over fiber systems transmitting sparse multi-band RF signals

Abstract

1. Introduction

2. Operation principle of the multi-band DPD technique for SCM-RoF system

3. Coefficients extraction

4. Experimental setup and results

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (4)

Tables (1)

Equations (12)

Optics Express