диафрагмированные волноводные фильтры / 042be49e-629d-4e89-947b-7ce2e6ae6808

Abstract—An instrumentation, measurement and postprocessing technique is presented to characterize transmitters by multiple input multiple output (MIMO) Volterra series. The MIMO Volterra series is decomposed as the sum of nonlinear single-variable self-kernels and a multi-variable cross-kernel. These kernels are identified by sample averages of the outputs using inputs of different sample periodicity. This technique is used to study the HW effects in a RF MIMO transmitter composed by input and output coupling filters (cross-talk) sandwiching a non-linear amplification stage. The proposed technique has shown to be useful in identifying the dominant effects in the transmitter structure and it can be used to design behavioral models and compensation techniques.

Index Terms—Amplifiers, behavioral modeling, concurrent, digital predistortion, linearization, MIMO, MIMO Volterra series.

I. INTRODUCTION

TRANSMISSIONS of multiple signals is one of the key enablers to overcome the challenges of increased data rate in wireless networks. These multiple signals are channelized

either through different frequency spectrum as in concurrent transmitters or in multiple antenna paths as in MIMO transmitters. Towards the deployment of compact multi-channel transmitters, the multiple signals are channelized within the same hardware (HW) causing RF leakage between them. Additionally, efficient amplification of wireless transmitters exhibit nonlinear effects. These two HW detrimental effects limit the system capacity and in consequence its characterization is of importance.

The characterization of the HW effects enables first to improve the transmitter design and secondly it paves the way to digital compensation techniques combating specific

Manuscript received August 24, 2015; revised November 16, 2015; accepted December 10, 2015. Date of publication March 25, 2016; date of current version April 6, 2016.

E. Zenteno was with the Department of Electronics, Mathematics, and Natural Sciences, the University of Gävle, Gävle 801 76, Sweden and was also with the Department of Signal Processing, Royal Institute of Technology KTH, Stockholm 114 28, Sweden, and is now with the Universidad Catolica San Pablo, Arequipa 01, Peru (e-mail: ezenteno@ucsp.edu.pe).

Z. A. Khan is with the Department of Electronics, Mathematics, and Natural Sciences, the University of Gävle, Gävle 801 76, Sweden, and also with the Department of Signal Processing, Royal Institute of Technology KTH, Stockholm 114 28, Sweden (e-mail: zanahn@hig.se).

M. Isaksson is with the Department of Electronics, Mathematics, and Natural Sciences, the University of Gävle, Gävle 801 76, Sweden (e-mail: min@hig.se).

P. Händel is with the Department of Signal Processing, Royal Institute of Technology KTH, Stockholm 114 28, Sweden (e-mail: ph@kth.se).

Color versions of one or more of the figures in this letter are available online at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/LMWC.2016.2525019

HW impairments improving the error performance. This letter is motivated by the fact that, the HW impairments can be discovered resourcing to behavioral modeling using the MIMO Volterra series [1]. However, this approach suffers from numerical stability and large computational complexity due to increased number of basis functions and requires testing different set of model parameters as nonlinearity orders and memory depths. So the question is how to retain the modeling properties of the MIMO Volterra series to discover HW effects at a fraction of its computational cost without resourcing to extensive testing. The answer is described in this letter using recent signal processing tools.

The letter proposes a method using input signals with different periodicity and can be used for multiple input single output (MISO) and MIMO systems such as concurrent and MIMO transmitters, respectively. The output of these systems can be decomposed as selfand cross-interactions of its inputs. With a careful choice of the set of input periodicities the singlevariable self-kernels and the multiple-variable cross-kernel can be identified yielding information about the HW effects of the system.

For clarity, for the rest of the letter a MIMO transmitter system is used. The i-th complex-valued output yi(n) (the baseband equivalent) in a K × K MIMO transmitter (in Fig. 1) modeled by the MIMO Volterra series [1] can be rewriten as

K

yi(n) = Hk,is [xk(n)] + Hic [x1(n), . . . , xK (n)] . (1)

k=1

That is, each output is composed by the sum of K singlevariable Volterra functions Hk,is [·], called self-kernels, and a K-variable function Hic[·] called cross-kernel. (1) Follows by an algebraic manipulation of the standard Volterra model. The

self-kernels Hk,is [xk] model the effect of the xk input on the i-th output which includes linear and nonlinear dynamic effects

associated with xk. The cross-kernel Hic[·] models the multiplicative interaction of the inputs at the i-th output. That is, it includes nonlinear dynamic effects.

Consider Tk samples as the k-th input signal periodicity. That is, xk (n − mTk) = xk (n), m Z. xk (n) is referred to as Tk-periodic. Note that, the output of a self-kernel has the sam-

ple period of its input signal. That is, for gk,is (n) = Hk,is [xk (n)], for m integer gk,is (n − mTk) = gk,is (n). Thus, the k-th selfkernel is also Tk-periodic. In fact, SISO Volterra systems obey

this property. Then, with inputs of different periodicity, each self-kernel has a distinct periodicity.

Fig. 1. A K × K MIMO transmitter with input and output coupling crosstalks and its measurement setup.

The sample period of the cross-kernel functions Hic[·] depends also on the inputs. For independent realistic communication signals with noise-like statistical properties, its sample period is the least common multiple of the set of input periods, denoted lcm({Tk}Kk=1). Hence, the outputs of the MIMO Volterra series are composed of signals with different periodicity. This work exploits this observation to decompose the MIMO Volterra series cf. (1).

II. INTEGER PERIODICITY

An intrinsic-integer periodic (IIP) decomposition can be used to uniquely separate a T -periodic signal y(n) into the sum of

d is an integer value proper divisor of T . E.g., with T = 10, d = {1, 2, 5, 10}. So, y(n) is decomposed with 4 periodic signals, that is, y(n) = v(1)(n) + v(2)(n) + v(5)(n) + v(10)(n). The decomposition (2) is given by the Ramanujan-sum-expansion obtained as a T -point circular convolution [3]

gcd(p,d)=1

y(n) of the Volterra series has a sample periodicity of T =

That is, the output y(n) is decomposed by its mean v(1)(n), K signals each with the period of an input signal (Tk-periodic) and one T -periodic signal. Note that this is the same periodicity encountered in the selfand cross-kernel in (1). Hence, it is reasonable to estimate the selfand cross-kernel by

Using distinct prime numbers in the set {Tk}K=1, the decomposition in (2) applied individually to each one of the terms in (1) shows that the self-kernels are perfectly separated from each other. This can be seen as its decomposed IIP signals are of different periodicity and hence orthogonal to each other [cf. (5)]. However, the cross-kernel has IIP components of the same periodicity of the self-kernels. Thus, they will distort the self-kernels. However, realistic amplifiers have a relatively low level for the cross-kernel compared to self-kernels. Hence, this effect is not dominating. Furthermore, the cross-kernels have an IIP T -periodic component that is not present in self-kernels and motivates the estimates in (7) and (8).

The Ramanujan-sum in (4) for a prime Tk reduces to [3]

Using (9) into (3) the k-th self-kernel can be estimated directly using coherent averaging over the period Tk. Averaging has been used in RF instrumentation to enhance dynamic range [4]. Once the self-kernels were estimated, the cross-kernel can be obtained by subtraction. This brings a computationally simpler method, (for n = 1, . . . , N)

Where, in (4) the integer sequence cd(n) is the d-periodic Ramanujan sum and gcd is the greatest common divisor. IIP signals v(d)(n) observe orthogonality. That is, for d1 = d2 [2]

lcm(d1,d2)

n=1

Using the set of sample periodicity of the inputs {Tk}Kk=1 to be distinct prime numbers the decomposition in (2) applied to the MIMO Volterra series in (1) separates the signals of different periodicity. With this “prime” condition, the output

The separated kernels can be used for studying the properties of the MIMO transmitters giving valuable feed-back about its HW imperfections and to develop behavioral models and compensation techniques. In particular, the self-kernels are singlevariable nonlinear functions for which parameter efficient models are mature [5]. This allows to devise models that are efficient in computational complexity and have good error performance. The cross-kernel Hic[·] is a multi-variable function. However, using the proposed method it can be studied independently from the self kernels analyzing its non-linearity order and dynamic contribution.

A 2 × 2 MIMO transmitter is used to evaluate the technique. The measurement setup used two SMBV100A vector signal generators (VSG) from Rohde & Schwarz and 2 acquisition paths sampled with an ADC ADQ214 from SP devices (cf. Fig. 1). The MIMO transmitter is built using 2 mini-circuits ZHL-42 amplifiers and dual couplers to mimic the effect of the cross-talks. Each cross-talk has a measured coupling factor of 18 dB between channels.

Three scenarios for the transmitter are considered: I) has only input cross-talk, II) has only output cross-talk and III) with both input and output cross-talk. For cases I) and II) the dual couplers were removed from the output and input respectively of the amplification stage cf. Fig. 1. The excitation signals were two independently generated noise-like signals using (distinct prime numbers) T1 = 2399 and T2 = 2411 samples, respectively. These signals were up converted to 2.14 GHz exciting the power amplifiers operated at 2 dB of output compression. The measurement collects T = T1 × T2 samples.

The mean square (in dB) of the estimated selfand crosskernels of the first channel k = 1 is presented in Table I. The different hardware effects exhibited in each scenario are indicated by the relative change of the mean square value. In particular, the cross-kernel H1c[·] is theoretically zero in the case II [6] which is indicated in Table I up to the measurement noise and observed by its normalized power spectral density in Fig. 2. Note that the self-kernels are estimated through averaging (10) which reduces the noise level [4], while the cross-kernel is obtained by an arithmetic difference (11) which does not reduce the noise. This is depicted in Fig. 2. For the case I a predominately linear self-kernel H2s,1[·] is observed while in cases II and III the same self-kernel has a clear non-linear contribution. Thus, the spectral analysis of the extracted kernels yields the dominant structure in the MIMO transmitter cf. Fig. 2. Finally, for both cases I and III, analysis made on the separated crosskernel H1c[·] show that this is predominately static and hence can be efficiently modeled despite of its higher dimensionality.

Table II compares the proposed technique with other modeling/DPD methods used in MIMO transmitters [7], [8]. Table II indicates which effects are considered in the MIMO structure and its relative complexity. This digital complexity increases when considering both input and output cross-talk effects. However, with this technique, both effects can be included and behavioral models of reduced complexity can be obtained.

IV. CONCLUSION

A technique to decompose the MIMO Volterra series to self and cross kernels is presented. The technique uses input signals

Fig. 2. Power Spectral density of y1 = H1s,1[x1] + H2s,1[x2] + H1c[x1, x2].

(a) Case I: only input cross-talk. (b) Case II: Only output cross-talk.

(c) Case III: Both input output cross-talk.

TABLE II

MODELING/DPD METHODS IN RF MIMO TRANSMITTERS

of different periodicity to decouple its effects at the output using sample averages. The technique was used to determine the dominant structure in a MIMO transmitter and can be exploited designing efficient behavioral models and digital compensation techniques.

REFERENCES

[1]E. Zenteno, S. Amin, M. Isaksson, P. Händel, and D. Rönnow, “Combating the dimensionality of nonlinear MIMO amplifier predistortion by basis pursuit,” in Proc. Eur. Microw. Conf. (EuMC), Oct. 2014, pp. 833–836.

[2]S.-C. Pei and K.-S. Lu, “Intrinsic integer-periodic functions for discrete periodicity detection,” IEEE Signal Process. Lett., vol. 22, no. 8,

pp.1108–1112, Aug. 2015.

[3]P. Vaidyanathan, “Ramanujan sums in the context of signal processing part I: Fundamentals,” IEEE Trans. Signal Process., vol. 62, no. 16,

pp.4145–4157, Aug. 2014.

[4]D. Wisell, D. Rönnow, and P. Händel, “A technique to extend the bandwidth of an RF power amplifier test bed,” IEEE Trans. Instrum. Meas., vol. 56, no. 4, pp. 1488–1494, Aug. 2007.

[5]A. Zhu, J. Pedro, and T. Brazil, “Dynamic deviation reduction-based volterra behavioral modeling of RF power amplifiers,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 12, pp. 4323–4332, Dec. 2006.

[6]S. Amin, P. Landin, P. Händel, and D. Rönnow, “Behavioral modeling and linearization of crosstalk and memory effects in RF MIMO transmitters,” IEEE Trans. Microw. Theory Tech., vol. 62, no. 4, pp. 810–823, Apr. 2014.

[7]S. Bassam, M. Helaoui, and F. Ghannouchi, “Crossover digital predistorter for the compensation of crosstalk and nonlinearity in MIMO transmitters,” IEEE Trans. Microw. Theory Tech., vol. 57, no. 5, pp. 1119–1128, May 2009.

[8]P. Suryasarman and A. Springer, “A comparative analysis of adaptive digital predistortion algorithms for multiple antenna transmitters,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 62, no. 5, pp. 1412–1420, May 2015.