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MeasurementandAnalysisofFrequency-DomainVolterraKernelsofNonlinearDynamic3x3MIMOSystems

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DOI:10.1109/TIM.2017.2664482

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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT 1

Measurement and Analysis of Frequency-DomainVolterra Kernels of Nonlinear Dynamic

3 × 3 MIMO SystemsMahmoud Alizadeh, Student Member, IEEE, Shoaib Amin, Student Member, IEEE,

and Daniel Rönnow, Senior Member, IEEE

Abstract— Multiple-input multiple-output (MIMO) frequency-domain Volterra kernels of nonlinear order 3 are experimentallydetermined in bandwidth-limited frequency regions. How theeffect of higher nonlinear orders can be reduced and how thisaffects the estimated errors are discussed. The magnitude andthe phase of the kernels are Kramers–Kronig consistent. Theself-kernels and cross-kernels have different symmetries, andthe kernels are therefore determined and analyzed in differentregions in the 3-D frequency space. By analyzing the propertiesalong certain paths in the 3-D frequency space, the blockstructures for the respective kernels are determined. Theseblock structures contain the significant blocks of the general blockstructures for the third-order kernels. The device under test isan MIMO transmitter for radio frequency signals.

Index Terms— Amplifier distortion, concurrent dual-band,cross modulation, digital predistortion, intermodulation,multiple-input multiple-output (MIMO), power amplifier.

I. INTRODUCTION

THE Volterra theory for nonlinear dynamic single-inputsingle-output (SISO) systems was formulated in the

late 19th century by Volterra [1] and further developed byWiener [2] in the 1958s and Schetzen [3] in the 1980s.A Volterra series can be used for modeling the behavior of anytime-invariant nonlinear dynamic causal system with fadingmemory [4]. It suffers from slow convergence and is often poorfor modeling strong nonlinearities [5]. The Volterra theoryhas been applied in recent decades for modeling the complexbehavior of (SISO) nonlinear dynamic systems [6], such aselectronic devices [1], [7]–[9], electromechanical systems [10],and communication systems [11]–[13].

There are several methods for identifying the time-domainVolterra kernels from experimental data [6], [14] as well asmethods for determining the SISO frequency-domain Volterra

Manuscript received July 12, 2016; revised September 28, 2016; acceptedNovember 12, 2016. The Associate Editor coordinating the review processwas Dr. Amitava Chatterjee.

M. Alizadeh and S. Amin are with the Department of Electronics,Mathematics and Natural Sciences, University of Gävle, SE-80176 Gävle,Sweden, and also with the Department of Signal Processing, KTH RoyalInstitute of Technology, SE-10044 Stockholm, Sweden (e-mail:mahali@kth.se; shoaib.amin@hig.se).

D. Rönnow is with the Department of Electronics, Mathematics andNatural Sciences, University of Gävle, SE-80176 Gävle, Sweden (e-mail:daniel.ronnow@hig.se).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIM.2017.2664482

kernels (also denoted generalized frequency response func-tions) [15]–[21]. Parametric models can be determined usingfrequency-domain techniques [22]. The structural informationof the block structure of the system can be obtained if thesymmetry properties of the frequency-domain Volterra kernelsare analyzed [23], [24].

Although much attention has been given to the Volterratheory for SISO systems, relatively little attention has beengiven to the Volterra theory for nonlinear dynamic multiple-input multiple-output (MIMO) systems. The Volterra theoryfor the latter system type has been formulated in the lastdecades [7], [25]–[27]. In addition to self-kernels, the prop-erties of which are the same as those of SISO systems.MIMO Volterra series also include cross-kernels [26]–[28]that are due to the nonlinear interaction of multiple inputsignals. The number of kernels of an MIMO system becomessignificantly higher than that of an SISO system. The cross-kernels also have lower symmetry than self-kernels.

The methods for determining the frequency-domain Volterrakernels of some block structures were reported in [25]. Thegeneral methods based on multitoned signal were reportedin [26], [27], and [29]. In [30], it was reported how thesystem parameters of an MIMO system could be determinedfrom the frequency-domain Volterra kernels. The determi-nation of MIMO frequency-domain Volterra kernels fromexperimental data has, to our knowledge, not been reported.In [31], the characterization of nonlinear multiple-inputsingle-output (MISO) system in time domain using concurrentdual-band two-tone signals was reported. A two-tone test isa finger print method and has the drawback that it does notexcite the third-order nonlinearities completely and it does notseparate the effects of kernels of different nonlinear orders.To excite third-order nonlinearity completely, a signal ofthree or more tones is required [23].

Radio frequency (RF) amplifiers are in many cases modeledby baseband (low-pass) equivalent discrete time behavioralmodels [32], [33] that are used to model the behavior in apassband around the center frequency; the model parametersand the input and output signals are complex valued. Themodels are full Volterra models or, more commonly, reducedversions that correspond to specific block structures. The blockstructures determined Volterra kernels could be compared withbaseband models for MIMO transmitters with input crosstalkand two input signals [34], [35].

0018-9456 © 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

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2 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT

In this paper, we describe a method for determiningthe third-order frequency-domain Volterra kernels of anMIMO system. The input channels are excited by combina-tions of one-, two-, and three-tone signals, the frequenciesof which are stepped. The determined Volterra kernels areanalyzed along the paths in the 3-D frequency space, and thesymmetry properties are used to determine the dominant blockstructures (Wiener, Hammerstein, or Hammerstein–Wiener).The effect on the third-order kernel from a higher nonlinearorder is compensated for in the method. We use the method tocharacterize RF amplifiers with crosstalk, which is a hardwareimpairment that may occur in MIMO transmitters [34], [35].

This paper is organized as follows. Section II brieflydescribes the time- and frequency-domain Volterra kernels forthe MIMO system followed by the Kramers–Kronig dispersionrelationship [36]. Test signals used in this paper are brieflydiscussed for the identification of self-kernels and cross-kernels. The symmetry properties of Volterra kernels arebriefly discussed. A description of the experimental setup isprovided in Section III. The measurement results are presentedand discussed in Section IV, and the conclusion is givenin Section V.

II. THEORY

We first go through the Volterra theory for MIMO systemsin Section II-A. We then describe the exciting signals usedin the experiments in Section II-B. The methods used fordetermining the kernels from experimental data are outlinedin Section II-C.

A. Volterra Theory of MIMO System

The Volterra theory for MIMO nonlinear dynamic systemsis described in [26] and [27]. It is an extension of thecorresponding theory for SISO systems [3]. An MIMO sys-tem with R input signals and M output signals couldbe described by M parallel MISO systems with R inputsignals and one output signal. We denote the r th inputsignal as u(r)(t) and the output signal of mth subsystemas y(m)(t). The output signals of the mth subsystem is writtenas

y(m)(t) =N∑

n=1

R∑

r1=1

R∑

r2=r1

. . .

R∑

rn=rn−1

y(m:r1,...,rn )n (t) (1)

where N is the maximum nonlinear order and

y(m:r1,...,rn)n (t) =

∫ ∞

−∞. . .

∫ ∞

−∞h(m:r1,...,rn)

n (τ1, . . . , τn)

u(r1)(t − τ1) . . . u(rn )(t − τn)dτ1 . . . dτn

(2)

where h(m:r1,...,rn )n (τ1, . . . , τn) is the nth nonlinear order time-

domain Volterra kernel of subsystem m for the input signalsr1, . . . , rn . If r1 = r2 = . . . = rn = r , the kernel iscalled a self-kernel, otherwise a cross-kernel. We assume thatthe self-kernels are symmetrized in the same way as for anSISO system [3] and that the cross-kernels are averaged [26];hence, r1 ≤ r2 ≤ · · · ≤ rn .

In the frequency-domain, the output signal Y m(ω) of themth subsystem is

Y (m)(ω) =N∑

n=1

R∑

r1=1

R∑

r2=r1

. . .

R∑

rn=rn−1

Y (m:r1,...,rn)n (ω) (3)

Y (m:r1,...,rn )n (ω)

= 1

(2π)n−1

∫ ∞

−∞. . .

∫ ∞

−∞H (m:r1,...,rn)

n (ω1 − μ1, . . . , μn−1)U(r1)(ω − μ1) . . .

U (rn)(μn−1)dμ1 . . . dμn−1 (4)

with

H (m:r1,...,rn )n (ω1, . . . , ωn)

=∫ ∞

−∞. . .

∫ ∞

−∞h(m:r1,...,rn)

n (τ1, . . . , τn)e− j (ω1τ1+...+ωnτn)dτ1 . . . dτn

(5)

being the n-dimensional Fourier transform (FT) of thenth-order Volterra kernel and U (r)(ω) is the FT of the inputsignal u(r)(t).

The system is real valued; hence, for any kernel [3]

H (m:r1,...,rn )n (ω1, . . . , ωn) = H (m:r1,...,rn)

n (−ω1, . . . ,−ωn)∗

(6)

where ∗ denotes the complex conjugate. We study a 3 × 3odd-order system and measure and analyze third-order kernels.We notice that, for a 3×3 system, there are nine linear kernels,nine self-kernels, eighteen 2 × 1 cross-kernels, and three3 × 1 cross-kernels. For the symmetrized third-order self-kernels, we have the symmetry properties

H (m:r,r,r)3 (ω1, ω2, ω3) = H (m:r,r,r)

3 (ω2, ω1, ω3) (7)

and for the other permutations of (ω1, ω2, ω3). For 2 × 1third-order cross-kernels

H (m:r1,r1,r2)3 (ω1, ω2, ω3) = H (m:r1,r1,r2)

3 (ω2, ω1, ω3) (8)

but different for all other permutations of (ω1, ω2, ω3). Forthird-order cross-kernels, when r1 �= r2 �= r3

H (m:r1,r2,r3)3 (ω1, ω2, ω3) �= H (m:r1,r2,r3)

3 (ω2, ω1, ω3) (9)

and all other permutations of (ω1, ω2, ω3), in the general case.In the Appendix, we derive the relations between the realvalued representation in (1)–(5) and a complex representationthat is common for narrowband communication systems.

We use the method in [3] for an SISO system for thesynthesis of a basic third-order Volterra operator from twomultiplications and five linear filters but extend it to cross-kernels. In Fig. 1, a basic third-order system is shown withthree input signals, u(r1), u(r2), and u(r3). Ha,q, . . . , He,q arelinear time-invariant systems and q is an index explained inthe following.

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ALIZADEH et al.: MEASUREMENT AND ANALYSIS OF FREQUENCY-DOMAIN VOLTERRA KERNELS 3

Fig. 1. Basic block structure of a third-order system.

The third-order kernel transform without symmetrizationis [3]

H (m:r1,r2,r3)3,q (ω1, ω2, ω3)

= Ha,q(ω1)Hb,q(ω2)

× Hd,q(ω1 + ω2)Hc,q(ω3)He,q(ω1 + ω2 + ω3). (10)

For a self-kernel, we have r1 = r2 = r3 = r and q = 1 [3]

H (m:r,r,r)3 (ω1, ω2, ω3)

= 1

6

∑

permutations of [ω1,ω2,ω3]H (m:r,r,r)

3,1 (ω1, ω2, ω3). (11)

For a 2 × 1 cross-kernel, we have r1 = r2 �= r3 and there arethree permutations of the input signals u(r1), u(r1), and u(r2)

in Fig. 1, which correspond to different systems. The permu-tations [u(r1), u(r2), u(r1)] and [u(r2), u(r1), u(r1)] result in thesame type of system, as interchanging Ha,q and Hb,q in Fig. 1does not result in different types of system when r1 = r2 �= r3.The basic block structure of a 2×1 kernel is the parallel of twosystems like that in Fig. 1, with input signals [u(r1), u(r1), u(r2)]and [u(r1), u(r2), u(r1)]. The symmetrized kernel is

H (m:r1,r1,r3)3 (ω1, ω2, ω3)

= 1

2

{H (m:r1,r1,r3)

3,1 (ω1, ω2, ω3)

+ H (m:r1,r1,r3)3,1 (ω2, ω1, ω3) + H (m:r1,r3,r1)

3,2 (ω2, ω3, ω1)

+ H (m:r1,r3,r1)3,2 (ω1, ω3, ω2)

}(12)

where q = 1 corresponds to the input signals[u(r1), u(r1), u(r2)] and q = 2 corresponds to the inputsignals [u(r1), u(r2), u(r1)]. For a 3 × 1 cross-kernel, we haver1 �= r2 �= r3, and there are six permutations of the inputsignals u(r1), u(r2), and u(r3) in Fig. 1. There are threedifferent systems: the permutations [u(r1), u(r2), u(r3)] and[u(r2), u(r1), u(r3)] result in the same type of system thatwe number q = 1, the permutations [u(r1), u(r3), u(r2)] and[u(r3), u(r1), u(r2)] in another type of system (q = 2), and thepermutations [u(r2), u(r3), u(r1)] and [u(r3), u(r2), u(r1)] in athird system (q = 3). The general kernel has three systems inparallel and cannot be symmetrized

H (m:r1,r2,r3)3 (ω1, ω2, ω3) = H (m:r1,r2,r3)

3,1 (ω1, ω2, ω3)

+ H (m:r2,r3,r1)3,2 (ω2, ω3, ω1)

+ H (m:r3,r1,r2)3,3 (ω3, ω2, ω1). (13)

The system is causal; hence, the real and imaginaryparts or amplitude and phase of the frequency-domain Volterrakernels are related to Kramers–Kronig relations that for the

latter case are [23], [36]

� Hn(σ ) = −2σ

πP

∫ ∞

0

ln|Hn(σ′)|

σ′2 − σ 2

dσ ′

= − 1

π

∫ ∞

0

dln|Hn(σ′)|

dσ ′ ln

∣∣∣∣σ ′+σ

σ ′−σ

∣∣∣∣dσ ′

ln|Hn(σ )|−ln|Hn(σ0)| = 2

πP

∫ ∞

0σ ′ � Hn(σ

′)[

1

σ′2 − σ 2

− 1

σ′2 − σ 2

0

]dσ ′ (14)

where σ is a path in the n-dimensional frequency space andP denotes the principal value.

B. Test Signals

We use three-tone signals and combinations of two- andone-tone signals to determine the frequency-domain Volterrakernels, as shown in Fig. 1. The input signals are bandwidthlimited around the carrier frequency ( fc = ωc/2π). Therefore,we disregard the even-order terms [23]. Moreover, only thefirst- and third-order responses are studied, as the first-orderresponse is the dominating one and the third-order responsetypically is the largest of the nonlinear terms. The test signalsfor the identification of self- and cross-Volterra kernels aredescribed in the following. A three-tone signal is required fordetermining self-kernels. The 2 ×1 kernels require a two-tonesignal in one channel and a single tone signal in another. The3 × 1 kernels require a single-tone signal in each channel.

In the following, we give equations for a self-kernelH (2:2,2,2)

3 . They are, of course, valid for any other self-kernel.In the same way, we give equations for the 2 × 1 cross-kernelH (2:2,2,3)

3 and the 3×1 cross-kernel H (2:1,2,3)3 . Naturally, these

could be used for any 2 × 1 or 3 × 1 cross-kernels.1) Three-Tone Test: To determine the third-order self-

Volterra kernels of a 3 × 3 nonlinear MIMO system [seeFig. 2(a)], the system is excited with a three-tone signal atinput channel 2. The three-tone signal is given as [23]

u(2)(t) = A[α′ cos(ω′t + φ′) + α′′ cos(ω′′t + φ′′) +α′′′ cos(ω′′′t + φ′′′)] (15)

where u(2)(t) denotes the input signal at the second inputof a 3 × 3 MIMO system. In (15), A is the amplitude,ω′, ω′′ and ω′′′ are the angular frequencies, and φ′, φ′′ andφ′′′ are the phases of the three tones, respectively, α′, α′′, andα′′′ are dimensionless constants. For simplicity, we assume thatα′ = α′′ = α′′′ = 1.

When excited with u(2)(t), the output of the second subsys-tem can be written as

y(2)(t) = y(2)1 (t) + y(2)

3 (t) (16)

where y(2)1 (t) denotes the linear output components of the

second subsystem and y(2)3 (t) describes the third-order output

components. Because we are interested in determining thethird-order self-Volterra kernels of a system, the third-orderoutput components of the second subsystem are

y(2)3 (t) =

K∑

k=1

Ak |H3,k|cos(ωk t + �k + � H3,k) (17)

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4 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT

Fig. 2. Illustration of test signals to determine self- and cross-Volterra

kernels of a 3×3 MIMO system. Determination of (a) self-kernel H (2:2,2,2)3 ,

(b) cross-kernel H (2:2,2,3)3 , and (c) cross-kernel H (2:1,2,3)

3 .

where K is the number of all third-order products, andAk, H3,k, ωk , and �k are amplitude, kernel response, fre-quency, and phase, respectively [23]. In this paper, the analysisof self-kernel is restricted to H (2:2,2,2)

3 (−ω′, ω′′, ω′′′); thisgives Ak = 3/2 A3.

2) 2 + 1 Tone Test: To analyze the third-order 2 × 1 cross-kernels, a two-tone signal in one channel and a single-tone inanother channel are used [see Fig. 2(b)]. The excitation signalsare

u(2)(t) = A[α′ cos(ω′t + φ′) + α′′ cos(ω′′t + φ′′)]u(3)(t) = Aα′′′ cos(ω′′′t + φ′′′) (18)

where u(2)(t) and u(3)(t) denote the two- and single-tonesignals at the second and third inputs of the MIMO system,respectively. In this test, we excite u(2)(t) and u(3)(t) at equalpowers as in three-tone test; therefore, α′ = α′′ = √

3/2 andα′′′ = √

3. The corresponding third-order output componentof the second subsystem becomes

y(2)3 (t) =

3∑

r1=2

3∑

r2=r1

3∑

r3=r2

y(2:r1,r2,r3)3 (t). (19)

Note that, in this paper, we restrict the analysisof 2 × 1 cross-kernels to H (2:2,2,3)

3 (−ω′, ω′′, ω′′′) andH (2:2,2,3)

3 (ω′, ω′′,−ω′′′). The third-order cross components atthe output of second subsystem are

y(2:2,2,3)3 (t) =

∑

k

3

2

(3

2

√3

)A3|H3,k|cos(ωk t + �k + � H3,k)

(20)

k indicates the terms corresponding to the frequencies at(−ω′, ω′′, ω′′′) and (ω′, ω′′,−ω′′′).

3) 1+1+1-Tone Test: 3×1 cross-kernels can be determinedby exciting three channels each with a single-tone signal,as shown in Fig. 2(c). The three single-tone signals are

u(1)(t) = Aα′ cos(ω′t + φ′)u(2)(t) = Aα′′ cos(ω′′t + φ′′)u(3)(t) = Aα′′′ cos(ω′′′t + φ′′′) (21)

where u(1)(t), u(2)(t), and u(3)(t) are three single-tone sig-nals applied to input channels 1, 2, and 3, respectively, andα′ = α′′ = α′′′ = √

3. Under these excitation signals,the corresponding third-order output components of the secondsubsystem can be described as

y(2)3 (t) =

3∑

r1=1

3∑

r2=r1

3∑

r3=r2

y(2:r1,r2,r3)3 (t). (22)

Notice that, when a 3 × 3 MIMO system is excitedby single-tone signals in each channel, the resulting third-order output components of mth subsystem will contain aself-component and 17 cross components. Therefore, in thispaper, the analysis is restricted to a 3 × 1 cross-kernel

of the form H (2:1,2,3)3 (−ω′, ω′′, ω′′′), H (2:1,2,3)

3 (ω′,−ω′′, ω′′′)and H (2:1,2,3)

3 (ω′, ω′′,−ω′′′). Similarly, the correspondingthird-order cross components in the output of the secondsubsystem are

y(2:r1,r2,r3)3 (t) =

∑

k

3

2(3

√3)A3|H3,k|cos(ωk t + �k + � H3,k)

(23)

where k indicates the terms at (−ω′, ω′′, ω′′′), (ω′,−ω′′, ω′′′),and (ω′, ω′′,−ω′′′).

C. Identification of Kernels

We identify the Volterra kernels by a standard least-squares (LS) technique for identifying the amplitude and phaseof spectral components of known frequencies of a sampledtime-domain signal [37].

For the self-kernels and cross-kernels, we form the equationsystems for each test case

y = Xθ (24)

that are solved in the LS sense as described in [37]. In (24),y is a column vector that contains the measured and sampledoutput signals of the respective test case

y = [y(m)(0) y(m)(Ts) · · · y(m)((N − 1)Ts)]T (25)

where N is the number of samples and Ts is the samplingtime. θ contains the estimated kernels⎡

⎢⎢⎢⎢⎢⎢⎢⎣

Aα′ H (m:r1)1 (ω′)

Aα′′ H (m:r2)1 (ω′′)...

32 A3α′α′′α′′′ H (m:r1,r2,r3)

3 (ω′, ω′′,−ω′′′)

⎤

⎥⎥⎥⎥⎥⎥⎥⎦

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ALIZADEH et al.: MEASUREMENT AND ANALYSIS OF FREQUENCY-DOMAIN VOLTERRA KERNELS 5

Fig. 3. Outline of the experimental setup.

where H1 and H3 are estimated kernels, α′ and α′′, and α′′′are different for the test cases in II-B, H1 is dimensionless,and H3 is in V −2. The matrix X has the size N × 12and is the regression matrix of the input signals and theirmultiplications [23]. For simplicity, we use only the threeterms that contain all ω′, ω′′, and ω′′′ in the following analysis.

The LS estimation technique has previously been usedin [23] and [31] for both frequency- and time-domain Volterrakernels for SISO and concurrent dual-band power amplifiers,respectively. The only requirement is to know the frequenciesof each spectral component. Because the input signals aregenerated at the known frequencies (ω′, ω′′, and ω′′′) andthe kernels are at known frequencies, this requirement issatisfied. The identified kernels at each output frequency willalso contain contributions from higher odd-order nonlinearterms [23] that is written in the form of polynomial modelin A2

H1(Q) = H1 + β1,3 H3 A2 + · · · + β1,Q HQ(A2)Q−1

2 , Q ≥ 1

(26)

H3(Q) = H3 + β3,5 H5 A2 + · · · + β3,Q HQ(A2)Q−3

2 , Q ≥ 3

(27)

where Q is the maximum nonlinear order, and β1,Q and β3,Q

are polynomial coefficients. The third-order Volterra kernelscan be identified at different amplitude levels and contributionsfrom higher nonlinear orders can be reduced by fitting apolynomial as in (27). The estimated third-order kernel is theconstant term, independent of A2 [23].

III. EXPERIMENT

The experimental setup is shown in Fig. 3. The vector signalgenerators (VSGs) were three Rohde & Schwarz SMBV100A.The analog-to-digital converter was 14-b SP-DeviceADQ214 digitized at 400-MHz sampling rate. The deviceunder test (DUT) was composed of three RF amplifiers,Mini-Circuits ZHL-42, with typical linear gain of 31.5 dB,

and with 1-dB compression point at an output of 30 dBm,which were coupled with a coupler on the input signals(see Fig. 3). The coupler causes crosstalk, which is a well-known effect in MIMO transmitters [35]. The coupler wasdesigned in microstrip technology using FR4 substrate (50 �).There were three transmission lines of width 15 mm andlength 65 mm. The coupling between an outer and the innerchannel was measured using a vector network analyzer (VNA)to be −13.5 dB, and between the two outer channels,it was −21.5 dB.

Three VSGs were synchronized in carrier frequency,reference frequency, and baseband digital clock. The generatedsignals were a three-tone signal, a 2 + 1-tone signals,and three single-tone signals, respectively, as mentionedin (15), (18), and (21) and as shown in Fig. 2(a)–(c), respec-tively. Two types of tones were investigated in this paper.In the first type, one of the tones was stepped, whereas theother two tones were fixed; it is termed the one-stepped-two-fixed type. The second type is termed the two-stepped-one-fixed type and has two stepped tones and one fixed.Table I shows the frequencies of the test signals. In each typeof tones, 1, 2, and 3 set of test signals were performed forself-kernel, 2×1 and 3×1 cross-kernels analysis, respectively.The power of the signals stepped from −5 to −15 dBm in thesteps of 0.5 dB. The output RF signals were downconvertedinto an intermediate frequency using mixers, Mini-CircuitsZX05-42MH-S, and thereafter filtered by bandpass filters. Themeasured signals were compensated by a constant gain inthe postprocessing part to compensate for losses in the usedcomponents. To improve the performance of the measurementsystem, 100 coherent averages were performed and then timealigned with the input signal in the postprocessing part [33].

IV. RESULTS

In Section IV-A, we describe how the kernels are identifiedfrom experimental data. In Section IV-B, we show how a self-kernel, a 2 × 1 cross-kernel, and a 3 × 1 cross-kernel areanalyzed to determine the corresponding block structures.

A. Kernel Estimation

A third-order kernel identified from experimental dataat one amplitude level may be affected by contributionsfrom higher nonlinear order kernels. As an example, Fig. 4shows the experimentally determined real and imaginary partsof H (2:2,2,2)

3 data versus input signal power (A2). Also shownare LS fitted polynomials as in (27) of different maximumnonlinear order of Q. A maximum nonlinear order of at least 7is needed to well fit the experimental data.

In Fig. 5, as an example the self-kernel H (2,2,2,2)3 is shown

along a path in the 3-D frequency space given by (− fc + 7.2,fc + fst, fc − fst) MHz, where fst is stepped from−10 to +10 MHz. The kernels are analyzed in detail in thefollowing. The kernel is shown as estimated using differentmaximum nonlinear orders, Q, in (27). The error bars in Fig. 5are obtained from the errors of the LS fit and are hencedue to random errors in the experimental data. Using ahigher Q in (27) results in a more pronounced structureof the curve. There is also a systematic shift in magnitude

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6 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT

TABLE I

SUMMARY OF THE PERFORMED EXPERIMENTS, FOR THE SELF-KERNEL r1 = r2 = r3 = 2, FOR THE 2 × 1 CROSS-KERNEL r1 = r2 = 2, AND r3 = 3 ANDFOR THE 3 × 1 CROSS-KERNEL r1 = 1, r2 = 2, AND r3 = 3. THE CENTER FREQUENCY IS fc = 2.14 GHz, fst IS STEPPED FROM −10 TO +10 MHz

IN THE STEPS OF 25 kHz AND fsh IS A FREQUENCY SHIFT OF −7.5, −2.5, +2.5, AND +7.5 MHz. THE CONSTANTS ARE IN MHz

Fig. 4. Real and imaginary parts of H (2:2,2,2)3 versus A2 at frequen-

cies (− fc + 7.2, fc − 10, fc + 10) MHz. Also shown are fitted curves fordifferent values of Q that include the effects of higher nonlinear orders.

Fig. 5. Magnitudes of the estimated self-kernel, H (2:2,2,2)3 , along the

frequency path (− fc +7.2, fc + fst, fc − fst) MHz, using Q as in (27). Theerror bars are obtained from the errors in the fitted polynomial coefficients.

with increasing Q; the contribution to the determine H3 fromterms of higher nonlinear order decreases. With increasing Q,the determined kernel becomes noisier. We use Q = 7

Fig. 6. Estimated phases of the self-kernel H (2:2,2,2)3 along the path

(− fc + 7.2, fc + fst, fc − fst) MHz, using Q = 7. Also shown are thephases obtained from the magnitudes in Fig. 5 using the Kramers–Kronigrelations in (14).

that gives the best compromise between random errors andmagnitude shift, i.e., a tradeoff between precision and accuracyas described in [23].

The magnitude of the seventh-order kernel in Fig. 5 isused to calculate the corresponding phase using (14). Theexperimental data are in a finite frequency range and themagnitude outside this range is set to zero. The obtained phaseis shown in Fig. 6 together with the experimentally determinedphase. Fig. 6 shows that our data are Kramers–Kronig consis-tent, which is a necessary condition to correctly determinefrequency-domain kernels. We see in Figs. 5 and 6 that thestructure in the magnitude has a corresponding structure inthe phase. We therefore analyze only the magnitude in thefollowing.

B. Analysis of Volterra Kernels

The basic block structures for the third-order kernels aregiven in Fig. 1 and (10)–(13). Here, we show how theexperimentally determined third-order kernels can be analyzedto determine which blocks give the dominating contributionto a kernel’s frequency dependence. We extend the techniqueto characterize the kernels along different frequency pathsin [23] and [24] to the analysis of cross-kernels.

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ALIZADEH et al.: MEASUREMENT AND ANALYSIS OF FREQUENCY-DOMAIN VOLTERRA KERNELS 7

Fig. 7. Frequency paths in three regions in the vicinity of (− fc, fc, fc), ( fc,− fc, fc), and ( fc, fc,− fc), respectively, along which we analyze thedetermined Volterra kernels. (a) Frequency paths of one-stepped-two-fixed tones as described in the legends. (b) Frequency paths of two-stepped-one-fixedtones as described in the legends.

For self-kernels H (m:r,r,r)3 , the symmetry properties in (7)

yields that the kernel is identical in the regions in the vicin-ity of (− fc, fc, fc), ( fc,− fc, fc), and ( fc, fc,− fc) shownin Fig. 7. We present data along the paths in the vicinity of(− fc, fc, fc) in Section IV-B1.

For 2 × 1 cross-kernels H (m:r2,r2,r3)3 , the kernel is iden-

tical along the paths in the vicinity of (− fc, fc, fc) and( fc,− fc, fc) but not ( fc, fc,− fc) [see (8)]. The 2 ×1 kernelis, thus, identical in two of the three regions in Fig. 7.In Section IV-B2, we present data along the paths in thevicinity of (− fc, fc, fc) and ( fc, fc,− fc).

For 3 × 1 cross-kernels H (m:r1,r2,r3)3 , the kernel can be

different in the vicinity of (− fc, fc, fc), ( fc,− fc, fc), and( fc, fc,− fc), respectively [see (9)], and we present data forall three frequency regions in Fig. 7 in Section IV-B3.

For the 2×1 and 3×1 cross-kernels, we present data alongthe paths for which the self-kernel is identical in the threefrequency regions in Fig. 7. The paths of the type one-stepped-two-fixed in Fig. 7(a) are obtained from the tests with only onefrequency being stepped, whereas the paths of the type two-stepped-one-fixed in Fig. 7(b) are obtained from tests with twostepped frequencies.

A path in the 3-D frequency space that is a straight line isdescribed by

fi = fc + fcoi + γi fsh + λi fst (28)

where i = 1, 2, 3 refers to the frequency axis, fc is the carrierfrequency, fcoi is a frequency distance from the carrier, andfsh is a shift in frequency, i.e., different values of fsh givedifferent paths within the same type of paths. γi = 1 whena frequency is shifted; otherwise, γi = 0. fst is stepped from−10 to +10 MHz. λi = 1 when a frequency is stepped; other-wise, λi = 0. The carrier frequency fc = 2.14 GHz, fcoi , i =1, 2, 3, is chosen as 0.3, 0.1, and −0.1 MHz, receptively.The frequency shift fsh gets four different values of −7.5,−2.5, +2.5, and +7.5 MHz to have four paths with a shiftof ±5 MHz. Fig. 7(a) shows all paths of type one-stepped-two-fixed and Fig. 7(b) shows all paths of type two-stepped-one-fixed, in the vicinity of (− fc, fc, fc), ( fc,− fc, fc), and( fc, fc,− fc), respectively.

Fig. 8. Magnitudes of the self-kernel H (2:2,2,2)3 along the given frequency

paths. The horizontal lines indicate the frequency shifts of the largest structure.

1) Self-Kernel H (2:2,2,2)3 : The self-kernel was determined

in the vicinity of (− fc, fc, fc) from a three-tone test.We first analyze the kernel along one-stepped-two-fixed paths,( f1, f2, f3) = (− fc−0.3− fsh, fc+0.1+ fsh, fc+ fst), wheref1 and f2 are fixed frequencies for each path and f3 is stepped.The magnitude of the self-kernel H (2:2,2,2)

3 along these pathsis shown in Fig. 8. The dominating structures of the curvesare minima in magnitude located around −7.2, −2.2, +2.8,and +7.8 MHz relative to fc, i.e., there are shifts of +5 MHzbetween the curves. The general frequency-domain kernel ofthe block structure in Fig. 1 is obtained from (10) and (11).Along the paths of the type one-stepped-two-fixed, the self-kernel H (2:2,2,2)

3 is

H3 = 1

6{[Ha,1(− fc − 0.3 − fsh)Hb,1( fc + 0.1 + fsh)

+Ha,1( fc + 0.1 + fsh)Hb,1(− fc − 0.3 − fsh)]×Hd,1(−0.2)Hc,1( fc + fst)

+[Ha,1( fc + 0.1 + fsh)Hb,1( fc + fst)

+Ha,1( fc + fst)Hb,1( fc + 0.1 + fsh)]×Hd,1(2 fc+0.1 + fsh + fst) Hc,1(− fc − 0.3 − fsh)

+[Ha,1( fc + fst)Hb,1(− fc − 0.3 − fsh)

+Ha,1(− fc − 0.3 − fsh)Hb,1( fc + fst)]×Hd,1(−0.3 − fsh+ fst)Hc,1( fc + 0.1 + fsh)}×He,1( fc − 0.2 + fst). (29)

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8 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT

Fig. 9. Magnitudes of the kernel H (2:2,2,2)3 along the given frequency paths.

The horizontal lines indicate the frequency shifts of two largest structures.

Only terms that are the functions of fst can give a frequencydependence of H3 along paths; other terms in (29) are constantalong this type of paths. The terms that are functions of( fc + fst) will not give any shift when fsh is shifted, as shownin Fig. 8. Only terms that are the functions of fsh and fstcan give a frequency dependence and frequency shift. TheHa,1, Hb,1, Hc,1, and He,1 terms can thus not contribute tothe frequency shifts and dependence seen in Fig. 8. The termHd,1(−0.3 − fsh + fst) gives a shift of fsh (or +5 MHz),which is in agreement with the shift of the structure in Fig. 8.This term corresponds to Hd,1 being excited at basebandfrequencies. The term Hd,1(2 fc + 0.1 + fsh + fst) would givea shift of − fsh (or −5 MHz), which is not seen in Fig. 8.This term corresponds to Hd,1 being excited at 2 fc, wherethe relative bandwidth is smaller than 0.5 % and hence hassmall frequency dependence. The frequency argument of termHd,1(−0.3 − fsh + fst) becomes zero at fst = fsh + 0.3. Themain structures in Fig. 8 therefore occur when the frequencyargument of Hd,1 is close to zero, i.e., at baseband frequencies.

The second type of paths, two-stepped-one-fixed,is ( f1, f2, f3) = (− fc − 0.3 − fsh, fc + fst, fc − fst),where f2 and f3 are stepped and f1 is fixed. Fig. 9 showsthe kernel H (2:2,2,2)

3 along these paths. Each curve hastwo clear structures that are shifted. The first is locatedat −7.2, −2.2, +2.8, and +7.8 MHz relative to fc and isshifted +5 MHz; the second one is located at +7.2, +2.2,−2.8, and −7.8 MHz relative to fc and is shifted −5 MHz.The self-kernel along this type of paths is obtainedfrom (10) and (11)

H3 = 1

6{[Ha,1(− fc − 0.3 − fsh)Hb,1( fc + fst)

+Ha,1( fc + fst)Hb,1(− fc − 0.3 − fsh)]×Hd,1(−0.3 − fsh + fst)Hc,1( fc − fst)

+[Ha,1( fc + fst) Hb,1( fc − fst) + Ha,1( fc − fst)

×Hb,1( fc + fst)]Hd,1(2 fc) Hc,1(− fc − 0.3 − fsh)

+[Ha,1( fc − fst) Hb,1(− fc − 0.3 − fsh)

+Ha,1(− fc − 0.3 − fsh)Hb,1( fc − fst)]×Hd,1(−0.3 − fsh − fst)Hc,1( fc + fst)}×He,1( fc − 0.3 − fsh). (30)

The only terms in (30) that may both be frequency dependentwhen fst is stepped and have structures that are shifted when

Fig. 10. Block structure of the self-kernel H (2:2,2,2)3 as determined from the

experimental data in Figs. 8 and 9.

fsh is changed are the terms Hd,1(−0.3 − fsh + fst) andHd,1(−0.3 − fsh − fst). The former causes a shift of fsh (or+5 MHz), which is seen in one of the structures in Fig. 9 at−7.2, −2.2, +2.8, and +7.8 MHz. The latter causes a shiftof − fsh (or −5 MHz), which is seen in Fig. 9 in the structureat +7.2, +2.2, −2.8, and −7.8 MHz. In the same way as forFig. 8, the terms Ha,1, Hb,1, Hc,1, and He,1 cannot cause thetype of shifts and frequency dependence seen in Fig. 9. Thus,the data in Figs. 8 and 9 show that the frequency dependenceof H (2:2,2,2)

3 is due to a term of the type Hd,1 in Fig. 1 thatis excited at baseband frequencies. These baseband effects areattributed to bias modulation or thermal or thermo electricmemory effects [38].

The functions Ha(·), Hb(·), and Hc(·) in Fig. 1 can be inter-preted as the models of the input matching networks and He(·)describes the output matching network [39]. The blocks Ha,Hb, Hc, and He do not contribute to the frequency dependenceof H (2:2,2,2)

3 for the DUT. Therefore, the simplified model thatdescribes the main behavior of H (2:2,2,2)

3 becomes

H3(ω1, ω2, ω3)

= 1

3{ Hd,1(ω1, ω2) + Hd,1(ω1, ω3) + Hd,1(ω2, ω3) } (31)

and the corresponding block structure is shown in Fig. 10.The block structure of the self-kernel corresponds to an input–output relationship y3(t) = u(t)× Hd,1[u2(t)]. A time-domaincomplex baseband model can be derived from the blockstructure in Fig. 10, using standard techniques [32], [33]. Thecomplex baseband equivalent third-order terms are

y(2:2,2,2)3 (n) = u(2)(n)

M∑

m=0

h3(m) |u(2)(n − m)|2 (32)

where u(2)(n) is the baseband discrete time equivalent ofu(2)(t), h3(m) is the model parameter, and M is the memorylength of Hd,1(·) for the physical system M → ∞. Themodel in (32) is known as the envelope-memory polynomialmodel [40]. It can be explained by a Hammerstein–Wienerstructure of a cascaded system N-H -N , where N is a staticnonlinearity and H is a linear dynamic system [23]. We didnot include the term u∗(n) (u(n − m))2 in (32), as thisterm corresponds to exciting Hd(·) at 2 fc. It was found byanalyzing (29) and (30) that this term does not contribute tothe frequency dependence and hence memory effects.

2) Cross-Kernel H (2:2,2,3)3 : We analyze H (2:2,2,3)

3 in thevicinity of (− fc, fc, fc) and ( fc, fc,− fc) along the pathsof the types one-stepped-two-fixed and two-stepped-one-fixed.To simplify the expressions, we omit the terms correspondingto Ha,q , Hb,q , Hc,q , and He,q in Fig. 1 in the followinganalysis. These terms are analyzed in the same way as for theself-kernel, and it was found that they did not contribute to

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ALIZADEH et al.: MEASUREMENT AND ANALYSIS OF FREQUENCY-DOMAIN VOLTERRA KERNELS 9

Fig. 11. Magnitudes of the 2 × 1 cross-kernel H (2:2,2,3)3 along the given

frequency paths in the vicinity of (a) (− fc, fc, fc) and (b) ( fc, fc,− fc).Horizontal lines: frequency shifts of the largest structure.

the frequency dependence of the analyzed kernel. We use theexpressions for the paths in the 3-D frequency space in (10)and (12) to obtain model expressions for the Volterra kernel.

The first paths of type one-stepped-two-fixed are (− fc −0.3 − fsh, fc + 0.1 + fsh, fc + fst) in the vicinity of(− fc, fc, fc). These paths give

H3 = 1

2{2Hd,1(−0.2) + Hd,2(2 fc + 0.1 + fsh + fst)

+ Hd,2(−0.3 − fsh + fst)}. (33)

Only Hd,2(−0.3 − fsh + fst) gives a shift of fsh (or +5 MHz)and hence describes the dominating structure in the curvesin Fig. 11(a). The Hd,1 term does not contribute to thefrequency dependence seen in Fig. 11(a). The term Hd,2(2 fc+0.1 + fsh + fst) is excited at 2 fc and does not cause anyfrequency dependence, by analogy with the self-kernel.

In a similar way, analyzing the kernel along paths ( fc +fst, fc + 0.1 + fsh, − fc + 0.1 − fsh) in the vicinity of( fc, fc,− fc) gives the same result that Hd,2 gives the fre-quency dependence seen in Fig. 11(b).

In the second type of paths, two-stepped-one-fixed, the pathsin the vicinity of (− fc, fc, fc) are ( f1, f2, f3) = (− fc −0.3−fsh, fc + fst, fc − fst). These paths give

H3 = 1

2{2Hd,1(−0.3 − fsh + fst)

+ Hd,2(2 fc) + Hd,2(−0.3 − fsh − fst)}. (34)

Hd,1(−0.3− fsh+ fst) gives a shift of fsh (or +5 MHz), whichis seen in Fig. 12(a) as the weaker structure at −7.2, −2.2,+2.8, and +7.8 MHz, respectively. The term Hd,2(−0.3 −fsh − fst) gives shifts of − fsh (−5 MHz), which is seen inthe more pronounced structure in Fig. 12(a) with positionsat +7.2, +2.2, −2.8, and −7.8 MHz, respectively. Both theterms Hd,1 and Hd,2 contribute to the frequency dependencein Fig. 12(a).

Fig. 12. Magnitudes of the 2 × 1 cross-kernel H (2:2,2,3)3 along the given

frequency paths in the vicinity of (a) (− fc, fc, fc) and (b) ( fc, fc,− fc).Horizontal lines: frequency shifts of the largest structure.

Fig. 13. Block structure of the 2 × 1 cross-kernel H (2:2,2,3)3 as determined

from the experimental data in Figs. 11 and 12.

Fig. 12(b) shows the results for the kernel along the( f1, f2, f3) = ( fc + fst, fc − fst, − fc + 0.1 − fsh) in thevicinity of ( fc, fc,− fc). By analyzing in a similar way,it gives that only Hd,2 is significant along this path.

The analysis showed that a block structure with two systemsin parallel with Hd,1 and Hd,2 is required to model the experi-

mental data for H (2:2,2,3)3 in the vicinity of both (− fc, fc, fc)

and ( fc, fc,− fc). In Fig. 13, such a block structure is shown.The magnitude level of the cross-kernel H (2:2,2,3)

3 in Fig. 11is ∼ 12 dB lower than the magnitude level of the self-kernelH (2:2,2,2)

3 in Fig. 8. The difference cannot be explained byanalyzing only H (2:2,2,3)

3 . We know, however, that it is due tothe crosstalk in the DUT that did not cause any significantfrequency dependence, as seen in the experimental data. Thisresult agrees with the effect of coupler attenuation measuredby the VNA. In Fig. 13, we have added blocks for Hc,1 andHa,2 that would model the effect of the crosstalk. For the DUT,they have only attenuations independent of frequency.

We can compare the block structure in Fig. 13 to behavioralmodels in the literature. Fig. 13 can be compared only to someterms in these models, as they have several cross terms ofdifferent nonlinear orders. A time-domain baseband behavioral

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10 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT

Fig. 14. Magnitudes of the 3 × 1 kernel H (2:1,2,3)3 along the given

frequency paths in the vicinity of (a) (− fc, fc, fc), (b) ( fc,− fc, fc), and(c) ( fc, fc,− fc). Horizontal lines: frequency shifts of the largest structures.

model can be derived from the block structure in Fig. 13 usingstandard techniques [32], [33]

y(2:2,2,3)3 (n)

=M1∑

m1=0

M2∑

m2=0

h3,1(m1, m2)|u(2)(n − m1)|2

× u(3)(n − m2) + u(2)(n)

M1∑

m1=0

M2∑

m2=0

h3,2(m1, m2)

× u(2)(n − m1)u∗(3)(n − m1 − m2) + u(2)(n)

M1∑

m1=0

M2∑

m2=0

× h3,3(m1, m2)u∗(2)(n − m1)u

(3)(n − m1 − m2) (35)

where h3,1(m1, m2) corresponds to the multiplication of thesignals after Hc,1(·) and Hd,1(·) in Fig. 13 and h3,2(m1, m2)and h3,3(m1, m2) correspond to the cascade of Ha,2(·) andHd,2(·). M1 is the memory length of Hd,1(·) and Hd,2(·)and M2 is the memory length of crosstalk seen in Hc,1(·) andHa,2(·). For the physical system M1 → ∞ and M2 → ∞,the truncation could be different, i.e., M1 �= M2. As above,the terms that excite Hd(·) at 2 fc are not included in (35).For the special case m1 = m2, the first term in (35)corresponds to the 2 × 2 parallel Hammerstein model [41].

Fig. 15. Magnitudes of the 3 × 1 cross-kernel H (2:1,2,3)3 along the given

frequency paths in the vicinity of (a) (− fc, fc, fc), (b) ( fc,− fc, fc), and(c) ( fc, fc,− fc). Horizontal lines: frequency shifts of the largest structures.

The second and third terms in (35) resemble the y(2:2,2,3)3

kernel of the generalized memory polynomial for nonlinearcrosstalk [34].

3) Cross-Kernel H (2:1,2,3)3 : We analyze the cross-kernel

H (2:1,2,3)3 in the vicinity of (− fc, fc, fc), ( fc,− fc, fc), and

( fc, fc,− fc), respectively. We omit Ha,q , Hb,q , Hc,q , andHe,q ; these terms were analyzed in the same way as forthe self-kernel and were found not to cause any frequencydependence. We use the expressions for the paths in the3-D frequency space in (10) and (13) to obtain model expres-sions for the Volterra kernel along these paths.

The first paths of type one-stepped-two-fixed, ( f1, f2, f3) =(− fc − 0.3 − fsh, fc + 0.1 + fsh, fc + fst), give

H3 = Hd,1(−0.2) + Hd,2(2 fc + 0.1 + fsh + fst)

+ Hd,3(−0.3 − fsh + fst). (36)

Only the term Hd,3(−0.3 − fsh + fst) gives shifts of fsh(or +5 MHz), as shown in Fig. 14(a).

Similarly, the behavior of the kernel seen in Fig. 14(b) isalong the paths ( fc+0.3+ fsh, − fc−0.1− fsh, fc + fst) in the( fc,− fc, fc) zone, where the dominating term correspondsto Hd,2. Fig. 14(c) shows the kernel along the paths ( fc +fst, fc +0.1+ fsh, − fc +0.1− fsh) in the ( fc, fc,− fc) zone,where by analysis it gives that Hd,3 is dominating.

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ALIZADEH et al.: MEASUREMENT AND ANALYSIS OF FREQUENCY-DOMAIN VOLTERRA KERNELS 11

Fig. 16. Block structure of the 3 × 1 cross-kernel H (2:1,2,3)3 as determined

from the experimental data in Figs. 14 and 15.

The first paths of type two-stepped-one-fixed are (− fc −0.3 − fsh, fc + fst, fc − fst). The model of H (2:1,2,3)

3 is

H3 = Hd,1(−0.3 − fsh + fst)

+ Hd,2(2 fc) + Hd,3(−0.3 − fsh − fst). (37)

The term Hd,1(−0.3 − fsh + fst) gives the shifts of fsh(or 5 MHz) seen in the weaker structure in Fig. 15(a), whereasthe term Hd,3(−0.3 − fsh − fst) gives the shifts of − fsh(or −5 MHz) seen in the stronger structure.

In a similar analysis, the kernel gives a frequency depen-dence on Hd,1 and Hd,2 along the paths ( fc + fst, − fc −0.1−fsh, fc − fst) in the vicinity of ( fc,− fc, fc) and a frequencydependence on Hd,2 and Hd,3 along the paths ( fc + fst, fc −fst, − fc + 0.1 − fsh) in the vicinity of ( fc, fc,− fc), whichare shown in Fig. 15(b) and (c), respectively.

Thus, the kernel H (2:1,2,3)3 is described by the block struc-

ture in Fig. 16(a) and (c) in the (− fc, fc, fc) frequencyzone. In the vicinity of ( fc,− fc, fc), the block structuresin 16(a) and (b) describe the behavior. The block structuresin Fig. 16(b) and (c) describe the behavior in the vicinityof ( fc, fc,− fc). To the best of our knowledge, there areno reports in the literature on complex baseband models for3 × 1 systems.

V. CONCLUSION

We propose a method for determining the frequency-domainMIMO Volterra kernels in a 3×3 MIMO system. A three-tonetest, a 2+1-tone test, and a 1+1+1-tone test were performed todetermine the self-kernels, 2×1 cross-kernels, and 3×1 cross-kernels, respectively. The DUT was three RF amplifiers withinput crosstalk. Due to different symmetries, the self-kerneland 2 × 1 and 3 × 1 cross-kernels were determined in one,two, and three frequency zones, respectively. We show how theeffects of higher nonlinear order terms can be compensated forand how Kramers–Kronig analysis can be used for consistencycheck. Errors were estimated from LS fits. In particular,we show how the kernels can be analyzed along certain pathsto determine the block structures that describe the main fea-tures of the kernels. Time discrete complex baseband modelswere determined from the block structures. The self-kernel

was found to correspond to the envelope memory polynomialmodel, which can also be derived from a Hammerstein–Wienerstructure. The 2 × 1 kernel was found to be a more generalform of the 2 × 2 parallel Hammerstein model. The presentedmethod could thus be used to determine digital predistortionalgorithms for RF MIMO transmitters.

The method for determining and analyzing the third-order MIMO Volterra kernels could be modified to thesecond-order kernels. Also, kernels of higher nonlinear orders(fourth, fifth, . . . ) could be determined; the analysis of thekernels along the paths in the multidimensional frequencyspace could be made, although the paths could not be plottedbut represented mathematically. The method for analyzingkernels could also be used for mechanical nonlinear dynamicMIMO systems.

APPENDIX

We derive the relationships between the time and frequencydomain Volterra kernels in the real valued representation usedabove and in the complex baseband representation used incommunication theory. We use the method for SISO systemsin [32] and extend it to the third-order MIMO systems.In addition to [32], we also give the relations for the frequencydomain kernels. In the derivations, it is assumed that thesystem is excited only in a passband around fc. We use . todenote the complex baseband representation.

For real valued signals and systems, the third-order termin time domain is given by (2) with n = 3. The frequencydomain representation is given by (4) and (5). We write theinput signals on complex form [32]

uri (t) = (1/2) [u(ri )(t) e jωct + u(ri )∗(t) e− jωct ] (38)

where i = 1, 2, 3 and u(ri )(t) is the complex signal. We put(38) into (2) with n = 3 and get

y(m:r1,r2,r3)3 (t)

= 1

23

∫ +∞

−∞

∫ +∞

−∞

∫ +∞

−∞h(m:r1,r2,r3)

3 (τ1, τ2, τ3) {e jωct

× [u(r1)(t − τ1)u(r2)(t − τ2)u

(r3)∗(t − τ3)e

jωc(−τ1−τ2+τ3)

+ u(r1)(t − τ1)u(r2)

∗(t − τ2)u

(r3)(t − τ3)ejωc(−τ1+τ2−τ3)

+ u(r1)∗(t − τ1)u

(r2)(t − τ2)u(r3)(t − τ3)e

jωc(+τ1−τ2−τ3)]+ e− jωct [. . . ] + e+3 jωct [. . . ] + e−3 jωct [. . . ]}. (39)

The terms multiplied by e− jωct are the complex conjugate ofthe e+ jωct terms. In the following, we omit the e±3 jωct terms,since they give contributions at 3 fc and not fc. Comparing(39) with (38), we see that the complex output signal aroundfc is

y(m:r1,r2,r3)3 (t)

= 1

23

∫ +∞

−∞

∫ +∞

−∞

∫ +∞

−∞h(m:r1,r2,r3)

3 (τ1, τ2, τ3)

× [u(r1)(t − τ1)u(r2)(t − τ2)u

(r3)∗(t − τ3)e

jωc(−τ1−τ2+τ3)

+ u(r1)(t − τ1)u(r2)∗(t − τ2)u

(r3)(t − τ3)ejωc(−τ1+τ2−τ3)

+ u(r1)∗(t − τ1)u

(r2)(t − τ2)u(r3)(t − τ3)e

jωc(+τ1−τ2−τ3)].(40)

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12 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT

For the self-kernel, r1 = r2 = r3 = r and h(m:r,r,r)3 is

symmetric under all permutations of (τ1, τ2, τ3) [see (7)] andthe three terms in (40) are identical. The real valued kernelcan be approximated by the complex kernel [32]

h(m:r1,r2,r3)3 (τ1, τ2, τ3)

= (1/2)h(m:r1,r2,r3)3 (τ1, τ2, τ3)e

jωc(τ1+τ2−τ3). (41)

The FT, (5), of (41) gives

H (m:r,r,r)3 (ω1, ω2, ω3)

= 2H (m:r,r,r)3 (ω1 + ωc, ω2 + ωc, ω3 − ωc) (42)

that is, the complex valued frequency domain kernel is propor-tional to the real valued kernel in the vicinity of (ωc, ωc,−ωc).

For the 2 × 1 cross kernel, r1 = r2 �= r3 and h(m:r1,r1,r3)3

is symmetric under permutations of (τ1, τ2) and two differentterms in (40) will contribute to the output signal. For thesetwo terms, the real valued kernel can be approximated by

h(m:r1,r1,r3)3 (τ1, τ2, τ3)

= (1/2) h(m:r1,r1,r3)3,i (τ1, τ2, τ3) e jωc(τ1+τ2−τ3)

h(m:r1,r1,r3)3 (τ1, τ2, τ3)

= (1/2) h(m:r1,r1,r3)3,ii (τ1, τ2, τ3) e jωc(τ1−τ2+τ3). (43)

The FT, (5), of (43) gives

H (m:r1,r1,r3)3,i (ω1, ω2, ω3)

= 2H (m:r1,r1,r3)3 (ω1 + ωc, ω2 + ωc, ω3 − ωc)

H (m:r1,r1,r3)3,ii (ω1, ω2, ω3)

= 2H (m:r1,r1,r3)3 (ω1 + ωc, ω2 − ωc, ω3 + ωc) (44)

that is, there are two different complex baseband frequencydomain kernels for the 2×1 cross-kernel and they correspondto the real valued kernel in two different regions in the3-D frequency space (see Fig. 7).

For the 3 × 1 cross kernel, r1 �= r2 �= r3 and h(m:r1,r2,r3)3 is

not symmetric and the three terms in (40) will all give differentcontributions to the output signal. The real valued kernel canfor the three terms be approximated by

h(m:r1,r2,r3)3 (τ1, τ2, τ3)

= (1/2) h(m:r1,r2,r3)3,i (τ1, τ2, τ3) e jωc(τ1+τ2−τ3)

h(m:r1,r2,r3)3 (τ1, τ2, τ3)

= (1/2) h(m:r1,r2,r3)3,ii (τ1, τ2, τ3) e jωc(τ1−τ2+τ3)

h(m:r1,r2,r3)3 (τ1, τ2, τ3)

= (1/2) h(m:r1,r2,r3)3,iii (τ1, τ2, τ3) e jωc(−τ1+τ2+τ3). (45)

The FT, (5), of (45) gives

H (m:r1,r2,r3)3,i (ω1, ω2, ω3)

= 2H (m:r1,r2,r3)3 (ω1 + ωc, ω2 + ωc, ω3 − ωc)

H (m:r1,r2,r3)3,ii (ω1, ω2, ω3)

= 2H (m:r1,r2,r3)3 (ω1 + ωc, ω2 − ωc, ω3 + ωc)

H (m:r1,r2,r3)3,iii (ω1, ω2, ω3)

= 2H (m:r1,r2,r3)3 (ω1 − ωc, ω2 + ωc, ω3 + ωc) (46)

that is, there are three different complex baseband fre-quency domain kernels for the 3 × 1 cross-kernel, eachcorresponding to one region in the 3-D frequency spacein Fig. 7.

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Mahmoud Alizadeh (S’14) received the M.Sc. degree in wireless commu-nication from the Electrical and Information Technology Department, LundUniversity, Lund, Sweden, in 2013. He is currently pursuing the Ph.D.degree with the Department of Signal Processing, KTH Royal Institute ofTechnology, Stockholm, Sweden, with a focus on digital signal processingfor communication systems, linearization techniques and characterization fornonlinear RF power amplifiers.

He is also with the Department of Electronics, Mathematics and NaturalSciences, University of Gävle, Gävle, Sweden.

Shoaib Amin (S’12) received the B.E degree in avionics engineering fromthe College of Aeronautical Engineering, National University of Sciences andTechnology, Islamabad, Pakistan, in 2008, the M.Sc. degree in electronicsfrom the University of Gävle, Gävle, Sweden, in 2011, and the Licentiatedegree in electrical engineering from the School of Electrical Engineering,KTH Royal Institute of Technology, Stockholm, Sweden, in 2015. He iscurrently pursuing the Ph.D. degree with the University of Gävle, andKTH Royal Institute of Technology.

His current research interests include signal processing techniques forcharacterization and mitigation of nonlinear dynamic effects in multibandmultichannel RF transmitters.

Daniel Rönnow (SM’04) received the M.Sc. degree in engineering physicsand the Ph.D. degree in solid-state physics from Uppsala University, Uppsala,Sweden, in 1991 and 1996, respectively.

He was involved in semiconductor physics with the Max-Planck-Institut fürFestkörperforschung, Stuttgart, Germany, from 1996 to 1998, and in infraredsensors and systems with Acreo AB, Stockholm, Sweden, from 1998 to 2000.From 2000 to 2004, he was a Technical Consultant and the Head of Researchwith Racomna AB, Uppsala. From 2004 to 2006, he was a University lecturerwith the University of Gävle, Gävle, Sweden. From 2006 to 2011, he wasa Senior Sensor Engineer with Westerngeco, Oslo, Norway, where he wasinvolved in signal processing and seismic sensors. In 2011, he became aProfessor in electronics with the University of Gävle. He has been an AssociateProfessor with Uppsala University since 2000. He has authored or co-authoredover 50 peer-reviewed papers and holds eight patents. His current researchinterests are RF measurement techniques, and linearization of nonlinear RFcircuits and systems.

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