Extending the dynamic range of RFreceivers using nonlinear equalization

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Citation Goodman, J. et al. “Extending the dynamic range of RF receiversusing nonlinear equalization.” Waveform Diversity and DesignConference, 2009 International. 2009. 224-228. © 2009 IEEE

As Published http://dx.doi.org/10.1109/WDDC.2009.4800349

Publisher Institute of Electrical and Electronics Engineers

Version Final published version

Citable link http://hdl.handle.net/1721.1/59342

Terms of Use Article is made available in accordance with the publisher'spolicy and may be subject to US copyright law. Please refer to thepublisher's site for terms of use.

Extending the Dynamic Range of RF Receiversusing Nonlinear Equalization

Joel Goodman, Benjamin Miller, Matthew Herman, Michael Vai and Paul MonticcioloMIT Lincoln Laboratory

Lexington, Massachusetts 02420Email: {jgoodman,bamiller,mherman,mvai,monticciolo}@ll.mit.edu

Abstract— Systems currently being developed to operate acrosswide bandwidths with high sensitivity requirements are limitedby the inherent dynamic range of a receiver’s analog and mixed-signal components. To increase a receiver’s overall linearity,we have developed a digital NonLinear EQualization (NLEQ)processor which is capable of extending a receiver’s dynamicrange from one to three orders of magnitude. In this paper wedescribe the NLEQ architecture and present measurements of itsperformance.

I. INTRODUCTION

In radar systems as well as many other applications (e.g.,emerging Ultra-WideBand (UWB) communications), there isa need for a large dynamic range relative to the backgroundnoise and all other distortions. The need for dynamic rangein radar returns is driven by the ratio of strong signals (suchas clutter) versus weaker signals (returns from small targets).Systems currently being developed to operate across widebandwidths with high sensitivity requirements are limitedby the inherent dynamic range of the receiver’s analog andmixed-signal components. Among these components (e.g.,LNA, mixer), the ADC commonly has the lowest dynamicrange [1]. A receiver’s deviation from its ideal ‘linear’ per-formance is commonly characterized by its spurious- and/orintermodulation-free dynamic range (SFDR and IFDR), whichis a frequency-domain measurement that determines the min-imum signal level that can be distinguished from distortioncomponents. The SFDR and IFDR of an ADC are typicallydominated by circuit-based (e.g., buffer amplifier, sample-and-hold) nonlinearities that are distinct from the nonlinear processof ideal quantization, which in principle can be circumventedwith processing gain [2]

In this paper we develop an NLEQ processor composed ofnonlinear polynomial filters to reduce polynomial distortionsgenerated by the LNA, mixer and ADC. We also addresspractical problems in identifying an equalization architecture,such as separating the distortions that are induced by theanalog signal generator (i.e., the excitation) from those thatare generated by the receiver itself.

The uniqueness of NLEQ is that it suppresses wide bandreceiver nonlinear distortions in a computationally efficientfashion. Existing approaches to achieving computationally

This work is sponsored by DARPA under Air Force contract FA8721-05-C-0002. Opinions, interpretations, conclusions and recommendations are those ofthe author and are not necessarily endorsed by the United States Government.

efficient polynomial filter architectures for RF compensation,principally developed to mitigate distortions generated bypower amplifiers in transmitters, limit the multidimensionalsignal space over which the architecture can suppress spectralregrowth and in-band spurs [3], [4]. In this paper, we developa technique to construct a polynomial filter architecture thatsearches over an unrestricted multidimensional signal spaceto select polynomial components that yield the highest equal-ization performance for a given computational complexity.In particular, we develop a coordinate system representationfor polynomial filters, and leverage compressed sensing tech-niques to identify a sparse polynomial representation of aninverse nonlinearity.

The rest of this paper is organized as follows. In SectionII, we develop a coordinate system representation used toconstruct an NLEQ architecture, and present an optimizationprocedure for selecting processing elements in that coordinatesystem. In Section III, we demonstrate the performance ofNLEQ on both Maxim and Analog Devices ADCs, and inSection IV we conclude with a brief summary.

II. NONLINEAR EQUALIZER CONSTRUCTION

To construct an NLEQ architecture, we will first developa horizontal coordinate system representation in a polynomialbasis, and then describe two separate optimization techniquesfor identifying a computationally efficient NLEQ architectureusing components from that basis.

A. Polynomial Basis

The nonlinear system response of an ADC can often bedescribed with the P th-order (truncated) polynomial seriesexpansion [5]

yNL(n) =P∑p=2

hp[x(n)], where

hp[x(n)]︸ ︷︷ ︸yp(n)

=Np−1∑m1=0

· · ·Np−1∑mp=0

hp(m1, . . . ,mp)p∏l=1

x(n−ml) (1)

where Np is the memory depth in each dimension of thepth-order Volterra kernel and hp(m1, . . . ,mp) are the pth-order kernel coefficients. The number of non-redundant terms

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in yNL(n) is∑Pp=2

(Np+p−1

p

). Using N samples of x(n),

equation (1) can be rewritten in matrix form as

yp = Xp(x)hp (2)

with

yp =[yp(n) . . . yp(n−N + 1)

]T(3)

where each of the(Np+p−1

p

)columns of Xp(x) can be

represented by[ ∏pl=1 x(n−ml) . . .

∏pl=1 x(n−ml −N + 1)

]T.

The full series can now be expressed in matrix form as

yNL = X(x)h (4)

with nonlinear convolution matrix X(x) =[X2(x) . . . XP (x)

]and h =

[hT2 . . . hTP

]T.

It was shown in [6] that a large class of nonlinear systemscan be approximated with arbitrarily small error using thepolynomial representation in (1). However, this comes withthe disadvantage that a relatively large number of parameters(factorial in N and p) are needed to represent systemswith modest polynomial order and memory. To reduce thecomputational complexity when using (1) to model ADCnonlinearities, we develop an efficient representation of (1) ina new coordinate system.

The kernels in equation (1) can be rewritten in terms ofelements of a horizontal coordinate system (HCS) in which apth-order processing element (PE) is formulated as

yHCSp (n, α2, . . . , αp) =Np+i1−1∑m=i1

h(m,α2, . . . , αp)x(n−m)︸ ︷︷ ︸h(n;α2,...,αp)?x(n)

p∏l=2

x(n−αl), (5)

where i1 is used to center the data over the taps of the filter.Hence, it is possible to sum all the pth-order HCS PEs toobtain the pth order kernel

yp(n) =Np+i2−1∑α2=i2

. . .

Np+ip−1∑αp=ip

yHCSp (n, α2, . . . , αp) (6)

where the variables ik in (5) are used to center the data over thetaps of the multidimensional filter. Equation (5) geometricallycorresponds to coefficients selected along a single horizontal(m) dimension while the other dimensions of the pth-orderkernel (α2 to αp) remain fixed. The HCS representation has avery appealing interpretation, which is that we can represent(1) as the sum of one-dimensional convolutions multiplied bythe product of time-delayed values of the input.

Let the data matrix associated with the jth HCS PE of orderpj be defined as XPEj with the cth column given by(

XPEj (x))c

= x(n− cj − i1,j) ◦ x(n− αj,2) ◦ · · ·· · · ◦ x(n− αj,pj

),

for cj = 0, 1, . . . , NPEj

taps − 1 (7)

Σ

h

AnalogIn

NonlinearEqualized

Output

-

Coefficients

XP

Nonlinearcombinations

of the data

ADCADC Delay

NLEQ

Fig. 1. Nonlinear equalization of ADC distortions by modeling andsubtraction

where x(n) =[x(n) x(n− 1) . . . x(n−N + 1)

]T, ◦

represents the Hadamard product, N is the number of samples,NPEj

taps is the number of filter taps in HCS processing elementj and pj ∈ {2, 3, . . . , P}. Using (7), we can formulate anapproximation to (1) in which

y(n) =[XPE1 XPE2 . . . XPEK

]

hPE1

hPE2

...hPEK

(8)

where XP =[XPE1 , . . . , XPEK

]is the data matrix associ-

ated with processing element set P . The construction in (8)both simplifies the computational burden of using sequentialestimation in architecture identification, described next, andprovides a regular structure for hardware implementation.

B. Nonlinear Equalizer Architecture Identification

Using (8), we can derive an equalizer architecture formitigating harmonic and intermodulation (nonlinear) distortiongenerated by an ADC, as illustrated in Fig. 1, where thenonlinear response of the ADC is estimated and subtractedfrom an appropriately delayed version of the ADC output. Theobjective of this section is to present techniques that enablethe construction of a polynomial equalizer using a minimumset of PEs by efficiently searching over the multidimensionalsignal space.

1) Forward-Backward Sequential Estimation in Architec-ture Identification: In this section we develop a sequentialalgorithm for selecting PEs that minimize the mean squareerror, ε. Let ε(y;P) = minh ‖y−XPh‖2 denote the modelingerror, where XP =

[XPE1 , . . . , XPEL

], P is the set of PEs in

the architecture with cardinality |P| = L, and y is the vector ofADC output samples. Let P ⊂ X , where X is the set of user-defined candidate processing elements with |X | � L, then thepseudo-code for sequentially selecting processing elements toconstruct an NLEQ architecture is shown in Fig. 2. The totalnumber of processing elements, |X |, can be adjusted accordingto computational considerations; the PEs that comprise a setare unique in their polynomial order, delay values and filter

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initialize P = ∅while ε(y;P) ≥ δ {

// Add Processing Elementswhile |P| ≤ k {p← arg minp∈X ε(y;P ∪ {p})P ← P ∪ {p}, X ← X\{p} };

// Remove Processing Elementswhile |P| ≥ m {p← arg minp∈P ε(y;P\{p})P ← P\{p}, X ← X ∪ {p} };

};

Fig. 2. Pseudo-code for forward-backward sequential estimation. Each newPE j from the set X that yields the best minimum mean square error (MMSE)performance in combination with the previous j − 1 PEs is added to the setP in the forward stage. In the backward stage, a PE is removed one at atime from P and placed back into X such that the PE removed has the leastimpact on MMSE performance.

coefficients. The parameter δ in the outer loop of the pseudo-code is a threshold for the MMSE; alternatively a fixed numberof iterations can be used. In either case, the parameters kand m, where m ≤ k, are user defined and can change fromiteration to iteration.

2) Global Estimation in Architecture Identification: Analternative to forward-backward sequential estimation is toformulate the problem of NLEQ architecture identification as aconstrained optimization, in which the constraints are imposedto insure a computationally efficient solution. In [7], basispursuit was used to find a sparse set of coefficients in a linearsystem identification problem. However, unlike linear systems,the size of the convolution matrix X(x) representing the non-linear combinations of the input data can grow prohibitivelylarge. One approach to reducing the dimensionality leveragesthe following theorem:

Theorem 1: Let X ∈ RN×M, with N � M , be a non-singular matrix whose columns correspond to the p-foldproducts of the input, and let y correspond to some vectorin RN . Then there exists a projection matrix P ∈ RM×Nwith X = PX ∈ RM×M, and hence an orthogonal projectionmatrix P⊥bX bX = (I − X(XT X)−1XT ), such that yTP⊥bX bX y =yTP⊥XXy, where y = Py.

Proof: Consider a matrix X with full column rank havingsingular value decomposition

X =[U U

] [ D0

]V T ∈ RN×M,

with U ∈ RN×M, and N �M . Further, consider the solutionsto the least squares problems h = arg minh ‖y − Xh‖and h∗ = arg minh ‖Py − PXh‖, where P is a M × Nmatrix that projects any N -dimensional vector v down to anM -dimensional space. Then if P = UT , h∗ = h.

It is possible to use the left singular vectors to reducethe dimensionality without any loss in performance; however,the computational cost of computing the SVD (singular valuedecomposition) of a large matrix, coupled with the fact that

Theorem 1 is only valid for N ≥M , is of very little practicalvalue. We propose two techniques for reducing the dimen-sionality in a computationally efficient fashion. To reduce thedimensionality of the rows, we leverage the following lemma[8].

Lemma 1 (Johnson-Lindenstrauss): For any 0 < ε < 1 andany integer n, let K be a positive integer such that K ≥4(ε2/2− ε3/3)−1lnn. Then for any set V of n points in RN ,there is a map f : RN → RK such that for u, v ∈ V

(1− ε) ‖u− v‖ ≤ ‖f(u)− f(v)‖ ≤ (1 + ε) ‖u− v‖ . (9)Extensions to the Johnson-Lindenstrauss lemma [9] have usedconcentration measure theory to show that if points in a vectorspace are projected onto a randomly selected subspace ofsuitably high dimension then the distances between the points(in a Euclidean sense) are approximately preserved. Therefore,we can construct a matrix Φ ∈ RK×N whose elements are arerandomly drawn from a Gaussian distribution (N (0, 1√

K)),

so that the projected matrix given by ΦX ∈ RK×M whereK < N does not impart a significant error during architectureidentification.

To further reduce the size of the matrix X , its columnscan be pruned by projection and subtraction. First, the Mcolumns of ΦX are unit normalized so that each column canbe considered as the coordinate of a point on the surface ofa unit hypersphere. From the set of M vectors, L vectors areselected one at a time, such that the mth vector chosen has thehighest correlation with Φy(m−1) after the m− 1 projectionsof the previously selected columns have been subtracted off,that is

Φy(1) = Φy −⟨(ΦX)i(1),Φy

⟩(ΦX)i(1)

Φy(2) = Φy(1) −⟨(ΦX)i(2),Φy(1)

⟩(ΦX)i(2)

...Φy(L) = Φy(L−1) −

⟨(ΦX)i(L),Φy(L−1)

⟩(ΦX)i(L).

(10)

In (10), the symbol (ΦX)i(m) is the mth column of ΦX thathas the highest correlation with Φy(m−1), whose column indexis given by i(m), and 〈x, y〉 is the dot product of x and y.The goal of projection and subtraction is to find the points onthe unit hypersphere with widest angular separation that havea significant projected component on the received data. Weuse projection and subtraction to reduce the dimensionalityof the data so that the subsequent constrained optimizationis computationally tractable. Defining the matrix Θ ∈ RM×Lsuch that XΘ keeps only the columns of X we get fromprojection and subtraction in (10), the constrained optimizationto find a sparse NLEQ architecture is given by

h = arg minheTh

s.t. ‖Φy − Φ [XΘ | −XΘ]h‖2 ≤ εh ≥ 0, (11)

where e = [1 1 . . . 1]T ∈ R2L and h+, h− ∈ RL, withh = [hT+ hT−]T and hopt = h+ − h−. Equation (11) is easilysolved using second order cone programming (SOCP) [10].The basis pursuit (BP) cost function, along with the constraint

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Noise

(Dither)

Po

we

r C

om

bin

er

Signal

Generators

Control Computer pNLEQ Processing

Logic Analyzer

(Data Capture)Filter

ADC

Temperature

Chamber

Fig. 3. A simplifi ed view of the the Lincoln Laboratory pNLEQ testbed.The testbed consists of three Agilent E8257D tone generators connected toan analog power combiner, an Agilent 16702B Logic Analyzer used for datacapture, a temperature chamber where the time-interleaved ADCs are seated,and a Windows-based PC running Matlab to control the instrumentation forexcitation and data capture.

that h ≥ 0, in (11) is an 1 norm that favors sparse solutions[7]. The scalar regularization parameter in the constraintbalances the residual 2 reconstruction error, that is, it ensuresthat the sparse solution hopt whose nonzero entries span thecolumns of ΦXΘ is in the cone of feasible solutions. Note thatunlike forward-backward sequential estimation, the columnsselected (non-zero entries of hopt) correspond to single tapprocessing elements, where individual PEs are not connectedwith any specifi c coordinate system (e.g., horizontal, etc.).

III. MEASURED RESULTS

A. Test Setup

To evaluate NLEQ performance, we used the MIT LincolnLaboratory NLEQ testbed depicted in Fig. 3. The analogoutputs of three Agilent E8257D tone generators were com-bined, filtered, and injected into an ADC that was seated in atemperature controlled chamber set to 20◦C. A Windows PCrunning MATLAB scheduled the tone generators, controlledthe Agilent 16702B logic analyzer that was used to capturedata at the output of the ADC, and transferred data from thelogic analyzer’s memory back to the PC’s hard drive. Wepresent NLEQ performance results using an 8-bit and a 14-bitADC: the Maxim MAX108, sampling at a rate of 1500 MS/sand the Analog Devices AD6645 sampling at a rate of 105MS/s.

TABLE ISOURCE EXCITATION AND VERIFICATION PARAMETERS.

ADC Excitation Verifi cation

Maxim MAX10839 one-tone sets65 two-tone sets

262 three-tone sets

350 one-tone sets280 two-tone sets

2340 three-tone sets

Analog Device AD664539 one-tone sets60 two-tone sets

262 three-tone sets

350 one-tone sets280 two-tone sets

2340 three-tone sets

B. Training and Verifi cation

We trained both devices with a series of one-, two-, andthree-tone sets. We spaced the tones across a 40 MHz bandof interest for the AD6645, and a 500 MHz band for theMAX108. The number and type of tone sets used to excitethe linear and nonlinear modalities in the ADCs are listed inTable I. In all cases, the NLEQ architecture’s computationalcomplexity was constrained so that it could be effi cientlyimplemented in hardware.

After identifying an NLEQ architecture with the techniquespresented in Section II-B.1, we cross-validated NLEQ perfor-mance using a sequence of verification tone sets. These one-,two-, and three-tone verification sets were entirely differentfrom the training sets used to derive the coeffi cients. Wequantify dynamic range by taking the mean of the individualSFDR/IFDR measurements for each of the verification tonesets. This is the mean dynamic range (MDR) performancemetric, which we measure in dBFS.

The individual tone generators did not impart intermod-ulation distortion, however, each tone generator did impartharmonic distortion. This makes it diffi cult to separate thedistortions generated by the RF receiver from those of theexcitation used during training. By operating in the secondNyquist zone (IF sampling), harmonic distortions fell out ofband and were filtered off by the anti-aliasing filter precedingthe ADC. Alternatively, if first Nyquist zone (base band) sam-pling were required, ignoring harmonic distortions in trainingand instead spacing two tones very close to one another tomimic harmonic distortion yielded satisfactory results.

C. Performance

Table II and Figures 4 and 5 illustrate the measured perfor-mance of NLEQ operating on data from the MAXIMMAX1088-bit 1.5 GSps ADC, and the Analog Device AD6645 14-bit105 MSps ADC. The MDR improvement of the MAX108 afterNLEQ is roughly 22 dB, with and equalizer complexity of 217operations per sample. The equalizer was constructed usingforward/backward sequential estimation which selected 8 HCSPEs on the forward pass. Only marginal improvement (< 1 dB)using backward iterations was achieved. Basis pursuit yieldedthe same performance/computational complexity results asforward sequential estimation, indicating the robustness ofthe forward sequential estimation process in selecting HCSPEs. The MDR improvement of the AD6645 is only 10 dB,however, as is evident from Fig. 5, the distortions are virtuallypushed down to the noise fl oor. The NLEQ architecture for theAD6645 required 9 HCS processing elements.

TABLE IIEQUALIZATION RESULTS

Device MDR(dBFS)

MDRImprovement

(dB)

Operationsper Sample

MAX108 -51 -73 217AD6645 -89 -100 147

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> 25 dB

NLEQ Performance on the MAX108

Fig. 4. Representative response of the Maxim 8-bit MAX108 to a two tone stimulus. The plot to the left represents the unequalized output of the ADC, withboth intermodulation and harmonic distortions present. The plot to the right represents the equalized response of the MAX108 after running NLEQ, whichdemonstrates roughly 25 dB improvement in dynamic range.

NLEQ Performance on the AD6645

60 65 70 75 80 85 90

-100

-80

-60

-40

-20

0

Frequency (MHz)

Power (dBFS)

Before NLEQ

Frequency (MHz)

Before NLEQ

60 65 70 75 80 85 90

-100

-80

-60

-40

-20

0

Frequency (MHz)

Power (dBFS)

After NLEQ

~10 dB

Pow

er (d

BFS

)

Frequency (MHz)

Pow

er (d

BFS

)After NLEQ

Fig. 5. Representative response of the 14-bit AD6645 to a three tone stimulus. The plot to the left represents the unequalized output of the ADC, withboth intermodulation and harmonic distortions present. The plot to the right represents the equalized response of the AD6645 after running NLEQ, whichdemonstrates roughly 10 dB improvement in dynamic range.

IV. SUMMARY

In this paper we developed a computationally efficientarchitecture for nonlinear equalization (NLEQ) in a new coor-dinate system to improve the dynamic range of RF receivers.We demonstrated a dynamic range improvement of over twoorders of magnitude on the 8-bit, 1.5 MSps Maxim MAX108,and an order of magnitude improvement in the dynamic rangeof the 14-bit, 105 MSps Analog Devices AD6645. In bothcases, the computational complexity of NLEQ was on theorder of 200 operations per sample. We also demonstrated thatNLEQ was capable of significantly improving the linearity ofa device (AD6645) that already started out with a very high(90 dB) dynamic range.

REFERENCES

[1] A. Abidi, “The path to the software defined radio receiver,” IEEE Journalof Solid State Circuits, pp. 954–966, 2007.

[2] F. Adamo et al., “A/D converters nonlinearity measurement and correc-tion by frequency analysis and dither,” IEEE Trans. on Instrumentationand Measurement, pp. 1200–1205, 2003.

[3] D.R. Morgan et al., “A generalized memory polynomial for digital pre-distortion of RF power amplifiers,” IEEE Trans. on Signal Processing.,pp. 3852–3860, 2006.

[4] L. Ding et al., “A robust digital baseband predistorter constructedusing memory polynomials,” IEEE Trans. Communications, pp. 159–165, 2004.

[5] V. Mathews, “Adaptive Volterra filters using orthogonal structures,”IEEE Signal Processing Letters, pp. 307–309, 1996.

[6] G. Palm and T. Poggio, “The Volterra representation and Wienerexpansion: Validity and pitfalls,” SIAM Journal of Applied Mathematics,pp. 195–216, 1977.

[7] E. Candes and T. Tao, “The Dantzig selector: Statistical estimation whenp is much larger than n.” To appear in Annals of Statistics.

[8] W. Johnson and J. Lindenstrauss, “Extensions of Lipschitz mappingsinto a Hilbert space,” Conference in Modern Analysis and Probability,pp. 186–206, 1977.

[9] D. Achlioptas, “Database friendly random projections,” Proceedings ofthe ACM Symposium on the Principles of Database Systems, pp. 274–281, 2001.

[10] F. Alizadeh and D. Goldfarb, “Second order cone programming.” RUT-COR Research Report, Nov. 2001.

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