spectral bias in adaptive beamforming with narrowband...

Spectral Bias in Adaptive Beamforming withNarrowband Interference

Brian D. Jeffs,Senior Member, IEEE, and Karl F. WarnickSenior Member, IEEE

Abstract— It is shown that adaptive canceling arrays whichtrack interference by regular updates of the beamformer weightscan introduce a spectral null at the excised interference fre-quency. This PSD estimation bias effect we call “spectral scoop-ing” is most prominent for narrowband interference (i.e. occu-pying only a few spectral bins at the desired PSD estimationresolution). Scooping is problematic in radio astronomy wherebias in either the weak signal or noise floor spectra can corruptthe observation. The mathematical basis for scooping is derived,and an algorithm to eliminate it is proposed. Both simulatedand real data experiments demonstrate the effectiveness oftheproposed algorithm.

Index Terms— Adaptive arrays, Interference suppression, Ra-dio Astronomy, Spectral analysis

I. I NTRODUCTION

A. Background

This paper considers using a sensor array to form a powerspectral density (PSD) estimate for a signal of interest (SOI)in the presence of strong narrowband interference. We assumethe processing architecture of Figure 1. Though the problemisgeneral, our motivation arises from radio astronomy (RA) ob-servation, where the SOI is very weak and there is frequentlydisabling man-made interference. The array could be a set ofmany separate bare antennas as proposed for some new largeRA instruments such as LOFAR [1]–[3], or a compact arrayused as an adaptive feed at the focal point of a large dishtelescope, as under study for the Green Bank Telescope [4]–[8]

We will show that adaptive array interference cancellationcan have an undesired effect of over-canceling to the pointof notching out the baseline noise or SOI spectrum. Thisphenomenon which we have dubbed “spectral scooping” canbe seen in Figures 2, 3, 6, and 7 in following sections. Inthese examples, interference has been effectively cancelled bythe adaptive spatial filter, but so has some of the noise andsignal spectrum at the same frequencies.

The problem is more prevalent when interference, eventransiently, has a strong very narrowband component (spanningonly a few PSD estimation bin widths) as may be encounteredwith some navigational beacons, pilot tones, dominant carriersduring quiet passages in AM modulation, and a variety of othermodulation schemes. The scooping behavior does not occurwhen adaptive beamforming weights are computed from exactarray covariance matrices, but it will be shown that sampleestimation error terms induce a coupling between a strong

This work was funded by National Science Foundation under grant numberAST - 0352705

Authors are with the Dept. of Elec. and Comp. Eng., Brigham YoungUniversity, Provo, UT 84602 USA ([email protected], [email protected]).

η0[n]

x1[n]

x2[n]

+

×

w0

×

w1

×

w2

×

wP-1

y[n] = wHx[n]

xP-1[n]

s[n]

θ

Space

signal

of interest

x0[n]

ηP-1[n]

Noise :

Interference

PSD Estimator

(e.g. periodogram)

Sy

d1[n]

Fig. 1. Block diagram of the basic array PSD estimation system.

interferer and noise plus SOI which causes this unexpectednotch in the desired PSD.

Of course, placing a null in the PSD biases the estimate andcan be as problematic as is incomplete interference canceling.For radio astronomy, cancellation biases have been a sourceof astronomers’ reluctance to adopt adaptive beamformingtechniques [9]–[11]. The focus of this paper is understandingthe causes of spectral scooping, and presenting a solution.

The remainder of the paper is organized as follows. SectionI-B and I-C describe mathematical models for the sensor arrayand array processing and spectral estimation architectures.Section II defines the “spectral scooping” effect which is thecentral subject of this paper, provides an approximate analysisof its origin, and introduces a proposed algorithm to mitigatethe effect. Section III presents both simulated and real dataexperimental results to demonstrate spectral scooping andper-formance of the proposed mitigation algorithm. Conclusionsand observations are found in Section IV. Finally, extendedderivations, mathematical notation, operators, and functiondefinitions are provided in the Appendix.

B. Signal Model

Consider theP element sensor array of Figure 1, whichproduces a lengthP ×1 complex baseband sample data vector

x[n] = as[n] +

Q∑

q=1

vq[n]dq[n] + η[n] (1)

wheres[n] is the SOI,η[n] is noise, anddq[n] is one ofQ“detrimental” interfering sources. Vectorsa andvq[n] are nor-malized array spatial responses tos[n] anddq[n] respectively.Assume thats[n], dq[n], and η[n] are mutually independentwide sense stationary zero mean random processes with vari-ancesσ2

s , σ2dq

, and covarianceσ2ηI respectively. Assumes[n]

is spatially fixed, soa is constant. Due to interferer motion,

2 SUBM. IEEE TRANSACTIONS ON SIGNAL PROCESSING, MAY 23, 2008

vq[n] varies, but overL time samples called the “short termintegration (STI)” window, it is approximately constant. Wewill treat vq[n] as a deterministic, though unknown, functionof time which depends on interferer motion dynamics andthe fixed element spatial response patterns. Examples of suchdeterministic motion include interference seen at a radio tele-scope from downlink transmissions of a man-made satellite,or from a fixed ground transmitter where apparent motion isdue to telescope re-pointing to track sidereal motion of a deepspace source of interest.

The single–complex–weight–per–sensor beamformer archi-tecture of Figure 1 assumes relatively narrowband array op-eration (or subband processing) such that BW≪ D/c whereBW is the processing bandwidth,D is the full array aper-ture diameter, andc is the speed of wave propagation. Atmicrowave frequencies, BW can be as much as 20 MHz ormore for a radio astronomical focal plane array. Thedq[n]are each assumed to occupy only a small fraction of the totalBW, thus spanning only a few frequency bins at the desiredPSD estimator spectral resolution scale. Each interferer mayhowever be at a distinct frequency. This near-single-frequencyinterference scenario is the condition that leads to spectralscooping during adaptive interference cancellation.

It is desired to estimate the frequency sampled temporalPSD of s[n] (possibly in combination with the noise PSD)without the corrupting effect of interferersdq[n]. One straight-forward approach is to compute a beamformer steered tos[n]followed by a sample PSD estimator, as in Figure 1. Let thebeamformer output be

y[n] = wHj x[n], j = ⌊n/L⌋ (2)

wherewj is an adaptive weight vector computed for thejthSTI.

A PSD estimate,Sy[k], can be formed fromy[n] usingWelch’s overlapping windowed averaged periodogram [12],

Sy[k] =γ

M

M−1∑

m=0

∣

∣

∣

∣

∣

N−1∑

n=0

g[n]y[n + m(N − O)]e−i2πkn/N

∣

∣

∣

∣

∣

2

(3)where g[n] is a length N spectral shaping window (e.g.Hamming). For a fixed beamformer withwj = w, (3) canbe expressed in matrix-vector form as

STy = [Sy[0], · · · , Sy[N − 1]]

=γ

M

M−1∑

m=0

∣

∣DFTN{yTm ⊙ gT }

∣

∣

⊙2

=γ

M

M−1∑

m=0

∣

∣DFTN{wHXmG}∣

∣

⊙2

=γ

M

M−1∑

m=0

∣

∣wHDFTN{XmG}∣

∣

⊙2, where (4)

yTm =

[

y[m(N − O)], · · · , y[m(N − O) + N − 1]]

Xm =[

x[m(N − O)], · · · ,x[m(N − O) + N − 1]]

g = [g[0], · · · , g[N − 1]]T , G = Diag{g}M is the number of DFT averaging windows used,O isthe overlap between successive windows, and the DFT (or

equivalently FFT) operates separately along all matrix rowsof its argument. Settingγ = 1/tr{GG} properly scales thePSD estimate.

Exploiting stationarity over an STI and assuming spatiallywhite (i.i.d.) noise, define the time dependent array autocor-relation matrix as

Rx[n] = E{x[n]xH [n]} ≈ Rx,j, j = ⌊n/L⌋

Rx,j = σ2saa

H +

Q∑

q=1

σ2dq

vq,jvHq,j + σ2

ηI

where vq,j = vq[jL]. Signal levels, variances (e.g.σ2s and

σ2dq

), signal to noise ratios (SNR), and interference to noiseratio (INR) in the sequel are defined at sampled complexbaseband rather than at the antenna.

The L-point, jth STI sample estimator ofRx,j is

Rx,j =1

L

(j+1)L−1∑

n=jL

x[n]xH [n] (5)

Note that we do not in general requireL = N , and that unlikethe DFT windows, STIs do not overlap.

C. Subspace Projection Adaptive Beamformer

Beamformer weights,wj , are computed fromRx,j suchthat interferers will be cancelled while steering a high gainmainlobe towards[n]. The beamformer is updated each STIto track interferer motion so that

wHj Vj ≈ 0 where

Vj = [v1,j , · · · ,vQ,j ].

Any number of algorithms can be considered, includingLCMV, MVDR, maximum SNR, and subspace projectionspatial filtering [13]–[20]. Though we have observed spec-tral scooping in each of these mentioned algorithms, in ourapplication we prefer subspace projection since its zero forc-ing properties often form deeper cancellation nulls. This isparticularly desirable in the RA where signals of interest aretens of dB below the noise level, so the following analysiswill focus on this architecture. The time-varying (on the STIscale) subspace projection weight vector is computed as

wj = Pjw (6)

wherew is a unit length deterministic beamforming weightvector with the desired quiescent (interference-free) responseand Pj is an estimate ofP⊥

Vj, the perpendicular subspace

projector forVj . When interference dominates, the partitionedSVD approach can be used to compute

Pj = I− Ud,jUHd,j (7)

whereUd,j contains the normalized eigenvectors correspond-ing to theQ largest eigenvalues in the decompositionRx,j =UjΛUH

j , Uj = [Ud,j | Us+η, j], andΛ is the diagonal matrixof ordered eigenvalues.

JEFFS AND WARNICK: UNBIASED ARRAY PSD ESTIMATION 3

II. SPECTRAL SCOOPING

Causes of spectral scooping are subtle. All cancellation isperformed by spatial processing across the channels usingwj , which remains constant over an entire STI. However,scooping is a time domain phenomenon that appears in theDFTs used to compute the PSD estimate. The cause is anunderlying coupling between the spatial interference cancelerand the temporal PSD estimate that arises through the sampleSTI covariancesRx,j.

We have observed scooping in a wide range of simulatedand real data experiments for PSD estimation with an inter-ference canceling array using several adaptive beamformingalgorithms. A simplified demonstration program to illustratescooping behavior is available in [21]. This MATLAB (R)script shows how scooping occurs with both LCMV andsubspace projection beamforming, and how the proposed al-gorithm corrects scooping.

The following section presents a approximation analysis togive insights into why scooping occurs from the perspectiveof subspace projection beamforming.

A. Analysis

To illustrate the mathematical basis for scooping, considera single narrowband interferer at frequencyωd. Figure 3shows scooping effects for both subspace projection and singleconstraint LCMV (i.e. MVDR) beamformers.

We assume that

σ2d ≫ σ2

η ≫ σ2s ,

σd σ2η/√

L ≫ σ2s (8)

These requirements may appear restrictive, but for the radioastronomy application of interest, the signal power is alwaysless than the noise power, andσ2

d ≫ σ2η is common. Also, the

second assumption is usually satisfied with typical parametervalues, for example withL = 1024, INR = 20 dB, and SNR =-10 dB. Scooping can also occur when these assumptions arenot strictly satisfied, but these will be used in developing thefollowing approximation analysis which illustrates the genesisof scooping.

Define the SOI plus noise signalz[n] = as[n] + η[n], so

x[n] = vj σd eiωdn + z[n], (9)

then

Rx,j = Rd,j + Rz = σ2d vjv

Hj + σ2

saaH + σ2

η I. (10)

Using (9) and (10) in (5), the sample covariance for STIj can

be expanded as

Rx,j =1

L

(j+1)L−1∑

n=jL

(

σ2d vjv

Hj + σd vj eiωdnzH [n]

+ σd z[n]vHj e−iωdn + z[n]zH [n]

)

= σ2d vjv

Hj +

σd

Lvj

(j+1)L−1∑

n=jL

zH [n] eiωdn

+σd

L

(j+1)L−1∑

n=jL

z[n] e−iωdn

vHj

+ σ2saa

H + σ2ηI + Ej

= Rd,j +σd

L

(

vj nHj + njv

Hj

)

+ Rz + Ej , where

(11)

nj = e−iωdjLL−1∑

n=0

z[n + jL] e−iωdn (12)

Sample error in the SOI plus noise covariance estimate isexpressed byEj , such that

Rz,j =1

L

∑

n

z[n]zH [n] = Rz + Ej .

Equation (11) indicates that the terms involvingnj or Ej

account for all sample estimation error inRx,j. They arethe only random components inRx,j. From (12),nj can berecognized as the vector discrete-time Fourier transform forSTI windowj of z[n], evaluated atω = ωd. As will be shownbelow, this observation is key to understanding the cause ofspectral scooping.

For Gaussians[n] and η[n], LRz,j is distributed Wishart,which impliesE{Rz,j} = Rz, E{Ej} = 0, and stdv{Ej} =σ2

z/√

L element-wise, whereσ2z = σ2

s + σ2η [22]. Also, it is

easily shown thatE{nj} = 0 and stdv{nj} = O(√

Lσz′ )element-wise, whereσ2

z′ = σ2η + Ss[kd]. The PSDSs[kd] of

s[n] is evaluated at binkd which contains the interferencefrequencyωd, so σ2

η ≤ σ2z′ ≤ σ2

z , with the upper boundbeing achieved only if signal and interference have overlappingspectra. Given that they are zero mean, we can use the standarddeviations ofnj andEj for a fair comparison of relative sizesin their expected contributions toRx,j with respect to the otherdeterministic components. Accordingly, an order analysisofterms inRx,j, taken in sequence as they appear in (11), yields

Rx,j = O(σ2d) + O

(

σd σz′√L

)

+ O(

σ2η

)

+ O(

σ2s

)

+ O(

σ2z√L

)

(13)

where the “order of” relationship is taken element-wise inRx,j. Given the stated assumptions,σd σz′/

√L is much

greater than bothσ2s andσ2

z/√

L, so we may neglect the finaltwo terms. Indeed, any combination of signal power levels andL such that the first two terms dominate the last two will leadto scooping. The third term corresponds toσ2

ηI, which canalso be neglected since its constant diagonal form does not


affect the eigenvectors used to compute projection matrixPj .Thus we defineRx,j by neglecting all terms inRx,j whichhave negligible influence onPj

Rx,j = σ2d

[

vj vHj +

1

Lσd

(

vj nHj + njv

Hj

)

]

. (14)

We now exploit this approximation to evaluate the structureof Pj , as computed fromRx,j. It will be shown that at theinterference frequency,Pj is approximately orthogonal to thecombined interference, SOI, and noise signal. This producesa scooping null.

It should be noted that thoughPj andnj are random, thescooping phenomenon occurs per realization. When computinga PSD estimate, these quantities come from the same, singleSTI window realization ofx[n]. We have used statisticalarguments to assess relative sizes (on average) of terms inRx,j so negligible components could be dropped to simplifyanalysis. Now applying these average scale relationships in asingle realization, the effect of cross terms (i.e.vjn

Hj ) which

dominate sample estimation error can be studied.Consider eigenvectorsuj,k, k = 1, 2, of Rx,j, which

closely approximate the corresponding dominant eigenvectorsuj,k of Rx,j. In the strong interferer case, the identifiedinterferer subspace is dominated by the two distinct terms in(14). Sincevj andnj are the only linearly independent vectorsin the expression, the first two eigenvectors of this rank twomatrix are seen by inspection to have the form

uj,k = vj + ξknj (15)

whereξk is a scalar to be determined. It is shown in AppendixA using the approach of [23] that the exact solution for thedominant eigenvector of (14) is given when

ξ1 =2/(Lσd)

1 +

√

(

1 + 2Re{ρ}α

)2

− 4(

|ρ|2

α2 + βα

)

− i4 Im{ρ}α

whereα = σ2d‖vj‖2, β = 1

L2 ‖nj‖2, andρ = σd

L vHj nj .

For strong interferenceση/σd → 0, implying that Re{ρ}/α,|ρ|/α, β/α, and Im{ρ}/α → 0. So

limση/σd→0

ξ1 = 1/(Lσd)

and the dominant eigenvector is

uj,1 ≈ uj,1 = vj +1

Lσdnj . (16)

Using (7), the projection matrix for interference canceling isPj ≈ I − uj,1u

Hj,1/(uH

j,1uj,1), which, given (16), will beorthogonal tovj + nj/(Lσd).

Now consider the subspace-projection-based PSD estimatorfor this scenario, and for simplicity use non-overlapping (O =0) DFT windows which match the STI window length (soN = L and m = j), and a rectangular window (G = I andγ = 1/L). With these parameters, substituting (6) into (4) andusing the single interferer and SOI plus noise signal of (9)yields

STy =

1

LM

M−1∑

j=0

∣

∣

∣

∣

wHPHj DFTL

{

vjσdeiωd[n+jL]

+z[n + jL]

}∣

∣

∣

∣

⊙2

.

(17)

−3 −2 −1 0 1 2 3

−40

−35

−30

−25

−20

−15

Frequency in rad/sample

Pow

er/fr

eque

ncy

(dB

/rad

/sam

ple)

Noise dip = 0 dB

Noise dip = −12 dB

Noise dip = −36 dB

Fig. 2. Spectral scooping and interference leakage effects. For all curvesz[n] = η[n], i.e. noise only with no SOI. For the solid curve noise istemporally white. For the dashed and dash-dot curves noise is colored, with adip in the spectrum of -12 and -36 dB respectively near the interferencefrequency, ωd = π/4. Total noise power was the same for all curves.Subspace projection beamforming was used to cancel interference. Notescooping is prominent in the white noise case, and is overwhelmed byinterference leakage in the deepest dip case. Total in-bandpower INR was+10 dB for all curves, the interferer was at−60◦ relative to broadside for aseven elementλ/2 spaced ULA,L = N = 256, andO = 128.

where the DFT is over0 ≤ n ≤ L − 1. Let ωk = 2πk/N bethe kth FFT frequency bin, and pickkd such thatωkd

= ωd,the interference frequency. Then evaluating the DFT in (17)at ωkd

and using (12) yields

Sy(kd) =1

LM

M−1∑

j=0

∣

∣

∣

∣

∣

wHPHj

L−1∑

n=0

(

vjσdeiωd[n+jL]

+ z[n + jL]

)

e−iωdn

∣

∣

∣

∣

2

=1

LM

M−1∑

j=0

∣

∣

∣wHPH

j

(

LeiωdjLσdvj + eiωdjLnj

)

∣

∣

∣

2

=1

LM

M−1∑

j=0

∣

∣

∣

∣

LeiωdjLσd

∣

∣

∣

∣

2 ∣∣

∣

∣

wHPHj

(

vj +1

Lσdnj

)∣

∣

∣

∣

2

=Lσ2

d

M

M−1∑

j=0

∣

∣

∣

∣

wHPHj

(

vj +1

Lσdnj

)∣

∣

∣

∣

2

(18)

≈ 0

which is approximately zero due to the orthogonality ofPj

andvj + nj/(Lσd). Therefore a null is placed atωd, in binkd of Sy. The desired, interference free PSD bin,Sz(kd),is replaced with a null. At frequenciesωk 6= ωd, the noisespectrum term from the DFT in (18) differs fromnj , soorthogonality withPj is lost andSz(k) is properly estimated.

We are aware of one condition where the stated assumptionscan be met but scooping will not be apparent. In (14) wehave neglectedEj using order arguments to simplify analysis,but in practice this term causes an error inPj that reducesthe interference rejection ratio (IRR). Since cancellation is


not perfect, interference leakage due toEj can “fill in” thescoop null, and even appear as a peak. For scooping to beevident, there must be something to scoop. In other words,the underlying SOI plus noise spectrumSz(kd) must be higherthan the residual interference level. Figure 2 illustratesbothscooping as predicted by (18), and conditions where theassumptions are satisfied, but interference leakage overwhelmsthe scoop null. As shown, this does not occur unless the levelfor Sz(ω) is much lower nearω = ωd than the average levelacross the band.

To summarize, spectral scooping is caused by some finitesample error terms inRx,j which appear as a cross correlationbetween interference and the signal plus noise sequencez[n].The beamformer combines the individual array element datastreams so that the residual interference component cancelsthe other signals present in the interference frequency bin.This result generalizes straightforwardly to other adaptivebeamforming algorithms such as LCMV and maximum SNR.

B. Correcting Spectral Scooping

Consider a subspace projection PSD estimator of the form

STy =

1

LM

M−1∑

j=0

∣

∣

∣wH(Pa

j )H DFTL {Xj}∣

∣

∣

⊙2

(19)

where like (17) we have assumedL = M , O = 0, andG =I, but have replacedPj with Pa

j which is computed usingonly part of the samples in the STI window. Specifically, letNj = {n | jL ≤ n ≤ (j + 1)L − 1} be theL–element setof all sample indices in thejth STI. PartitionNj into non-overlapping subsetsN a

j and N bj with La and Lb elements

respectively such thatNj = N aj ∪ N b

j , N aj ∩ N b

j = φ (theempty set), andL = La + Lb. Sample order is arbitrary. Wemay then define

Rax,j =

1

La

∑

n∈Naj

x[n]xH [n] (20)

which is a sample estimate for STIj computed from anLa

sample subset of the STI window. Following (15) and Ap-pendix A, it can be shown thatRa

x,j has dominant eigenvectorua

j,1 ≈ vj +1/(Laσd)naj , wherena

j is defined as in (12), butsumming only over samplesn ∈ N a

j .

Let projection matrixPaj be computed fromRa

x,j as

Paj = I− ua

j,1(uaj,1)

H/((uaj,1)

H uaj,1). (21)

It is shown in Appendix B that this projection matrix is closelyapproximated by

Paj ≈ I− vj v

Hj −

vj(naj )H + na

j vHj

Laσd‖vj‖

where vj = vj/‖vj‖. Using Paj in place ofPj in (18), the

projection product becomes

Paj

(

vj +1

Lσdnj

)

≈(

I − vjvHj −

vj(naj )H + na

j vHj

Laσd‖vj‖

)

(vj +1

Lσdnj)

= vj − vj −na

j

Laσd+

naj + nb

j

Lσd−

vHj nj

Lσdvj

− (nj)H vj

Laσdvj −

(vHj nj)n

aj + ((na

j )Hnj)vj

LaLσ2d‖vj‖

≈nb

j − Lb

Lana

j −(

vHj nj + L

La(na

j )H vj

)

vj

Lσd(22)

where we have usednj = naj + nb

j , L − La = Lb, and thefinal term from the previous line was dropped in (22) due todivision by largeLaL.

Substituting (22) into (18) and taking expectation yields

E{Sy(kd)}

≈ 1

LM

M−1∑

j=0

E

{∣

∣

∣

∣

wHnbj −

Lb

LawHna

j

−(

vHj nj +

L

La(na

j )H vj

)

wH vj

∣

∣

∣

∣

2}

≈ 1

LM

M−1∑

j=0

E

{

∣

∣

∣

∣

wHnbj −

Lb

LawHna

j

∣

∣

∣

∣

2}

≈ 1

LM

M−1∑

j=0

wHE{

nbj(n

bj)

H}

w +

L2b

L2a

wHE{

naj (na

j )H}

w

≈ 1

LM

M−1∑

j=0

wHwLbσ2z′ + wHw

L2b

Laσ2

z′

≈ Lb

Laσ2

z′ =L − La

Laσ2

z′ , 0 < La ≤ L. (23)

The second line follows becausewH vj is small. Notethat ‖w‖ = ‖vk‖ = 1 and vj is the normalized arrayresponse vector for the interferer. Thus in the typical scenariowhere the interferer is observed in the sidelobe pattern ofthe quiescent beamformer,wH vj is 20 dB or more belowthe mainlobe response. In line three we have assumedna

j

and nbj are uncorrelated. This is not strictly true since they

represent Fourier transforms of disjoint sample sets ofz[n] =as[n] + η[n]. Typically s[n] is colored, and thoughη[n]dominates inz[n], we have not assumed it is temporally white.However, in the usual scenario the correlation time ofz[n] ismuch less than largeL, so choosingN a

j and N bj as non-

interleaved sets (e.g. first and last halves of the STI windowrespectively) yields highly uncorrelatedna

j andnbj . Line four

exploits spatially i.i.d. noise across array elements, andthefact thatE{|na

j |2} = La σ2z′ , and E{|nb

j |2} = Lb σ2z′ . Line

five relies on‖w‖ = 1 and some algebra.For La = Lb = L/2 the estimator in (23) is approximately

unbiased by interference scooping, withE{Sy(kd)} = σ2z′ .


−3 −2 −1 0 1 2 3

−25

−20

−15

−10

−5

0

5


Pow

er/fr

eque

ncy

(dB

/rad

/sam

ple)

Corrected subspace proj.

Fixed beamformer

Subspace proj.

LCMV

Fig. 3. Spectral scooping and the correction algorithm. Narrowband interfer-ers are atω ≈ −0.5 andω ≈ 0.3, each with +10 dB INR relative to total in-band noise power. The sawtooth shaped SOI is at0 ≤ ω ≤ 1. Conventionalbeamforming did not cancel interference. Subspace projection and LCMVbeamformers both show significant spectral scooping, though typically theLCMV null is not quite as deep. Insets show detail of the corrected PSD.

Note thatσ2z′ is the desired SOI plus noise PSD atωd. This is

the basis for our proposed algorithm to eliminate scooping.On the other hand, forLa = L, Lb = 0, E{Sy(kd)} =0, which is the full scooping case predicted by (18). Aresidual interference peak corrupts the PSD whenLa < L/2.These behaviors predicted by (23) are verified in simulationexperiments presented below.

The scooping bias corrected estimator has the disadvantagethat only half of the data is used in forming the adaptivebeamformer weight,wj (though all data is used in the DFTcomputation). Thuswj has increased estimation error jitter.The following algorithm partially mitigates this limitation anduses all the data to compute projections matrices. LetN a

j

contain the firstL/2, and N bj the lastL/2 sample indices

in STI j. FormPaj andPb

j usingRax,j andRb

x,j computed asin (20) fromN a

j andN bj respectively. Then

STy =

γ

2M

M−1∑

m=0

(

∣

∣

∣wHPa

j DFTN{XmG}∣

∣

∣

⊙2

+∣

∣

∣wHPb

jDFTN{XmG}∣

∣

∣

⊙2)

(24)

j = ⌊m(N − O)/L⌋where we have allowed STI and FFT windows to be of unequallength, and FFT windows to overlap.

III. R ESULTS

A. Simulation Experiments

Figure 3 presents a computer simulated scenario using aP = 7 element uniform line array with half wavelengthspacing. The SOI is located at an azimuth of5◦ with apeak (across frequency) signal to noise ratio of 5 dB. TheSOI is a Gaussian narrowband random process with a PSD

0 100 200 300 400 500−15

−10

−5

0

5

10

15

La

dB r

el. n

oise

pow

er

Scoop depth vs subwindow length, La

Experimental resultAnalytical model

Fig. 4. Scooping depth in the interference frequency bin as afunction ofSTI window subset size,La. Plots are relative to the desired PSD estimateof the true noise floor PSD level. No SOI is present. Negative dB valuesindicate a scoop; positive values show incomplete interference cancellation.The “analytical model” curve is for equation (23), and the “experimentalresult” is obtained with 100 Monte Carlo trials of subspace projection arrayPSD estimates for each value ofLa. STI window length wasL = 512.

shaped by bandpass filtering, and appears as a decliningramp structure with scalloped top in the figures. Two movingnarrowband interferers were present, one inside the SOI band,and one outside. The initial azimuths were−60◦ and 70◦,with angular rates of2 × 10−5 and−6 × 10−5 degrees persample respectively. A total of5×105 samples were used, withnon-overlapping STIs of lengthL = 256 and rectangularlywindowed (G = I) FFTs of lengthN = 256. Scoopingis clearly evident in the subspace projection beamformer,distorting both the noise floor and SOI PSD estimates. Thealgorithm of (24) was used to obtain the corrected curve,which completely eliminates scooping while fully cancelingthe interferer. Similar results (not shown) were obtained withthe LCMV beamformer by usingRa

x,j alone in computingwj .

Figure 4 compares the scoop depth predicted by (23) withnumerical results from 100 Monte Carlo trials. In the simula-tions the subspace projection PSD estimator of (19) was usedwith Pa

j computed from STI subwindows of varying length,La. A single stationary interferer at -60◦ was observed with20 dB INR at the 7 element,λ/2 spaced uniform line array.STI lengthL and FFT window lengthN were both 512, withFFT windows overlapped 50%.

Good agreement between the curves is found untilLa →L, at which point (23) over-predicts scoop depth due toapproximations made in the analysis. Interferer leakage, orequivalently low IRR, due theEj error term is not taken intoaccount in the approximate analysis. However, both curvescross 0 dB (i.e. no scooping bias) as predicted forLa = L/2 =256.


(a)

(b)Fig. 5. The array feed test platform. a) Feed array mounted onthree meterdish reflector. b) Close up of seven element feed.

B. Real Data Experiments

This section presents results from an experiment with aseven element L-band (≈ 1600 MHz) array feed mounted atthe focal plane of a three meter diameter parabolic dish. Thistest platform at Brigham Young University (BYU) is usedfor rapid prototyping to study signal processing algorithmsfor radio astronomy. Figure 5 shows the antenna system andhexagonal array feed. The inter-element spacing of 0.6 wave-length provides a “fully sampled” array [4], [24]. Developmentof the BYU array is reported in [6], [25].

Figures 6 and 7 presents PSD estimation results for a realdata experiment. Array signals were acquired by a seven chan-nel analog receiver, digitized at 1.25 Msamp/sec/channel,andstreamed to disk for post processing. The SOI and interferencesignals were from CW microwave function generators withantennas located on neighboring buildings. The SOI was seenat boresight and the moving interferer was seen in the dishsidelobe pattern at approximately30◦ azimuth. Interference-to-signal power ratio was 113 dB at the sources, but theinterferer was approximately three times further from the

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−65

−60

−55

−50

−45

−40

−35

−30

−25


Pow

er/fr

eque

ncy

(dB

/rad

/sam

ple)

Corrected subspace proj.Fixed beamformerSubspace proj.

Fig. 6. Real data experiment for a 7 element array feed and 3m dish. The SOIis at ω = 0.0, corresponding to 1571.3 MHz. The 1571.35 MHz interfererappears atω = −0.5. Note the 10 dB deep subspace projection scoopingnull, which is eliminated by the proposed algorithm.

−2 −1.5 −1 −0.5 0 0.5 1 1.5

−55

−50

−45

−40

−35

−30

−25


Pow

er/fr

eque

ncy

(dB

/rad

/sam

ple)

Corrected subspace proj.Fixed beamformerSubspace proj.

Fig. 7. Scooping for the seven-element array feed using 50% overlappingwindows, and Hamming window shaping for the FFT calculations. Note thatscooping is less prominent than the non-overlapping, rectangular windowedcase of Figure 6 for the same data samples, but the proposed algorithm stillimproves the PSD estimate.

dish than was the SOI. Data was processed using (24), withlength L = N = 128 non-overlapping rectangular windows.Additional physical details for the experiment are found in[26].

Figure 6 clearly demonstrates that spectral scooping does infact occur in real adaptive cancellation scenarios. We observedscooping in virtually all of our real sample data with nar-rowband interference, and even in some relatively broadbandFM interference cases. The proposed algorithm effectivelycorrects the PSD bias. In cases (as in the figure) where asmall residual interference “leak through” occurs, its height istypically less than the original scooping null depth. Figure 7shows how a tapered spectral shaping window (e.g. Hamming)


does reduce scooping depth when compared to using therectangular window which was assumed in the analysis ofSection II.

IV. CONCLUSIONS

We have demonstrated, both mathematically and experimen-tally, the existence of previously unknown PSD estimation biaswhich we have dubbed “spectral scooping.” This null in thesignal plus noise spectrum occurs while using adaptive arraybeamforming to spatially cancel an interferer who’s bandwidthis spans just a few frequency bins of the PSD estimator.Though beamformer weights operate on the array spatial re-sponse and are not generally considered capable of affectingthe temporal-spectral content of a signal, the fact that weightsare updated periodically with a data dependent calculationopens an avenue for modification of the estimated spectrum. Itwas shown that scooping is an artifact of estimation error intheshort-term integration (STI) sample covariance matrix. Thissample error includes cross product terms between interferenceand noise plus signal which couple into the PSD estimator.Scooping does not occur when adaptive beamformer weightsare computed from snapshot exact covariance matrices, evenfor time-varying interference spatial signatures.

A correction algorithm was introduced that exploits a spe-cific relationship between the scoop depth and the percentageof samples in the covariance estimation window which areused to calculate beamformer weights,wj . We have found thatscoop depth is reduced (though not eliminated) when taperingwindows are used forG, whenL 6= N , and whenO 6= 0 soDFT widows overlap. We recommend using these parametersettings in conjunction with the proposed correction algorithm.

Scooping is problematic only in a somewhat specializedobserving scenario, i.e. withi) quite narrowband interference,ii) regularly updated spatial adaptive filtering,iii) specificrelationships for the various signal powers and STI length asin (8), andiv) when a spectral notch in the SOI is detrimen-tal to signal processing goals. However, these conditions doarise in real-world scenarios, and we have encountered themrepeatedly in realistic radio astronomical observing situations.The effect is surprising, and we believe practitioners of arraysignal processing should be aware if its potential corruption totheir experiments and applications which use canceling arraybeamformers.

A simple demonstration code is available at [21] for readersto familiarize themselves with the scooping phenomenon andoperation of the proposed correction algorithm.

APPENDIX

A. Exact Eigen Solution for a Rank Two STI Covariance

The first two eigenvectors of (14) are seen by inspection ofto have the form

uj,k = vj + ξknj , k = 1, 2. (25)

The exact solution for the dominant eigenvector ofRx,j isfound as follows [23]. Using (16), the eigen equation can be

expressed as

λuj = Rx,juj

λ(vj + ξnj) =[

σ2dvj vH

j +σd

L

(

vj nHj + njv

Hj

)]

×(vj + ξnj)

λvj + λξnj =(

α + ρ∗ + (ρ + β)Lσd ξ)

vj

+

(

α

Lσd+ ρξ

)

nj

Where α = σ2d‖vj‖2, β = 1

L2 ‖nj‖2, and ρ = σd

L vHj nj .

Matching coefficients ofvj and nj respectively and solvingfor λ yields the two equations

λ = α + ρ∗ + (ρ + β)Lσd ξ, and

λ =α

Lσd ξ+ ρ.

Thus Lσdξ = αλ−ρ , which when substituted into the first

equation leads to

λ2 − (α + ρ∗ + ρ)λ − αβ + |ρ|2 = 0.

This is solved with the quadratic formula to yield

λ =α + 2Re{ρ} ±

√

(α + 2Re{ρ})2 − 4(|ρ|2 − αβ)

2.

Substituting this result intoξ = αLσd(λ−ρ) and taking the pos-

itive radical term produces the needed parameter to determinethe dominant eigenvector:

ξ1 =2/(Lσd)

1 +

√

(

1 + 2Re{ρ}α

)2

− 4(

|ρ|2

α2 + βα

)

− i4 Im{ρ}α

B. Projection Matrix for Partial STI Window

We find a closed form close approximation for the projec-tion matrix computed from a sub-window ofLa samples fromSTI j. Compute the dominant eigenvector,ua

j,1, for Rax,j. Let

Paj = I − ua

j,1(uaj,1)

H/((uaj,1)

H uaj,1), which by substituting

from (16) can be expanded as

Paj ≈ I −

(vj + 1Lσd

naj )(vj + 1

Lσdna

j )H

‖vj + 1Lσd

naj ‖2

≈ I −(vj + 1

Lσdna

j )(vj + 1Lσd

naj )H

‖vj‖2

≈ I −vjv

Hj

‖vj‖2−

najv

Hj

Laσd‖vj‖2

−vj(n

aj )H

Laσd‖vj‖2−

naj (na

j )H

L2aσ

2d‖vj‖2

Paj ≈ I − vjv

Hj −

vj(naj )H + na

j vHj

Laσd‖vj‖

wherevj = vj/‖vj‖ and in the final approximation the lastterm was neglected since it is very small due to division byL2

a.


C. Notation

The following notation, operators and identities have beenused in the paper:

1) z : bold, lower case denotes a column vector.2) A : bold, upper case denotes a matrix.A = [a1,a2, · · · ,aN ].3) a, B, α : italic, non-bold, Roman or Greek denotes a

scalar quantity.4) I : identity matrix of appropriate size.5) i : =

√−1.

6) AT : transpose ofA.7) A∗ : complex conjugate ofA.8) AH : complex conjugate (Hermitian) transpose ofA.9) |A|⊙2 : element-wise magnitude squared.

10) diag{A} : extract diagonal ofA to form a columnvector.

11) Diag{z} : diagonal matrix withz appearing along thediagonal.

12) tr{A} : trace ofA.

13) A ⊙ B =

a1,1b1,1 · · · a1,Nb1,N

.... . .

...aM,1bM,1 · · · aM,NbM,N

, element-

wise, or Hadamard product.14) Sx : DFT based power spectral density vector for finite

length random sequencexj . This notation is an excep-tion to rule (2).

15) DFTN{A} : N - point discrete Fourier transform alongrows of M × N A.

16) E{A} : expected value of randomA.17) stdv{X} : standard deviation ofX .18) ⌊a⌋ : floor operation, rounding toward zero.19) A : estimate ofA.20) O(·) : “on the order of.”21) ∈ : “is an element of.”22) ∪, ∩ : Set union and intersection respectively.

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[3] S. van der Tol, B.D. Jeffs, and A.-J. van der Veen, “Self calibrationfor the LOFAR radio astronomical array,”IEEE Transactions on SignalProcessing, vol. 55, no. 9, pp. 4497–4510, Sept. 2007.

[4] J.R. Fisher and R.F. Bradley, “Full sampling array feedsfor radiotelescopes,”Proceedings of the SPIE, Radio Telescopes, vol. 4015, pp.308–318, 2000.

[5] Chad K. Hansen, Karl F. Warnick, Brian D. Jeffs, J. Richard Fisher, andRichard Bradley, “Interference mitigation using a focal plane array,”Radio Science, vol. 40, doi:10.1029/2004RS003138, no. 5, June 2005.

[6] J.R. Nagel, K.F. Warnick, B.D. Jeffs, J.R. Fisher, and R.Bradley,“Experimental verification of radio frequency interference mitigationwith a focal plane array feed,”Radio Science, vol. 42, 2007, RS6013,doi:10.1029/2007RS003630.

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[8] R. Maaskant, M. V. Ivashina, R. Mittra, and N. T. Huang, “ParallelFDTD modeling of a focal plane array with Vivaldi elements onthehighly parallel LOFAR BlueGene/L supercomputer,” inProc. IEEEAntennas and Propagation Symposium, Albuquerque, NM, June 20-252006, pp. 3861 – 3864.

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[12] P.D Welch, “The use of the fast fourier transform for theestimation ofpower spectra,” IEEE Trans. Audio Electroacoustics, vol. AU-15, pp.70–73, June 1970.

[13] H.L. Van Trees,Detection, Estimation, and Modulation Theory, Part IV,Optimum Array Processing, John Wiley and Sons, 2002.

[14] B.D. Van Veen and K. M. Buckley, “Beamforming: A versatile approachto spatial filtering,” IEEE ASSP Mag., pp. 4–24, Apr. 1988.

[15] A. Leshem, A.-J. van der Veen, and A.-J. Boonstra, “Multichannelinterference mitigation techniques in radio astronomy,”AstrophysicalJournal Supplements, vol. 131, no. 1, pp. 355–374, 2000.

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Brian D. Jeffs (M’90–SM’02) received B.S. (magnacum laude) and M.S. degrees in electrical engineer-ing from Brigham Young University in 1978 and1982 respectively. He received the Ph.D. degree fromthe University of Southern California in 1989. He iscurrently an associate professor in the Departmentof Electrical and Computer Engineering at BrighamYoung University, where he lectures in the areas ofdigital signal processing, digital image processing,and probability theory. Current research interestsinclude array signal processing for radio astronomy,

RF interference mitigation, MIMO wireless communications, and digitalimage restoration. Previous employment includes Hughes Aircraft Companywhere he served as a sonar signal processing systems engineer in the anti-submarine warfare group. Dr. Jeffs was a Vice General Chair for IEEEICASSP-2001, held in Salt Lake City Utah. He was a member of the executiveorganizing committee for the 1998 IEEE DSP Workshop, and served severalyears as chair of the Utah Chapter of the IEEE Communicationsand SignalProcessing Societies.


Karl F. Warnick (S’95–M’98–SM’04) receivedthe B.S. degree (magna cum laude) with Univer-sity Honors and the Ph.D. degree from BrighamYoung University in 1994 and 1997, respectively.From 1998 to 2000, he was a Postdoctoral Re-search Associate in the Center for ComputationalElectromagnetics at the University of Illinois atUrbana-Champaign. Since 2000, he has been afaculty member in the Department of Electrical andComputer Engineering at Brigham Young, where heis currently an Associate Professor. He has authored

a book, Problem Solving in Electromagnetics, Microwave Circuits, andAntenna Design for Communications Engineering (Artech House). Researchinterests include array antenna systems, computational electromagnetics,remote sensing, and inverse scattering. Dr. Warnick was a recipient of theNSF Graduate Research Fellowship, Outstanding Faculty Member award forElectrical and Computer Engineering in 2005, and the BYU Young ScholarAward (2007), and served as Technical Program Co-Chair for the 2007 IEEEInternational Symposium on Antennas and Propagation.

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