Broadband Beamforming

Bhaskar D RaoUniversity of California, San Diego

Email: brao@ucsd.edu

Broadband Signals

Narrow band signals: f (t− τ)e jωc (t−τ) ≈ f (t)e jωc (t−τ), for τ << T = 12B

The signal becomes broadband if the bandwidth becomes large and thedelay across the array can no longer be considered small compared to thesampling interval

Signals such as speech signals are naturally baseband signals and arebroadband in nature

Delay and Sum beamformer

Delay the output of the sensors appropriately and enable constructiveaddition of the copies to get SNR improvement over white noiseIn the presence of an interferer will use FIR filters at the output to cancelor suppress interference

Time-Delay Estimation: Preliminaries1

rx1x2 [m] = E (x1[n + m]x∗2 [n]) and from LTI systems theory we know thecross-power spectrum is given by

Sy1y2 (e jω) =∞∑

m=−∞ry1y2 [m]e−jωm = H1(e jω)H∗2 (e jω)Sx1x2 (e jω)

Sy1y1 (e jω) is the power spectrum which is real and positive.If we are given data, we can divide the data into L blocks of length Nand estimate the cross-power spectrum using the FFT

1Carter, G. Clifford. Coherence and time delay estimation: an appliedtutorial for research, development, test, and evaluation engineers. IEEE, 1993.

Time Delay Estimation

y1[n] = r1(nT ) = s[n] + z1[n]

y2[n] = r2(nT ) = s[n − D] + z2[n]

Assuming the noises are uncorrelated, ry2y1 [m] = rss [m − D].

D = arg maxm

ry2y1 [m] = arg maxm

1

2π

∫ π

−πSy2y1 (e jω)e jωmdω

The peak is sharp if s[n] is a white noise sequence and in the absence ofreverberation/multipath.

Generalized Cross-Correlation (GCC) Methods

D = arg maxm

1

2π

∫ π

−πφ(e jω)Sy2y1 (e jω)e jωmdω

Cross-Correlation Method: φ(e jω) = 1The purpose of φ(e jω) is to whiten the signal and/or weight thespectrum based on the SNR in the frequency domain. Note thatSy2y1 (e jω) = Sss(e jω)e−jωD .PHAse Transform (PHAT): φ(e jω) = 1

|Sy2y1(e jω)|

Smoothed COherence Transform (SCOT): φ(e jω) = 1√Sy1y1

(e jω)Sy2y2(e jω)

Maximum-Likelihood (ML): φ(e jω) = 1|Sy2y1

(e jω)||γy2y1

(e jω)|2

1−|γy2y1(e jω)|2 where

γy2y1 (e jω) =Sy2y1 (e jω)√

Sy1y1 (e jω)Sy2y2 (e jω), 0 ≤ |γy2y1 (e jω)| ≤ 1,∀ω

is the coherence function.

Adaptive Broadband Beamformer2

2Frost, Otis Lamont. ”An algorithm for linearly constrained adaptive arrayprocessing.” Proceedings of the IEEE 60.8 (1972): 926-935.

Wideband BF Formulation

Number of sensors: N = K , number of taps, i.e. order of FIR filters is J.

If we denote the input measurement vector as x[n], then the data in thetap delay line is [x[n], x[n − 1], . . . , x[n − J + 1]]

Weights: Using vector notation, the weights in the tap delay line are[W0,W1, . . . ,WJ−1] , where Wk = [wk,0,wk,1, . . . ,wk,N−1].

Key Question: How to characterize and design such an array?

Characterization: Frequency-Wavenumber Response

Extension of the MPDR formulation: Look direction constraint plusminimize output power

Frequency Wavenumber Response

If the source signal is s(t) = e jΩt from direction k, then the signalreceived at sensor l is delayed relative to the origin by τl .

xl(t) = s(t) = e jΩ(t−τl ) or xl [n] = xl(nT ) = e jΩ(nT−τl ) = e jω(n−τl )

where ω = ΩT and τl = 1T τl .

Output of sensor l is given by

xl [n]→ Hl(z)→ yl [n] = e jω(n−τl )Hl(ejω)

The array output

y [n] =N−1∑l=0

yl [n] =N−1∑l=0

e jω(n−τl )Hl(ejω) = e jωn

N−1∑l=0

e−jωτlHl(ejω) = e jωnB(ω, k)

B(ω, k) =∑N−1

l=0 e j−ωτlHl(ejω) is the frequency-wavenumber response of

array

Time-Domain Formulation

Beamformer output

y [n] = W T0 x[n] +W T

1 x[n−1] + . . .+W TJ−1x[n−J + 1] =

J−1∑k=0

W Tk x[n−k]

Constraints: Define 1 = [1, 1, . . . , 1]T , a vector in RN .

N−1∑m=0

wk,m = fk or 1TWk = fk , k = 0, 1, . . . , J − 1

This is equivalent to look direction transfer function constraint of

Hd(z) = F (z) = f0 + f1z−1 + . . .+ fJ−1z

−(J−1) =J−1∑m=0

fmz−m

Frost Beamformer

Stacking all the BF weights and the constraints into large vectors ofdimension NJ, we have

y [n] = WTX[n], where W = [W T0 ,W

T1 , . . . ,W

TJ−1]T

and X[n] = [xT [n], xT [n − 1], . . . , xT [n − J + 1]]T .The output power is E (y2[n]) = WTRxxW where Rxx = E (X[n]XT [n]).The constraints are given by

CTW = f or

1T 0 . . . 00 1T . . . 0...

......

...0 0 . . . 1T

W =

f1f2...

fJ−1

The optimization problem is

minW

WTRxxW subject to CTW = f

Using Lagrange multipliers, we can show

Wo = R−1xx C(CTR−1

xx C)−1f

GSC structure

Because of the symmetric nature of the constraints with respect to time1TWk = fk , i.e. the constraint matrix 1 is the same, the constraint canbe implemented first followed by filtering. Our Wq = 1

N [1, 1, . . . , 1]T andour B matrix is an orthonormal set of vectors orthogonal to Wq. IfN = 8, a choice for B is from the Haar basis

B =

1 0 0 0 1 0 1−1 0 0 0 1 0 10 1 0 0 −1 0 10 −1 0 0 −1 0 10 0 1 0 0 1 −10 0 −1 0 0 1 −10 0 0 1 0 −1 −10 0 0 −1 0 −1 −1

GSC Implementation of Adaptive Broadband BF3

3Griffiths, Lloyd, and C. W. Jim. ”An alternative approach to linearlyconstrained adaptive beamforming.” IEEE Transactions on antennas andpropagation 30.1 (1982): 27-34.

Broadband via Narrowband processing

Narrow-band modeling at the output of each band

x(ωc , n) = V(ωc , ks)Fs [ωc , n] +D−1∑l=1

V(ωc , kl)Fl [ωc , n] + Z[ωc , n]

Assumptions

I Fs [ωc , n], Fl [ωc , n], l = 1, ..,D − 1, and Z[ωc , n] are zero mean

I E (|Fs [ωc , n]|2) = ps(ωc), and E (|Fl [n]|2 = pl(ωc), l = 1, . . . ,D − 1,and E (Z[ωc , n]ZH [ωc , n]) = σ2

z (ωc)I

I All the signals/sources are uncorrelated with each other and overtime: E (Fl [ωc , n]F ∗m[ωc , p]) = pl(ωc)δ[l −m]δ[n − p] andE (Fl [ωc , n]F ∗s [ωc , p]) = 0

I The sources are uncorrelated with the noise:E (Z[ωc , n]F ∗l [ωc ,m]) = 0

I We also need properties at two distinct center frequencies ωc1 andωc2 . Usually assumed uncorrelated, e.g.E (Fl [ωc1 , n]F ∗m[ωc2 , p]) = 0

FFT implementation

Some Issues to Address

I Source recovery: MPDR or other beamformers on each bandfollowed by inverse FFT. No serious impediments here

I DOA estimation: How do we combine the bands to find an estimateof the DOA. Can independently compute DOA in each band andthen combine. Can do DOA estimation by utilizing all the bandstogether in a synergistic manner. SINR may be different in eachband.

I Element spacing: The wavelength has a range (λmin, λmax). How tospace the sensors? Non-uniform spacing is an option with someelements at λmin/2 spacing.

Independent Component Analysis (ICA)

Bhaskar D RaoUniversity of California, San Diego

Email: brao@ucsd.edu

ICA

Instantaneous Mixing Model: Assume N sensors, D sources with D < N,that are linearly mixed

x[n] = As[n]+Z[n]or

x1[n]x2[n]

...xN [n]

=

a11 a12 . . . a1Da21 a22 . . . a2N

......

......

N1 aN2 . . . aND

s1[n]s2[n]

...sD [n]

+Z[n]

Problem: Both mixing matrix A and sources s[n] are unknown. Frommeasurements of x[n], n = 0, 1, ..., L− 1, recover A and s[n].Assumptions:

I Sources are zero mean, i.e. E (sl [n] = 0, l = 1, ..,D.

I Sources are uncorrelated, i.e. E (s[n]sT [n]) = diag(pl).

I Sources are independent, i.e. p(s1, s2, .., sD) = ΠDl=1pl(sl).

I Sources are non-Gaussian distributed except for possibly one.

ICA relies on independence and non-Gaussian source assumption.

ICA and Applications1

More general model is Convolutive Mixing: xl [n] =∑D

k=1 hlk [n] ∗ sk [n]

ICA has several practical applications:

I The ”cocktail party problem”: separation of voices or music orsounds

I Sensor array processing, e.g. radar

I Biomedical signal processing with multiple sensors: EEG, ECG,MEG, fMRI

I Telecommunications: e.g. multiuser detection in CDMA

I Financial and other time series

I Noise removal from signals and images

I Feature extraction for images and signals

I Brain modelling

1Hyvarinen, A., Karhunen, J., & Oja, E. (2004). Independent componentanalysis (Vol. 46). John Wiley & Sons.

Cocktail Party Problem

ICA and EEG Processing2

2Makeig, S., Bell, A. J., Jung, T. P., & Sejnowski, T. J. (1996).Independent component analysis of electroencephalographic data. In Advancesin neural information processing systems (pp. 145-151).

Principal Component Analysis(PCA)

PCA is a data representation/ compression technique: Finds an efficientlow-dimensional orthonormal basis to represent a random vector in amean squared sense.

Rxx = E (x[n]xH [n]) = APAH + σ2z I, where P = diag(pl)

Eigen-decomposition should allow us to find the number of sources Dand the noise variance.Let us assume N = D and σ2

z = 0.Problem: Find an orthonormal basis W for a p-dimensional subspace(p ≤ N), i.e W ∈ CN×p and W HW = Ip×p, such that E (|W Hx[n]|2) ismaximized.

maxW∈CN×p,W HW=Ip×p

W HRxxW

Solution (PCA): W = [q1, ..,qp], where ql is an eigenvector of Rxx

corresponding to the lth largest eigenvalue. The approximation for x[n] isobtained as

x[n] = WW Hx[n]

PCA cont’d

If E = [q1, ..,qN ], then Rxx = EΛEH , is an eigen-decomposition whereΛ = diag(λl) contains the ordered eigenvalues, i.e. λ1 ≥ λ2... ≥ λN ≥ 0.

Then B = Λ−12EH is a whitening matrix, i.e. BRxxB

H = IN×N andy[n] = Bx[n] has uncorrelated components.If B is a whitening matrix, QB is also a whitening matrix where Q is anorthonormal matrix and yQ [n] = QBx[n] also has uncorrelatedcomponents.Implications

I Gaussian sources are not separable: If the sources (s[n]) areGaussian, y[n] and yQ [n] represent independent Gaussian sources.Non-unique

I Uncorrelated property is not enough to separate sources

Need Independent and Non-Gaussian sources: The goal is to find Q, theremaining unknown.