Direction of Arrival Estimation: MUSIC,ESPRIT, Spatial Smoothing

Bhaskar D RaoUniversity of California, San Diego

Email: brao@ucsd.edu

Reference Books and Papers

1. Optimum Array Processing, H. L. Van Trees

2. Stoica, P., & Moses, R. L. (2005). Spectral analysis of signals (Vol.1). Upper Saddle River, NJ: Pearson Prentice Hall.

3. Schmidt, R. (1986). Multiple emitter location and signal parameterestimation. IEEE transactions on antennas and propagation, 34(3),276-280.

4. Roy, Richard, and Thomas Kailath. ”ESPRIT-estimation of signalparameters via rotational invariance techniques.” IEEE Transactionson acoustics, speech, and signal processing 37.7 (1989): 984-995.

5. Rao, Bhaskar D., and K. S. Arun. ”Model based processing ofsignals: A state space approach.” Proceedings of the IEEE 80.2(1992): 283-309.

Narrow-band Signals

x(ωc , n) = V(ωc , ks)Fs [n] +D−1∑l=1

V(ωc , kl)Fl [n] + Z[n]

Assumptions

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

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

z I

I All the signals/sources are uncorrelated with each other and overtime: E (Fl [n]F ∗

m[p]) = plδ[l −m]δ[n − p] and E (Fl [n]F ∗s [p]) = 0

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

Circular Gaussian Noise

Since the random variables are complex, we need some care in dealingwith them.Let Z[n] = ZR [n] + jZI [n]. We will need to model the joint properties ofthe random vectors ZR [n] and ZI [n].

I ZR [n] and ZI [n]) are zero mean random vectors. This impliesE (Z[n]) = 0.

I ZR [n] and ZI [n]) are uncorrelated with each other, i.e.E (ZR [n]ZT

I [n]) = 0.

I E (ZR [n]ZTR [n]) = E (ZI [n]ZT

I [n]) = 12σ

2z I.

I Equivalently E (Z[n]ZH [n]) = σ2z I and E (Z[n]ZT [n]) = 0.

If we assume they are jointly Gaussian, this results in Z[n] beingCircularly Gaussian.

Direction of Arrival (DOA) Estimation

x[n] =D−1∑l=0

VlFl [n] + Z[n], n = 0, 1, . . . , L− 1

Here we have chosen to denote Vs by V0 for simplicity of notation. NoteVl = V(kl). We will refer to the L vector measurements as L snapshots.

Goal: Estimate the direction of arrival kl or equivalently(θl , φl), l = 0, ...,D − 1.Can also consider estimating the source powers pl , l = 0, 1, ...,D − 1, andnoise variance σ2

z .

ULA :V(ψ) = [e−j N−1

2 ψ, . . . , e jN−1

2 ψ]T = e−j N−12 ψ[1, e jψ, e j2ψ, ..., e j(N−1)ψ]T

Note for a ULA, V(ψ) = JV∗(ψ), where J is all zeros except for onesalong the anti-diagonal. Also note J2 = I.

ULA and Time Series

ULAx0[n]x1[n]

...xN−1[n]

=D−1∑l=0

1

e jψl

...e j(N−1)ψl

e−j N−12 ψlFl [n]+Z[n], n = 0, 1, . . . , L−1

Similar to a time series problems composed of complex exponentialsx [n] =

∑D−1l=0 cle

jωln + z [n], n = 0, 1, . . . , (M − 1).In vector form, the time series measurement is

x [0]x [1]

...x [M − 1]

=D−1∑l=0

1

e jωl

...e j(M−1)ωl

cl +

z [0]z [1]

...z [M − 1]

The sum of exponentials problem with M samples is equivalent to a ULAwith M sensors and one (vector) measurement or one snapshotIn array processing we have a limited number of sensors N but cancompensate potentially with L snapshots.The time series problem is equivalent to one snapshot but a potentiallylarge array if the number of samples M is large.

Signal and Noise Subspaces

Subspace methods: Based on eigen-decomposition of a covariance matrix

Eigen-decomposition of Ss : Note Ss = VSFVH

Ss has rank D, is Hermitian symmetric and positive semi-definite.Hence Ss has D non-zero eigenvalues and corresponding orthornomaleigenvectors

Ss = [q1,q2, . . . ,qD ]

λs1 0 . . . 00 λs2 . . . 0...

......

...0 0 . . . λsD

qH1qH2...qHD

= EsΛsEHs

where Es ∈ CN×D , and Λs = diag(λi ). Note EHs Es = ID×D and

Ps = EsEHs is an orthogonal projection matrix onto R(Es), i.e. Ps = PH

s

and Ps = P2s .

Important Observation: R(V) = R(Es) and R(Es) is referred to asSignal subspace

Subspaces Cont’d

Choose En = [qD , . . . ,qN ] such that EHn En = I(N−D)×(N−D) and

EHn Es = 0(N−D)×D . Defining E = [Es ,En] ∈ CN×N , we have

EHE = EEH = IN×N

Important Observation: R(En) ⊥ R(V) = R(Es). R(En) is referred asNoise subspace

Ss = EsΛsEHs = [Es ,En]

[Λs 00 0

] [EHs

EHn

]= E

[Λs 00 0

]EH

σ2z I = σ2

zEEH = E (σ2

z I) EH = E

σ2z 0 . . . 0

0 σ2z . . . 0

......

......

0 0 . . . σ2z

EH

In fact E can be replaced by any orthonormal matrix in the expansion forσ2z I. For subspace methods, choosing a matrix compatible with Ss is

more useful.

Observations

Sx = Ss + σ2z I

= Es (Λs + σ2z ID×D) EH

s + σ2zEnE

Hn

=D∑l=1

(λsl + σ2z )qlq

Hl + σ2

z

N∑l=(D+1)

qlqHl

I Sx has eigenvalues λl = λsl + σ2z , l = 1, . . . ,D, and

λl = σ2z , l = (D + 1), . . . ,N

I The number of sources D can be identified by examining themultiplicity of the smallest eigenvalue which is expected to be σ2

z

with multiplicity (N − D)

I The eigenvectors corresponding to the signal subspaces and noisesubspaces are readily available from the eigen decomposition of Sx ,i.e. Es and En are easily extracted.

MUltiple SIgnal Classification (MUSIC)

MUSIC uses the noise subspace spanned by the ”noise” eigenvectors En.

Since R(V) = R(Es) ⊥ R(En), we haveVl ⊥ R(En), l = 0, 1, . . . , (D − 1) or VH

l En = 0, l = 0, 1, . . . , (D − 1)Many options to use this orthogonality property. All methods start withSx , usually estimated from the data.

MUSIC Algorithm (also known as Spectral-MUSIC)

I Perform an eigen-decomposition of Sx

I Estimate D and the signal subspace Es and the noise subspace En

I Compute the spatial MUSIC spectrum as follows:

PMUSIC (k) =1

|VH(k)En|2

Note that |VH(k)En|2 = VH(k)EnEHn V(k) = VH(k)PnV(k) where

Pn = EnEHn

I The peaks in PMUSIC (k) gives us the DOA.

MUSIC Example

ULA and MUSIC

ULA : V(ψ) = e−j N−12 ψ[1, e jψ, e j2ψ, ..., e j(N−1)ψ]T . Will drop e−j N−1

2 ψ forthis discussion for simplicity and because it does not matter.

|VH(ψ)En|2 =N∑

l=D+1

|VH(ψ)ql |2 =N∑

l=D+1

|ql(e jψ)|2

where ql(e jψ) = VH(ψ)ql =∑N−1

m=0 ql(m)e−jmψ, withql = [ql [0], ql [1], ..., ql [N − 1]]T

ql(e jψ) is the Fourier transform of the impulse response of a filter formedfrom the eigenvector ql

Defining B(e jψ) =∑N

l=D+1 |ql(e jψ)|2. The MUSIC spectrum is

PMUSIC (ψ) = 1B(eψ)

, and it will have peaks at ψl , l = 0, 1, . . . , (D − 1)

corresponding to the DOAThen B(e jψl ) = 0, l = 0, 1, . . . , (D − 1)

Root-MUSIC

With B(e jψ) =∑N

l=D+1 |ql(e jψ)|2, in the z-domain we have

B(z) =∑N

l=D ql(z)q∗l ( 1z∗ ). Alternately

B(z) =∑N

l=D+1 Bl(z) where

Bl(z) = ql(z)q∗l ( 1z∗ ) =

∑N−1l=−(N−1) rql [m]z−m,

Each of the Bl(z) represents a Moving Average Spectrum and the sumB(z) is also a Moving Average spectrum

B(z) =N−1∑

l=−(N−1)

rq[m]z−m = Hrm(z)H∗rm(

1

z∗),

where rq[m] is an autocorrelation sequence and Hrm(z) is obtained byspectral factorization

B(e jψl ) = 0, l = 0, 1, . . . , (D − 1) implies Hrm(z) will have D of it’s zerosat e jψl , where ψl corresponds to the DOAThe remaining (N − 1− D) zeros will be inside the unit circle assumingwe use a minimum-phase spectral factor.

Root-MUSIC Algorithm

I Perform an eigen-decomposition of Sx

I Estimate D and the signal subspace Es and the noise subspace En

I Compute ql(z) =∑N−1

m=0 ql [m]z−m, l = D + 1, . . . ,N − 1.

I Compute B(z) =∑N

l=D+1 Bl(z) where

Bl(z) = ql(z)q∗l ( 1z∗ ) =

∑N−1l=−(N−1) rql [m]z−m

I Perform a spectral factorization to find a minimum phase factorHrm(z), i.e. B(z) = Hrm(z)H∗

rm( 1z∗ ).

I The D roots on/close to the unit circle are identified and theirargument used to estimate the DOA

Root-MUSIC and Spectral-MUSIC Example

Minimum Norm Method

In principle, any vector from the noise subspace can be used to find theDOA.

Let

[1g

]be any vector in the noise subspace, i.e.

[1g

]∈ R(En).

Then VHl

[1g

]= 0, l = 0, 1, . . . , (D − 1).

One can compute the following spatial spectrum

P(k) =1

|VH(k)

[1g

]|2

and locate the DOA from the peaks.

Min-Norm: Choose a vector

[1g

]in the null-space with minimum

norm, i.e. ‖g‖2 being minimum.

Minimum Norm Method

Now EHs

[1g

]= 0. Alternately, if EH

s = [e1, E2], then

E2g = −e1 or gmin−norm = −E+2 e1

This can be efficiently computed by noting that

E+2 = EH

2 (E2EH2 )−1 = EH

2 (I− e1eH1 )−1

= EH2 (I +

1

1− ‖e1‖2e1e

H1 )

This is obtained using the matrix inversion lemma, leading to

gmin−norm = − 1

1− ‖e1‖2EH

2 e1

Estimation of Signal Parameters via Rotation InvarianceTechniques (ESPRIT)

ESPRIT exploits the structure in the signal subspace R(Es) = R(V).We will first consider a ULA and then extend to the general case.Structure in the Signal Subspace

V =

1 1 . . . 1

e jψ0 e jψ1 . . . e jψD−1

......

......

e j(N−1)ψ0 e j(N−1)ψ1 . . . e j(N−1)ψD−1

=

[V 1

xxx

]=

[xxxV2

]

V1 and V2 are first and last (N − 1) rows of V respectively.

V2 = V1Ψ where Ψ =

e jψ0 0 . . . 0

0 e jψ1 . . . 0...

......

...0 0 . . . e jψD−1

Ψ is a diagonal matrix whose diagonal entries (eigenvalues) aree jψl , l = 0, 1, . . . , (D − 1)

ESPRIT

The structure in V extends to Es

Es =

[E1

xxx

]=

[xxxE2

]= VT =

[V1Txxx

]=

[xxxV2T

]where E1 ∈ C (N−1)×D , E2 ∈ C (N−1)×D , and T ∈ CD×D is a nonsingularmatrix.

E2 = V2T = V1ΨT = E1T−1ΨT = E1Φ

where Φ = T−1ΨT ∈ CD×D and whose eigenvalues aree jψl , l = 0, 1, . . . , (D − 1) because of the similarity transformation.

ESPRIT algorithm

I Perform an eigen-decomposition of Sx

I Estimate D and the signal subspace Es and the noise subspace En

I From Es determine E1 as the first (N − 1) rows and E2 as the last(N − 1) rows of Es

I Compute Φ by solving E2 = E1Φ using a Least Squares Approach,i.e. Φ = E+

1 E2.

I Perform an eigen-decomposition of Φ

I The argument of the D eigenvalues are used to estimate the DOA

As in the minimum norm method, E+1 can be obtained efficiently using

the matrix inversion lemma and the computation of Φ can be simplified.Another option is to use a Total Least Squares approach, as opposed to aLeast Squares approach, to estimate Φ

ESPRIT: ULA and Doublets

ESPRIT with Doublets: Theory

If the array manifold for the first array is v1(k), and the array manifoldfor the second array is v2(k),then for a plane with wavenumber kl , wehave v2(kl) = v1(kl)e−jωcτl . The overall array manifold has the form

V(kl) =

[v1(kl)v2(kl)

]=

[v1(kl)

v1(kl)e−jωcτl

]Now consider D plane waves. The signal subspace

V = [V0,V1, . . . ,VD−1] =

[v1(k0) v1(k1) . . . v1(kD−1)v2(k0) v2(k1) . . . v2(kD−1)

]=

[V1

V2

]=

[v1(k0) v1(k1) . . . v1(kD−1)

v1(kl)e−jωcτ0 v1(kl)e−jωcτ1 . . . v1(kl)e−jωcτD−1

]=

[V1

V1Ψ

]An ESPRIT algorithm can be developed now with

Es =

[E1

E2

]with E2 = E1Φ

Coherent Sources

Sx = Ss + σ2z I = VSFV

H + σ2z I

The dimension D of the signal subspace depends on the rank of thesource covariance matrix SF .

If the sources are highly correleated, SF can be ill-conditioned or singular.This can make the dimension of the signal subspace less than D and inparticular when an estimates of Sx is used.

Spatial Smoothing for ULAs: A techniques to overcome this rankdeficiency problem.

Spatial Smoothing

We will consider the ULA as two sub-arrays as in ESPRIT: subarray 1 isthe first (N − 1) sensors and subarray 2 is the last (N − 1) sensors.Since the data is given by x[n] = VF[n] + Z[n], we have the data fromthe two sub-arrays given by

x1[n] = V1F[n] + Z1[n] and x2[n] = V2F[n] + Z2[n]

Covariance of the two subarrays are given by

S(1)x = V1SFV

H1 + σ2

z I and S(2)x = V2SFV

H2 + σ2

z I = V1ΨSFΨHVH1 + σ2

z I

Spatially Smoothed Covariance

S(s)x =

1

2

(S(1)x + S(2)

x

)=

1

2

(V1SFV

H1 + σ2

z I + V2SFVH2 + σ2

z I)

=1

2

(V1SFV

H1 + V1ΨSFΨHVH

1

)+ σ2

z I

= V1

(SF + ΨSFΨH

2

)VH

1 = V1SFVH1

SF = SF +ΨSF ΨH

2 is more likely to have rank D restoring the signalsubspace dimension in the spatially smoothed covariance matrix

Spatial Smoothing Cont’d

We can generalize this idea by viewing the ULA as P sub-arrays: subarray1 is the first (N − P + 1) sensors, and subarray 2 is the next (N − P + 1)sensors and so on.Since the data is given by x[n] = VF[n] + Z[n], we have the data fromthe l sub-array given by

xl [n] = VlF[n] + Zl [n], l = 1, 2, ..,P with Vl = V1Ψl−1

Covariance of the lth sub-array is given by

S(l)x = VlSFV

Hl + σ2

z I = V1Ψl−1S(ΨH)l−1V1 + σ2z I

Spatially Smoothed Covariance

S(s)x =

1

P

P∑l=1

S(l)x = V1SFV

H1

SF = 1P

∑Pl=1 Ψl−1SF (ΨH)l−1 is more likely to have rank D restoring the

signal subspace dimension in the spatially smoothed covariance matrix

Covariance Estimation

Given L snapshots

Sx =1

L

L−1∑n=0

x[n]xH [n] =1

LDDH , where D = [x[0], x[1], . . . , x[L− 1]]

Smoothed Covariance Estimate

S(s)x =

1

P

P∑l=1

S(l)x =

1

PL

P∑l=1

DlDHl where Dl = [xl [0], xl [1], . . . , xl [L− 1]]

and xl [n] = [xl−1[n], xl [n], . . . , xN−P+l−1[n]]T .For P = 2, we have x1[n] = [x0[n], x1[n], . . . , xN−2[n]]T andx2[n] = [x1[n], x2[n], . . . , xN−l [n]]T .Alternatively, we can express the smoothed covariance estimate as

S(s)x =

1

L

L−1∑n=0

Ds [n]DHs [n], where Ds [n] = [x1[n], x2[n], . . . , xP [n]]

Time Series: Sum of Exponentials

x [n], n = 0, 1, . . . ,M − 1: L = 1, one snapshot, array size N = M, whereM is the number of samples containing D exponentials.To compute a covariance matrix, we have to do smoothing.

Ds =

x [0] x [1] . . . x [P − 1]x [1] x [2] . . . x [P]

......

......

x [M − P] x [M − P + 1] . . . x [M − 1]

The smoothed covariance can be computed a S

(s)x = 1

PDsDHs .

Since we need rank of the matrix to be greater than D, we need P > Dand M − P + 1 > D. This requires M > 2D.

Instead of an eigen-decomposition of S(s)x , one can do a SVD of Ds .

Numerically superior.Can use MUSIC, ESPRIT or any subspace method to estimate thefrequencies ωl .

Forward-Backward (FB) Smoothing for ULAs

For a ULA, we have JV∗(ψ) = V(ψ). Hence

JS∗xJ = JV∗S∗

FVTJ + σ2

zJIJ = VS∗FV

H + σ2z I

The signal subspace of JS∗xJ is also spanned by V and so subspace

methods MUSIC etc can be applied to JS∗xJ.

The FB covariance estimate

SFBx =

1

2(Sx + JS∗

xJ) =1

2V(SF + S∗

F )VH + σ2z I

Subspace methods can be readily applied to SFBx The FB covariance

matrix has a better condition number and so improves performance.The FB approach can be applied to spatial smoothing, i.e.12

(S

(s)x + JS

∗(s)x J

)and also to the time series [Ds , JD∗

s ].