# The Polynomial - Space Of The Music

## MUSIC Super-Resolution DOA Estimation

- The Degree Of The Polynomial
- Add The Polynomials
- Find The Polynomial
- The Polynomial - Space Of The Music Man

*MUltiple SIgnal Classification* (MUSIC)is a high-resolution direction-finding algorithm based on the eigenvaluedecomposition of the sensor covariance matrix observed at an array.MUSIC belongs to the family of subspace-based direction-finding algorithms.

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### Signal Model

The signal model relates the received sensor data to the signalsemitted by the source. Assume that there are *D* uncorrelatedor partially correlated signal sources, *s _{d}(t)*.The sensor data,

*x*,consists of the signals, as received at the array, together with noise,

_{m}(t)*n*.A sensor data snapshot is the sensor data vector received at the

_{m}(t)*M*elementsof an array at a single time

*t*.

$$\begin{array}{l}x(t)=As(t)+n(t)\\ s(t)=[{s}_{1}(t),{s}_{2}(t),\dots ,{s}_{D}(t)){]}^{\prime}\\ A=[a({\theta}_{1})a({\theta}_{2})\dots a({\theta}_{D})]\end{array}$$

*x(t)*is an*M*-by-1vector of received snapshot of sensor data which consist of signalsand additive noise.*A*is an*M*-by-*D*matrixcontaining the arrival vectors. An arrival vector consists of therelative phase shifts at the array elements of the plane wave fromone source. Each column of*A*represents the arrivalvector from one of the sources and depends on the direction of arrival,*θ*._{d}*θ*isthe direction of arrival angle for the_{d}*d*th sourceand can represents either the broadside angle for linear arrays orthe azimuth and elevation angle for planar or 3D arrays.*s(t)*is a*D*-by-1vector of source signal values from*D*sources.*n(t)*is an*M*-by-1vector of sensor noise values.

An important quantity in any subspace method is the *sensorcovariance matrix*,*R _{x}*,derived from the received signal data. When the signals are uncorrelatedwith the noise, the sensor covariance matrix has two components, the

*signalcovariance matrix*and the

*noise covariancematrix*.

where *R _{s}* isthe

*source covariance matrix*. The diagonalelements of the source covariance matrix represent source power andthe off-diagonal elements represent source correlations.

For uncorrelated sourcesor even partially correlated sources, *R _{s}* isa positive-definite Hermitian matrix and has full rank,

*D*,equal to the number of sources.

The signal covariance matrix, *AR _{s}A^{H}*,is an

*M*-by-

*M*matrix, also withrank

*D < M*.

### The Degree Of The Polynomial

An assumption of the MUSIC algorithm is that the noise powersare equal at all sensors and uncorrelated between sensors. With thisassumption, the noise covariance matrix becomes an *M*-by-*M* diagonalmatrix with equal values along the diagonal.

Because the true sensor covariance matrix is not known, MUSICestimates the sensor covariance matrix, *R _{x}*,from the

*sample*sensor covariance matrix. Thesample sensor covariance matrix is an average of multiple snapshotsof the sensor data

where *T* isthe number of snapshots.

### Signal and Noise Subspaces

Because *AR _{s}A^{H}* hasrank

*D*, it has

*D*positive realeigenvalues and

*M – D*zero eigenvalues.The eigenvectors corresponding to the positive eigenvalues span the

*signalsubspace*,

*U*.The eigenvectors corresponding to the zero eigenvalues are orthogonalto the signal space and span the

_{s}= [v_{1},...,v_{D}]*null subspace*,

*U*.The arrival vectors also belong to the signal subspace, but they areeigenvectors. Eigenvectors of the null subspace are orthogonal tothe eigenvectors of the signal subspace. Null-subspace eigenvectors,

_{n}=[u_{D+1},...,u_{N}]*u*,satisfy this equation:

_{i}Therefore the arrivalvectors are orthogonal to the null subspace.

When noise is added, the eigenvectors of the sensor covariancematrix with noise present are the same as the noise-free sensor covariancematrix. The eigenvalues increase by the noise power. Let *v _{i}* beone of the original noise-free signal space eigenvectors. Then

shows that the signalspace eigenvalues increase by *σ _{0}^{2}*.

The null subspace eigenvectors are also eigenvectors of *R _{x}*.Let

*u*be one of the nulleigenvectors. Then

_{i}with eigenvalues of *σ _{0}^{2}* insteadof zero. The null subspace becomes the

*noise subspace*.

MUSIC works by searching for all arrival vectors that are orthogonalto the noise subspace. To do the search, MUSIC constructs an arrival-angle-dependentpower expression, called the MUSIC pseudospectrum:

When an arrival vectoris orthogonal to the noise subspace, the peaks of the pseudospectrumare infinite. In practice, because there is noise, and because thetrue covariance matrix is estimated by the sampled covariance matrix,the arrival vectors are never exactly orthogonal to the noise subspace.Then, the angles at which *P _{MUSIC}* hasfinite peaks are the desired directions of arrival. Because the pseudospectrumcan have more peaks than there are sources, the algorithm requiresthat you specify the number of sources,

*D*, as aparameter. Then the algorithm picks the

*D*largestpeaks. For a uniform linear array (ULA), the search space is a one-dimensionalgrid of broadside angles. For planar and 3D arrays, the search spaceis a two-dimensional grid of azimuth and elevation angles.

### Root-MUSIC

For a ULA, the denominator in the pseudospectrum is a polynomialin $${e}^{ikd\mathrm{cos}\phi}$$,but can also be considered a polynomial in the complex plane. In thiscases, you can use root-finding methods to solve for the roots, *z _{i}*.These roots do not necessarily lie on the unit circle. However, Root-MUSICassumes that the

*D*roots closest to the unit circlecorrespond to the true source directions. Then you can compute thesource directions from the phase of the complex roots.

### Spatial Smoothing of Correlated Sources

When some of the *D* source signals are correlated, *R _{s}* isrank deficient, meaning that it has fewer than

*D*nonzeroeigenvalues. Therefore, the number of zero eigenvalues of

*AR*exceedsthe number,

_{s}A^{H}*M – D*, of zero eigenvalues forthe uncorrelated source case. MUSIC performance degrades when signalsare correlated, as occurs in a multipath propagation environment.A way to compensate for correlation is to use spatial smoothing.

### Add The Polynomials

*Spatial smoothing* takes advantage ofthe translation properties of a uniform array. Consider two correlatedsignals arriving at an *L*-element ULA. The sourcecovariance matrix, *R _{s}* isa singular 2-by-2 matrix. The arrival vector matrix is an

*L*-by-2matrix

### Find The Polynomial

$$\begin{array}{l}{A}_{1}=\left[\begin{array}{c}1\\ {e}^{ikd\mathrm{cos}{\phi}_{1}}\\ \vdots \\ {e}^{i(L-1)kd\mathrm{cos}{\phi}_{1}}\end{array}\begin{array}{c}1\\ {e}^{ikd\mathrm{cos}{\phi}_{2}}\\ \vdots \\ {e}^{i(L-1)kd\mathrm{cos}{\phi}_{2}}\end{array}\right]=\left[a\left({\phi}_{1}\right)a\left({\phi}_{2}\right)\right]\end{array}$$

for signals arrivingfrom the broadside angles *φ _{1}* and

*φ*.The quantity

_{2}*k*is the signal wave number.

*a(φ)*representsan arrival vector at the angle

*φ*.

You can create a second array by translating the first arrayalong its axis by one element distance, *d*. Thearrival matrix for the second array is

$${A}_{2}=\left[\begin{array}{c}{e}^{ikd\mathrm{cos}{\phi}_{1}}\\ {e}^{i2kd\mathrm{cos}{\phi}_{1}}\\ \vdots \\ {e}^{iLkd\mathrm{cos}{\phi}_{1}}\end{array}\begin{array}{c}{e}^{ikd\mathrm{cos}{\phi}_{2}}\\ {e}^{i2kd\mathrm{cos}{\phi}_{2}}\\ \vdots \\ {e}^{iLkd\mathrm{cos}{\phi}_{2}}\end{array}\right]=\left[{e}^{ikd\mathrm{cos}{\phi}_{1}}a\left({\phi}_{1}\right){e}^{ikd\mathrm{cos}{\phi}_{2}}a\left({\phi}_{2}\right)\right]$$

where the arrival vectorsare equal to the original arrival vectors but multiplied by a direction-dependentphase shift. When you translate the original array *J –1* moretimes, you get *J * copies of the array. If you forma single array from all these copies, then the length of the singlearray is *M = L + (J – 1)*.

In practice, you start with an *M*-elementarray and form *J* overlapping subarrays. The numberof elements in each subarray is *L = M – J + 1*.The following figure shows the relationship between the overall lengthof the array, *M*, the number of subarrays, *J*,and the length of each subarray, *L*.

For the *p*th subarray, the source signal arrivalmatrix is

$$\begin{array}{c}{A}_{p}=\left[{e}^{ik\left(p-1\right)d\mathrm{cos}{\phi}_{1}}a\left({\phi}_{1}\right){e}^{ik\left(p-1\right)d\mathrm{cos}{\phi}_{2}}a\left({\phi}_{2}\right)\right]\\ =\left[a\left({\phi}_{1}\right)a\left({\phi}_{2}\right)\right]\left[\begin{array}{cc}{e}^{ik\left(p-1\right)d\mathrm{cos}{\phi}_{1}}& 0\\ 0& {e}^{ik\left(p-1\right)d\mathrm{cos}{\phi}_{2}}\end{array}\right]={A}_{1}{P}^{p-1}\\ P=\left[\begin{array}{cc}{e}^{ikd\mathrm{cos}{\phi}_{1}}& 0\\ 0& {e}^{ikd\mathrm{cos}{\phi}_{2}}\end{array}\right].\end{array}$$

The original arrivalvector matrix is postmultiplied by a diagonal phase matrix.

The last step is averaging the signal covariance matrices overall *J* subarrays to form the averaged signal covariancematrix, *R ^{avg}_{s}*.The average signal covariance matrix depends on the smoothed sourcecovariance matrix,

*R*.

^{smooth}### The Polynomial - Space Of The Music Man

$$\begin{array}{l}{R}_{s}^{avg}={A}_{1}\left(\frac{1}{J}{\displaystyle \sum _{p=1}^{J}{P}^{p-1}{R}_{s}{\left({P}^{p-1}\right)}^{H}}\right){A}_{1}^{H}={A}_{1}{R}^{smooth}{A}_{1}^{H}\\ {R}^{smooth}=\frac{1}{J}{\displaystyle \sum _{p=1}^{J}{P}^{p-1}{R}_{s}{\left({P}^{p-1}\right)}^{H}}.\end{array}$$

You can show that thediagonal elements of the smoothed source covariance matrix are thesame as the diagonal elements of the original source covariance matrix.

$${R}_{ii}^{smooth}=\frac{1}{J}{\displaystyle \sum _{p=1}^{J}{\left({P}^{p-1}\right)}_{im}{\left({R}_{s}\right)}_{mn}{\left({P}^{p-1}\right)}_{ni}{H}^{}}=\frac{1}{J}{\displaystyle \sum _{p=1}^{J}{R}_{s}}={\left({R}_{s}\right)}_{ii}$$

However, the off-diagonal elements are reduced. The reductionfactor is the beam pattern of a *J*-element array.

$${R}_{ij}^{smooth}=\frac{1}{J}{\displaystyle \sum _{p=1}^{J}{e}^{ikd(p-1)\left(\mathrm{cos}{\phi}_{1}-\mathrm{cos}{\phi}_{2}\right)}}{\left({R}_{s}\right)}_{ij}=\frac{1}{J}\frac{\mathrm{sin}\left(kdJ\left(\mathrm{cos}{\phi}_{1}-\mathrm{cos}{\phi}_{2}\right)\right)}{\mathrm{sin}\left(kd\left(\mathrm{cos}{\phi}_{1}-\mathrm{cos}{\phi}_{2}\right)\right)}{\left({R}_{s}\right)}_{ij}$$

In summary, you can reduce the degrading effect of source correlationby forming subarrays and using the smoothed covariance matrix as inputto the MUSIC algorithm. Because of the beam pattern, larger angularseparation of sources leads to reduced correlation.

Spatial smoothing for linear arrays is easily extended to 2Dand 3D uniform arrays.