1. AGC DSP AGC DSP Professor A G Constantinides©1 Eigenvector-based Methods A very common problem in spectral estimation is concerned with the extraction of uncorrelated sinusoids from noise The so-called eigen-decomposition methods are amongst the best at high SNR They are used also extensively in array signal processing eg for the estimation of the Direction of Arrival (DoA).(Measurements of spatial frequencies is equivalent to direction finding)

2. AGC DSP AGC DSP Professor A G Constantinides©2 Eigenvector-based Methods These techniques can resolve frequencies that are closely spaced and hence are often referred to as “super- resolution” methods

3. AGC DSP AGC DSP Professor A G Constantinides©3 Mathematical Background The signal model: Assume that the signal is given by Then where

4. AGC DSP AGC DSP Professor A G Constantinides©4 Mathematical Background Set

5. AGC DSP AGC DSP Professor A G Constantinides©5 Mathematical Background Then we can write the autocorrelation matrix as where

6. AGC DSP AGC DSP Professor A G Constantinides©6 Mathematical Background And hence Where And The vector space is the signal subspace of

7. AGC DSP AGC DSP Professor A G Constantinides©7 Mathematical Background If then has rank Let the eigenvalues of be Corresponding to the normalised eigenvectors Then and

8. AGC DSP AGC DSP Professor A G Constantinides©8 Mathematical Background Since is off rank then And hence The eigenvectors are the “principal eigenvectors” of

9. AGC DSP AGC DSP Professor A G Constantinides©9 Mathematical Background An important result is “The principal eigenvectors span the signal subspace” Thus given a sequence of observations we can determine the autocorrelation matrix and its eigenvectors. Knowing the first eigenvectors we can determine the space in which the signals reside even though at this point we do not know their frequencies.

10. AGC DSP AGC DSP Professor A G Constantinides©10 Mathematical Background The noise model: We assume that we observe signal contaminated additively, by a stationary, zero mean, white noise, independent of it Then

11. AGC DSP AGC DSP Professor A G Constantinides©11 Mathematical Background From the above we have Moreover with

12. AGC DSP AGC DSP Professor A G Constantinides©12 Mathematical Background We have Clearly is of rank ie full rank Let its eigenvalues be where the first are While the rest are all ewual to the variance of noise

13. AGC DSP AGC DSP Professor A G Constantinides©13 Mathematical Background Thus we can write The space Is called the noise subspace. Important result: Any vector in the signal subspace is orthogonal to the noise subspace

14. AGC DSP AGC DSP Professor A G Constantinides©14 Pisarenko The Pisarenko harmonic decomposition exploits the orthogonality of two subspaces directly. Let the number of sinusoids (modes) be known Set so that the noise is spanned a single vector

15. AGC DSP AGC DSP Professor A G Constantinides©15 Mathematical Background Note that must be orthogonal to all the signal subspace vectors With We have

16. AGC DSP AGC DSP Professor A G Constantinides©16 Mathematical Background The last equation is a polynomial in and hence its which all lie on the unit circle correspond to the frequencies of the sinuoidal signal. The amplitudes are obtained from the autocorrelation relationships of the observations as given earlier. The noise strength is given from the last eigenvalue of the same autocorrelation matrix.

17. AGC DSP AGC DSP Professor A G Constantinides©17 MUSIC Multiple Signal Classification relies on the same principle of orthogonality. Let

18. AGC DSP AGC DSP Professor A G Constantinides©18 MUSIC Now define the function Clearly.Hence its reciprocal is infinite Thus the reciprocal of the above function exhibits peaks at the input frequencies. So that whenever (one of the input requencies), then for any vector in the noise subspace The signal strengths can be computed as in the Pisarenko Harmonic Decomposition

19. AGC DSP AGC DSP Professor A G Constantinides©19 MUSIC The quantity below is known as the MUSIC spectrum.