Subspace-based Estimators - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 211

where the d m are the coefficients of vector d and with:

) 2 (

( )

− 1

()

j 2 π f

[8.31]

S f = Pe

MN MN

It was shown in [KUM 83] that, by choosing the vector d of minimum norm,
with 1 as first component, the zeros of the estimating polynomial of P complex sine
waves in noise associated with the noise were reduced inside the unit circle.

Searching the vector d of minimum norm which belongs to esp ( V b ) means
searching the vector w so that:

dVw =

b

[8.32]

( )

2

()

H

J wd = λ − d e 1minimum

1

where e 1 is a vector of first component equal to 1 with all the other components
equal to zero. Minimizing:

( )

()

HH HH

1

J w = wVVw +λ wVe − 1 [8.33]

bb b

H

with respect to w , by noting that the vectors of V b being orthonormal, VV = I ,

bb

leads to:

λ

H

wVe =− [8.34]

b 1

2

The condition of first component equal to 1 gives:

2

λ=−

eVVe

[8.35]

HH
1 bb 1

Finally, we obtain:

H

VV e

1
HH
1

bb

[8.36]

d =

eVVe

bb 1

Thus, the frequencies included in the signal can be directly obtained by searching
the zeros of the polynomial of degree M - 1, P MN ( z ). It is shown in [KUM 83] that
the P zeros corresponding to the peaks in [8.29] are situated on the unit circle and
that the M - P other zeros remain limited inside the unit circle.