Subspace-based Estimators - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 200

where δ m-m ' is the Kronecker symbol. If the α i are non-correlated (for example, if
α i = A i e j φ i with the determinist A i and the random φ i uniformly distributed on [0, 2π]
and independent two by two), then x ( k ) is stationary and its correlation coefficients
are:

p

( )

( )

∑

2

e π

j 2 f mmT − ' 2

γ mm −= ' σ + σδ [8.3]

i e

xx α i m − m '

i = 1

and the correlation matrix of order M Γ is written:

xx

() ( )

γ 0 γ M − 1

⎡ ⎤

xx xx

⎢ ⎥

Γ = ⎢ [8.4]

⎥

xx

( ) ( )

⎢ ⎥

γ M − 1* γ 0

⎣ ⎦

xx xx

(* indicates the conjugate of a complex number). We will consider this case in the
following.

In practice, it is not

x Γ that will be used but its estimate:

() ( )

⎡ γ ˆ 0 γ ˆ M − 1 ⎤

xx xx

⎢ ⎥

ˆ

Γ = ⎢ [8.5]

xx

⎥

( ) ( )

⎢ ⎥

γ ˆ M − 1* γ ˆ 0

⎣ ⎦

xx xx

where:

( 1/2

N −

1

()

∑

( ) ( )

ˆ

γ m = x n + m x * n [8.6]

xx

N

( )

nN =− − 1/2

and N is supposed to be odd. This estimation of

x Γ supposes the ergodicity of the

signal x ( k ).

8.1.2. Concept of subspaces

The methods presented in what comes next are based on the decomposition of
the observation space in two subspaces, the signal subspace and the noise subspace.
Actually, by introducing the vector x ( k ) of the M observations

{ x ( k ), x ( k + 1), …, x ( k + M - 1)}, we easily verify that: