Let *y(k)* be a *p*^{th}-order autoregressive process disturbed by an additive noise *b(k)* which is independent of the driving process *u(k)* i.e.:

In Chapter 2, we saw that the noisy observation *z(k)* can itself be considered as a *p*^{th}-order autoregressive process. In fact, we can easily show that:

To understand and analyze the influence of the noise on the estimation of the AR parameters, Kay proposes a comparison between the spectral flatness ξ_{y} of the process *y(k)* and that of the observation *z{k)*, i.e. ξ_{z} [1]. For any process *x*, the spectral flatness is defined as follows:

where *S _{xx}*(ω) and

When *x(k)* is an autoregressive process, it can be seen as a sequence *w(k)* filtered by a filter whose transfer function is *H(z)* = 1/ *A(z)*. Since the poles of *H(z)* necessarily lie inside the unit circle in the *z*-plane, the filter with transfer function *A(z)* satisfies the following criterion^{4}:

This implies that:

However, ...

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