the signal spectrum has a given a priori shape.

Let the spectrum shape is $\langle (a_l^m)^2 \rangle =Q^2 F_{lm}$. If the amplitude is $Q$, the likelihood function may be described as: $$f(a_{lm},Q|F_l) = \prod_{lm} \frac{1}{\sqrt{2 \pi Q^2 F_{lm}}}
 \exp \left( - \frac{ a_{lm}^2 } { 2 Q^2 F_{lm} } \right),$$

The maximum likelihood estimation $Q_{\rm ML}$ may be found if solve the equation maximizing the likelihood function: $$\frac{\partial \ln f(Q|F_l)}{\partial Q^2} = 0$$ $$Q^2_{\rm ML} = \frac {\sum (a^2_{lm}/F_{lm})}{ M }, \qquad
 M=\sum_{l,m} 1 \,$$

where $M$ is the number of the analyzed harmonics.

The parameter is evident having $\chi^2_M$ distribution with $M$ degree of freedom, $Q^2_{\rm ML} \sim \chi^2_M$. The amplitude estimation calculated using the variance on the sphere $ \sigma^2
$ is: $$Q^2_{\rm POWER}=\frac{\sum \Delta T^2_l}{\sum \langle \Delta
T^2_l \rangle |{Q=1}}
= \frac{\sigma^2}{\sigma^2_{th}|Q=1} \sim \chi^2_{N_{eff}} \, .$$

It is clear that: $N_{eff}= \frac{(\sum_{lm} \Delta T^2_l)^2}{\sum_{lm} (\Delta
T^2_l)^2} \le  M$. So for the Harrison-Zel’dovich spectrum $N_{eff} \approx 100$, $ M \approx 700$ пpи $l_{max} =25$. The maximum likelihood estimation accuracy is about 5% and the total power estimation is about 14%.

Using the approach we may determine the signal spatial spectrum parameters in the presence of noise. Maximum likelihood 