will always periodogram produce correct frequencies?

Question

let say we have some data,some signal written in this form

$x(t)=s(t)+\epsilon(t)$

where $s(t)$ is pure signal and $\epsilon(t)$ is white noise with mean zero and some variance ,question is that if we have enough sample data,should periodogram always estimate frequencies correctly?let say we have following sinusoidal data

$y[t]=A_1(sin(\omega_1*t+\phi_1)+A_2*sin(\omega_2*t+\phi_2)+....+A_p*sin(\omega_p*t+\phi_p)$+$z(t)$

i know that there is problem of variance increasing related to periodogram,but will it affect on frequency estimation?also i know that resolution of periodogram is $1/N$,but let us suppose that two frequency are not so close to each other,then will periodogram produce correct frequency estimation?variance is not important for me know

Answer 1

A periodogram created from a finite number of samples using an DFT is not really a "correct" frequency estimator, as much as it is a (leaky) frequency energy binning sieve, where the width of each bin is set by the DFT length and the window function applied.

The width and spacing of each bin may or may not be narrow enough for you to consider an error of up to half that size to still result in a useful frequency estimate. Various interpolation methods that post-process DFT results may allow more precise estimates of spectral peaks than the bin spacing.

Answer 2

You will get good frequency information from a periodogram if, as you say, you have enough data to resolve the frequencies to the desired resolution.

It is important that your signal be band limited to 1/2 the sample rate of the data for your periodogram.