Using a detailed Galactic model in which we consider distinct contributions from the bulge, thin disc, thick disc, and halo, and a population synthesis approach, we determined the birth rates, numbers, and period distributions of double white dwarfs (DWDs) within each component. In the Galaxy as a whole, our model predicts the current birth rate of DWDs to be 3.21x10^{-2} yr^{-1}, the local density to be 2.2x10^{-4} pc^{-3} and the total number to be 2.76x10^{8}. Assuming SNIa are formed from the merger of two CO white dwarfs, the SNIa rate should be 0.0013 yr^{-1}. The frequency spectra of DWD strain amplitude and number distribution are presented as a function of galactic component, DWD type, formation channel, and metallicity. We confirm that CO+He and He+He white dwarf (WD) pairs should dominate the GW signal at very high frequencies (log f Hz^{-1} > -2.3), while CO+CO and ONeMg WD pairs have a dominant contribution at log f Hz^{-1} < -2.3. Formation channels involving two common-envelope (CE) phases or a stable Roche lobe overflow phase followed by a CE phase dominate the production of DWDs detectable by LISA at log f Hz^{-1} > -4.5. DWDs with the shortest orbital periods will come from the CE+CE channel. The Exposed Core plus CE channel is a minor channel. A number of resolved DWDs would be detected, making up 0.012% of the total number of DWDs in the Galaxy. The majority of these would be CO+He and He+He pairs formed through the CE+CE channel.
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