Context : The gravitational wave ( GW ) background in the range 0.01 - 30 mHz has been assumed to be dominated by unresolved radiation from double white dwarf binaries ( DWDs ) .
Recent investigations indicate that , at short periods , a number of DWDs should be resolvable sources of GW .

Aims : To characterize the GW signal which would be detected by LISA from DWDs in the Galaxy .

Methods : We have constructed a Galactic model in which we consider distinct contributions from the bulge , thin disc , thick disc , and halo , and subsequently executed a population synthesis approach to determine the birth rates , numbers , and period distributions of DWDs within each component .

Results : In the Galaxy as a whole , our model predicts the current birth rate of DWDs to be 3.21 \times 10 ^ { -2 } yr ^ { -1 } , the local density to be 2.2 \times 10 ^ { -4 } pc ^ { -3 } and the total number to be 2.76 \times 10 ^ { 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 .

Conclusions : 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 } } \leq - 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 .