  Let $A$ be a finite dimensional hereditary algebra over a finite field, and
let $m$ be a fixed integer such that $m=0$ or $m>2$. In the present paper, we
first define an algebra $L_m(A)$ associated to $A$, called the $m$-periodic
lattice algebra of $A$, and then prove that it is isomorphic to Bridgeland's
Hall algebra $\mathcal {D}\mathcal {H}_m(A)$ of $m$-cyclic complexes over
projective $A$-modules. Moreover, we show that there is an embedding of the
Heisenberg double Hall algebra of $A$ into $\mathcal {D}\mathcal {H}_m(A)$.
