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In this article, we consider a harmonic analysis of locally compact groups and introduce a generalization of the classical cross-Wigner distribution defined on G × G by W_F (ψ,φ)(g,ξ)=_Gξ(h)ψ(τ_1(g,h))φ(τ_2(g,h))dμ(h).We construct the so-called Weyl-Heisenberg frame on a locally compact communicative Hausdorff group and establish its properties. Thus, we show that assume Λ and Γ are closed cocompact subgroups of G and G, respectively, then, for a given window φ∈ L^2(G), either both systems {m_γτ_λφ}_λΛ,γΓ and {m_κτ_vφ}_κΛ^⊥,vΓ^⊥ are Gabor systems in L^2 (G), simultaneously, with the same upper bound, or neither {m_γτ_λφ}_λΛ,γΓ nor {m_κτ_vφ}_κΛ^⊥,vΓ^⊥ comprises a Gabor system. Finally, pseudo-differential operators on locally compact groups are studied, we establish that assuming a pseudo-differential operator A_a corresponds to the symbol aW^∞,1_τ,1oι^-1 (G×G) then A_a is bounded operator W^p,q_τ(G)→ W^p,q_τ(G). , number=4, journal=Journal of Nonlinear Modeling and Analysis, year=2024, month=Dec., pages=1031–1045