a = 6;  b  =  6;

lattice1  =  
        Table[Sign[-0.5 + 
              Sin[(2*Pi/a)*i]*Sin[(2*Pi/b)*j]],  {i,  1,  
            128},  {j,  1,  128}];
ListDensityPlot[lattice1,  Mesh  ->  False];

fourspec = Abs[Fourier[lattice1]];
temp1  =  
        Transpose[
          Join[fourspec[[Range[65, 128],  Range[1, 128]]],  
            fourspec[[Range[1, 64],  Range[1,  128]]]]];
recipspace  =  
        Transpose[
          Join[temp1[[Range[65, 128],  Range[1, 128]]],  
            temp1[[Range[1, 64],  Range[1,  128]]]]];
ListDensityPlot[recipspace,  PlotRange  ->  {0,  20},  
        Mesh  ->  False];