In the present work, we study stability characteristics of synthetic profiles prototypical of coannular jets, using a spatial, inviscid and compressible formulation of linear stability theory. The focus of our study is to understand the effect of the wake of the nozzle that separates the primary jet from the secondary jet, on the characteristics of different instability modes. Some of the parameters under focus are Mach number of the primary jet (M₁), wake deficit (W_amp), and temperature ratio (T₀₁/T_∞). One of the other aims of this work is to compare our results with that of the 2-D planar mixing layers separated by a splitter plate - for which counteracting effect of compressibility and the wake of the splitter plate, on the instability characteristics, are known. Our results show that, for circular jets, unlike 2-D planar mixing layers, the vorticity introduced by the wake fails to overcome the stabilization effect caused by compressibility. Heating the jet, i.e., increasing the temperature ratio T₀₁/T_∞, may increase or decrease the maximum growth rate depending on the exact combination of the flow parameters and the particular stability mode.