The limitations of homogeneous uniform Electromagnetic Bandgap (EBG) structures are well known. For broad operating bandwidths thicker EBGs are required. For VHF and UHF frequency bands total EBG plus antenna thickness could be prohibitively large. This report presents a unique concept to design and develop broadband EBG structures. Based on the analogy of microwave filter theory a new class of Non-Uniform Aperiodic (NUA) EBG structure is proposed. Under the proposed scheme the dimensions of the patches and their inter-element distances are varied according to a taper. It is shown that a broadband (more than an octave bandwidth) directional dipole antenna can be developed with total EBG plus antenna thickness of 1/30th of the free-space wavelength. Experimental validation of the proposed concept is also presented.