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Ultra-Wideband (UWB) communication systems have been developed for short distance, high data rate communications. To avoid interfering with the existing systems in the same environment, very short duration pulses used by these systems should satisfy a predefined spectral mask. Data rate of UWB systems can be increased by using multiple pulse shapes simultaneously. Orthogonality of the simultaneously used pulse shapes simplifies the receiver design. In this work, design of orthogonal pulse shapes which satisfy the spectral mask is modelled as an optimization problem. First, it is converted to a convex optimization problem by constraining the pulse shapes to lie in a subspace spanned by the Hermite-Gaussian (HG) functions. Then the optimal solution is obtained. It is shown that a larger pulse shape set can be designed compared to the existing approaches, and hence, a higher data rate can be achieved.