208 IEEE TRANSACTIONS ON COMMUNICATIONS LETTERS, VOL. 2 , NO. 8, AUGUST 1998 The Modified Gaussian: A Novel Wavelet with Low Sidelobes with Applications to Digital Communications F. Dovis, M. Mondin, Member, IEEE, and F. Daneshgaran, Member, IEEE Abstract— In this letter we present a wavelet with very low sidelobes whose spectral occupancy in the frequency domain is controlled by a parameter that can assume any positive real value. The associated scaling function is derived from the Gaussian waveform. Due to its good spectral characteristics, this function is well suited as an elementary shaping pulse for digital modulation. Index Terms—Modulation, shaping pulse, wavelets. I. INTRODUCTION T HE USE OF the scaling functions and wavelets as elementary waveforms for modulation has been proposed in several recent works [1], [2]. The focus of this letter is exclusively on dyadic scaling functions and wavelets. Further- more, in this letter we shall keep referring to wavelets with the understanding that, in actuality, two pulses are involved in any dyadic multiresolution analysis; namely, the scaling function and the wavelet. For the sake of the discussions that follow, what we need to know about the wavelets is that in a dyadic multiresolution analysis the scaling function satisfies the scaling equation where is the set of integers, and the mother wavelet satisfies a similar equation The sequence is called the scaling vector, while the sequence is denoted as the wavelet vector. The functions and , satisfying the above relations, individually sat- isfy the Nyquist I criterion and are, therefore, shift orthogonal with respect to integer shifts. Furthermore, these two pulses Manuscript received October 7, 1997. The associate editor coordinating the review of this letter and approving it for publication was Prof. H. V. Poor. This work was supported in part by the Italian National Research Council (CNR). F. Dovis and M. Mondin are with the Dipartimento di Elettronica, Politec- nico di Torino, Torino, Italy (e-mail: dovis@polito.it; mondin@polito.it). F. Daneshgaran is with the Electrical and Computer Engineering Depart- ment, California State University, Los Angeles, CA 90032 USA (e-mail: fdanesh@calstatela.edu). Publisher Item Identifier S 1089-7798(98)06463-1. and their integer shifts are mutually orthogonal to each other so that in effect, the two pulses and their shifts span orthogonal frequency channels (the scaling function is a baseband Nyquist pulse, and the wavelet a bandpass Nyquist pulse). An undesirable feature commonly observed in wavelets found in the literature in connection with their use as ele- mentary shaping pulses for modulation is that their spectral occupancy is not parametrically defined and can vary from one wavelet family to another. Hence, when selecting a suitable wavelet shaping pulse in a communication system, it is neces- sary to examine a particular family of wavelets to determine its suitability given the available bandwidth. It would be desirable to have a parametrically defined wavelet, having a feature like the square root raised-cosine shaping pulse, whose spectral occupancy and excess bandwidth are controlled (although to a limited extent) by the rolloff parameter. In fact, the square root raised-cosine pulse satisfies the dyadic scaling equation and is, therefore, a scaling function, but only for values of the rolloff parameter less than one-third. To the best of our knowledge, there are no parametrically defined wavelets with good spectral characteristics with a parameter whose value can vary over the entire positive real line. However, in this letter, we present such a wavelet. II. THE MODIFIED GAUSSIAN The wavelet presented here is obtained by applying the orthogonalization trick [3] to a Gaussian waveform. It is known that this trick preserves some properties of the original function and, if successful, permits the creation of a new family of wavelets in the multiresolution analysis toolbox. The use of the Gaussian waveform as a starting point in the application of the orthogonalization trick is motivated by the fact that this function has excellent time–bandwidth product near the uncertainty bound [4], in addition to being parametrically defined with the parameter being the variance of the pulse. Note that the original Gaussian waveform is not shift-orthogonal and its use as an elementary pulse in a coherent modulation scheme causes intersymbol interference, even on a channel with flat spectrum. In what follows, we describe the construction of the modi- fied Gaussian wavelet using the procedure presented in [3]. 1) Choose a function with good decay characteristics in both the time and frequency domains satisfying the 1089–7798/98$10.00  1998 IEEE