A method for constructing orthonormal basis functions with good time-frequency localization

In this paper we derive an explicit, single expression for a complex-valued, orthonormal basis well localized in time-frequency domain. We construct it from a single real function Φ(x) which is a Gaussian divided by the square root of a Jacobi theta θ 3 function. Then we simplify Φ(x) to the form of inverse square root of a Jacobi theta θ 3 function. We show that the shape of Φ(x) can be changed from Gaussian-like to rectangular-like with a single parameter. The basis generating function Φ(x) and its Fourier transform Φ have exponential decay. We also show how to modify a standard I and Q processor to compute complex-valued time-frequency expansion coefficients.