The main goal of the formulation was to modify the classical Fourier kernel (complex exponentials) in order to get the required frequency resolution in the frequency domain. A simple idea is to warp the frequency scale of the complex exponential functions by a proper inner function. If, e.g., a uniform resolution spectrum is transformed by using this new warped basis and resynthesized with a uniform (nonwarped) basis, then a new spectrum of nonuniform resolution is created. The new frequency resolution depends on the actual inner function used in the warping.
The next problem was the orthogonalization of the frequency warped bases. It is quite easy to see, that the bases needs the derivative of the inner function as a"weight". In other words, if all the functions are normalized by the quare rooth of this weight, then they are orthonormalized. The FAM class was introduced in ICASSP'90 [1].
The FAM class is defined by:
where a is the order of the function and the inner function v(f) (nu of f ) defines the frequency warping.
In 1993 M.Mohan Sondhi (AT&T Bell Labs / Lucent Technologies) pointed out [2] that a similar formulation has been published earlier by E. Masry, K. Steiglitz and B. Liu in 1968 [3]. However, their important study is not widely known and there seems to be few furher studies and applications made which are directly related to this pioneering work. The only difference between the FAM class and the MSL class [3] is that the latter uses an integral function in the exponent instead of the nu of f function of FAM and the square root of the integrand forms the weighting function instead of the square root of the derivative of nu of f.
It was already mentioned abowe that the FAM class is generated from complex exponentials by using axis warping. This topic was analyzed in detail by R. G. Baraniuk and D. L. Jones [4].
The concept FAMlet was introduced in 1992 [5]. FAMlets are inverse Fourier transformed FAM functions. Thus FAMlets form orthonormal bases, too.