Unto K. Laine

Short History

The FAM class of orthonormal functions was created in 1983 by the author. The aim was to develop new effective signal processing methods (transforms) for an auditory type of spectral analysis. The classical Fourier analysis with uniform frequency resolution is not well suited to perceptual audio signal processing since the human auditory system utilizes an approximately logarithmic frequency scale.

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.


[1] Laine U. K., Altosaar T., An Orthogonal Set of Frequency and Amplitude Modulated (FAM) Functions for Variable Resolution Signal Analysis. Proc. of ICASSP-90, Vol. 3, pp. 1615-1618, Albuquerque, New Mexico, April 3-6, 1990.

[2] M. M. Sondhi, personal e-mail communication, March 2, 1993.

[3] E. Masry, K. Steiglitz and B. Liu, Bases in Hilbert Space Related to the Representation of Stationary Operators, SIAM J. Appl. Math., Vol 16, No. 3, pp. 552-562, May 1968.

[4] R. G. Baraniuk, D. L. Jones, Unitary Equivalence: A New Twist on Signal Processing, IEEE Transactions on Signal Processing, vol. 43, no. 11, pp. 2269-2282, October 1995. Abstract

[5] Laine U. K.: Famlet, to be or not to be a wavelet, IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis, Victoria, British Columbia, Canada, Oct. 4-6, pp. 335-338, 1992.

FAM and FAMlet publications

Other related publications


Created: 21.1.98

Last modified: 26.10.98 ©ukl