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.

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Created: 21.1.98

Last modified: 26.10.98 ©ukl