Research on FAM and FAMlet Classes and Their Applications

Unto K. Laine

1. Introduction

FAM functions are defined by the equation,

(1)

where a is the order of the function and the inner function v(f) (nu of f ) defines the frequency warping of these complex exponentials.

In the discrete case a is an integer which defines the order of the function. These functions are special types of Frequency-Amplitude Modulated (FAM) complex exponentials and they form orthonormal sets when a is a real integer.

FAM class is a family of many known orthonormal bases. When, e.g.,

• v(f) = f the functions reduce to the classical Fourier kernel,
• v(f) = arctan(f) we get (slightly modified) Laguerre functions and,
• v(f) = arccos(f) we get Chebyshev functions.

The generative v(f) can be almost any well behaving, smooth function. Thus the FAM class includes principally infinitely many orthonormal sets.

A FAM transform can be produced by using a FAM set as a kernel in integral transformation. In a typical signal processing problem FAM functions are defined in the frequency domain (as above). A coresponding set of orthonormal time domain functions can be produced by inverse Fourier-transforming a FAM set . These time-domain functions are called FAMlets.

The research on FAM and FAMlet classes can be devided into two levels: basic and applied research.

2. Basic Research on FAM & FAMlet classes

On the basic level the FAM operator and related differential equations have been studied. The resulted differential equation is very close to that written for a nonuniform transmission-line. These equations are used, e.g., in quantum mechanics (WKB approximarion) and to model the basilar membrane movements in the human cochlea.

Examples of other topics studied:

• Orthogonal auditory filterbank generation by using FAM transform
• Adaptation of v(f) for optimal FAM & FAMlet sets (data compression)
• Application of allpass filters for FAMlet transform
• Fast algorithms for FAMlet transform
• Critical sampling and FAMlet transform (matrix formulations)

3. Applied Research on FAM & FAMlet classes

FAM and FAMlet classes have been applied in different areas of perceptual audio signal processing and even in hearing research:

1. FAMbac, FAMlet based audio coding project

In this part FAMlet techniques and related frequency warping methods (WLP) are applied to a new type of perceptual audio coder prototype. Presently, WLP-based methods are the most promising ones. Our new residual processing algorithm together with the auditory WLP processing leads to high compression rate with high quality. At the best the compression rate is about 1:30 which means about half bit per sample (per channel).

2. Study on auditory temporal resolution

Bark-FAMlet clicks have also been used to study human perception. In a pilot work Bark-FAMlets of different order (and length of 0.5 - 5 ms) were played in natural or time reversed direction to define the sensitivity of the perception to these phase-only changes in these stimuli. In two other experiment both FAMlets had the same direction (natural/time-reversed), however, they were of different order (and length). First publication of this work is coming in the ICASSP-96 conference.

3. Generalized Linear Prediction (GLP)

A new idea of Generalized Linear Prediction (GLP) has grown from the basis of FAM studies. GLP gives a new tool to adapt the classical LP according to the frequency properties of the signal and according to the problem in question. GLP is able to "focus" its power of modeling to any limited frequency range of interest and can give a detailed model of that range without limiting the signal bandwidth. A related patent is pending.

4. Study of auditory brain stem responses by Bark-FAMlet clicks

This work is carried out in Turku University Central Hospital (Dr. Altti Salmivalli). Bark-FAMlets are produced by approximating the psycoacoustic Bark-frequency scale by a proper inner function (generative function) g(x) in the corresponding FAM functions. The study has shown that many of the members of the Bark-FAMlet set are able to produce more clear brain stem response than the conventionally used simple rectangular pulse. A publication of these results is coming.
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The study continues in both basic and applied levels. On the basic level, e.g., the solving of nonuniform transmission line equation with FAM functions (related to cochlear modeling and WKB-approximation) is of interest. On the level of applications Bark-FAMlets are further applied to hearing research. One of the topics will be binaural effects applied to spatial hearing.

Short history

A Mathematica notebook is awailable for Bark-FAMlet generation and experimentation. The notebook also includes a short introduction to FAMlets.

[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] Laine U. K.: A new high resolution time-Bark analysis method for speech. Proc. of the XIIth Int. Conference of Phonetic Sciences, Vol. 2, pp. 402-405, Aix-en-Provence, France 1991.

[3] Laine U. K., Karjalainen M. and Altosaar T.: Time-frequency and multiple-resolution representations in auditory modeling. Paper summaries of 1991 IEEE ASSP Workshop on Applications of Signal Processing to Audio and Acoustics, Paper 8. Session 1., New Paltz, USA 1991.

[4] Laine U. K.: Analysis of short fragments of speech using complex orthogonal auditory transform (COAT). ESCA Workshop "Comparing Speech Signal Representations", Sheffield, England April 7-9 1992.

[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.

[6] Laine U. K.: Speech analysis using complex orthogonal auditory transform (COAT). Proc. of 1992 Int. Conf. on Spoken Language Processing, Banff, Alberta, Canada, Oct. 12-16 1992.

[7] Laine U. K.: MSE filter design and spectrum parametrization by orthogonal FAM Transform. Proc. of the ISCAS-93, Chicago, Illinois, I pp. 148-151, 1993.

[8] Laine U. K., Karjalainen M., Altosaan T., Warped linear prediction (WLP) in speech and audio processing. Proc. ICASSP-94, Adelaide, South Australia, III pp. 349-352, 1994.

[9] Laine U. K., Generalized linear prediction based on analytic signals. Proc. ICASSP-95, Detroit, MI, pp. 1701-1704 , 1995.

[10] Laine U. K. and Huotilainen M.: A study on auditory resolution using Bark-FAMlet clicks. To be published at ICASSP-96, Atlanta, GA, 1996.

[11] Kähkönen E., Salmivalli A., Laine U. K., Uusipakka E. and Johansson R.: FAMlet - a new stimulus for BRA (to be published), 1995.