(1)
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.,
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.
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:
FAM and FAMlet classes have been applied in different areas of perceptual audio signal processing and even in hearing research:
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).
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.
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.
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.
A Mathematica notebook is
awailable for Bark-FAMlet generation and experimentation. The
notebook also includes a short introduction to FAMlets.
See also the FAM & FAMlet related publications.
[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.