(*^

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  FAMlets
     
    v. 1.0  
   (17.8.- 20.11.1995)
     
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Unto K. Laine

Senior Scientist, Dr. Tech.

Helsinki University of Technology
Otakaari 5 A, 02150 Espoo
FINLAND

Unto.Laine@hut.fi
;[s]
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Short introduction to the FAM class and Bark-FAMlet functions
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¥ Purpose of the program

This program generates a set of complex FAM functions defined in the frequency domain.  Their phase characteristic follows the Hz-to-Bark mapping.  The associated time domain FAMlet functions are then produced by inverse Fourier transforming the FAM functions.  

The generated Bark-FAMlet functions are useful tools for auditory based signal processing and for studies of human auditory perception.

¥ What FAM stands for

Shortly, FAM stands for Frequency-Amplitude Modulated complex exponentials.  They were found by the author of this notebook around 1983.  More serious research on this topic started in the autumn of 1989.  The first publication of these studies was made in 1990 ICASSP meeting [1].  Later the method was applied to auditory based speech analysis [2-4, 6].  Also related FAM and FAMlet transforms were studied [5, 7].  Closely related developments are Warped Linear Prediction [8] (WLP) and Generalized Linear Prediction [9] (GLP), where the classical theory of LP was extended to analytic signals and more general predictor structures. 

The terminology for the FAM functions was chosen by thinking their general shape especially when defined in the time domain.  The main idea behind these functions is that we first warp the complex exponential functions thus introducing a type of frequency modulation ("FM", mathematically: by replacing the x variable by an inner function g(x)), and then amplitude modulate them  in order to make them orthogonal or even orthonormal ("AM", mathematically:  by using a proper weighting function). 

Definition for FAM functions:

FAM[a ,x] = am(x) Exp[a  I fm(x)] = Sqrt[dg(x)/dx] Exp[a  I g(x)], 

where a  is the order of the function and g(x) defines an inner function, or the warping. The "AM"-part is formed from the first derivative of the g(x).

The equation above shows that the FAM function consists of two parts: an "AM"-part and an "FM"-part, which are interdependent according to the FAM idea. This terminology describes the corresponding features of the functions when they are defined in the time domain.  However, when the FAM functions are defined in the frequency domain the "AM"-function is equal to the magnitude spectrum of all the FAM functions, and the "FM"-function (or the inner function) now defines the phase characteristic of the FAM of order one. The order zero FAM is a zero-phase function (magnitude only) and the higher order ones have a phase multiple of that of the first order one (multiple of the phase defined by the"FM"-function).

¥  FAMily!

FAM is a class, or a family  of orthonormal functions.  This class has many known members, like Fourier functions ("AM" equals one, no "FM"-property , i.e., g(x)=x), Laguerre (or slightly modified, when "FM" equals to the arctan(x)), Chebyshev ("FM" equals to the arccos(x)), Legendre, etc.  FAM sets can form a kernel in orthogonal transforms.  Many known transforms can be generated: Fourier-, Scale- etc.

Up to now the most interesting results are found when the FAM functions are defined in the frequency domain.  The corresponding time domain functions called FAMlets are finally produced by inverse Fourier transforming the frequency domain FAM functions.  The name FAMlet was chosen to indicate that they have similar use as the wavelets in certain areas of time-frequency analysis and perceptual audio processing.

¥ Bark-FAMlets shortly

1. They form an orthonormal set of functions (signals).
2. They have an identical power spectrum defined by the term dg/df on the Hz-scale.
3. They have equal energy over each critical band, thus forming a set of clicks with an unform masking property.
4. They are compact clicks in time, the length increasing linearly with order a.
5. The zeroth order FAMlet is a symmetric pulse. The higher order ones are similar to frequency chirps, the chirp rate decreasing with increasing order a.
6. The average rate of the frequency chirp of the Bark-FAMlets is uniform over the middle part of the Bark-scale (over the Bark channels 5-15).  The chirp rate decreases linearly with the increasing order of the functions.  Thus the higher order FAMlets have longer duration in time.

¥ Present status and the future of the FAMlet research

Bark-FAMlets have recently been used to study the human auditory phase resolution.  In a preliminary study made subjects were able to perceive quality differences in Bark-FAMlet clicks of less than 1ms in duration when played in a normal or time-reversed direction.  The first publication of the listening experiment will be presented in the coming ICASSP-96 conference [10].

Bark-FAMlets have been also used as a stimulus in measuring the human auditory brain stem responses.  This study has been going on about two years now together with Turku University Central Hospital (Dr. Altti Salmivalli).  Interesting new results have been achieved and the first publication will be released soon [11].

In the summer -95 a new audio coding project started in the HUT Acoustics Laboratory.  In this project the main goal is to study how the FAMlets can be applied to the broad band perceptual audio coding.

On the basic research side there are many lines going on.  One of them is related to the cochlea modeling.  So called WBK approximation has been used to solve the wave equation for the cochlea.  FAM functions seem to give a new approximative solution. 

¥ Context of the program

In the following we first define the Hz-to-Bark warping, derive a function for the "AM"- and "FM"-part of the FAMs and then generate the corresponding FAM functions.  The related FAMlets are finaly produced by inverse Fourier transform.

The first part of this notebook describes the derivation of the functions in detail.  At the end of the notebook a compact function for FAMlet generation is given.  NOTE that the notebook must be first evaluated in order to get the function working (it needs other functions like the Hz-to-Bark mapping).

The notebook includes also some sounds made of Bark-FAMlets.

¥ Summary of FAM and FAMlet publications and related topics 
  (From the very begining up to the summer -95)
 
[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] KŠhkšnen E., Salmivalli A., Laine U. K., Uusipakka E. and Johansson R.: FAMlet - a new stimulus for BRA (to be published), 1995.

 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 
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NOTE
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 (* The blue parts are evaluated automatically when 
 the notebook is started.  If not yet done do  
 "Evaluate Initialization" now from the menu "Action".
 
 If you want to study the derivation of the FAMlets in 
 detail follow the notebook and evaluate manually 
 all the red lines!  The black lines are optional.
 
 Free parameters:
 
 bw = bandwidth used in kHz (equals to SampleRate/2)
 value 22.05 chosen in the example below.
 
 np = number of samle points in the FAMlets.
 
 Note that this must be carefully chosen in order to
 avoid the possible folding phenomenon.  In other words 
 the value of this parameter depends on the number of 
 FAMlets generated (the highest order needed).  
 An experimental rule is that:
  
 np >= 8 * highest FAMlet order (nf).
 
 nf = number of FAMlets, or the highest order generated.
 
 GOOD LUCK!
 
 * UKL *)
;[s]
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Some preliminaries
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*)
pi=N[Pi];
(*
;[s]
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Hz-to-Bark mapping
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*)
bark[f_]:= Sign[f] (13 ArcTan[0.76 Abs[f]] +
 3.5 ArcTan[(f/7.5)^2]);
(*
;[s]
2:0,1;69,0;70,-1;
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Derivation of the "AM"-function
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am2=D[13 ArcTan[0.76 x] + 3.5 ArcTan[(x/7.5)^2],x]
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9.88/(1 + 0.5776*x^2) + 
  (0.1244444444444444445*x)/
   (1 + 0.0003160493827160493828*x^4)
;[o]
    9.88            0.124444 x
------------- + ------------------
            2                    4
1 + 0.5776 x    1 + 0.000316049 x
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*)
am[f_]:=Sqrt[9.88/(1. + 0.5776*f^2) + 
0.1244445*Abs[f]/(1. + 0.000316049382716*f^4)];
(*
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Derivate tables for "AM" and "FM"
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Define bw = bandwidth in kHz and np = number of points
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bw=22.050;
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np=256;
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am0=am[v]/.v->0;
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am0
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3.143246729100342196
;[o]
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AM-table
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st=2 bw/np;amT=Table[am[v]/am0,{v,-bw+st,bw,st}];
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Length[amT]
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256
;[o]
256
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FM-table
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b0=bark[bw];
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2:1,12,10,Courier,1,12,0,0,0;1,12,10,Courier,1,12,65535,0,0;
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fmT=Table[N[bark[v]/b0],{v,-bw+st,bw,st}];
;[s]
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Visualize the "AM" and "FM" functions
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ListPlot[amT,PlotJoined->True,
PlotRange->{{0,np},{1,0}}]
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The Unformatted text for this cell was not generated.
Use options in the Actions Preferences dialog box to 
control when Unformatted text is generated.
;[o]
-Graphics-
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Prepare functions for FAM generation
:[font = text; inactive; preserveAspect; ]
Define number of functions (nf) and prepare the "AM" and "FM" functons for FAM and
FAMlet generation.
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nf=24;
;[s]
2:0,1;6,0;7,-1;
2:1,12,10,Courier,1,12,0,0,0;1,12,10,Courier,1,12,65535,0,0;
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amT=RotateLeft[amT,np/2-1];
;[s]
2:0,1;27,0;28,-1;
2:1,12,10,Courier,1,12,0,0,0;1,12,10,Courier,1,12,65535,0,0;
:[font = input; preserveAspect; groupLikeNormal; ]
fmT=RotateLeft[fmT,np/2-1];
;[s]
2:0,1;27,0;28,-1;
2:1,12,10,Courier,1,12,0,0,0;1,12,10,Courier,1,12,65535,0,0;
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Table of Warped complex exponentials for FAMs
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exT=Exp[I pi fmT];
;[s]
1:0,1;19,-1;
2:0,12,10,Courier,1,12,0,0,0;1,12,10,Courier,1,12,65535,0,0;
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Generate table of FAMs
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famT=Table[amT exT^k ,{k,0,nf-1}];
;[s]
2:0,1;34,0;35,-1;
2:1,12,10,Courier,1,12,0,0,0;1,12,10,Courier,1,12,65535,0,0;
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General::spell: 
   Possible spelling error: new symbol name "famT"
     is similar to existing symbols {amT, fmT}.
:[font = input; preserveAspect; groupLikeNormal; ]
(*Don't worry,Mathematica just do not like
similar variable names!*)
;[s]
3:0,0;14,1;25,0;69,-1;
2:2,12,10,Courier,1,12,0,0,0;1,12,10,Courier,3,12,0,0,0;
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Generate table of FAMlets
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famlet=Table[InverseFourier[famT[[k]]],{k,1,Length[famT]}];
;[s]
2:0,1;59,0;60,-1;
2:1,12,10,Courier,1,12,0,0,0;1,12,10,Courier,1,12,65535,0,0;
:[font = input; preserveAspect; groupLikeNormal; ]
(*Just an example...*)
:[font = input; preserveAspect; groupLikeNormal; ]
ListPlot[Re[famlet[[5]]],PlotJoined->True,
PlotRange->{{1,128},{2,-2}}]
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The Unformatted text for this cell was not generated.
Use options in the Actions Preferences dialog box to 
control when Unformatted text is generated.
;[o]
-Graphics-
:[font = subsubsection; inactive; preserveAspect; groupLikeNormal; ]
Store the functions in a file
:[font = input; preserveAspect; groupLikeNormal; ]
ow=OpenWrite["famlets.m"]
:[font = input; preserveAspect; groupLikeNormal; ]
Write[ow,famlet];Close[ow]
:[font = subsection; inactive; preserveAspect; groupLikeNormal; fontColorRed = 65535; fontColorBlue = 65535; ]
Play a little!
:[font = input; preserveAspect; groupLikeNormal; ]
Max[Abs[famlet[[22]]]]
:[font = output; output; inactive; preserveAspect; groupLikeNormal; ]
0.8571013408250572515
;[o]
0.857101
:[font = input; preserveAspect; groupLikeNormal; ]
ListPlay[famlet[[22]],PlayRange->{0.9,-0.9},
SampleRate->2000 bw]
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%!
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% Start of sampled sound
1 string
44100. 256 8
{ currentfile 1 index readhexstring pop }
 false 1 sampledsound
7F7F7F7F7F807F807F7E8959D80AB3D11B4DE1C2331A8EF0D05C091D7CD9F9D3843A141C467F
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pop
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The Unformatted text for this cell was not generated.
Use options in the Actions Preferences dialog box to 
control when Unformatted text is generated.
;[o]
-Sound-
:[font = input; preserveAspect; groupLikeNormal; ]
ListPlay[Reverse[famlet[[22]]],PlayRange->{.9,-.9},
SampleRate->2000 bw]
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{ currentfile 1 index readhexstring pop }
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The Unformatted text for this cell was not generated.
Use options in the Actions Preferences dialog box to 
control when Unformatted text is generated.
;[o]
-Sound-
:[font = input; preserveAspect; groupLikeNormal; ]
(* Do You hear any difference between these two 
signals above? *)
;[s]
3:0,0;6,1;9,0;67,-1;
2:2,12,10,Courier,1,12,0,0,0;1,12,10,Courier,1,12,65535,0,0;
:[font = input; preserveAspect; groupLikeNormal; ]
(* If your answer is yes, 
then Ohm did not hit the point! - as we all know! *)
;[s]
3:0,0;32,1;35,0;80,-1;
2:2,12,10,Courier,1,12,0,0,0;1,12,10,Courier,1,12,65535,0,0;
:[font = input; preserveAspect; groupLikeNormal; ]
(* The length of these pulses is about 
210(samples)/44.1 kHz = 4.8 ms, however, in listening 
tests made the best listeners are able to hear the 
difference even with pulses of about 1 ms in length! 
The first publication about auditory phase resolution 
studied with Bark-FAMlets will be published at ICASSP-96 
conference [10]*)
:[font = subsection; inactive; preserveAspect; groupLikeNormal; ]
FAMlet generator as a Mathematica function
;[s]
3:0,0;22,1;33,0;43,-1;
2:2,16,12,Times,1,14,0,0,0;1,16,12,Times,3,14,0,0,0;
:[font = text; inactive; preserveAspect; ]
Now we can collect the details developed above to form an Mathematica  function.

NOTE that you have to initialize this notebook first in order to get this function properly evaluated!
;[s]
3:0,0;58,1;69,0;185,-1;
2:2,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0;
:[font = input; initialization; preserveAspect; groupLikeNormal; ]
*)
famletG[bwf_,npf_Integer,nff_Integer]:=(
am0=am[v]/.v->0;b0=bark[bwf];
st=2 bwf/npf;amTf=Table[am[v]/am0,{v,-bwf+st,bwf,st}];
fmTf=Table[N[bark[v]/b0],{v,-bwf+st,bwf,st}];
amTf=RotateLeft[amTf,npf/2-1];fmTf=RotateLeft[fmTf,npf/2-1];
exTf=Exp[I pi fmTf];
famTf=Table[amTf exTf^k ,{k,0,nff-1}];
famletf=Table[InverseFourier[famTf[[k]]],{k,1,nff}];
Do[famletf[[k]]=RotateRight[Re[famletf[[k]]],10],
{k,1,nff}];Return[Print["FAMLETS DONE!"]]);

(*
;[s]
1:0,1;441,-1;
2:0,12,10,Courier,1,12,0,0,0;1,12,10,Courier,1,12,0,0,65535;
:[font = input; preserveAspect; groupLikeNormal; ]
Clear[famletG]
:[font = subsubsection; inactive; preserveAspect; groupLikeNormal; ]
An example how to use famletG function
:[font = input; preserveAspect; groupLikeNormal; ]
(* bw = 22.05, np = 256, nf=10 *)
:[font = input; preserveAspect; groupLikeNormal; ]
famletG[22.05,256,10];
;[s]
2:0,1;22,0;23,-1;
2:1,12,10,Courier,1,12,0,0,0;1,12,10,Courier,1,12,65535,0,0;
:[font = print; inactive; preserveAspect; groupLikeNormal; ]
FAMLETS DONE!
:[font = input; preserveAspect; groupLikeNormal; ]
Length[famletf]
:[font = output; output; inactive; preserveAspect; groupLikeNormal; ]
10
;[o]
10
:[font = input; preserveAspect; groupLikeNormal; ]
ListPlay[famletf[[9]],PlayRange->{1.5,-1.5},
SampleRate->2000 bw]
:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; groupLikeNormal; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; ]
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The Unformatted text for this cell was not generated.
Use options in the Actions Preferences dialog box to 
control when Unformatted text is generated.
;[o]
-Sound-
:[font = subsubsection; inactive; preserveAspect; groupLikeNormal; ]
Test of the orthogonality
:[font = input; preserveAspect; groupLikeNormal; ]
(*Compute a correlation matrix*)
:[font = input; preserveAspect; groupLikeNormal; ]
cT=Table[famletf[[k]].famletf[[j]],{k,1,10},{j,1,10}];
:[font = input; preserveAspect; groupLikeNormal; ]
ma=Max[cT]
:[font = output; output; inactive; preserveAspect; groupLikeNormal; ]
29.07177380551882339
;[o]
29.0718
:[font = input; preserveAspect; groupLikeNormal; ]
(*Not orthonormal.  To make them orthonormal notmalize 
with Sqrt[Max[cT]]*)
:[font = input; preserveAspect; groupLikeNormal; ]
fu=famletf[[5]]/Sqrt[ma];
:[font = input; preserveAspect; groupLikeNormal; ]
fu.fu
:[font = output; output; inactive; preserveAspect; groupLikeNormal; ]
0.9999999999999999995
;[o]
1.
:[font = input; preserveAspect; groupLikeNormal; ]
(*Now orthonormal*)
:[font = input; preserveAspect; groupLikeNormal; ]
Min[cT]
:[font = output; output; inactive; preserveAspect; groupLikeNormal; ]
-0.0005171872683814913804
;[o]
-0.000517187
:[font = input; preserveAspect; ]
(* Almost orthogonal, too! *)
:[font = input; preserveAspect; ]
 
:[font = input; preserveAspect; groupLikeNormal; ]
ListDensityPlot[cT,PlotRange->{{0,10},{0,10},{-.02,.02}}]
:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; groupLikeNormal; pictureLeft = 34; pictureWidth = 282; pictureHeight = 282; ]
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:[font = output; output; inactive; preserveAspect; groupLikeNormal; ]
The Unformatted text for this cell was not generated.
Use options in the Actions Preferences dialog box to 
control when Unformatted text is generated.
;[o]
-DensityGraphics-
:[font = input; preserveAspect; groupLikeNormal; ]
(* This is principally an identity matrix which shows 
that the famlets are orthogonal with a high accuracy *)
:[font = subsubsection; inactive; preserveAspect; groupLikeNormal; ]
Just in the case...
:[font = input; preserveAspect; groupLikeNormal; ]
In the case you need only every stp-th function
use this!
:[font = input; initialization; preserveAspect; groupLikeNormal; ]
*)
famletG2[bwf_,npf_Integer,nff_Integer,stp_Integer]:=(
am0=am[v]/.v->0;b0=bark[bwf];
st=2 bwf/npf;amTf=Table[am[v]/am0,{v,-bwf+st,bwf,st}];
fmTf=Table[N[bark[v]/b0],{v,-bwf+st,bwf,st}];
amTf=RotateLeft[amTf,npf/2-1];fmTf=RotateLeft[fmTf,npf/2-1];
exTf=Exp[I pi fmTf];
famTf=Table[amTf exTf^(k stp) ,{k,0,nff-1}];
famletf=Table[InverseFourier[famTf[[k]]],{k,1,nff}];
Do[famletf[[k]]=RotateRight[Re[famletf[[k]]],10],
{k,1,nff}];Return[Print["FAMLETS DONE!"]]);

(*
;[s]
1:0,1;460,-1;
2:0,12,10,Courier,1,12,0,0,0;1,12,10,Courier,1,12,0,0,65535;
:[font = input; preserveAspect; groupLikeNormal; ]
Clear[famletG2];
:[font = input; preserveAspect; groupLikeNormal; ]
famletG2[22.05,1024,10,10];
;[s]
1:0,1;28,-1;
2:0,12,10,Courier,1,12,0,0,0;1,12,10,Courier,1,12,65535,0,0;
:[font = print; inactive; preserveAspect; groupLikeNormal; ]
FAMLETS DONE!
:[font = input; preserveAspect; groupLikeNormal; ]
Max[Abs[famletf[[2]]]]
:[font = output; output; inactive; preserveAspect; groupLikeNormal; ]
2.380923862624224928
;[o]
2.38092
:[font = input; preserveAspect; groupLikeNormal; ]
ListPlay[famletf[[4]],PlayRange->{2.4,-2.4},
SampleRate->2000 bw]
:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; groupLikeNormal; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; ]
%!
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;[o]
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(***  THE STUDY CONTINUES  ***)
:[font = input; preserveAspect; center; fontColorRed = 65535; ]
(***  ALL COMMENTS CONCERNING THIS NOTEBOOK 
AND THE TOPIC ARE WELLCOME!  ***)
^*)