WAVEFORM   

OBJECT-> functional

Plot[(Cos[1 x]),{x,-3.1,3.1},PlotRange->All]

Pmywaveform=Cos[1 x]

L=1
T=1

xx=1

Plot[Cos[(2 Pi/ L)  xx-(2 Pi/T) tt],{tt,-3.1,3.1},PlotRange->All]

OBJECT identified as 1/t = frequency dependent object=functional

When I write 1/T= f  I know what wavefornm object it is. 

Object recognition. NOTE information is in t space (including phase, relative position of object and other objects)

Clear[m,n]

∫ Sin[3 x]*Sin[ 2 x]\[DifferentialD]x

(∫_(-Pi)^Pi)( Sin[  3 x])*Sin[2 x]\[DifferentialD]x

(∫_(-Pi)^Pi)( Sin[  3 x])*Sin[3 x]\[DifferentialD]x

(∫_(-Pi)^Pi)( Cos[  3 x])*Cos[2 x]\[DifferentialD]x

(∫_(-Pi)^Pi)( Cos[  3 x])*Cos[3 x]\[DifferentialD]x

more objects

OBJECT OVERLAPPING -finding Nemo or Waldo :)



((∑_(i=1)^700)(i Sin[i x]))*Sin[i x]

(∫_(-Pi)^Pi)((∑_(n=1)^700)(n Sin[n x]))*Sin[10 x]\[DifferentialD]x
$Aborted

(∫_(-Pi)^Pi)( Sin[x])*Sin[x]\[DifferentialD]x
π

(∫_(-Pi)^Pi)( Sin[x])*Sin[x]\[DifferentialD]x
∫ Sin[n x]*Sin[x]\[DifferentialD]x
((-n Cos[n x] Sin[x]+Cos[x] Sin[n x])/(-1+(n^2)))
∫ Sin[n x]*Sin[ m x]\[DifferentialD]x
((Sin[(m-n) x])/(2 (m-n)))-((Sin[(m+n) x])/(2 (m+n)))

THIS IS WHAT FT actually does,playing with integrals for an array of objects
How:




OBJECT OVERLAPPING -finding Nemo or Waldo :)

FourierTransform[Sin[t],t,ω]

what is happening with sin we saw that
TAKE only say time part

ftq=Table[Cos[60 x],{x,-3.3,3.3, 0.001}]

ListPlot[ftq,Joined->True]

m

It is a waveform with some frequency (note scale here is not time)

Fourier[ftq]

ListPlot[Abs[Fourier[ftq]],PlotRange->{{0,100},{0,20}}, Joined->True]

Now we see the transform from waveform in time spae to FT in frequency space. The larger area the more overlap and recognition. Normalized overlap

What if objec is hidden among other objects?
Object recognition.  Compare objects, find the most overlapping feartures

Many objects.....

Plot[(∑_(i=1)^100)(Cos[i x]),{x,-0.1,0.1},PlotRange->All]

even  more ....

Plot[(∑_(i=1)^700)(Cos[i x]),{x,-0.1,0.1},PlotRange->All]

Convert function to points Plot->ListPlot:

ftq=Table[(∑_(i=1)^700)(Cos[i x]),{x,-0.1,0.1, 0.001}];

ListPlot[ftq,Joined->True,PlotRange->All]

ListPlot [Abs[Fourier[ftq]], Joined->True]

what does FT do ? recognizes objects by overlapping them with known objects

plots intensity for each waveform 's frequency, and shows spectrum of those frequencies.

Now FT (NOTE because of the math symmetry, take 1/2 out

ListPlot[InverseFourier[Fourier[ftq]],PlotRange->All,Joined->True]

ListPlot[Abs[Fourier[ftq]],PlotRange->True,Joined->True]

ListPlot[Abs[Fourier[ftq]],PlotRange->True,Joined->True]

ListPlot[Abs[Fourier[ftq]],PlotRange->{{0,100},{0,300}},Joined->True]

Plot[(∑_(i=1)^700)(Sin[i x]),{x,0,0.1},PlotRange->All]

Plot[(∑_(i=1)^700)(i Sin[i x]),{x,0,0.1},PlotRange->All]

The spectrum of all frequencies or the overal of all waveforms present the white noise!!!!