In my student days we didn't have lasers to give coherent monochromatis light. It was still magic to be able to flip from frequency domain to real world. The beauty of the images, and the rapidly developing application of X-rays to crystallography were very attractive to me. A year or two later and I built a mains powered He-Ne gas laser. Now we can buy high-intensity solid-state laser cheaply on the 'net.

The screen shots below show the analysis and re-synthesis of four waves.

Beats shows 'sin 30 +sin 33'. Pulse is a low mark/space rectangular wave. Square wave approximation was generated from the known series of terms.

. . . .

Example LB code- uses a supplied colour-scale.

' *********************************************************** ' ** ** ' ** 1DFFT.bas 27/02/2017 tenochtitlanuk ** ' ** beats ** ' *********************************************************** global n, pi ' globally available pi, number of data values- a multiple of 2- n =1024 ' globally available real and imaginary data for main and sub sections dim realData( n), imagData( n) ' the FFT operates in-place, changing the data to its transform. pi = 4 *atn( 1) WindowWidth =1080 WindowHeight = 720 nomainwin open "FFT demo." for graphics_nsb as #wg #wg "trapclose quit" #wg "goto 20 100 ; down ; size 1 ; fill cyan ; color white" for i =1 to n 'if i >508 and i <516 then realData( i) =1 else realData( i) =0.05 for k = 1 to 3 step 2 realData( i) =realData( i) +1 /k *sin( i /n *k *10 *2 *pi) next k imagData( i) =0 #wg "color red ; line "; 20 +1024 *i /n; " 80 "; 20 +1024 *i /n; " "; 80 -int( 25 *realData( i)) next i call fft -1 ' ie compute FFT not inverse- 1 for fwd FFT, -1 for reverse #wg "size 5" for i =1 to n ' but symmetrical on x axis so really only want left-hand half. #wg "color red ; line "; 20 +1024 *i /n; " 250 "; 20 +1024 *i /n; " "; 250 -int( 0.18 *realData( i)) #wg "color darkblue ; line "; 20 +1024 *i /n; " 350 "; 20 +1024 *i /n; " "; 350 -int( 0.18 *imagData( i)) next i call fft 1 ' ie compute inverse #wg "size 1" for i =1 to n ' but symmetrical on x axis so really only want left-hand half. #wg "color red ; line "; 20 +1024 *i /n; " 550 "; 20 +1024 *i /n; " "; 550 -int( 25 *realData( i)) next i #wg "color black ; up ; goto 10 80 ; down ; goto 1034 80" for v =250 to 550 step 100 if v <>450 then #wg "up ; goto 10 "; v; " ; down ; goto 1034 "; v next v wait sub fft isi j =1 for i =1 to n if i <j then call exchangeTerms i, j m =n /2 while j >m j =j -m m =int( ( m +1) /2) wend j =j +m next i mmax =1 while mmax <n istep =2 *mmax for m =1 to mmax theta =pi *isi *( m -1) /mmax wr =cos( theta) wi =sin( theta) for i =m to n step istep j =i +mmax tempr =wr *realData( j) -wi *imagData( j) tempi =wr *imagData( j) +wi *realData( j) realData( j) =realData( i) -tempr imagData( j) =imagData( i) -tempi realData( i) =realData( i) +tempr imagData( i) =imagData( i) +tempi scan next i next m mmax =istep wend if isi =-1 then exit sub ' else normalise values for i =1 to n realData( i) =realData( i) /n imagData( i) =imagData( i) /n next i end sub sub exchangeTerms i, j temp =realData( i) realData( i) =realData( j) realData( j) =temp temp =imagData( i) imagData( i) =imagData( j) imagData( j) =temp end sub sub quit h$ close #h$ end end sub..which needs this colour-scale bmp saved in the same directory folder.

Starting with a circular aperture the result of the two series of 1D transforms is the 2D result shown. Note that to agree with the way we see diffraction the results need re-assembling by re-ordering the four quarters, as done below . . .

. . .

' *********************************************************** ' ** ** ' ** 2DFourier9.bas 05/03/2017 tenochtitlanuk ** ' ** beats ** ' *********************************************************** nomainwin global n, pi ' globally a<ailable pi, number of data <alues- a multiple of 2- n = 128 ' globally a<ailable real and imaginary data for main and sub sections dim realData( n), imagData( n) ' the FFT operates in-place, changing the data array to its transform. dim imageArray$( n, n) ' holds R and I <alues as a cs< string dim ColLookUp$( 256) pi = 4 *atn( 1) UpperLeftX = 10 UpperLeftY = 10 WindowWidth =1300 WindowHeight = 700 graphicbox #w.gb1, 20, 10, 600, 570 graphicbox #w.gb2, 640, 10, 600, 570 open "2D Fourier Transform demo." for window as #w #w "trapclose quit" #w.gb1 "down ; fill 40 40 180 ; size 1 ; color darkgray" #w.gb2 "down ; fill 40 40 40 ; size 1 ; color darkgray" loadbmp "scr", "colRange.bmp" #w.gb1 "drawbmp scr 0 20 ; size 1" gosub [fillColLookUp] gosub [initImageArray] for row =1 to n for col =1 to n ' copy a row of data to array for computing its FFT realData( col) =<al( word$( imageArray$( row, col), 1, ",")) imagData( col) =<al( word$( imageArray$( row, col), 2, ",")) next col call fft -1 for col =1 to n ' do FFT and copy back to image array imageArray$( row, col) =str$( realData( col)) +"," +str$( imagData( col)) amp =amp( ( realData( col)^2 +imagData( col)^2)^0.5) ' int( 100 *log( 1 +( realData( col)^2 +imagData( col)^2)^0.5)) #w.gb2 "backcolor "; ColLookUp$( amp) #w.gb2 "color "; ColLookUp$( amp) call sqDraw 2, row, col next col next row call screenGrab "X" for col =1 to n for row =1 to n ' copy a row of data to array for computing its FFT realData( row) =<al( word$( imageArray$( row, col), 1, ",")) imagData( row) =<al( word$( imageArray$( row, col), 2, ",")) next row call fft -1 for row =1 to n imageArray$( row, col) =str$( realData( row)) +"," +str$( imagData( row)) amp =amp( ( realData( row)^2 +imagData( row)^2)^0.5) ' int( 100 *log( 1 +( realData( row)^2 +imagData( row)^2)^0.5)) #w.gb2 "backcolor "; ColLookUp$( amp) #w.gb2 "color "; ColLookUp$( amp) call sqDraw 2, row, col next row next col 'call screenGrab "Y" wait sub screenGrab i$ for x =0 to 256 step 256 for y =0 to 256 step 256 #w.gb2 "getbmp scr "; x +50; " "; y +20; " 256 256" bmpsa<e "scr", i$ +str$( x) +str$( y) +".bmp" next y next x end sub sub fft isi j =1 for i =1 to n if i

Code will be up shortly as a 2D_FFT.zip