Fourierov integral

Primjer snimanja signala, numeričkog računanja Fourierovog integrala, i rekonstrukcije signala nakon rezanja frekvencija.



In [1]:

    
using PortAudio, SampledSignals, LibSndFile
using FileIO: load, save, loadstreaming, savestreaming
using QuadGK



In [2]:

    
PortAudio.devices()









    Out[2]:





13-element Array{PortAudio.PortAudioDevice,1}:
 PortAudio.PortAudioDevice("Microsoft Sound Mapper - Input", "MME", 2, 0, 44100.0, 0, 0.09, 0.09, 0.18, 0.18)
 PortAudio.PortAudioDevice("Microphone (Realtek High Defini", "MME", 2, 0, 44100.0, 1, 0.09, 0.09, 0.18, 0.18)
 PortAudio.PortAudioDevice("Microsoft Sound Mapper - Output", "MME", 0, 2, 44100.0, 2, 0.09, 0.09, 0.18, 0.18)
 PortAudio.PortAudioDevice("Speaker/Headphone (Realtek High", "MME", 0, 2, 44100.0, 3, 0.09, 0.09, 0.18, 0.18)
 PortAudio.PortAudioDevice("Speakers ()", "Windows WDM-KS", 0, 2, 44100.0, 4, 0.01, 0.01, 0.08533333333333333, 0.08533333333333333)
 PortAudio.PortAudioDevice("Microphone (Realtek HD Audio Mic input)", "Windows WDM-KS", 2, 0, 44100.0, 5, 0.01, 0.01, 0.04, 0.04)
 PortAudio.PortAudioDevice("Mic in at front panel (black) (Mic in at front panel (black))", "Windows WDM-KS", 2, 0, 44100.0, 6, 0.01, 0.01, 0.04, 0.04)
 PortAudio.PortAudioDevice("Speakers (Realtek HD Audio output)", "Windows WDM-KS", 0, 2, 44100.0, 7, 0.01, 0.01, 0.04, 0.04)
 PortAudio.PortAudioDevice("Stereo Mix (Realtek HD Audio Stereo input)", "Windows WDM-KS", 2, 0, 48000.0, 8, 0.01, 0.01, 0.04, 0.04)
 PortAudio.PortAudioDevice("Output (@System32\\drivers\\bthhfenum.sys,#4;%1 Hands-Free HF Audio%0\r\n;(iPhone))", "Windows WDM-KS", 0, 1, 8000.0, 9, 0.01, 0.01, 0.08533333333333333, 0.08533333333333333)
 PortAudio.PortAudioDevice("Input (@System32\\drivers\\bthhfenum.sys,#4;%1 Hands-Free HF Audio%0\r\n;(iPhone))", "Windows WDM-KS", 1, 0, 8000.0, 10, 0.01, 0.01, 0.08533333333333333, 0.08533333333333333)
 PortAudio.PortAudioDevice("Headset Earphone (@System32\\drivers\\bthhfenum.sys,#2;%1 Hands-Free AG Audio%0\r\n;(JBL Charge 3))", "Windows WDM-KS", 0, 1, 8000.0, 11, 0.01, 0.01, 0.08533333333333333, 0.08533333333333333)
 PortAudio.PortAudioDevice("Headset Microphone (@System32\\drivers\\bthhfenum.sys,#2;%1 Hands-Free AG Audio%0\r\n;(JBL Charge 3))", "Windows WDM-KS", 1, 0, 8000.0, 12, 0.01, 0.01, 0.08533333333333333, 0.08533333333333333)

Nakon što ste pronašli vaše uređaje, možete otkomentirati sljedeću ćeliju u kojoj se ulaz s mikrofona direktno prenosi na zvučnik. Nakon toga treba ponovo pokrenuti jezgru.



In [3]:

    
#=
stream = PortAudioStream(2, 2)
try
    # cancel with Ctrl-C
    write(stream, stream)
finally
    close(stream)
end
=#

Ukoliko želite snimiti svoj tekst, zamijenite mikrofon i izlaz s vašim uređajima, i izvedite sljedeće četiri čelije.



In [6]:

    
# stream = PortAudioStream("Microphone (BRIO 4K Stream Edit","Microsoft Sound Mapper - Output")









    Out[6]:





PortAudioStream{Float32}
  Samplerate: 44100.0Hz

  2 channel sink: "Microsoft Sound Mapper - Output"
  2 channel source: "Microphone (BRIO 4K Stream Edit"



In [7]:

    
# Namjestite duljinu snimke (nemojte pretjerivati)
# buf = read(stream, 2s)









    Out[7]:



In [8]:

    
# Zatvorite stream
# close(stream)



In [9]:

    
# Spremite vašu snimku
# save("myvoice.wav", buf)



In [4]:

    
glas=load("myvoice.wav")









    Out[4]:



In [5]:

    
length(glas)









    Out[5]:





176400



In [6]:

    
glas.samplerate









    Out[6]:





44100.0



In [7]:

    
typeof(glas)









    Out[7]:





SampleBuf{Float32,2}



In [8]:

    
# Napravimo mono snimku
glas_mono=similar(glas)









    Out[8]:



In [9]:

    
glas_mono.data=sum(glas.data,dims=2)
glas_mono.samplerate=glas.samplerate









    Out[9]:





44100.0



In [10]:

    
save("myvoice_mono.wav", glas_mono)



In [11]:

    
glas_mono.data









    Out[11]:





88200×1 Array{Float32,2}:
 -0.00015258789
 -9.1552734f-5
 -6.1035156f-5
 -9.1552734f-5
 -3.0517578f-5
  6.1035156f-5
  0.00015258789
  9.1552734f-5
  0.0
  0.0
  0.0
  3.0517578f-5
  0.0
  ⋮
 -3.0517578f-5
 -0.0011291504
 -0.0039367676
 -0.006164551
 -0.007843018
 -0.009490967
 -0.011474609
 -0.0146484375
 -0.018371582
 -0.020141602
 -0.0211792
 -0.023773193



In [12]:

    
# Pogledajmo signal
using Plots



In [14]:

    
S=glas_mono.data
plot(S)









    Out[14]:



In [103]:

    
plot(T,S)









    Out[103]:



In [18]:

    
# Trapezna formula, približno. Računamo kosinusni i sinusni spektar
using LinearAlgebra
A(λ::Float64)=(cos.(λ*T)⋅S)*((T[end]-T[1])/(length(T)*π))
B(λ::Float64)=(sin.(λ*T)⋅S)*((T[end]-T[1])/(length(T)*π))









    Out[18]:





B (generic function with 1 method)



In [19]:

    
# Nacrtajmo kosinusni spektar i odlučimo gdje koje frekvencije ćemo zadržati
plot(B,300,5000)









    Out[19]:



In [20]:

    
# Ograničimo spektar, opet koristimo približnu formulu
λ₀=600.0
λ₁=4500.0
step=0.5
Λ=range(λ₀,λ₁,step=step)
# Ovo traje cca 1 minutu, ali se računa samo jednom
Aλ=A.(Λ)
Bλ=B.(Λ)









    Out[20]:





7801-element Array{Float64,1}:
 -1.4276545343692649e-5
  6.9369216861432245e-6
  2.112717985256516e-5
  3.123817843795039e-5
  4.1020107290376104e-5
  4.6898213248294996e-5
  3.876846443879242e-5
  1.0373092159497102e-5
 -3.040583150749619e-5
 -6.26014173311179e-5
 -6.625478447345567e-5
 -3.870254324568669e-5
  1.4653952013808025e-6
  ⋮
  1.7794981642116358e-5
  9.992730104962406e-7
 -1.277551704963596e-5
 -1.5925982443486848e-5
 -4.188816410633976e-6
  1.9864473329428913e-5
  4.641590648579104e-5
  6.2524350825799e-5
  5.901390212558618e-5
  3.584836468909258e-5
  2.2516323865110587e-6
 -2.8636014669379933e-5



In [167]:

    
g(x::Float64)=(Aλ⋅cos.(x*Λ)+Bλ⋅sin.(x*Λ))*step









    Out[167]:





g (generic function with 2 methods)



In [21]:

    
# Rekonstruirajmo signal s odrezanim frekvencijama
g(x::Float64)=(Aλ⋅cos.(x*Λ)+Bλ⋅sin.(x*Λ))*step
G=g.(T)









    Out[21]:





88200-element Array{Float64,1}:
  0.000184805327142117
  0.00022355499304577928
  0.00026082494707742234
  0.0002962364964993873
  0.00032942895326896885
  0.0003600634641776046
  0.00038782662073811507
  0.0004124338106432171
  0.00043363227525326325
  0.0004512038405850413
  0.0004649672926284134
  0.0004747803714866872
  0.00048054136277459234
  ⋮
  0.0014759800471028286
 -0.00017988954098602
 -0.0017398810480251034
 -0.003200230175866396
 -0.00455783291565055
 -0.005810252841406547
 -0.006955723797088994
 -0.007993148002196426
 -0.008922089635965701
 -0.009742763993387377
 -0.010456022339221955
 -0.011063332617785724



In [22]:

    
# Usporedimo grafički
plot(T,[S G],layout=(2,1))









    Out[22]:



In [23]:

    
# Spremimo signal s odrezanim frekvencijama i poslušajmo
glas_mono_odrezan=deepcopy(glas_mono)
glas_mono_odrezan.data[:,1]=map(Float32,G)
save("myvoice_odrezan.wav", glas_mono_odrezan)



In [24]:

    
glas_mono_odrezan









    Out[24]: