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 Processamento de Sinais
Processamento de Sinais
 
Introdução
 
O Maple 17 oferece novas ferramentas de processamento de sinais para a análise e manipulação de dados da freqüência e no domínio do tempo. Ela pode ser utilizada para diversas aplicações, tais como a criação de um espectrograma de fala, a remoção de ruído de sinais com chiados, e identificar a periodicidade dos dados.
 
 
Este pacote inclui ferramentas para:
 
  • Cosseno, Fast Fourier e wavelet transforms
  • Janelas Bartlett, Blackman, Kaiser, Hann e Hanning
  • Geração de sinal
  • Correlação cruzada, autocorrelação, estatísticas de dados e upsampling/downsampling
  • Filtros FIR, IIR e Butterworth

 

 
TELA TÍPICA DOS CÁLCULOS DA FOLHA DE DOCUMENTO DO MAPLE:
 
 
> with(SignalProcessing[Engine])
 







 
 

 

Application: Imaging a Speech Spectrogram

Import Wave File and Manipulate Data into Overlapping Slices

>
 
> filename := cat(kernelopts(datadir), kernelopts(dirsep),
filename := cat(kernelopts(datadir), kernelopts(dirsep),
 
C:\Program Files\Maple 17\data\signalprocessing\maplesim.wav  
 
> data := AudioTools:-Read(filename)
 
data := Vector[column](%id = 4361768386)  
 
>
 
11025  
 
> len := evalf(`/`(`*`(numelems(data)), `*`(samplingRate)))
 
.7462131519  
 

Slice the data into segments with 256 samples. Each slice has a 50% overlap with the previous slice (that is, each slice starts at the half-way point of the previous slice). 

> samps := 256; -1
 
> nTimes := `+`(floor(`+`(`/`(`*`(2, `*`(numelems(data))), `*`(samps)))), `-`(1))
 
63  
 
> sample := Matrix(samps, nTimes, proc (i, j) options operator, arrow; data[`+`(i, `*`(`/`(1, 2), `*`(`+`(j, `-`(1)), `*`(samps))))] end proc, datatype = float[8]); 1
 
sample := Vector[column](%id = 4524928962)  
 

 

Calculate the Spectrogram Data 

Filter the data, slice by slice. 

> a := `<|>`(`<,>`(seq(HannWindow(sample[() .. (), i]), i = 1 .. nTimes))); -1
 

Calculate FFT of time segment consecutively. 

> FFTa := `<|>`(`<,>`(seq(FFT(a[i, () .. ()]), i = 1 .. nTimes))); -1
 

Calculate Power Spectrum of each slice consecutively. 

> spectra := `<|>`(`<,>`(seq(`~`[sqrt](PowerSpectrum(FFTa[i, () .. ()])), i = 1 .. nTimes))); -1
 

Convert spectrum into decibels. 

> dB := proc (x) options operator, arrow; `+`(`*`(20, `*`(log10(x)))) end proc; -1
 
> spectra := map(dB, spectra); -1
 

Strip out the repeated data. 

> sim := LinearAlgebra:-SubMatrix(Re(spectra), 1 .. nTimes, 1 .. `+`(`*`(`/`(1, 2), `*`(samps)), 1)); -1
 
> nFreqs := LinearAlgebra:-ColumnDimension(sim)
 
129  
 

 

The frequencies go from 0 to half the sampling rate. Hence the frequencies are in steps of (in Hz): 

> `*`(samplingRate, `*`(`/`(`+`(`*`(2., `*`(nFreqs))))))
 
42.73255814  
 

 

Scale the spectra so that the values are between 0 and 255. 

> minSim := min(sim)
 
HFloat(-111.18888523873858)  
 
> maxSim := max(sim)
 
HFloat(4.43801796851939)  
 
> scale := proc (i) options operator, arrow; `/`(`*`(`+`(`*`(255, `*`(i)), `-`(`*`(255, `*`(minSim))))), `*`(`+`(maxSim, `-`(minSim)))) end proc; -1
 

Consecutive rows represent slices in time, while columns contain the spectra at each time. 

> simScaled := `~`[scale](sim); 1
 
simScaled := Vector[column](%id = 4524921090)  
 

 

Plot the Spectrogram and Waveform 

> commonPlotOpts1 := labelfont = [Helvetica], labeldirections = [horizontal, vertical]; -1
 
> p1 := listdensityplot(`+`(`-`(simScaled)), style = patchnogrid, smooth = true, tickmarks = [[], [seq(i = `*`(round(`+`(`*`(10, `*`(samplingRate, `*`(i, `*`(`/`(`+`(`*`(2000., `*`(nFreqs)))))))))), `*`...
p1 := listdensityplot(`+`(`-`(simScaled)), style = patchnogrid, smooth = true, tickmarks = [[], [seq(i = `*`(round(`+`(`*`(10, `*`(samplingRate, `*`(i, `*`(`/`(`+`(`*`(2000., `*`(nFreqs)))))))))), `*`...
p1 := listdensityplot(`+`(`-`(simScaled)), style = patchnogrid, smooth = true, tickmarks = [[], [seq(i = `*`(round(`+`(`*`(10, `*`(samplingRate, `*`(i, `*`(`/`(`+`(`*`(2000., `*`(nFreqs)))))))))), `*`...
 
> p2 := listplot([seq([`/`(`*`(i), `*`(samplingRate)), data[i]], i = 1 .. numelems(data))], thickness = 0, gridlines, axes = boxed, labels = [
p2 := listplot([seq([`/`(`*`(i), `*`(samplingRate)), data[i]], i = 1 .. numelems(data))], thickness = 0, gridlines, axes = boxed, labels = [
 
> display(Array(`<|>`(`<,>`(p1, p2))), aligncolumns = [1])
 
Plot_2d
Plot_2d
 

Application: Filtering Audio 

Import Speech Sample 

> filename2 := cat(kernelopts(datadir), kernelopts(dirsep),
filename2 := cat(kernelopts(datadir), kernelopts(dirsep),
 
C:\Program Files\Maple 17\data\signalprocessing\MapleSimMono11025.wav  
 
> originalSpeech := AudioTools:-Read(filename2)
 
originalSpeech := Vector[column](%id = 4524921794)  
 

 

Plot Waveform and Power Spectrum 

> commonPlotOpts2 := titlefont = [Helvetica, 12], labelfont = [Helvetica], labeldirections = [horizontal, vertical], axis = [gridlines = [color =
commonPlotOpts2 := titlefont = [Helvetica, 12], labelfont = [Helvetica], labeldirections = [horizontal, vertical], axis = [gridlines = [color =
 
> samplingRate := attributes(originalSpeech)[1]
 
11025  
> duration := evalf(AudioTools:-Duration(originalSpeech))

 

4.504671202  

 

> p1 := plots:-listplot([seq([`/`(`*`(i), `*`(samplingRate)), originalSpeech[i]], i = 1 .. numelems(originalSpeech))], thickness = 0, gridlines, axes = boxed, title =
p1 := plots:-listplot([seq([`/`(`*`(i), `*`(samplingRate)), originalSpeech[i]], i = 1 .. numelems(originalSpeech))], thickness = 0, gridlines, axes = boxed, title =
p1 := plots:-listplot([seq([`/`(`*`(i), `*`(samplingRate)), originalSpeech[i]], i = 1 .. numelems(originalSpeech))], thickness = 0, gridlines, axes = boxed, title =
p1 := plots:-listplot([seq([`/`(`*`(i), `*`(samplingRate)), originalSpeech[i]], i = 1 .. numelems(originalSpeech))], thickness = 0, gridlines, axes = boxed, title =

 

> plots:-display(p1)

 

Plot_2d  

 

> fq := FFT(originalSpeech[1 .. `^`(2, 15)]); -1

 

> psq := PowerSpectrum(fq); -1

 

> ps1 := plots:-pointplot([seq([`/`(`*`(i, `*`(samplingRate)), `*`(`^`(2, 15))), psq[i]], i = 1 .. `+`(`*`(`/`(1, 2), `*`(`^`(2, 15)))))], thickness = 0, color = black, gridlines, connect = true, title ...
ps1 := plots:-pointplot([seq([`/`(`*`(i, `*`(samplingRate)), `*`(`^`(2, 15))), psq[i]], i = 1 .. `+`(`*`(`/`(1, 2), `*`(`^`(2, 15)))))], thickness = 0, color = black, gridlines, connect = true, title ...
ps1 := plots:-pointplot([seq([`/`(`*`(i, `*`(samplingRate)), `*`(`^`(2, 15))), psq[i]], i = 1 .. `+`(`*`(`/`(1, 2), `*`(`^`(2, 15)))))], thickness = 0, color = black, gridlines, connect = true, title ...
ps1 := plots:-pointplot([seq([`/`(`*`(i, `*`(samplingRate)), `*`(`^`(2, 15))), psq[i]], i = 1 .. `+`(`*`(`/`(1, 2), `*`(`^`(2, 15)))))], thickness = 0, color = black, gridlines, connect = true, title ...

 

> plots:-display(ps1)

 

Plot_2d  

 

 

Apply IIR Butterworth or Chebyshev Filter 

Apply Filter 

> fc := 800; -1
 
> taps := GenerateButterworthTaps(9, `/`(`*`(fc), `*`(samplingRate)), 'filtertype' = 'lowpass', normalise = true)
taps := GenerateButterworthTaps(9, `/`(`*`(fc), `*`(samplingRate)), 'filtertype' = 'lowpass', normalise = true)
 
taps := Vector[column](%id = 4524922882)  
 
> filteredSpeech := InfiniteImpulseResponseFilter(originalSpeech, taps); -1
 

 

View Before and After Power Spectrum and Waveform 

> FFTfilteredSpeech := FFT(filteredSpeech[1 .. `^`(2, 15)]); -1
 
> PSfilteredSpeech := PowerSpectrum(FFTfilteredSpeech); -1
 
>


 
> plots:-display(Array(`<,>`(`<|>`(ps1, ps2))))
 
Plot_2d Plot_2d
 
 
> p2 := plots:-listplot([seq([`/`(`*`(i), `*`(samplingRate)), filteredSpeech[i]], i = 1 .. numelems(originalSpeech))], thickness = 0, gridlines, axes = boxed, color = black, title =
p2 := plots:-listplot([seq([`/`(`*`(i), `*`(samplingRate)), filteredSpeech[i]], i = 1 .. numelems(originalSpeech))], thickness = 0, gridlines, axes = boxed, color = black, title =
p2 := plots:-listplot([seq([`/`(`*`(i), `*`(samplingRate)), filteredSpeech[i]], i = 1 .. numelems(originalSpeech))], thickness = 0, gridlines, axes = boxed, color = black, title =
 
> plots:-display(Array(`<,>`(`<|>`(p1, p2))), view = [0 .. duration, -1 .. 1])
 
Plot_2d Plot_2d
 
 

 

Apply FIR Filter 

Apply Filter 

> flow := 200; -1
 
> fhigh := 700; -1
 
>
 
> filteredSpeech := FiniteImpulseResponseFilter(originalSpeech, taps); -1
 

 

View Before and After Power Spectrum and Waveform 

> FFTfilteredSpeech := FFT(filteredSpeech[1 .. `^`(2, 15)]); -1
 
> PSfilteredSpeech := PowerSpectrum(FFTfilteredSpeech); -1
 
>


 
> plots:-display(Array(`<,>`(`<|>`(ps1, ps2))))
 
Plot_2d Plot_2d
 
 
> p2 := plots:-listplot([seq([`/`(`*`(i), `*`(samplingRate)), filteredSpeech[i]], i = 1 .. numelems(originalSpeech))], thickness = 0, gridlines, axes = boxed, color = black, title =
p2 := plots:-listplot([seq([`/`(`*`(i), `*`(samplingRate)), filteredSpeech[i]], i = 1 .. numelems(originalSpeech))], thickness = 0, gridlines, axes = boxed, color = black, title =
p2 := plots:-listplot([seq([`/`(`*`(i), `*`(samplingRate)), filteredSpeech[i]], i = 1 .. numelems(originalSpeech))], thickness = 0, gridlines, axes = boxed, color = black, title =
 
> plots:-display(Array(`<,>`(`<|>`(p1, p2))), view = [0 .. duration, -1 .. 1])
 
Plot_2d Plot_2d



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