Sound Basics #

Five Key Concepts #

1. Decibels
2. Clipping & gain
3. ADC: Sample rate & bit depth
4. Fast Fourier Transform (FFT)
5. Spectrograms: time/frequency trade-off

1: Decibels #

Decibels #

Decibels are always the log of a ratio

$$10 \log_{10}{\left( \frac{\text{Power}}{\text{Reference Power}} \right)}$$

$$20 \log_{10}{\left( \frac{\text{Amplitude}}{\text{Reference Amplitude}} \right)}$$

2: Clipping and Gain #

3: Analogue to digital conversion #

3: Analogue to digital conversion #

4: Fourier transform #

4: Fourier transform #

Maximum frequency in spectrograms = sample rate / 2 = Nyquist

4: Fourier transform #

Function FFT($array){
    $Len = $array.Count
    If($Len -le 1){Return $array} 
    $Len_Over_2 = [Math]::Floor(($Len/2))
    $Output = New-Object Complex[] $Len

    $EvenArray = @()
    $OddArray  = @()

    For($i = 0; $i -lt $Len; $i++){
        If($i % 2){ $OddArray  += $array[$i] }
        Else      { $EvenArray += $array[$i] }
    }

    $Even = FFT($EvenArray)
    $Odd  = FFT($OddArray)

    For($i = 0; $i -lt $Len_Over_2; $i++){
        $Twiddle = [Complex]::Exp(
            [Complex]::ImaginaryOne*[Math]::Pi*($i*-2/$Len)
        ) * $Odd[$i]
        $Output[$i]             = $Even[$i] + $Twiddle
        $Output[$i+$Len_Over_2] = $Even[$i] - $Twiddle
    }

    Return $Output
}

5: Spectrograms: time/frequency trade-off #

5: Spectrograms: time/frequency trade-off #