Guerino Mazzola (Fall 2014©): Introduction to Music Technology
III Digital Audio
III.9 (We Oct 29)
Phase vocoder for tempo and pitch changes
Guerino Mazzola (Fall 2014©): Introduction to Music Technology
Phase Vocoder
This algorithm is built in order to enable tempo and pitch changes of a
digital sound file.
Tempo change: Same music, but played at a different tempo
Pitch change: Same tempo, but transposed pitch.
The algorithm was first described by James L. Flanagan and R.M.
Golden in a paper “Phase Vocoder” published in Bell System
Technical Journal, Vol. 45, No. 9, p. 1493, November 1966.
James L. Flanagan
Phase Vocoder
Guerino Mazzola (Fall 2014©): Introduction to Music Technology
The basic technique starts with so-called resampling.
amplitude
time
Δ
amplitude
time
Δ’
frequencies’ = frequencies. Δ/Δ’
problem: sound changes dramatically! Why???
Phase Vocoder
Guerino Mazzola (Fall 2014©): Introduction to Music Technology
The fundamental idea here is this:
1. Construct a new sample with longer duration + same pitch ⇒
back to original duration + higher pitch via resampling
2. Construct a new sample with shorter duration + same pitch ⇒
back to original + lower pitch via resampling.
So basically we are dealing with the time change problem!
The procedure is that we first cover the original signal by a sequence of
sound frames of equal length, but in order to grasp their commonalities, we
choose overlapping frames. Typically this is achieved by 75%, and the
frame duration is typically 1/20 sec (corresponding to 20 Hz fundamental
frequency for finite Fourier).
amplitude
time
Guerino Mazzola (Fall 2014©): Introduction to Music Technology
Phase Vocoder
The idea is to work on these frames, processing them on the frequency
space, and then generating a synthesis sound by adding these new
frames with different overlapping times and thereby changing the
tempo of the overall signal:
Guerino Mazzola (Fall 2014©): Introduction to Music Technology
Phase Vocoder
A frame is generated from the original sample by multiplying it with a
Hanning window function H(t)
amplitude
time
amplitude
time
amplitude
time
Guerino Mazzola (Fall 2014©): Introduction to Music Technology
Phase Vocoder
amplitude
time
D
First step of the algorithm: Analysis
The frame is the transformed to frequency representation via FFT.
The fundamental frequency is of course f = 1/frame duration = 1/D. The highest
frequency is fs = n.f, so that we have n frequency intervals from 0 to fs(n-1)/n Hz.
Attention: n has nothing to do with the original sample frequency of the signal!!
The temporal delay of ¼ frame has then 2n/4 (temporal) samples; this number is
called analytical hop size hopa. In other words, we have the equation
hopa/2fs = 2n/(4.2nf) = D/4 = Da = analytical hop time between successive frames.
Phase Vocoder
Guerino Mazzola (Fall 2014©): Introduction to Music Technology
What is the problem now?
The reproduction of the frames with different distances causes phase problems:
synthetical hop time
analytical hop time
Da
Ds
Phase Vocoder
Guerino Mazzola (Fall 2014©): Introduction to Music Technology
sin(2πft)
Second step (processing) : We look at the phase problem:
sin(2πf(t+Da)) = sin(2πft+ΔΦ)
ΔΦ can be calculated, omit this!
ΔΦ = 2πf.Da , f = ΔΦ/2πDa = “true frequency”.
Frame i1
Third step (synthesis): Replace Da by the
synthetic frame distance Ds and then set a new
phase of frame i
ΔΦs, i = ΔΦs, i-1 + 2πf.Ds
Frame
i
take sinusoidal signal
component
where the (i-1)th phase has been calculated by
recursion.
Correct the complex coefficients of the FFT
transform accordingly. FFTransform back,
multiply each frame by a Hamming curve and
add it all.