Calculation of a constant Q spectral transform
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Judith C. Brown
Journal of the Acoustical Society of America,1991
Jain-De,Lee
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INTRODUCTION
CALCULATION
RESULTS
SUMMARY
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The work is based on the property that, for sounds made up of harmonic frequency components
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The positions of these frequency components relative to each other are the same independent of fundamental frequency
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The conventional linear frequency representation
◦Rise to a constant separation◦Harmonic components vary with fundamental frequency
The result is that it is more difficult to pick out differences in other features
◦ Timbre◦Attack◦Decay
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The log frequency representation
◦Constant pattern for the spectral components◦Recognizing a previously determined pattern becomes a
straightforward problem
The idea has theoretical appeal for its similarity to modern theories
◦ The perception of the pitch–Missing fundamental
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To demonstrate the constant pattern for musical sound◦ The mapping of these data from the linear to the logarithmic
domain Too little information at low frequencies and too much
information at high frequencies
For example
◦Window of 1024 samples and sampling rate of 32000 samples/s and the resolution is 31.3 Hz(32000/1024=31.25)
The violin low end of the range is G3(196Hz) and the adjacent note is G#3(207.65 Hz),the resolution is much greater than the frequency separation for two adjacent notes tuned
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The frequencies sampled by the discrete Fourier transform should be exponentially spaced
If we require quartertone spacing
◦ The variable resolution of at most ( 21/24 -1)= 0.03 times the frequency
◦A constant ratio of frequency to resolution f / δf = Q
◦Here Q =f /0.029f= 34
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Quarter-tone spacing of the equal tempered scale ,the frequency of the k th spectral component is
The resolution f / δf for the DFT, then the window size must varied
fk = (21/24)k fmin
Where f an upper frequency chosen to be below the Nyquist frequency
fmin can be chosen to be the lowest frequency about which Information is desired
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For quarter-tone resolution
Calculate the length of the window in frequency fk
Q = f / δf = f / 0.029f = 34
Where the quality factor Q is defined as f / δfbandwidth δf = f / QSampling rate S = 1/T
N[k]= S / δfk = (S / fk)Q
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We obtain an expression for the k th spectral component for the constant Q transform
Hamming window that has the form
1
0
}/2exp{][][][N
n
NknjnxnWkX
1][
0
]}[/2exp{][],[][
1][
kN
n
kNQnjnxnkWkN
kX
W[k,n]=α + (1- α)cos(2πn/N[k])
Where α = 25/46 and 0 ≤ n ≤ N[k]-1
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Constant Q transform of violinplaying diatonic scale pizzicato from G3 (196 Hz) to G5(784 Hz)
Constant Q transform of violinplaying D5(587 Hz) with vibratoConstant Q transform of violin glissando from D5 (587 Hz) to A5 (880Hz)Constant Q transform of flute playing diatonic scale from C4 (262 Hz) to C5 (523 Hz) with increasing amplitude
Constant Q transform of piano playing diatonic scale from C4 (262 Hz) to C5(523 Hz)The attack on D5(587 Hz) is also visible
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Straightforward method of calculating a constant Q transform designed for musical representations
Waterfall plots of these data make it possible to visualize information present in digitized musical waveform