Calculation of a constant Q spectral transform

Calculation of a constant Q spectral transform

Judith C. Brown

Journal of the Acoustical Society of America,1991

Jain-De,Lee

INTRODUCTION

CALCULATION

RESULTS

SUMMARY

Outline

The work is based on the property that, for sounds made up of harmonic frequency components

INTRODUCTION

The positions of these frequency components relative to each other are the same independent of fundamental frequency

INTRODUCTION

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

INTRODUCTION

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

INTRODUCTION

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)

INTRODUCTION

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

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

INTRODUCTION

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

CALCULATION

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

For quarter-tone resolution

Calculate the length of the window in frequency fk

CALCULATION

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

We obtain an expression for the k th spectral component for the constant Q transform

Hamming window that has the form

CALCULATION

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

CALCULATION

CALCULATION

RESULTS

RESULTS

RESULTS

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

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

SUMMARY