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

Outline • INTRODUCTION • CALCULATION • RESULTS • SUMMARY

INTRODUCTION • 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) • 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

INTRODUCTION • 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

CALCULATION • 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)kfmin Where f an upper frequency chosen to be below the Nyquist frequency fmincan be chosen to be the lowest frequency about which Information is desired

CALCULATION • 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 / δf bandwidth δf = f / Q Sampling rate S = 1/T N[k]= S / δfk = (S / fk)Q

CALCULATION • We obtain an expression for the k th spectral component for the constant Q transform • Hamming window that has the form W[k,n]=α + (1-α)cos(2πn/N[k]) Where α= 25/46 and 0 ≤ n ≤ N[k]-1

CALCULATION

RESULTS

RESULTS 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 Constant Q transform of violin playing diatonic scale pizzicato from G3 (196 Hz) to G5(784 Hz) Constant Q transform of violin playing D5(587 Hz) with vibrato Constant 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

SUMMARY • 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

Calculation of a constant Q spectral transform