Report

Judith C. Brown Journal of the Acoustical Society of America,1991 Jain-De,Lee INTRODUCTION CALCULATION RESULTS SUMMARY The work is based on the property that, for sounds made up of harmonic frequency components The positions of these frequency components relative to each other are the same independent of fundamental frequency 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 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 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 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 Quarter-tone spacing of the equal tempered scale ,the frequency of the k th spectral component is 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 The resolution f / δf for the DFT, then the window size must varied For quarter-tone resolution 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 Calculate the length of the window in frequency fk N[k]= S / δfk = (S / fk)Q We obtain an expression for the k th spectral component for the constant Q transform N 1 X [k ] W [ n ] x[ n ] exp{ j 2 kn / N } n0 X [k ] 1 N [k ] N [ k ] 1 W [ k , n ] x[ n ] exp{ j 2 Qn / N [ k ]} n0 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 Constant Q transform of piano playing diatonic violin flute playing glissando scale playing from diatonic C4 from (262 scale pizzicato to(880Hz) C5(523 from D5 diatonic D5(587 scale (587 Hz) Hz) toHz) C4 with A5 (262 vibrato Hz)from toHz) C5 The G3 (196 attack ontoD5(587 G5(784Hz) Hz) is also visible (523 Hz)Hz) with increasing amplitude 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