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Fast Algorithm Of Frame Approximation Of Music Signal And Single-pitch Melody Identification Using SVM

Posted on:2012-08-15Degree:MasterType:Thesis
Country:ChinaCandidate:X D LiFull Text:PDF
GTID:2178330332999604Subject:Computational Mathematics
Abstract/Summary:PDF Full Text Request
With the development of the information era, intelligent analysis of music and semanticunderstanding attract the attention of most scholars. Especially music search based on Internethas a broad market, corporations such as Yahoo, Google have built their own research teams.Moreover, music analysis is the base of music signal processing, the research about it involvesdi?erent subjects such like music, mathematics, physics and psychology and so on.Music is produced by the vibration of musical instruments. Besides the whole body, 1/2, 1/3and 1/4 ... of the body vibrate simultaneously, the sounds produced by these parts are calledharmonic overtones. The physical concept of pitch is frequency, when an instrument plays a note,the usual pitch includes the fundamental tone and overtones. And the frequency of any overtoneis integral times of that of fundamental tone. The link between note and pitch is provided bythe temperament. Now"Twelve-tone equal temperament"is widely accepted. It regulates thepitch of adjacent semitones as a geometric series and the common ratio is 2 112, the fundamentalfrequency of the standard note (A) accepted internationally is 440Hz, thus we can compute allthe semitones, which means we can know all the notes. Based on this, we consider using stablebases to analyze music signals.Let be the set of frequencies, with total number K:(?)and(?) here (?) is B spline function with order m. Then we have the following theoremTheorem (?) is a set of stable bases.and (?) have the following properties:Proposition Let(?)is linearly independent of entire axis, then (?) is also linearlyindependent of entire axis.Similarly, let(?)here(?)is Gauss function, which is(?)using the notation above, we can obtain the following theoremTheorem If a function (?) can be represented as(?)here k,n 2, and there are 1 2 such that(?)then (?)n Z is a set of stable bases.Similarly, we have(?)are all sets of stable bases respectively,Φis either B spline or Gauss function.As we know, music signal can be treated as the sum of trigonometric functions, which meansthat a music signal has the form (?) here are the sets of coe?cient, is the set of frequency. So based on the theory that weobtained, we use the following bases to approximate music signals.Gauss function bases:(?) spline function bases:(?)here nm ( ) is the B spline function with order , the coordinate of the left endpoint of itssupport .Now we have the approximation problem: Given that a music signal , choose a set of basesΦn( ) and coe?cients n to construct(?)Use to approximate , we can apply least square method to solve the problem, which meansis the solution of following equation(?)After derivation, we can get the equation set(?)solve it to get n, then we can obtain .Since the property of B spline, if the distance of translation is more than 2 , the relevantelements in matrix and are all 0, so obviously, and are band matrices,the bandwidthis direct proportional with the number of frequencies and the the order of spline. Meanwhile, forthe property of prompt dropping of Gauss function, and in Gauss model can be analogouslyset as band matrices arti?cially.Let ij and ij represent the elements of th row and column of and respectively. After being rearranged, the model has the following form(?)from this we know that it's a band matrix, we can use fast algorithm to improve the computationspeed.Moreover, if we choose the following bases(?)then we have new approximation form(?)After derivation, we can get a periodic block-tridiagonal matrix(?)and we can use two-parameter fast algorithm to speed up.If we change the form of ( ) so that(?)here is either Gauss function or B spline. Let(?)and we know that ( ) is the amplitude of signal. If we discrete time into N samples, (?), then we can get the amplitude of any frequency at any moment in matrix(?)In , the th row is a vector that contains amplitude information of all frequencies that occurredin the signal.Theoretically, for matrix which is generated by single-pitch melody, each row only containsthe amplitude information of one note. It means amplitudes of the notes which are not beingplayed are all 0. According to the music theory, di?erent combinations of fundamental tone andovertones is just the distinguish feature for instruments. So the non-zero elements of each rowof correspond a note in a reverse way. After identifying the note of every moment, we gatherthe number of every note to get their lasting time, and use logical transformation, we can obtainthe score.For most identi?cation situations with unknown rules, a powerful classi?cation tool is SupportVector Machine(SVM). It's a set of related supervised learning methods that analyze data andrecognize patterns, used for classi?cation and regression analysis. Because we don't know thefeature of every instrument, we can input as training data into SVM to construct a modelthat we can classify notes.To be more adaptable, we choose several di?erent sets of parameters to approximate signal,such as Fourier transformation threshold value, distance of translation of bases. And experimentsshowed model that we obtain by the approximation can work perfectly in single-pitch melody.The rate of right prediction is 100%, and because of the cross zone generated by the reductionof previous note and rise of next note, the error of lasting time is 0.5%-1.6%, this result can beused in real-time precessing when the tempo of music is under 200.We propose a new format in approximating music signal in this paper, and get the preciseapproximation by only picking the frequencies in the signal. And since getting the amplitudeinformation of every frequency, we can identify the note of any moment by using SVM. Lotsof experiments showed that our method is feasible and correct, providing a reliable way in converting music into score.
Keywords/Search Tags:time-frequency analysis, stable bases, music approximation, SVM, note identi-cation
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