AN EFFICIENT TWO-STAGE IMPLEMENTATION OF HARMONIC MATCHINGPURSUIT

Chris Duxbury, Nicolas Chétry, Mark Sandler and Mike Davies

DSP & Multimedia GroupDepartment of Electronic Engineering

Queen Mary, University of LondonMile End Road, E1 4NS London, UK

chris.duxbury@elec.qmul.ac.uk, nicolas.chetry@elec.qmul.ac.uk

ABSTRACT

We introduce an algorithm implementation for the de-composition of quasi-steady state audio signals using har-monic matching pursuits. Specifically, we propose an initiallow-resolution pitch analysis followed by a high resolutionharmonic grain extraction based on local complex interpo-lation within the spectral domain. We describe the imple-mentation of the algorithm and illustrate its applications tomusical analysis of monophonic signals before finally dis-cussing possible improvements that could lead to the designof an iterative multi-pitch harmonic analysis system.

1. INTRODUCTION

Recent frameworks based on hybrid representation of au-dio signals attempt to decompose an input waveform as alinear combination of several distinct components. In suchschemes, the stationary part is extracted from the originalsignal and separated from the noisy [1] or noisy plus tran-sient [2] part. Resulting signals can then be processed indi-vidually, taking into account their own spectral, temporal orstochastic characteristics.

In this paper, we have focused on the modeling of theharmonic stationary component of audio signals. Such wave-forms can be obtained by removing the transients [3], whichoccur during the attack of a note and present strong time lo-calisation together with non-stationary properties. Diverseapplications can be found in signal compression, musicalanalysis, musical modeling and automatic music transcrip-tion.

1.1 Harmonics, pitch and fundamental frequency

Many musical instruments produce harmonically relatedsounds, made up of several sinusoidal components, hav-ing their frequencies approximately integer multiples of thenote’s fundamental frequency. These sinusoidal componentsare also called harmonics, or partials, and give the soundits timbral characteristic, whilst the fundamental frequencygenerally gives the sound its pitch. However, in many cases,the fundamental frequency may not be present, without hav-ing an effect on the perceived pitch [4]. As an illustration,figure 1(a) shows a missing partial, and figure 1(b) shows aweak fundamental frequency component. These two soundsare however perceived as if the latter frequency componentswere unaltered.

Because of these issues, simply choosing the position ofthe lowest frequency high energy sinusoidal component as a

Figure 1: Spectral plots of both (a) low and (b) high vio-lin notes showing respectively a missing partial and a weakfundamental frequency.

possible fundamental frequency may not be relevant in termsof meaningful grain extraction.

The low-resolution pitch analysis stage presented in sec-tion 2 helps to select the harmonic series which will removethe maximum energy from the spectra.

1.2 Matching pursuit using harmonic grains

Introduced in [5], the iterative harmonic matching pursuit al-gorithm approximates a signal with a linear combination ofelementary waveforms, also called harmonic atoms. At eachiteration, the best matching function is chosen from a redun-dant dictionary of harmonic Gabor functions. The residual isthen similarly processed until the pre-defined energy-basedstop criterion is satisfied.

Harmonic matching pursuit is an extension of the moregeneral matching pursuit decomposition introduced in [6].By considering dictionaries of harmonic atoms, such algo-rithms allow the extraction of higher level objects, which, inthe case of audio signals, can lead to a meaningful musicalinterpretation (e.g note detection).

We present here an efficient dual resolution spectral-based approach that significantly reduces the computationalrequirements but still maximises the energy extracted at eachstage:• Low resolution harmonic energy analysis: the harmonic

energy is calculated for each potential fundamental fre-quency. The harmonic series corresponding to the max-

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imum harmonic energy is chosen. A rough value of thefundamental frequency is then estimated from the bin lo-cation of the fundamental and its partials.

• High resolution harmonic grain extraction: using theabove value of the fundamental frequency, a local com-plex interpolation within the FFT frame is performedin order to determine more accurate values of the fun-damental and its partials frequencies together with theircorresponding amplitudes and phases. The resulting har-monic grain is then synthesised and subtracted from theoriginal grain.

2. LOW-RESOLUTION HARMONIC ENERGYANALYSIS

2.1 Harmonic energy calculation

Within an FFT frame, a frequency domain component con-tributes to the harmonic energy of the series if it is the max-imum component within the corresponding harmonic win-dow. This window increases in width linearly with the par-tial number. To be more specific, let us consider an FFTframe with a resolution of 20 Hz and a harmonic series cor-responding to the fundamental frequency bin 100− 120 Hz.The first partial could appear anywhere between 200 and 240Hz, corresponding to a harmonic window twice the width ofthe resolution value. Similarly, the next partial could appearanywhere between 300 and 360 Hz, i.e. three times the ini-tial window width. The harmonic window width (in bins) asa function of the partial number p is given by:

ν(p) = p, p = 1, ...,ρ

where p is the partial number; p = 1 represents the funda-mental and ρ − 1 the desired number of partials excludingthe fundamental. Its value depends on the input signal andthe sampling frequency. In practice, values around ρ = 15for a 44.1 kHz sampled signal gave satisfactory results withour test signals.

The harmonic energy Λ(k) is calculated for each poten-tial fundamental index k as:

Λ(k) = |X(k)|2 +ρ

∑p=2

maxν|X(kp+ν(p)−1)|2

where X is the complex FFT of the input frame. The har-monic series with the highest energy is then retained. It cor-responds to the fundamental index k = k0.

The selected series is used to calculate an approximationof the fundamental frequency:

f̃0 =1ρ

ρ

∑p=1

fk0,pp

where fk0,p is the frequency corresponding to the maximumamplitude over the considered window ν(p):

fk0,p = fmaxν |X(k0 p+ν(p))|This new measure of fundamental is used to select the

most relevant partial bins, by rounding the expected partialposition

f̃p = p f̃0, p = 1, ...,ρto the nearest bin value. These bins values are used in thehigh resolution harmonic grain extraction stage described insection 3.

3. HIGH-RESOLUTION GRAIN EXTRACTION

3.1 Frequency domain interpolation

Prior to the grain extraction, frequencies and correspondingphases of the selected fundamental and partials are interpo-lated in order to counterbalance the FFT finite resolution andtherefore to maximise the energy extracted at each iteration.

Zero-padding the time domain input frame could achievethis goal but is clearly computationally inefficient as only thefundamental and the considered partials must be synthesised.

The following technique is identical to a zero-padding,but is applied locally to a region of the FFT around the con-sidered peak (typically the width of the harmonic window asdefined in section 2.1, extended by 2, 3 or 4 bins). It hasthe advantage of interpolating both phase and amplitude to-gether.

Figure 2: Illustration of a harmonic grain extraction withina FFT frame. (a) original FFT frame. (b) FFT of the ex-tracted grain after interpolation. (c) FFT of the residualwithout interpolation. (d) FFT of the residual after inter-polation (ξ = 20).

1. The interpolation function is calculated using the FFT ofa zero padded rectangular window (by a factor ξ ). Thesevalues are calculated once and saved in a table.

2. For each input frame, the frequency bins required for theinterpolation (the fundamental and its related partials) areextracted as vectors and upsampled by the interpolationfactor ξ .

3. The vectors are then convolved with the interpolationfunction values, achieving a local, complex domain in-terpolation.

Figures 2(c) and 2(d) show the advantages of using theinterpolation in terms of the extracted energy per FFT frame.

The harmonic grain is then synthesised in the time do-main and subtracted from the original grain. In the case ofmonophonic signals (one instrument), the algorithm stops. Ifnot, the residual is processed similarly, until no more har-monic series are detected.

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3.2 Orthogonality of harmonic grains

As shown in [5], the windowed sinusoids used in the algo-rithm are not necessarily orthogonal. Re-orthogonalisation,while possible, would be much too computationally expen-sive. In order to quantify the possible effects of using non-orthogonal basis functions, we compared the energy mea-sures taken from the harmonic matching pursuits algorithmwith the same decomposition using sinusoids which had beenre-orthogonalised after selection using the QR orthogonal-triangular decomposition. The results for a violin tone areshown in figure 3. It can be seen that for the frame lengthconsidered here, the error due to non-orthogonality is trivialfor all but the very low frequencies (in the case where thefundamental frequency is below 100 Hz).

500 1000 1500 2000 2500

20

40

60

a) F

FT

500 1000 1500 2000 25000

5000

10000

15000

b) H

arm

. En.

500 1000 1500 2000 2500

10−10

100

Frequency

c) A

bsol

ute

Diff

.

Non−orthogonalOrthogonal

Figure 3: Effects of orthogonalising sinusoidal componentsof a harmonic grain on the harmonic energy measure. TheFFT is shown in (a), with the two different measures of har-monic energy given in (b) and the absolute difference be-tween the two values in (c). Frequencies are in Hertz.

3.3 Transient/Steady-State (TSS) separation

The harmonic matching pursuit algorithm is designed tomodel the slowly time varying steady state component of au-dio signals. Noisy high frequency time localised informa-tion, like transients or hard onsets, cannot be studied usingharmonic grains. Further, their presences in the input sig-nal may introduce artifacts as a virtual pitch (and possiblya whole harmonic series) can be detected during the low-resolution analysis stage. For this reason, a pre-treatmentis applied in order to separate both transient and steady statesignals from the original waveform. Details and examples ofsuch TSS implementation can be found in [3].

4. ALGORITHM IMPLEMENTATION ANDRESULTS

Tests audio signals (trumpet and jazz guitar pieces) weresampled at 44.1 kHz. The analysis stage was performed onsuccessive frames of 2048 samples weighted by a Hanningwindow. Using such a window clearly sacrifices temporallocalisation. A hop size of 25% (i.e. 512 samples) was there-fore used to retain some signal timing information. ξ wasset to 20 and the local interpolation is applied on a 7 binsvector (the selected peak plus 3 bins on each side). A fi-nal point regarding robust implementation of this algorithm

is to ignore the first three bins corresponding to the frequen-cies 0−65 Hz, as issues related to the lack of orthogonalitybecome more preponderant in that frequency range as stipu-lated in section 3.2. However, this should not induce severeartifacts as it is quite uncommon for musical signal to con-tain notes at such low frequency. Interpolated amplitudes,frequencies and phases are then used to synthesise the grainin the time domain using the oscillator bank approach (see[7] for details).

Figure 4 is an example of pitch extraction for a mono-phonic trumpet signal. The algorithm accurately restitutesthe evolution of the fundamental frequency as a function oftime. Figure 5 is an illustration of the complete application ofthe harmonic matching pursuits on a monophonic jazz guitarpiece after having removed the transients [3]. Figure 5 (e)represents the time waveform residual after substraction ofthe synthesized signal from the original. One can notice thatthe transients are still slightly present, but nevertheless muchmore attenuated than if the whole original signal was usedduring the decomposition. Finally, figure 6 is an exampleof notes extraction. An artificial mixture of two piano noteswithout overlapping harmonics (B-250 Hz and F-350 Hz) hasbeen synthesised. Two successive iterations of the algorithmare needed to decompose the input frame in two harmonicgrains corresponding to the two individual notes. Residualerrors (figures 6(c) and 6(f)) mainly contain noise and infor-mal listening tests did not show any perceivable differencesbetween the originals and extracted notes.

Fre

quen

cy

Time0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

0

500

1000

1500

2000

50 100 150 200 250 300100

200

300

400

500

Hop number

Ext

ract

ed fr

eq. o

f fun

dam

enta

l (H

z)

Figure 4: Extracted pitches (fundamental frequencies) as afunction of time for a harmonic monophonic trumpet piece.

5. CONCLUSION AND FUTURE WORK

We introduced in this paper an efficient two-stage implemen-tation of the matching pursuits algorithm based on a har-monic grain extraction within the spectral domain.

We first showed the advantages of considering harmonicsobjects in terms of:

• high level musical interpretation: as the extracted graincontains a complete harmonic series, it can be usefullycharacterised as a distinct musical note.

• computational efficiency: at each iteration, a completeseries of sinusoidal components is extracted, as opposedto sinusoidal matching pursuit, where only one peak ispicked at a time, thus requiring far fewer iterations.

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Figure 5: Harmonic matching pursuits complete analy-sis/synthesis decomposition of a monophonic jazz guitar sig-nal. (a) spectrogram of the original signal. (b) originalwaveform. (c) spectrogram of the resynthesised using a sin-gle harmonic grain per frame. (d) resynthesised waveform.(e) left over time domain residual.

Then, we introduce a local interpolation scheme, equiv-alent to a local zero-padding in order to obtain accurate fre-quency, amplitude and phase values of the selected harmonicpeaks.

This implementation performs well for stationary audiosignals. The quality of the extraction is good even though wemake no assumptions about the type of instrument which isplayed.

Improvements, however, are needed to make the algo-rithm more robust and applicable to a wider range of signals.Firstly, problems arise with polyphonic audio mixtures con-taining overlapping harmonics. This is a common drawbackof all the spectral analysis techniques. In such a case, har-monics corresponding to a given pitch may be affected toanother harmonic series, thus introducing some false notesin the considered grain. This can be overcome for exampleby making assumptions about the harmonic distributions [8].Secondly, if the signal is not purely steady state, the residualis shaped into harmonics which may introduce artifacts in theextracted signal. Particular attention should therefore be paidregarding the quality of the TSS decomposition.

REFERENCES

[1] Xavier Serra, Musical Signal Processing, chapter Musi-cal Sound Modeling with Sinusoids plus Noise, pp. 91–123, Swets and Zeitlinger Publishers, 1997.

[2] L. Daudet and B. Torrésani, “Hybrid representations for

Figure 6: Example of two piano notes separation. (a) and(d) are the original waveforms (respectively B-250Hz and F-350Hz), (b) and (e) the resynthesised signals, (c) and (f) thecorresponding residual errors.

audiophonic signal encoding,” Signal Processing, vol.82, 2002.

[3] C. Duxbury, M. Davies, and M. Sandler, “Extraction oftransient content in musical audio using multiresolutionanalysis techniques,” in Proc. Digital Audio Effects Con-ference (DAFX,’01), 2001.

[4] B.C.J. Moore, An Introduction to the Psychology ofHearing, Academic Press, 1997.

[5] R. Gribonval and E. Bacry, “Harmonic decompositionof audio signals with matching pursuit,” IEEE Trans.Signal Process., vol. 51, no. 1, pp. 101–111, jan 2003.

[6] S. Mallat and Z. Zhang, “Matching pursuit with time-frequency dictionnaries,” IEEE Trans. Signal Process-ing, vol. 41, pp. 3397–3415, Dec 1993.

[7] X. Amatriain, J. Bonada, and A. Loscos, DAFX - DigitalAudio Effects, chapter Spectral Processing, pp. 373–438,Udo Zölzer, 2002.

[8] A. P. Klapuri, “Mutipitch estimation and sound separa-tion by the spectral smoothness principle,” in IEEE In-ternational Conference on Acoustics, Speech and SignalProcessing, ICASSP, 2001.

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AboutCurrent paperPresentation sessionAbstractAuthorsChris DuxburyNicolas ChetryMark SandlerMike Davies