Homomorphic Filtering for Extracting Javanese Gong Wave Signals Matias H.W. Budhiantho1 and Gunawan Dewantoro2 Department of Electronics and Computer Engineering1,2 Satya Wacana Christian University Salatiga, Indonesia [email protected] [email protected] Abstract—The Gong is a unique percussion instrument of instrument sounds using higher order spectra such as Gamelan because of its wave-like sounds after being struck Bispectrum and Trispectrum. Experimental results on a widely immediately. These special sounds correspond to the used dataset shows that higher order spectra based features characteristics of each gong which is originated from different improve the recognition accuracy, when combined with places. This phenomenon might raises from the primary beat of conventional features such as Mel Frequency Cepstral the partials and or mismatched overtone frequencies. However, Coefficient (MFCC), Cepstral, Spectral and Temporal there has been a lack of studies that rigorously locate the beat features. Nineteen western musical instruments covering four and/or a set of beat frequencies the better understand the families with full pitch range have been used for perceptible beat. This study attempts to trace the beat-showing experimentation. Tsai et al. [5] proposes a background occurrences in the quefrency domain employing a set of accompaniment removal approach for SID by exploiting the homomorphic operations. First, acoustic measurement was conducted and further analyzed using homomorphic operations underlying relationships between solo singing voices and their to obtain its cepstral profile. The rapidly-varying frequency part accompanied versions in cepstrum. Such a transformation was filtered out to show the location of a set of beats. The reflects the cepstrum variations of a singing voice before and cepstrum also show a prominent peak at 0.01 s, which after it is added with accompaniments. Gaikwad et al. [6] The corresponds to the harmonicity of 47.6 Hz. This is in agreement presents automatic classification of Indian Classical with the measured fundamental frequency of Gong Kempul. instruments based on spectral and MFCC features using well trained back propogation neural network classifier. Musical Keywords—gong kempul; homomorphic filtering; cepstrum; instruments such as Harmonium, Santoor and Tabla are beat frequency considered for experimentation. The spectral features such as amplitude and spectral range along with Mel Frequency I. INTRODUCTION Cepstrum Coefficients are considered as features. Caetano [7] proposed a method to obtain a smooth function that Sound of Javanese Gong plays an important role in approximately matches the main peaks of the waveform using Javanese culture. In gamelan music orchestra, gong sound true envelope estimation, dubbed true amplitude envelope. instructs, marks and ends certain parts of a gamelan True envelope is a cepstral smoothing technique that has been composition [1]. The gong sound is also used to declare the shown to outperform traditional envelope estimation opening and closing of important religious and secular events techniques both in accuracy of estimation and ease of order or various rituals. The roaring wavelike sound of the gong is selection. True amplitude envelope gives a reliable estimation associated by Javanese with Bima’s giggle that creates a that follows closely sudden variations in amplitude and avoids grandeur yet calming feeling. Bima is known as a bold but ripples in more stable regions with near optimal order honest and just hero, a great legend in Javanese puppet selection depending on the fundamental frequency of the shadow (wayang) stories. The number of wavelike sound signal. Bispo et el. [8] presents a new approach based on repetition cycles in a best sounding gong can be as many as 12 cepstral analysis to identify and cancel the feedback path. The to 13. A gong that cannot produce a wavelike sound is only known previous cepstrum based formulations are taken in considered to be only “howling” [2]. consideration and reviewed, and the process is now A temporal and spectral description of Javanese gong completely derived. A new method is proposed and simulation kempul was studied in [3]. The gong has a set of results showed that it is able to present an improvement in the approximately harmonic partial with fundamental frequency at estimation of the feedback path when compared to the state- about 93 Hz. The partials, both harmonic and nonharmonic, of-art method. Kelly et al. [9] presents a method of beat together producing roaring sound. However, it was not headphone/earphone equalization based upon deconvolution clear the origins of the beats since it was difficult to detect the of the headphone impulse response from other acoustic filters partials at low frequency. Bhalke et al. [4] presents in the processing chain. The methods presented are thus classification and recognition of monophonic isolated musical applicable to areas such as spatial audio, where input signals 978-1-4799-7447-4/14/$31.00 ©2014 IEEE are processed with binaural impulse responses. The extraction π 1 ω of low order from higher order acoustic impulse responses is c (t) = log | X ( jϖ ) | e j t dω (4) x π ∫ justified based upon an application of the theory pertaining to 2 −π the clustering of the zeros of random coefficient polynomials about the unit circle. Fraile et al. [10] This paper approaches Since the cepstrum is based on only the Fourier transform the problem of inverse filtering by homomorphic prediction. magnitude, it is not invertible, i.e., x(t) cannot be recovered While not favoured much by researchers in recent literature, from cx(t). However, the cepstrum is easier to compute than such an approach offers two potential advantages: it does not the complex cepstrum. require previous pitch detection and it does not rely on any assumptions about the spectral enevelope of the glottal signal. B. Homomorphic Deconvolution Its performance is herein assessed and compared to that of an The principle of generalized superposition as stated for a adaptive inverse filtering method making use of synthetic system require that, if H is the system transformation, then for voices produced with a biomechanical voice production any two inputs x (t) and x (t) and any scalar c, model. Results indicate that the performance of the inverse 1 2 Δ = filtering based on homomorphic prediction is within the range H{x1 (t) x2 (t)} H{x1 (t)} ◊ H{x2 (t)} (5) +of that of adaptive inverse filtering and, at the same time, it has a better behavior when the spectral envelope of the glottal and signal does not suit an all-pole model of predefined order. H{c D x (t)}= c • H{x (t)} (6) Hasani et al. [11] uses homomorphic filtering to produce time- 1 1 domain intensity envelopes of the heart sounds and separates where Δ is a rule for combining inputs with each other (e.g., the sounds into four overlapping parts:the first heart sound, the addition, multiplication, convolution, etc) and ◊ is a rule for systolic period, the second heart sound and the diastolic combining inputs with scalars. Similarly, D will denote a rule period. This study incorporate homomorphic filtering to obtain for combining system outputs and • is a rule for combining the fast-varying frequency part of gong signal. The paper is outputs with scalars. Such systems which obey a generalized organized as follow: Section II introduces the homomorphic principle of superposition are referred to as homomorphic systems that led to the cepstral analysis fundamentals; Section systems since they can be represented by algebraically linear III describes the spectral analysis of the gong signal under (homomorphic) mappings between input and output signal Matlab; in Section IV, the obtained cepstral separation are spaces. Any homomorphic systems can be decomposed as a presented and discussed. Finally, Section V concludes the cascade of three homomorphic systems, as indicated in Fig. 1, paper emphasizing the beat frequency(ies) of the gong which with the first system depending only on the input operation Δ , give the audience a well-described perceptible beat sounds. the last system depending only on the output operation ◊ and the middle system corresponding to a linear system. II. HOMOMORPHIC SYSTEMS [12] Δ ◊ A. Complex Cepstrum + + + + − The word “cepstrum” has four interchanging letters in the L 1 DΔ ∧ ∧ D◊ word “spectrum” because in general we find ourselves x(t) x(t) y(t) y(t) operating on the frequency side in ways customary on the time side and vice versa. Consider a stable sequence x(t) whose Fourier transform is H ∠ ω Fig. 1. Canonic representation of homomorphic systems. X ( jω) =| X ( jω) | e j X ( j ) , (1) The system DΔ [.] is referred to as the characteristic system the complex cepstrum corresponding to x(t) is defined to be − ∧ Δ 1 associated with the input operation , and D◊ [.] is the the stable sequence x(t) whose Fourier transform is: inverse of DΔ [.], the characteristic system associated with the ∧ output operation ◊ . Such decomposition will be very useful in X ( jω) = log X ( jω) (2) applying homomorphic systems to deconvolution. In particular, DΔ [.] can be chosen with convolution as the Therefore, the complex cepstrum of x(t) is given by the inverse Fourier transform integral as: operation of superposition at the input and addition as the operation of superposition at the output. As in (2), if the π ∧ complex logarithm is appropriately defined, when X(jω)= 1 jωt x(t) = log X ( jϖ )e dω (3) X1(jω)+X2(jω), then 2π ∫ −π ∧ ∧ ∧ ω = ω + ω = ω + ω X ( j ) log X1 ( j ) log X 2 ( j ) X1 ( j ) X 2 ( j ) (7) In contrast to the complex cepstrum, the cepstrum cx(t) of a signal (sometimes referred to as the real cepstrum) is defined and as the inverse Fourier transform of the logarithm of the magnitude of the Fourier transform; i.e., ∧ ∧ ∧ x(t) = x1(t) + x2 (t) (8) The operation that define the complex cepstrum is shown in sound card then interfaced and digitized this signal in order Fig.2.