Efficient Multi-resolution Sinusoidal Modeling

Transcription

1 ANDERSON AND CLEMENTS 1 Efficient Muti-resoution Sinusoida Modeing David V. Anderson and Mark A. Cements Abstract Muti-resoution sinusoida modeing is a method of representing audio signas as a sum of discrete sinusoids having various durations, and time-varying frequencies, phases, and ampitudes. This paper discusses the motivation for the mutiresoution sinusoida mode (MRSM) and various structures used to compute it. A new fiter-bank anaysis structure is presented that is simper than previous methods. Finay, a matchingpursuit anaysis method is presented for more efficienty modeing time-varying signas. I. Introduction A system that modifies speech and audio signas for human isteners requires a suitabe signa representation on which to work. Common representations for genera speech or audio signas incude the Fourier transform of the signa, auto-regressive moving-average modes, sub-band signas or waveet coefficients, and the time sampes themseves. Each of these representations emphasizes certain signa characteristics and they a represent trade-offs in the time resoution and frequency resoution at which the signa is portrayed. The sinusoida mode (SM), as deveoped by Quatieri and McAuay [1], for speech provides a way of representing an audio signa in a sparse manner as a sum of discrete sinusoids [1]. The sinusoids in the SM are continuay varying in ampitude, frequency, and phase; unike the fixed, unconnected sinusoida basis functions used in short-time Fourier anaysis. The SM is more fexibe than most other signa modes in that it is easy to represent the time evoution of each frequency component in a signa using the SM. Representing a signa using the SM is a ossy process, i.e., in most cases the signa cannot be reproduced exacty from the mode represention; however, the SM can be used to reproduce speech and audio signas which are neary indistinguishabe from the originas. The SM was originay designed for use in speech coding, and it can be used to achieve arge coding gains. However, it was soon discovered that the parametric method of representing speech in terms of discrete sinusoids was usefu in speech modification. The SM has since been used for time scae modification of speech [], [3], [4], [5], music modeing [6], [7], [8], [9], pitch modification of speech [], [1], co-channe interference suppression [11], [1], background noise suppression [13], speech synthesis [14], [1], singing voice synthesis [1], music synthesis and modification [15], [9], hearing compensation [16], and peak to RMS ratio reduction [17]. One of the probems with the sinusoida mode is that it does not do we at modeing wideband audio. This is not seem to be due to a fundamenta probem with the sinusoida mode itsef, but with the anaysis methods used. This work reviews some recent innovations to the SM in extending it to use a muti resoution anaysis. Severa muti resoution anaysis methods are reviewed and severa new methods are presented. Georgia Institute of Technoogy, Atanta, Georgia, USA. E-mai: david.anderson@ece.gatech.edu, mark.cements@ece.gatech.edu A. Sinusoida Mode for Speech II. Sinusoida Mode A great strength of the SM is that it is we matched to speech signas and can represent the important auditory characteristics with very few parameters. These parameters can then be manipuated to modify certain signa characteristics. The SM has an added advantage over most other speech signa representations in that it more accuratey modes a broader cass of signas. The SM has been successfuy used with muti-taker waveforms, music, speech in a musica background, and other signas [1], [18]. A.1 Voiced Speech The sinusoida transform is based on a speech production mode in which the excitation waveform is a sum of sinusoids of the form L(t) e(t) = a (t) cos [Ω (t)]. (1) =1 The instantaneous phase, Ω (t), of the th component is reated to its instantaneous frequency, ω (t), by Ω (t) = t ω (ξ)dξ + φ. () Note that the sinusoids vary in ampitude and frequency as a function of time and that the number of sinusoids used, L(t), aso varies with time. The excitation is then subject to the combined gotta waveform and voca tract response. This system response is aso time-varying and may be represented by H(ω; t) = M(ω; t) exp [jψ(ω; t)]. (3) It is convenient to track each component of the excitation signa separatey. Therefore, for the th frequency component we express the associated voca tract magnitude and phase as and M (t) = M[ω (t); t] (4) ψ (t) = ψ [ω (t); t] (5) respectivey. The speech signa which resuts from passing the excitation signa through the voca tract is then where and L(t) s(t) = A (t) exp θ (t) (6) θ (t) = =1 A (t) = M (t)a (t) (7) t ω (ξ)dξ + φ + ψ (t). (8) The ψ (t) and φ are often umped together and phase expression becomes Each θ (t) = t ω (ξ)dξ + ˆψ (t). (9)

2 ANDERSON AND CLEMENTS A. Unvoiced Speech One shortcoming of the sinusoida mode for speech is that it is best suited for voiced speech. However, if there are enough sinusoids, and if they are cose enough in frequency, it is possibe to represent the noise-ike unvoiced speech adequatey [1]. This is one of the issues addressed ater in this work. It is aso possibe to represent other signas such as muti-taker waveforms and music by increasing the number of sinusoids used and/or by using one of the aternate methods mentioned beow. A.3 Anaysis In practice the DFT is used to estimate the parameters in equations 6 and 9. Parameters are determined by finding peaks in the DFT spectrum of the zero padded signa segment and noting the corresponding frequency, ampitude, and phase. The estimates are typicay updated every 5- msec. To improve the accuracy of the peak picking process, the segment of interest is windowed with a window whose ength is at east -3 periods of the frequencies of interest. For high fideity reproduction and in situations where there is no singe pitch (as in mutipe takers or music) or with additive noise an even onger window is desirabe. A.4 Synthesis The individua sinusoids associated with each segment are caed partias. Since the SM parameters are ony estimated once per segment, it is necessary to match the partias in adjacent segment to form a continuous sinusoid. Interpoation is then used to obtain a smooth evoution of the sinusoid between segments. The ampitude is usuay interpoated ineary and the frequency and phase are treated together and interpoated so that they are maximay smooth [18]. Segment k of the discrete time output of the sinusoida mode is expressed as x (k) [n] = L (k) 1 = ( A (k) [n] cos where the foowing terms are defined: L (k) φ (k) ) [n] the number of sinusoida partias, the index to the sinusoida partia, A (k) [n] the interpoated ampitude of the partia, and φ (k) [n] the interpoated instantaneous phase vaue of the partia. (1) The fina SM output signa for segment k is then given by summing the appropriate signa bocks, x (k) [n], as [ ] x[n] = x (k) n n (k) (11) k where n (k) is the time index at the beginning of segment k. It is aso possibe to perform the synthesis using an overap/add technique. Overap/add synthesis does not do any expicit matching of tracks between segments; the interpoation between segments is performed by windowing the synthesized time waveform with a trianguar window that is at east twice the segment ength and overapping and adding with the previous segment(s). One probem with the overap/add technique is that it does not preserve waveform onsets as we as the matched partia synthesis. Another probem is that the overap/add method of interpoation can resut in phase canceations. This method works best if the SM parameters are estimated fairy often, at east every 1 msec. A.5 Variations There are many variations on the basic sinusoida transform as described above. These incude coding a residua separatey [15], [19], [], doing an anaysis-by-synthesis or matching pursuit anaysis [6], [1], and using harmonic signa pus noise modes [], [], [3]. B. Muti-Resoution Sinusoida Mode B.1 Motivation Perceptua Considerations The primary perceptua characteristic of human audition utiized by the SM is simutaneous auditory masking. Simutaneous masking describes the tendency of intense sounds or peaks in the short-term spectrum to mask out or hide softer sounds that are cose in frequency [4]. Thus, it is possibe to retain ony the spectra peaks whie the ess intense sounds that are adjacent to peaks may be ignored without any perceivabe change in the audio signa. The SM does this by encoding ony the peaks, reconstructing the signa by representing them as sinusoids with smoothy varying phase and ampitude [5]. In addition to auditory masking, there are other perceptua or psychoacoustic phenomena which may be expoited in the design of an audio coder. One of these is frequency resoution in the human auditory system. Resoution in perception is often measured in terms of just noticeabe differences (JNDs). Frequency JNDs are approximatey ogarithmic in frequency the ear can resove ower frequencies more cosey than it can resove high frequencies. Another perceptua phenomenon that often pays a roe in audio signa processing is tempora masking. Tempora masking occurs when an intense sound hides or masks ess intense sounds preceding or foowing it. An exampe commony encountered in signa compression is the arger quantization noise preceding a transient in a segment. In sinusoida modeing there is often a pre-echo generated when the ampitudes of the partias are ineary interpoated between segments. Ideay, the segments wi be cose enough together so that the pre-echo is masked. C. MRSM Motivation The muti resoution sinusoida mode (MRSM) offers improvements over the reguar SM by addressing the frequency resoution aspect of auditory perception. D. A Mutitude of MRSM Anaysis Methods The input to the MRSM anaysis stage is an audio signa, the output is a ist of frequencies, ampitudes, phases, and durations of sinusoids which mode the signa. There are various ways to perform the MRSM anaysis which wi be discussed beow. The goa of each of these methods is to produce an estimate of the sinusoids which represent the signa, with the high frequency sinusoids being updated frequenty and the ow frequency sinusoids updated ess often but with better frequency resoution as mentioned above. The one method discussed estimates the sinusoids using ony DFTs of various engths. The other three of the methods discussed use fiter-banks to break the signa into sub-bands which are then anayzed to estimate the sinusoids. D.1 Direct DFT Based MRSM Anaysis One method of performing MRSM anaysis is to directy use DFTs (using FFTs) of different engths [6]. The idea here is to manuay tie the time-frequency pane as shown in Figure 1. This method suffers from two difficuties. First, the computationa compexity is very high. Second, it is desirabe to use

3 ANDERSON AND CLEMENTS 3 windowed DFTs, but then the bases used don t competey cover the space [7]. 1 The peaks are encoded by recording their cacuated ampitudes, phases, frequencies, and durations. Duration is proportiona to the ength of the DFT used to estimate the peak; when waveet based anaysis is used it corresponds to sampe rate of the sub-band in which the peak was detected. If N is the duration of a sinusoid in the highest frequency band and L is the ength of the shortest DFT, then the onger DFTs wi have engths L d, d =, 1,,..., and the corresponding sinusoids wi have durations of N d. of Figure 5. The synthesis bank consists ony of an up-samper and anti-imaging fiter for the ow-pass fiter branch. Note that the fiter bank is a perfect reconstructing fiter bank regardess of the choice for H b (z). However, when using the fiter bank in sinusoida modeing, the synthesis bank is not used, so the perfect reconstructing aspect of the fiter bank is moot. The fiter bank is impemented in a tree fashion with the anaysis fiter outputs being fed into a sinusoida mode anaysis bocks (see Figure 3). Fig. 1. Discrete waveet tiing of the time frequency pane for a four band case Fig. 3. The genera anaysis structure used for the MRSM. The sinusoida mode parameters are extracted from the sub-band signas. The sinusoida mode anaysis window ength is the same for each Sin. Mod. bock but the effective window ength is onger for the ower frequency bands. D. Waveet Based MRSM Anaysis Initia attempts at MRSM anaysis were buit around a DFT anaysis on the outputs of waveet-ike fiter banks [6], [8]. It has been suggested that waveets are particuary good at modeing the frequency response of the human auditory system [9]. Waveets cosey approximate the higher resoution in frequency exhibited by the human auditory system at ower frequencies, and the higher resoution in time exhibited by the human auditory system at higher frequencies. This MRSM anaysis method uses waveet ike anaysis to provide a course frequency separation, then fine spectra determination is accompished with the DFT. The waveet based MRSM anaysis introduces aiasing that woud normay be canceed in a synthesis bank; however, when cacuating the MRSM the aiasing must be canceed manuay [6]. This causes it aso to have high computation requirements. Fig.. An anaysis bank for the MRSM as used by Levine [3] (shown with the synthesis bank). This fiter bank is oversamped in the high frequency band and the ow frequency cutoff is chosen to be beow so that there is itte or no aiasing. D.3 Oversamped Fiter Bank MRSM Anaysis I A better method for MRSM anaysis is based on oversamped fiter-banks [3]. Instead of using criticay samped fiter banks and then canceing aiased components, as with the waveet method, the oversamped fiter-bank method can eiminate or severey reduce aiasing in the frequencies of interest [31]. Thus, with good fiter design, the aiasing canceation requirement can be dropped. The outputs of the fiter-bank are oversamped by a factor of two; therefore, the compexity growth is not as severe as with the mutipe DFT method. Levine [3] proposed using a tree structured fiter-bank with each branch as shown in Figure. This structure was originay proposed by Fiege [31] as a modification to the Lapacian pyramid often used in muti-rate image processing. Fiters H d (z) and H i(z) are haf-band ow-pass fiters and they may be chosen so that H d (z) = H i(z). Fiter H b (z) eiminates, or attenuates to an arbitrary degree, aiased components introduced by the decimation foowing H d (z). Eimination of the aiasing is accompished by choosing H b (z) so that the aiased portion of the signa is in the stop band of H b (z) as shown in the top 1 In other words, seecting bins from different ength windowed DFTs to tie the time frequency does not permit perfect reconstruction. The oversamped fiter bank anaysis approach is usefu because it eiminates the compexity growth of the mutipe DFT method and the aiasing associated with criticay samped fiter banks (waveets). However, there are some disadvantages to the fiter bank used by Levine. The fiter cut-off frequency, f c, of H b (z) must satisfy f c < (f τb + f τd ) (1) where f τb and f τd are the transition bandwidths of H b (z) and H d (z) respectivey. Another disadvantage of the fiter bank structure introduced above is that it is designed for use as a perfect reconstructing fiter bank and it is more compex than needed. D.4 Oversamped Fiter Bank MRSM Anaysis II Since the synthesis portion of the fiter bank is not used and the idea of perfect reconstruction is obiterated by the operation of sinusoida modeing, the fiter bank structure may be simpified. A better structure is shown in Figure 4. In this fiter bank, H (z) is chosen to have a cut-off ess than such that H(z) for Ω, where z = ejω. This structure is much simper but, after the decimation, it can no onger be part of a perfect reconstructing system. However, as mentioned above, perfect reconstructing fiter banks are not necessariy the most appropriate for use as preprocessors for a sinusoida modeer. This new system aso has the advantage that the ow-pass signa can have a higher cut-off frequency for a given compexity as shown in Figure 5. Fig. 4. An improved anaysis bank for the MRSM. The stop-band of H (z) is the range Ω. This two channe fiter bank is used in a tree structure to provide (neary) octave band anaysis. E. Mutipe Bases Anaysis In performing sinusoida mode anaysis, the goa is to find the time varying sinusoids that best mode a signa. However, stationary sinusoids are usuay used as basis vectors for this anaysis. Ideay, the anaysis procedure woud optimay choose a sma number of time varying sinusoids from a arge set of sinusoida bases that vary over time in both ampitude and frequency. In this way a more accurate and compact mode of the signa coud be produced. Such an optimization probem has enormous compexity so a suboptima soution is sought. The fiter structure in Figure 4 can be part of a near-perfect reconstructing fiter bank, but not a perfect reconstructing fiter bank.

4 ANDERSON AND CLEMENTS 4 H d (z 1/ H b (z) ) f b H (z 1/ ) H (z 1/ ) H d ( z 1/ ) H d (z 1/ ) H ( z 1/ ) some dictionary of bases, D, which minimizes a residua r[n] at each iteration. The resuting approximation at iteration L is expressed as L 1 xˆ L[n] = α k b m(k) (13) k= where b m(k) is the basis chosen at the k th iteration. The residua is given by r L[n] = x[n] xˆ L[n]. (14) Note that the residua can be computed recursivey as r k+1 = r k α k b m(k) (15) where, for notationa convenience, r k [n] is represented as r k. After each iteration, k, the residua, r k+1 [n], is ony minimized if it is orthogona to the chosen basis vector, b m(k). If b m = 1 then the α k which minimizes r k+1 is f Fig. 5. The bandwidth of the ow frequency bands in Levine s structure and the proposed structure are compared. The proposed structure (bottom) resuts in a neary haf-band spit whie the upper structure resuts in a one-third/two-third spit for fiters of reasonabe compexity. Fig. 6. The anaysis structure used for the MRSM. The high frequency band is oversamped by a factor of two and the ow frequency band has a cutoff beow so that there is itte or no aiasing. E.1 Matching Pursuit Matching pursuit is a sub-optima method for finding a sparse representation of a signa given an over-compete set of bases [3], [33]. In a matching pursuit anaysis, the signa is modeed with successive approximations, each approximation adding a singe additiona basis contribution. At each step, the residua energy is minimized. The iteration is stopped after the residua or error energy is brought beow some threshod or some predetermined number of bases have been used. This method is simiar to the anaysis by synthesis procedure used by George [34], [6] with the exception that the basis functions are restricted to compex sinusoids. E. Matching Pursuit and Perception Matching pursuit is a greedy agorithm and, in genera, the resuting mode is not optima in terms of the residua energy for number of basis vectors used. However, if the basis vectors are we chosen, it is we suited to audio signa representation. This is because the ear is not very sensitive to noise (modeing error) that is quiet and/or spectray diffuse. In matching pursuit, the high energy spectra components tend to be modeed first, and sma spectra components may not even be modeed. Thus, priority is given to those signa components to which the ear is most sensitive. Aso, the overa energy in each critica band is very cose to that of the origina signa. If just peak picking is used, the energy may be noticeaby different as the broad and narrow peaks may have the same maximum vaue but they contain different amounts of energy. In [7] the matching pursuit process is modified to expicity take advantage of perception. E.3 Procedure Given some signa x[n], the idea with matching pursuit is to iterativey create a representation based on the bases, b m, in α k = r i, b m(k). (16) The best b m for use a the k th iteration is b m(k) = arg max b m, r k (17) b m D and it is found by minimizing r k+1. In short, α k is cacuated for each possibe b m(k) and the basis function which produces the argest α k is used. E.4 Reducing Compexity The matching pursuit agorithm as described above requires the cacuation of a arge number of inner products. However, the computation compexity may be reduced by iterativey updating the inner products: r k+1, b m = r k, b m α k b m(k), b m. (18) If enough memory is avaiabe, b n, b m may be precomputed and stored for use in Equation 18. E.5 Seection of the Basis Vectors Gabor dictionaries are often used for signa modeing because of the optima time-frequency representation [3]. However, when modeing a specific type of signa it is often usefu to choose basis vectors which better fit the signas to be modeed. Severa different choices of dictionaries are discussed beow. E.6 Frames of Compex Exponentias Frame is a mathematica term that refers to a basis set with some specia properties [35]. Specificay, Definition 1: A frame is a set of vectors φ n such that A x φ n, x B x for a x (19) A > and B > are the frame bounds. Definition : A tight frame is a frame that has A = B. Unike the orthonorma bases usuay used in signa decomposition, frames do not have the requirement of inear independence. Thus, frames can be thought of as over-samping in the transform domain. Figure 7 iustrates the difference between a basis set and a frame set in R. Fig. 7. Exampes of an orthogona basis set (a), and a frame set (b) in R.

5 ANDERSON AND CLEMENTS 5 E.6.a Compex Exponentia Frames. In sinusoida modeing, compex exponentias are usuay used in the anaysis. In order to get better frequency resoution, the DFT is typicay padded with zeros to provide more frequency terms. Padding with zeros and taking the orthogona projection onto the compex exponentia basis vectors is equivaent to simpy projecting the signa segment onto a tight frame of compex exponentia vectors of the same ength as the signa segment (see [36, pp 56-63]). Anaysis with compex exponentia basis functions is often performed by simpy picking the peaks in the DFT spectrum. Peak picking is ikey to resut in poor signa modeing when sinusoids are custered in the signa since they may med into a singe arge spectra peak. Using matching pursuit anaysis with the typica compex exponentia basis vectors tends to reduce modeing error by better modeing cose sinusoids and by reducing the expicit modeing of spurious peaks due to side-obes. E.7 Damped Sinusoids Goodwin [1], [33] has expanded the anaysis basis set to a dictionary of damped compex exponentia basis vectors. This is particuary usefu in modeing music since there are many sharp attacks with gradua decays. There are aso some efficient ways of impementing matching pursuit anaysis with damped compex exponentia basis vectors that make it an appeaing choice. E.8 Chirped Sinusoids Sinusoida modeing assumes that the sinusoids are continuay varying in ampitude and frequency. By using a dictionary of chirped sinusoids it is possibe to directy account for frequency varying signa components in the anaysis process. Note that the chirped sinusoids form a compete basis [37] so for each chirp rate used, a compete basis is added to the dictionary of basis vectors. The chirped basis vectors used in this work were of the form j kn e k (k) = w s[n] N e k e j N k ( k n+ k N n ) () where k < N k, (1) n < N, () k { β, β,, β, β}, (3) { 1 and w s[n] = N n < N,. (4) N n < N k The amount of frequency moduation in the chirp is determined by β; when β = 1, k = β produces a chirp whose instantaneous frequency range spans a singe DFT bin width. Note, each vaue of β yieds a frame or orthogona basis (depending on the vaue of N k ). The chirped sinusoid dictionary is convenient because it is easy to compute the basis projections and the inner products between basis vectors. The initia projection of the signa onto each basis vector of the basis vectors is computed using the FFT as shown in r, e k (k) = 1 N N k 1 n= w s[n]x[n]e j N k (kn k n+ k N n ). (5) The matrix of basis vector inner products, e κ(k), e k (k), can be efficienty computed and stored because it is highy structured. The foowing vaues are defined to simpify the resut: Let d k = k κ, (6) d k = k κ, and (7) Φ(d k, d k ) = e k (k), e κ(k). (8) Then Φ(d k, d k ) = 1 N From this we see that N 1 ( ws[n] e j N n k n= d k d k + d k (N 1) n ). (9) Φ(d k, d k ) = Φ ( d k, d k ) (3) and that Φ(d k, d k ) is ony a function of the differences k κ and k κ. The sinusoida mode parameters are then chosen as A k = N α k, Φ k = α k, and ω k is taken from the frequency of b m(k). F. Synthesis Synthesis is performed in a manner very simiar to that used with the reguar sinusoida mode. The partias in each subband are used to create sub-band signas that are then combined into the fu-band signa. The fina MRSM output signa for segment k is then given by summing the appropriate signa segments, x (k,b) [n], as [ ] x[n] = x (k,b) n n (k,b) (31) b k where n (k,b) is the time index at the beginning of frame k in sub-band b. III. Concusions and Comments We have discussed severa methods of muti-resoution sinusoida modeing and have presented two innovations improving upon current methods. The new MRSM has been successfuy used in various signa enhancement tasks incuding fast acting dynamic range contro for hearing compensation, peak to RMS ratio reduction of speech for communications, and background noise suppression [7]. In each of these cases the MRSM outperforms traditiona sinusoida mode and Fourier based techniques. References [1] Robert J. McAuay and Thomas F. Quatieri, Speech anaysis/synthesis based on a sinusoida representation, IEEE Transactions on Acoustics, Speech, and Signa Processing, vo. 34, no. 4, pp , August [] T. F. Quatieri and R. J. McAuay, Shape invariant time-scae and pitch modification of speech, IEEE Transactions on Signa Processing, vo. 4, no. 3, pp , March 199. [3] T. F. Quatieri, R. B. Dunn, and T. E. Hanna, Time-scae modification of compex acoustic signas, Proceedings of the IEEE Internationa Conference on Acoustics, Speech, and Signa Processing, pp. I13 16, Apri [4] E. B. George and M. J. T. 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