ON BEDROSIAN CONDITION IN APPLICATION TO CHIRP SOUNDS

Transcription

1 15th European Signal Processing Conference (EUSIPCO 7), Poznan, Poland, September 3-7, 7, copyright by EURASIP ON BEDROSIAN CONDIION IN APPLICAION O CHIRP SOUNDS E. HERMANOWICZ 1 ) ) and M. ROJEWSKI Faculty of Electronics, elecommunications and Informatics, Gdańsk University of echnology, 11/1 Narutowicza St., 8-95 Gdańs Poland 1 ) ) Multimedia Systems Department, hewa@eti.pg.gda.pl. eleinformation Networks Department, ro@ssound.eti.pg.gda.pl ABSRAC A ( e ) DF spectrum for envelope a[ A [ SF amplitude spectrum for envelope a[ Arg( ) principal argument D global measure of envelope and phasor SF amplitude spectra overlap F frequency in Hz sampling period in seconds U ( e ) DF spectrum for u[ It is assumed that for a certain class of sounds there exists a relationship between their pitch and their complex dynamic representation CDR (real-valued log-envelope and instantaneous angular frequency). he CDR processing appears a powerful means for pitch shifting of chirp-like sound signals fulfilling practically the assumption of the Bedrosian theorem. Scaling the CDR components can be used for entertainment as well as, e.g., in a generator of different melodies for calling signals, where the sound of each note in a melody is derived from a short recording of a voice of a chosen creature. For this aim in this paper a concept of a Bedrosian chirp is proposed whose local instantaneous amplitude and instantaneous complex phasor spectra are separated. A quantitative measure to estimate and predict the quality of sound processing using the CDR scaling is defined and its utility verified on the basis of experiments with real-life audio chirps. γ [ complex phasor λ [ instantaneous level (log-envelope) ϕ [ instantaneous phase κ CDR rescaling coefficient ω = πf normalised angular frequency ω [ instantaneous angular frequency Γ ( e ) DF spectrum for phasor γ [ Γ [ SF amplitude phasor spectrum for γ [ Abbreviations AM amplitude modulation AM FM AM FM representation of an arbitrarily modulated signal DDS direct digital synthesis DF discrete Fourier transform DF discrete-time Fourier transform DSP digital signal processing CDR complex dynamic representation FIR finite impulse response FM frequency modulation HF Hilbert filter HS Hilbertian signal H Hilbert transformer IAE instantaneous amplitude estimator IFE instantaneous frequency estimator PA phase accumulator SF short-time Fourier transformation Notation a [ instantaneous amplitude c [ real part of phasor γ [ d [k] local measure of envelope and phasor SF amplitude spectra overlap n number of a discrete-time sample x [ real-valued discrete-time signal 1. INRODUCION his paper is a continuation of our previously published papers [1] and [] devoted to sound processing using complex dynamic representation. It is based on the Bedrosian theorem formulated in 1963 [3] see also [4], [5], [6], [7]. We apply here the Bedrosian theorem to the processing of real-life, audible signals, rather than synthesised. he Bedrosian theorem is used in a way generalised for all instants of time of the signal under processing. he theorem is not treated binary (satisfied or not satisfied) but a quantitative measure of the degree of its fulfilment is proposed. It concerns the condition of non-overlapping spectra: the envelope spectrum and phasor spectrum, in the AM FM signal model.. HE BEDROSIAN HEOREM he Bedrosian theorem states that the Hilbert transform of the product of two signals with non-overlapping spectra equals the product of the low-pass term by the Hilbert transform of the high-frequency term. In other words, only the high-frequency term is transformed. Let x[, n =, 1,, K be a real-valued discrete-time sound signal having the form x[ = a[ cosϕ[ (1) 7 EURASIP 11

2 15th European Signal Processing Conference (EUSIPCO 7), Poznan, Poland, September 3-7, 7, copyright by EURASIP where a[ is the instantaneous amplitude (real-valued envelope) and ϕ [ is the instantaneous phase of x[. his form governs all discrete-time signals having an arbitrary modulation pattern. It is an approach typical of and adopted from telecommunications. he right-hand side of (1) is the so-called AM FM representation of the signal x[, where a [ is the AM factor and cosϕ [ is the FM factor. he complex signal u [ = x[ + jh { x[ } = a[ exp( jϕ[ ) () where j = 1 and H { x[ } is the Hilbert transform of x[, we further call the Hilbertian signal (HS). Here the ideal H stands for the discrete Hilbert transformer (H) which is normally defined using its frequency response as given by j ω = j, π ω H ( e ) (3) j, ω π where ω = πf stands for the digital (normalised) angular frequency, F stands for the physical frequency in Hz, is the uniform sampling period and j is the imaginary unit. he instantaneous amplitude of u [ is a [ = u[ and { u[ } the instantaneous phase of u [ is ϕ[ = arg. he instantaneous angular frequency of u[ we define as the first backward derivative of ϕ [ in the following way = ϕ[], n = ω[ (4a) ϕ[ ϕ[ n 1], n > Assuming that there exists a relationship between the pitch of sound signal and its complex angular frequency, we can transpose (detune) the sound by rescaling its complex dynamic representation CDR: real-valued envelope λ[ = ln a[ and instantaneous angular frequency ω [. 3. HE CDR PROCESSING (4b) he CDR processing used in this paper and shown in Fig. 1 is based on the following mapping of the instantaneous level λ [ and instantaneous angular frequency ω [ λ[, ω[ λ [ = κλ[ + λ, ω [ κω[ (5) κ κ = Based on this relation we obtain a new CDR having the components: λ κ [ and ω κ [. his CDR modification, which constitutes the core-processing bloc results in pitch shifting of a given sound signal x[ with a pitch modification (scaling) factor κ >. As a result of the CDR scaling in accordance with (5) not only the pitch but also the instantaneous level λ [ responsible for audibility of the sound is changed. In order to counteract the latter, the maximal value of the instantaneous level change has to be compensated for by adding to κλ [ a correction term sated for by adding to κλ [ a correction term λ ( κ), computed by using λ ( κ (1 κ) ) = λmax where λ max stands for the maximal level of the primary (input) signal. κ x [ λ [ λ κ [ x κ [ CDR mapper CDR scaler CDR demapper (), (4a), (4b) (5) DDS ω [ ω κ [ Figure 1 Pitch-shifting of x[ by κ via CDR processing. Hence, in Fig. 1, firstly the input signal x[ is filtered by the complex Hilbert filter HF. Next the filtered signal is mapped into its CDR { λ, ω}. he CDR components are extracted using the IAE instantaneous amplitude estimator, ln( ) and IFE instantaneous frequency estimator blocks. Further on both CDR components are multiplied by the same coefficient κ having a positive value as above and the instantaneous level κλ[n ] undergoes the abovementioned correction by λ ( κ). Finally, after this remapping, performed as shown in (5), the new CDR, {λ κ, ωκ }, is demapped into the target pitch-shifted sound signal record { x κ } using the DDS direct digital synthesis see, for example, [1], [11]. he main processing block of the DDS is PA phase accumulator [1]. Driven by ω κ [ n ] the PA wraps the instantaneous phase to the interval ϕ p[ [ π, π ) n (principal phase wind). 4. COMMENS ON HE BEDROSIAN CONDIION We can split the HS u [ into two factors: a real-valued envelope a [ and complex phasor γ[ = exp( jϕ[ ). For scaling the CDR, thus pitch-shifting, the fulfilment of the condition that stems from the Bedrosian theorem is required that the spectra of real-valued envelope and complex phasor are non-overlapping (thus are separable). For such signals the following Bedrosian identity holds where H { a[ c[ } = a[ H { c[ } = a[ γ[ (6a) = Re{ γ[ } = cos( ϕ[ ]) c[ n (6b) is real-valued. he above-mentioned identity refers to global spectra [5]. It means that the theorem concerns the whole signal because it says about the envelope and phasor spectra. his limits the range of possible applications of the CDR processing. However, for a much wider class of signals local, instantaneous spectra are separable. We call such signals the Bedrosian chirps (always locally Bedrosian). In this class the 7 EURASIP 1

3 15th European Signal Processing Conference (EUSIPCO 7), Poznan, Poland, September 3-7, 7, copyright by EURASIP songs of many species of birds and voices of some species of mammals are involved. A[ Γ [ In the angular frequency domain the product a[ γ [ d [ = (1) on the right-hand side of (6a) corresponds to circular convolution A[ + Γ [ U ( e ) = A( e ) Γ( e ) a[ γ[ (7) where d stands for the distance (metrics). he global measure of the spectra: A ( e ) and Γ ( e ), of its factors: a [ and is defined here as given by γ [, respectively. Because u [ is the HS, the spectrum A[ Γ [ U ( e ) = for π ω. hus in general the spectra n= D = A( e ) and Γ( e ) resulting from de-convolution of A[ + Γ [ U ( e ) overlap. In other words these spectra at least partially n= n= (11) cover up. It means that the envelope a [ and phasor his formula constitutes our definition of the global degree of envelope and phasor amplitude spectra overlap. It is the γ [ may be non-orthogonal, i.e., they may be correlated ratio of mutual energy of each of factors under summation to linearly dependent. he product of spectra formulated in the the average of energies of each of factors. he smaller is D, following way the weaker is correlation between the envelope and phasor. A ( e ) Γ ( e ) { a[ } { γ [ } (8) Having at disposal a sufficiently long signal record we where the asterisk stands for linear convolution, confirms this thesis. he right-hand side of (8) represents the sequence of mutual correlation of the envelope and phasor. Separation of their spectra means that the product in (8) is of zero value. hus the correlation between phasor and envelope does not exist. can divide it into frames and examine by block-by-block technique, computing the D values for consecutive frames. Generally these values belong to the interval D, 1 >. he values of D close to inform about strong separability of envelope and phasor amplitude spectra overlap. Consequently one can deduce good quality of the transposed sound. Opposite, the values of D close to 1 enable to predict poor 5. HE BEDROSIAN CONDIION FOR CHIRPS quality of the transposition result obtained by CDR scaling. he Bedrosian condition for chirps [8], especially for audio chirps otherwise called the pitched sounds, can be formulated as follows. here exists such an angular frequency ω (, π ), for which c and A C ( e c c ) = for ω ( ω, ω ) ( e c c ) = for ω [ ω, ω ] (9a) (9b) where C ( e ) is the spectrum of c[ (6b). his can be used for processing of sounds, which in practice fulfil the conditions (9), thus are prone to the CDR scaling aimed at pitch-shifting. his disposition means good quality of the results of pitch-shifting for, e.g., many species of palatable birds songs. heir instantaneous spectra (SFs short-time Fourier transforms and spectrograms [9]) for Bedrosian chirps show visible separability. he ridges in the spectrograms are disjoined. 6. HE DEGREE OF ENVELOPE AND PHASOR SPECRUM OVERLAP Let A [ and Γ[, k =,1, K, be the absolute values of SFs of the envelope a [ and phasor γ [, both with n =, 1, K,, k stands for the bin (spectrum sample) number, K is the number of bins and N is the number of samples of the investigated record u [. We define the local (instantaneous) degree of the envelope a[ and phasor γ [ amplitude spectra overlap in the following way 7. RESULS OF EXPERIMENS WIH EXEMPLARY AUDIO CHIRPS We performed our experiments in the MALAB environment. Figs. and 3 show the time-frequency performance of two different audio chirps using processing as described in Sect. 3. In Fig. we present the spectrograms (from top to bottom): of a duck chirp, two spectrograms of its deconvolution into AM and FM factors, the spectrogram as a result of resynthesis of the of original chirp based on the abovementioned AM and FM factors and finally the spectrogram obtained by remapping the CDR after scaling by κ (for e.g. κ =1). he order of spectrograms in Fig. 3 repeats that from Fig. but it was canary, which voiced the chirp analysed in Fig. 3 (see Fig. 5). For the latter we observe in respective spectrograms that the AM and FM factors for canary practically do not overlap as needed. Comparing the results we see that these spectrograms differ in that for canary they have typical ridges (except that for the AM factor). Moreover, the canary chirp from Fig. 3 having the smaller D value (see able 1) is better remapped after scaling, as expected, than the duck chirp from Fig. for which the value of D is much bigger. Fig. 4 presents the amplitude spectra of those chirps the original and recovered from respective AM and FM factors. Its role is to convince the reader about invertibility of CDR processing. In order to compute the local and global measures (1) and (11) multiplication of respective spectrogram values was done. Some global results obtained for exemplary sound records the chirps taken from available Internet bases are presented in able 1 in %. 7 EURASIP 13

4 15th European Signal Processing Conference (EUSIPCO 7), Poznan, Poland, September 3-7, 7, copyright by EURASIP Figure Spectrograms of original duck.wav chirp, its AM and FM factors, and spectrograms of the chirp recovered from its AM and FM factors and remapped back after scaling by κ >. Figure 3 Spectrograms of original canary.wav chirp, its AM and FM factors, and spectrograms of the chirp recovered from its AM and FM factors and remapped back after scaling by κ >. 7 EURASIP 14

5 15th European Signal Processing Conference (EUSIPCO 7), Poznan, Poland, September 3-7, 7, copyright by EURASIP Figure 4 Spectra of original and resynthesised duck.wav and canary.wav chirps, respectively, recovered from their AM and FM factors. able 1. D the global degree of envelope and phasor amplitude spectra overlap No of file Name of file 1 Canary.wav Nightingale.wav Cookoo.wav Barn_owl.wav Duck.wav hank_you.wav Sorry.wav Figure 5 Analysed chirps: duck.wav and canary.wav. REFERENCES [1] E. Hermanowicz and M. Rojewski, Sound processing using complex dynamic representation, in Proc. of 5 European Signal Processing Conference EUSIPCO 5, D in % Hanning SF Antalya, urkey, September 5-8, 5. window length [] E. Hermanowicz and M. Rojewski: Pitch shifter based length on complex dynamic representation and direct digital synthesis, Bulletin of the Polish Academy of Sciences, echnical Sciences, vol. 54, No. 4, 6. [3] E. Bedrosian, A product theorem for Hilbert transforms, Proc. of the IEEE, vol. 51, pp , May [4] S.L. Hahn, Hilbert ransforms in Signal Processing, Artech House, [5] D. Vakman, Signals, Oscillations and Waves. A Modern Approach, Artech House, CONCLUSIONS [6] A.H. Nuttall, On the quadrature approximation to the In this paper we assumed that for a certain class of sounds there exists a relationship between their pitch and their complex dynamic representation CDR (real-valued logenvelope and instantaneous angular frequency). he CDR processing appeared a powerful means for pitch shifting, especially for chirp-like sound signals fulfilling practically the assumption of the Bedrosian theorem. For this aim a concept of a Bedrosian chirp was proposed whose local instantaneous amplitude and instantaneous complex phasor spectra are separated. A quantitative measure to estimate and predict the quality of sound processing using the CDR scaling was defined and its utility verified on the basis of experiments with real-life audio chirps. Scaling the CDR components can be used for entertainment as well as, e.g., in a generator of different melodies for calling signals, where the sound of each note in a melody is derived from a short recording of a voice of a chosen creature. Hilbert transform of modulated signals and E. Bedrosian reply, Proc. of the IEEE, vol. 54, pp , October [7] Y. Xu and D. Yan, he Bedrosian identity for the Hilbert transform of product function, Proceedings of American Mathematical Society, vol. 134, No 9, pp , September 6. [8] P. Flandrin, Chirps everywhere, CNRS Ecole Normale Superiere de Lyon. [9] S.K. Mitra, Digital Signal Processing. A Computer- Based Approach, McGrawHill, International Edition 1, Chapter 11. [1].F. Quatieri, Discrete-ime Speech Signal Processing. Principles and Practice. Prentice Hall PR,. [11] V. F. Kroupa (ed.), Direct Digital Frequency Synthesizers, IEEE Press, New Yor EURASIP 15