LP-BLIT: BANDLIMITED IMPULSE TRAIN SYNTHESIS OF LOWPASS-FILTERED WAVEFORMS

Transcription

1 Proceedngs of the 2 th Internatonal Conference on Dgtal Audo Effects (DAFx-7), Ednburgh, UK, September 5 9, 27 LP-BLIT: BANDLIMITED IMPULSE TRAIN SYNTHESIS OF LOWPASS-FILTERED WAVEFORMS Sebastan Kraft, Udo Zölzer Department of Sgnal Processng and Communcatons Helmut-Schmdt-Unversty Hamburg, Germany sebastan.kraft@hsu-hh.de ABSTRACT Usng bandlmted mpulse tran (BLIT) synthess, t s possble to generate waveforms wth a confgurable number of harmoncs wth an equal ampltude. In contrast to the snc-pulse, whch s typcally used for bandlmtng n BLIT and only allows to set the cutoff frequency, a Hammerch pulse can be tuned by two ndependent parameters for cutoff frequency and stop band roll-off. Replacng the perfect lowpass snc-pulse n BLIT wth a Hammerch pulse, t s possble to drectly synthesse a multtude of sgnals wth an adjustable lowpass spectrum.. INTRODUCTION Subtractve sound synthess wth analogue synthessers requres generc and spectrally rch harmonc waveforms. Tradtonally, these are rectangular, trangular and sawtooth waveforms. Ther flat spectrum s successvely shaped by flters untl t matches the expectatons of the muscan. These flters usually have a lowpass characterstc wth a varable cutoff frequency, adjustable resonance peak and a slope of 2 or 24 db per octave. One major problem of dgtal subtractve synthess s the fact that trval mplementatons of the requred basc waveforms leads to massve alasng. The creaton of bandlmted oscllators has been an actve research topc ever snce dgtal sound synthess became of nterest. An extensve summary and dscusson of varous methods can be found n [] and [2]. The latter also ntroduced the Poly- BLEP approach (detaled n [3]) whch today s a standard method to create hgh qualty bandlmted waveforms due to ts low computatonal cost and easy mplementaton. Subtractve synthess s a straghtforward approach when buldng analogue synthessers. Today, many dgtal synthessers mmc an analogue subtractve workflow, manly due to the fact that muscans have been used to t for decades and the use of flters to shape the sound s at the same tme ntutve, smple and versatle. However, for the performng muscan the exact synthess method does not matter. It s more mportant that the synthesser allows to create muscally soundng sgnals whch can be ntutvely controlled by only a few but powerful parameters [4]. From an engneerng pont of vew t does not make sense to put a lot of effort nto the generaton of alasng-free waveforms wth harmoncs up to half the samplng frequency and then to remove most of the hgh frequency content wth a lowpass flter. A method to drectly synthesse the desred sgnal spectra would allevate the effort for ant-alasng strateges as such a sgnal wll contan less hgh frequency content from the begnnng. Addtve synthess as a dscrete summaton of ampltude-weghted sne waves would offer full control to the creaton process of perfectly bandlmted sgnals but due to the resultng computatonal complexty t s rarely used n practce. The dscrete summaton formula (DSF) from Moorer [5] are a set of closed form solutons to dscrete harmonc seres (nfnte or fnte). They allow a drect synthess of harmonc sgnals wth a specfed number of partals and an exponentally ncreasng or decreasng spectral envelope. By the combnaton of dfferently parametrsed DSF one can create further complex spectral envelopes. Although the DSF are more effcent than a dscrete addtve synthess they stll requre a consderable amount of trgonometrc functon evaluatons. Bandlmted mpulse tran (BLIT) synthess [] s another approach to create lowpass sgnals and reles on the fact that an mpulse tran exhbts a flat spectrum wth an nfnte amount of harmoncs. When the mpulse tran s convolved wth a snc-functon, a perfect bandlmted sgnal wll be obtaned, whereas the frequency of the snc-pulse determnes the cutoff frequency. In practcal applcatons, the nfnte length snc-functon has to be wndowed and lmted to a reasonable length and the computatonally expensve convoluton s replaced by a summaton of overlappng snc-pulses wth a pulse dstance proportonal to the fundamental frequency (sum of wndowed snc-functon BLIT synthess []). A wndowed snc-functon wll loose ts perfect lowpass characterstc and stop-band rpple occurs. The selecton and parametrsaton of the wndow and ts length nfluences the fnal lowpass shape [6]. BLIT synthess was further developed and optmsed n varous aspects (e.g. n [7, 8]), but the created sgnals have been always used as a bandlmted nput for subtractve workflows. To the knowledge of the authors, a drect BLIT synthess of sgnals wth real-tme confgurable lowpass spectra has not been nvestgated so far. The Hammerch pulse [9] was ntroduced as a pulse shape flter [] for transmsson systems and ts spectral shape can be tuned by two ndependent parameters for cutoff frequency and stop band roll-off. Replacng the snc-pulses n BLIT wth Hammerch pulses allows to drectly synthesse sgnals wth adjustable lowpass spectra. The fundamental frequency, cutoff frequency and stop band roll-off are monotonc parameters and can be modulated smoothly. In a sound synthess applcaton, the few parameters and nherent lmtaton to lowpass spectra reduces complexty n the user nterface and offers the muscan an ntutve and less techncal access. Nevertheless, a wde varety of sounds smlar to subtractve synthessers can be created wthout addtonal flterng. Moreover, unque sounds can be generated for example by modulatng the flter roll-off whch would not be possble wth classcal analogue synthessers. The dea to create a versatle oscllator wth Hammerch pulse shapes was already brefly descrbed n []. In ths paper, the focus wll be on a detaled dscusson of such a lowpass bandlmted mpulse tran (LP-BLIT) oscllator n the context of sound synthe- DAFX-255

2 Proceedngs of the 2 th Internatonal Conference on Dgtal Audo Effects (DAFx-7), Ednburgh, UK, September 5 9, 27 ss applcatons. Throughout the followng Secton 2, the prncple of BLIT synthess and the ntegraton of the Hammerch pulse wll be explaned. In Secton 3 we wll gve some examples how to create more complex spectra by combnaton of several oscllator outputs, leaky ntegraton and parameter modulaton before Secton 4 provdes a short dscusson and concluson. hh(t).5 α =.5 α =.33 α =. 2. IMPULSE TRAIN SYNTHESIS A contnuous-tme pulse tran wth an nter-pulse dstance T d(t) = δ(t lt ) () exhbts a spectrum D(jω) = δ(ω lω ), ω = 2π T (2) wth harmoncs at multples of ω. The fact that the harmonc nterval ω s drectly related to T can be utlsed to synthesse a sgnal wth an nfnte amount of harmoncs and a fundamental frequency F = /T. Samplng of d(t) for dgtal mplementatons would lead to massve alasng due to ts nfnte bandwdth. Applyng an deal lowpass flter wth a cutoff frequency ω c = 2πf c h(t) = sn(ωct) ω ct results n a bandlmted sgnal ( ) ω H(jω) = rect 2ω c (3) = d(t) h(t) (4) whch can then be sampled at a sample rate f s > 2f c wthout alasng []. In the frequency doman, the convoluton from Eq. 4 wll create a harmonc spectrum X(jω) = H(jω) D(jω) ( ) = H(jω) δ(jω lω ) weghted wth the spectral shape H of the flter. By replacng or modfyng the flter, t s possble to synthesse any bandlmted harmonc sgnal wth an arbtrary spectral shape. Instead of a computatonally expensve tme-doman convoluton, a drect summaton of tme-shfted mpulse responses = d(t) h(t) = h(t lt ) (6) wll lead to an dentcal result. Ths s partcularly useful f the mpulse response can be smply calculated wth a closed-form equaton as t s the case wth the snc-functon. Other pulses wth a lowpass characterstc, as for example the rased cosne pulse, offer a more detaled adjustment of ther frequency response than the snc-functon. The Hammerch pulse was ntroduced n [] as a pulse shape flter for transmsson systems. Its mpulse and frequency response s gven by h H(t) = α sn(ωct) snh(α ω ct) H H(jω) = [ ( )] ω ωc tanh 4f c 4αf c (5) (7) (8) HH(f) n db t f c (a) Tme doman f/f c (b) Frequency doman α =.5 α =.33 α =. Fgure : Hammerch pulse for varous values of α. and permts an ntutve adjustment of the lowpass characterstc by two ndependent parameters for cutoff frequency ω c and stop band slope α. Reasonable parameter ranges are < α < and ω c > 2πF. For small values of α, the mpulse response sn(ωct) lm hh(t) = = snc(ω α ct) (9) ω ct converges towards a snc-functon and for α the resultng stop band slope as well as the pulse wdth converges to zero. Exemplary mpulse and frequency responses are depcted n Fg. for selected values of α. The -6 db pont s farly accurate set by ω c, whereas the lnear slope beyond ths pont s only controlled byα. Fgure 2 depcts a tme-doman pulse tran after convoluton wth a Hammerch mpulse response together wth the correspondng spectrum. The parameters of the Hammerch flter were chosen as f c = 4F and α =.4. All harmoncs follow the shape of the pulse spectrum as t was expected based on Eq Dscrete-tme mplementaton wth fnte pulse length By samplng the Hammerch pulse from Eq. 7 wth a sample rate f s, the dscrete-tme pulse h H(n) = α sn(ωcn) snh(α Ω cn), Ωc = 2πfc, () f s s obtaned. The cutoff frequency f c = N h F can also be expressed as a multple of the fundamental frequency, whereas N h determnes the number of harmoncs before the flter starts to roll off. Ths yelds a pulse h H(n) = α sn(n h Ω n) snh(α N h Ω n), Ω = 2πF, () f s DAFX-256

3 Proceedngs of the 2 th Internatonal Conference on Dgtal Audo Effects (DAFx-7), Ednburgh, UK, September 5 9, 27.5 d(t) h H (t) 5 5 Sngle Double X(f) n db t/t (a) Tme doman f/f (b) Frequency doman Fgure 2: Pulse tran convolved wth a Hammerch pulse (f c = 4F and α =.4). wth parameters for the number of harmoncs, flter slope and fundamental frequency beng drectly accessble from a synthesser applcaton. Two facts have to be consdered for a dscrete-tme mplementaton compared to the contnuous-tme dervaton n the prevous secton. Frst, the theoretcally nfnte length of the pulse has to be lmted n order to avod an nfnte amount of overlappng pulses n the sum from Eq. 6. Ths lmtaton of the mpulse length s equvalent to a wndowng of the mpulse response and leads to a dstorton of the pulse spectrum. The shape of the resultng error can be optmsed by applyng a smooth wndow to both ends of the pulse [6]. The number of overlappng pulses s a trade-off between computatonal complexty and how close the actual spectral envelope wll match the theoretc spectrum gven n Eq. 8. Second, even f the cutoff frequency of the Hammerch pulse s below f s/2, alasng may occur due to the flat spectral roll-off after f c dependng on the actual selecton of α. Hence, t s necessary to lmt the combnaton of the parameters N h and α such that the resultng stop band attenuaton n Eq. 8 falls below an acceptable level at half the sample rate. Regardng computatonal complexty, the calculaton of the sne and hyperbolc sne n the Hammerch pulse s the lmtng factor. Both functons could be approxmated by the Taylor seres sn(x) = x x3 3! + x5 K 5! +... = k= ( ) k x 2k+ (2k +)! snh(x) = x+ x3 3! + x5 K 5! +... = x 2k+ (2k +)! k= (2) (3) where the number of evaluated terms K determnes the accuracy. For the perodc sne, the argument has to be wrapped to a range x [ π/2,π/2] to mnmze the requred order of the Taylor Number of Taylor termsk (a) Maxmum error Number of Taylor termsk (b) Power normalsed MSE Fgure 3: Error after the approxmaton of h H(n) wth K Taylor seres terms for sngle and double precson floatng pont mplementatons. polynomal. The nverse factoral can be calculated n advance and usng the Horner scheme for the evaluaton of the polynomal, only K addtons and multplcatons are requred for each Taylor seres. The standard Matlab sn and snh mplementatons and the sngle and double precson Taylor approxmatons were compared for the calculaton of a Hammerch pulse. The respectve maxmum error as well as the power normalsed mean square error are shown n Fg 3. Based on these results t appears that 7 terms are already suffcent to acheve an error below - db and for more than terms the numercal resoluton lmt of sngle precson floatng pont numbers wll be reached. 3. SOUND SYNTHESIS EXAMPLES 3.. Combnaton of oscllators Recursve flters are usually used n subtractve synthess due to computatonal constrants but wll lead to a frequency-dependent phase shft of each harmonc and a predctable combnaton of several oscllators wthout unwanted partal cancellatons s dffcult. The pulse shapng n BLIT corresponds to a lnear-phase FIR lowpass flterng, hence all harmoncs are stll n phase after the flterng and t s straghtforward to add or subtract multple oscllator outputs n a predctable manner to create more complex spectra. Let us defne a sgnal x (n) that has a fundamental frequency F = F whch s an nteger multple of another sgnalx(n) but both share the same spectral envelope. In ths case, the dfference between x(n) and x (n) x D(n) = x(n) x(n) = d(n) h d(n) H(n) [ = d(n) d(n) h H(n) ] h H(n) (4) yelds a sgnal where every -th harmonc s cancelled. For = 2, whch s equvalent to usng a bpolar mpulse tran as source sgnal [], a sgnal x O(n) wth only odd harmoncs remans (Fg. 4 b). DAFX-257

4 Proceedngs of the 2 th Internatonal Conference on Dgtal Audo Effects (DAFx-7), Ednburgh, UK, September 5 9, t/t f/f X(f) n db xsaw(t) t/t f/f Xsaw(f) n db (a) (a) xo(t) t/t f/f XO(f) n db xrect(t) t/t f/f Xrect(f) n db (b) (b) xe(t) t/t f/f XE(f) n db xtr(t) t/t f/f Xtr(f) n db (c) (c) Fgure 4: Unpolar pulse tran wth full number of harmoncs (a), bpolar pulse tran wth only odd harmoncs (b) and sgnal wth even harmoncs (c) as a sum of two dfferent unpolar pulse trans. Usng Hammerch pulses wth dfferent parameters for each oscllator offers further possbltes. A sgnal wth even harmoncs x E(n) = x (n)+ d2(n) 2 h H(n), (5) as depcted n Fg. 4 b), can be constructed from the sum of one pulse tran d 2(n) wth twce the fundamental frequency and arbtrary flter and another sngle harmonc sgnal x (n) and fundamental frequency F Standard waveforms It was shown n [] that the standard waveform (rectangular, sawtooth and trangular) can be created wth a smple ntegraton of mpulse tran sgnals. In our case, a lowpass fltered sawtooth x saw(n) = x(n) h I(n) (6) s obtaned by convolvng a bandlmted mpulse tran sgnalx(n) wth an ntegrator mpulse responseh I(n). To avod accumulaton of an error constant n the ntegraton, t s usually recommended to use a leaky ntegrator. A second order leaky ntegrator wth zero DC gan was proposed by [2] H I(z) = π γ + 2 ( z ) 2γz +γ 2 z 2 (7) Fgure 5: Sawtooth (a), rectangular (b) and trangular waveforms (c) created by a combnaton of oscllators and leaky ntegraton. and conssts of a cascaded frst order leaky ntegrator and a onepole hghpass. The parameterγ = exp(2πf ci /f s) defnes the cutoff frequency of the hghpass (typcallyf ci < 2 Hz) and thereby the crossover pont between leaky and non-leaky ntegraton. The rectangular waveform x rect(n) = x O(n) h I(n) (8) s obtaned by leaky ntegraton of a sgnal wth only odd harmoncs. Fnally, the trangular sgnal x tr(n) = x rect(n) h I(n) = x O(n) h I(n) h I(n) (9) s an ntegrated rectangular sgnal, or two-tmes ntegrated sgnal wth odd harmoncs. Fgure 5 depcts exemplary bandlmted sawtooth, rectangular and trangular sgnals whch were obtaned by leaky ntegraton Modulaton All pulse parameters can be drectly modulated n a sound synthess applcaton. Fgure 6 a) vsualses a fundamental frequency sweep rangng from2 Hz up to7 khz wthn H = 5 andα =.8 at a sample rate of 44. khz. Alasng s kept at a low level by constantly checkng and lmtng the parameters N H and α n dependency of the current fundamental frequency. A step-wse modulaton of the number of harmoncs s shown n Fg. 6 b). DAFX-258

5 Proceedngs of the 2 th Internatonal Conference on Dgtal Audo Effects (DAFx-7), Ednburgh, UK, September 5 9, ACKNOWLEDGMENTS Frequency n khz Magntude n db The authors would lke to thank the anonymous revewers for ther extensve and valuable feedback whch helped to mprove ths paper n many aspects. 6. REFERENCES Frequency n khz Tme n seconds (a) Sweep from 2 Hz to 7 khz wth N H = 5 and α = Tme n seconds (b) Step-modulated number of harmoncs (N H = 3,5,9,3). Fgure 6: Example spectrograms showng a modulaton of the fundamental frequency and number of harmoncs. 4. CONCLUSION Usually, n bandlmted mpulse tran (BLIT) synthess, snc-pulses are used to flter a pulse tran and to obtan a spectrum wth a defned number of harmoncs of equal magntude. In ths paper t was proposed to replace the snc-pulse wth a Hammerch pulse as ts spectral shape can be drectly controlled by two ndependent parameters for cutoff frequency and flter roll-off. The closed form equaton for the Hammerch pulse can be evaluated per sample, does not requre the creaton of a wavetable and an mmedate modulaton of the pulse parameters s possble. As all harmoncs n a pulse are n phase, dfferently confgured oscllators can be easly combned to create more complex spectral shapes. It was shown how to synthesse spectra wth odd or even harmoncs and together wth a leaky ntegrator, varous standard waveforms (rectangular, trangular, sawtooth) can be created. The Hammerch pulse consderably expands the BLIT prncple to become a fullfeatured synthess procedure and despte the restrcton to lowpass spectra, a wde varety of useful sounds and waveforms can be created wthout addtonal flterng. The possbltes and lmtatons of ths new waveform generaton algorthm stll have to be explored n practcal muscal applcatons. Frst tests wth a real-tme modulaton of the pulse parameters were qute promsng. In partcular the smple nterface wth only a few but very expressve parameters supports an ntutve and creatve workflow. For the future t mght be n partcular nterestng to fnd further pulse shapes whch can be calculated and parametrsed n a smlar fashon as the Hammerch pulse but exhbt a dfferent frequency response, e.g. hghpass, bandpass or resonant lowpass. 4 2 Magntude n db [] Tm Stlson and Julus Smth, Alas-free dgtal synthess of classc analog waveforms, n Proc. of the Int. Computer Musc Conference (ICMC), 996. [2] Vesa Välmäk and Antt Huovlanen, Antalasng oscllators n subtractve synthess, IEEE Sgnal Processng Magazne, vol. 24, no. 2, pp. 6 25, 27. [3] Vesa Välmaäk, Juss Pekonen, and Juhan Nam, Perceptually nformed synthess of bandlmted classcal waveforms usng ntegrated polynomal nterpolaton, The Journal of the Acoustcal Socety of Amerca, vol. 3, no., pp. 974, 22. [4] John Lazzaro and John Wawrzynek, Subtractve Synthess wthout Flters, n Audo Anecdotes II - Tools, Tps, and Technques for Dgtal Audo, pp [5] James A. Moorer, The Synthess of Complex Audo Spectra by Means of Dscrete Summaton Formulas, Journal of the Audo Engneerng Socety, vol. 24, no. 9, pp , 976. [6] Juss Pekonen, Juhan Nam, Julus O. Smth, Jonathan S. Abel, and Vesa Välmäk, On Mnmzng the Look-Up Table Sze n Quas-Bandlmted Classcal Waveform Oscllators, n Proc. of the 3th Int. Conference on Dgtal Audo Effects, 2. [7] Juhan Nam, Jonathan S. Abel, and Julus O. Smth, Effcent Antalasng Oscllator Algorthms Usng Low-Order Fractonal Delay Flters, IEEE Transactons on Audo, Speech and Language Processng, vol. 8, no. 4, pp , 2. [8] Stéphan Tassart, Band-lmted mpulse tran generaton usng sampled nfnte mpulse responses of analog flters, IEEE Transactons on Audo, Speech and Language Processng, vol. 2, no. 3, pp , 23. [9] Edwn Hammerch, A Generalzed Samplng Theorem for Frequency Localzed Sgnals, Samplng Theory n Sgnal and Image Processng, vol. 8, no. 2, pp , 27. [] Edwn Hammerch, Desgn of Pulse Shapes and Dgtal Flters Based on Gaussan Functons, 29. [] Udo Zölzer, Ptch-based dgtal audo effects, n Proc. of the 5th Int. Symposum on Communcatons, Control and Sgnal Processng (ISCCSP), 22. [2] El Brandt, Hard Sync wthout Alasng, n Proc. of the Int. Computer Musc Conference (ICMC), 2. DAFX-259