IMPROVED POLYNOMIAL TRANSITION REGIONS ALGORITHM FOR ALIAS-SUPPRESSED SIGNAL SYNTHESIS

Transcription

1 IMPROVED POLYNOMIAL TRANSITION REGIONS ALGORITHM FOR ALIAS-SUPPRESSED SIGNAL SYNTHESIS Dániel Ambrits and Balázs Bank Budaest University of Technology and Economics, Det. of Measurement and Information Systems, H-52 Budaest, Hungary ABSTRACT One of the building blocks of virtual analog synthesizers is the oscillator algorithm roducing simle geometric waveforms, such as saw or triangle. An imortant requirement for such a digital oscillator is that its sectrum is similar to that of the analog waveform, that is, the heavy aliasing that would result from a trivial modulo-counter based imlementation is reduced. Until now, the comutationally most efficient oscillator algorithm with reduced aliasing was the Polynomial Transition Regions (PTR) method. This aer shows that the efficiency can be increased even further by eliminating the hase offset of the PTR method. The new Efficient PTR (EPTR) algorithm roduces the same outut as the PTR method, while requires roughly 3% less oerations, making it the most efficient alias-reduced oscillator algorithm u to date. In addition to resenting an EPTR sawtooth algorithm, the aer extends the differentiated arabolic wave (DPW) triangle algorithm to the case of asymmetric triangle waves, followed by an EPTR imlementation. The new algorithm rovides continuous transition between triangle and sawtooth signals, while still requires low comutational ower.. INTRODUCTION Analog synthesizers roduced in the 6s and 7s are still very oular among musicians for their characteristic timbre, and the sound of these classic synthesizers has become an inherent art of many modern musical genres. However, the original synthesizers are hard to find, exensive, and usually do not rovide sufficient control (e.g., via MIDI) as required by today s musicians. Therefore, some comanies rovide modern analog synthesizers with digital control, but an even more cost-effective solution is to simulate the analog signal chain via digital signal rocessing. The first such synthesizer was the Clavia NordLead, which aved the way for virtual analog synthesis. For an excellent overview on related research, see []. In an analog synthesizer the signal flow starts with an oscillator generating geometric waveforms, such as square, sawtooth, triangle, sine, and sometimes a noise generator Coyright: c 23 Dániel Ambrits et al. This is an oen-access article distributed under the terms of the Creative Commons Attribution 3. Unorted License, which ermits unrestricted use, distribution, and reroduction in any medium, rovided the original author and source are credited x x x 4 Figure. Sectrum of the ideal, the trivial and (c) the DPW sawtooth signals. The dashed line is the enveloe of the ideal sectrum. is also rovided. Then this signal is fed to a filter that is controlled by enveloe generators, low frequency oscillators (LFO), etc., and finally its gain is adjusted by an amlifier, again controlled by an enveloe and LFO. This aer concentrates on the first art, that is, the digital modeling of the oscillator. The trivial otion for creating a digital relica of a geometric signal is to generate samles that corresond to the samling of the analog waveform. In the case of the sawtooth signal, this results in a simle modulo counter, which can be imlemented very effectively. However, as exected, this results in a heavy aliasing due to the nonbandlimited nature of the analog signal from which it originates [2]. This is dislayed in Fig., together with the sectrum of the ideal (analog) sawtooth in Fig.. A general remedy to the roblem of aliasing is oversamling, that is, running the modulo counter at a significantly higher samling rate, and then decimating. However, this leads to a considerable increase of comutational comlexity. Therefore, secial algorithms have been develoed that reduce the aliased comonents while still kee the comutational requirements low. Note that it is not required to eliminate aliasing comletely, since aliased comonents below a certain level are inaudible due to masking effects [3]. (c)

2 The aroaches include waveform generation based on a band-limited imulse train [4, 5] and band-limited ste function [2, 6, 7], and the distortion and filtering of sine waves [8]. The simlest, yet still ractically usable method is the differentiated arabolic wave (DPW) algorithm [9], that is based on the sectral tilt modification of the continuous-time signal before samling. Later the higher-order extension of the method has also been resented [], roviding better alias suression at the exense of larger comlexity. By noting that the DPW algorithm modifies only the samles around the discontinuity of the analog signal, a more efficient imlementation is ossible. This algorithm is called Polynomial Transition Regions (PTR), and is based on recomuting correction olynomials for the samles in the transition, while the linear regions of the signal are offset by a constant value []. This aer resents an even more efficient version of the PTR algorithm, which will be called EPTR throughout the aer. The method is based on the fact that the offset of DPW and PTR waveforms comared to the trivial (modulocounter generated) waveform is due to a hase shift of the DPW and PTR signals. When this hase shift is removed, the linear regions of the waveform can be taken simly as the trivial waveform values, eliminating the need for an extra addition. For the sawtooth signal this leads to the reduction of the number of oerations by around 3%. By modulating the ulse width of a square wave, very interesting sonic variations can be created. Accordingly, many classic and virtual analog synthesizers offer this kind of PWM signal. A similarly interesting effect can be a- chieved by modulating the symmetry of triangle waves. This way the triangle signal can be continuously transformed into a sawtooth waveform. Two of the rare examles generating triangle waves with variable symmmetry are the Moog Little Phatty and Sub Phatty analog synthesizers [2]. This aer first extends the DPW algorithm for the case of asymmetric triangle waves, then rovides a highly efficient imlementation by the use of the new EPTR algorithm. The rest of this aer is organized as follows. Section 2 reviews the DPW algorithm for the case of the sawtooth signal and rovides an extension for the case of the asymmetric triangle wave. This is followed by the basic idea of the PTR method in Sec. 3, while Sec. 4 rooses the new EPTR algorithm for the saw and asymmetric triangle waves. Finally, Sec. 5 comares the comutational comlexity of the DPW, PTR and the EPTR algorithms. 2. DIFFERENTIATED POLYNOMIAL WAVEFORM ALGORITHM First, let us consider the stes of generating an alias-suressed signal with the Differentiated Polynomial Waveform (DPW) algorithm. In the Nth-order method the continuous signal is integrated N times. This is equivalent to rocessing the samled signal with an N th-order olynomial waveshaer []. As a result, the sectrum of the sawtooth signal decreases by 6N db er octave instead of 6 db. This way the aliasing is significantly reduced. Then the (d) Figure 2. Generation of the triangle signal with variable symmetry. The trivial signal is first rocessed with the x 2 function. The result is then multilied with (c) a scaled rectangular wave to get (d) the arabolic waveform. Finally differentiating and scaling roduce (e) the desired alias-suressed waveform. signal is differentiated N times in the discrete-time domain to restore its sectral tilt, which means filtering with the ( z ) N transfer function. Finally it is scaled to the desired amlitude []. For N = 2 the integral of a iecewise linear signal is a iecewise arabolic (x 2 /2) function. The trivial sawtooth signal corresonds to a eriodic counter ranging from to. The alias-suressed sawtooth signal can be generated by squaring the signal, then differentiating the resulted iecewise olynomial waveform and finally scaling by a sufficient value [9]. While only symmetrical triangle wave is considered in the literature [], it can be easily extended to asymmetric case. The method is exlained by using Fig. 2. The gradient of the ascending region is A >, then the gradient of the descending region is B = A/(A ) <. Following the generation of the symmetric triangle signal for N = 2 [], the trivial waveform Fig. 2 is first rocessed with the x 2 function, giving Fig. 2. (In the general case, when the value of the eak is not, the waveshaer would be x 2 2 eak.) Then it is multilied with a rectangular waveform with /A duty cycle so that the arabolic regions of the signal are alternately ositive and negative. This rectangular wave is generated accord- (c) (e)

3 ing to the counting direction of the trivial signal which, for the symmetric triangle holds the value when the trivial waveform is ascending and when it is descending. Note that by using this ± rectangular wave the absolute values of the eaks are in both regions but the width of these arabolic regions are not the same. Thus at the transition of two successive regions the gradients are different and this would cause jums in the differentiated signal. This roblem can be solved by scaling the regions with / A and / B factors so that the transition is smooth (see Fig. 2(c) and (d)). This ste is the only difference between the symmetric and asymmetric case. Therefore the waveshaers are (x 2 )/A for the ascending region and (x 2 )/B for the descending region. Then the olynomial waveform is differentiated and multilied by a scaling factor. 3. POLYNOMIAL TRANSITION REGIONS ALGORITHM The signal generated with the N th-order DPW algorithm differs from the trivial waveform by only N samles er eriod. The differing samles are in the transition region, the linear sections only have an offset. This means unnecessary additional comutation, since the integration and differentiation is comuted even for the linear regions. The Polynomial Transition Regions algorithm was introduced to decrease the comutation cost based on this observation. In the PTR method the samle values are derived in a closed form for each section, and the final signal is generated from the trivial signal using these general forms []. To show that the comutational cost can be reduced even further, the linear section is discussed for N = 2. Two successive samles in the linear section are [n ] = 2AT and [n] =, where is the current value of the trivial signal generator, A is the gradient of the section. (A =, when the signal increases from to during one eriod. If A <, the signal decreases.) T = f /f s, where f is the fundamental frequency of the signal and f s is the samling frequency. For N = 2 the waveshaer is x 2 according to the DPW algorithm, then the differentiation and scaling leads to: y[n] = ([n])2 ([n ]) 2 = AT. () For the linear section an addition oeration is required for each samle. The value of the offset can be both ositive and negative deending on whether the signal is ascending or descending, thus also a branch oeration is needed. The AT offset reresents a half samle delay from the trivial generator as seen in Fig. 3. This delay comes from the behavior of the discrete differentiation. We will see in the next section that a more efficient algorithm can be derived by eliminating this half samle delay, and so the need for the addition oeration. [n-] [n-.5] [n] y[n] -AT [n-.5] [n+.5] [n]=y[n] Figure 3. Discrete-time differentiation using the trivial signal (dark dots) and the waveform shifted with half samle (ligth dots). 4. EFFICIENT POLYNOMIAL TRANSITION REGION ALGORITHM 4. Eliminating the Half Samle Delay in the Linear Region The simlest way to avoid the unnecessary comutation in the linear region is using the trivial generator as the outut. This is equivalent to using the differentiation on the adjacent samles which are at half samle distance from the origin as seen in Fig. 3. The [n.5] and [n +.5] are the values of the continuous signal halfway between the samles. y[n] = ([n +.5])2 [n.5]) 2 ) = = ( + AT ) 2 ( AT ) 2 =. (2) In other words, the PTR algorithm is alied on a trivial signal which is with half samle in advance to the desired signal. Therefore also the samles in the transition region should be calculated from this shifted trivial waveform. However, this does not require that these samles are known during the wave generation, an exlicit form of the correction can be calculated in advance. The transition region is a one samling time wide section. The shifted trivial signal causes that the samles to be corrected can be found before or after the break. So unlike in the PTR algorithm where this section was the [,] samle interval after the discontinuity, here it can be found in the [-.5,.5] interval with the transition in the centre. When we detect that the trivial signal generator is in this region, the osition of the samle must be insected and the correction must be alied according to the result. In the next section various tyes of transitions are derived for N = Sawtooth 4.2. Derivation of the sawtooth wave generation The continuous signal with A gradient jums from the value max to min as can be seen in Fig. 4. When the samle of the discrete trivial waveform is [n] = before the discontinuity, the next samle is [n + ] = + 2AT ( max min ). These samles are rocessed by the x 2 waveshaer. The correction deends on whether the samle to be corrected is before or after the discontinuity of the continuous waveform.

4 AT [n-] transition region [n-.5] [n]= [n+.5] [n+] max Figure 4. Correction of the sawtooth signal. During the derivation the half-samle delayed signal (light dots) is used instead of the trivial signal (dark dots). [n] is corrected to the desired value (dashed dot).. The samle is before the discontinuity ( > max AT ), as in Fig. 4 According to Section 4. the calculation must be alied using the adjacent samles at half samle distance from the outut samle. The trivial samle has a value of [n] =, as seen in Fig. 4. The revious samle [n.5] = AT, the next samle would be +AT but it is higher than the maximum value, so [n +.5] = + AT ( max min ). Alying the DPW algorithm leads to the desired outut: min y A [n] = ([n +.5])2 ([n.5]) 2 = = ( + AT + min max ) 2 ( AT ) 2. (3) For a sawtooth ranging from to, we have A =, max =, and min =. In this secial case the calculations lead to y[n] = T + T. (4) 2. The samle is after the discontinuity ( < min +AT ) The next samle is [n +.5] = + AT. The revious samle can be found before the discontinuity (since AT < min ), so it has a value of [n.5] = AT + ( max min ). y B [n] = ([n +.5])2 ([n.5]) 2 = ( + AT ) 2 ( AT + max min ) 2. (5) For the usual sawtooth signal A =, max = and min =. In this secial case the calculations lead to y[n] = T T +. (6) AT AT [n-] transition region [n-.5] y[n] [n] [n+.5] Figure 5. EPTR algorithm of the sawtooth signal The EPTR sawtooth wave algorithm max The PTR algorithm assumes that the trivial sawtooth signal is given, that is, the trivial signal generation and correcting the samles in the transition region are handled searately []. If we were doing the same, then two branch oerations would be required: one for detecting the jum of the trivial signal and one for finding the transition region. However, a comutationally simler algorithm can be realized by merging the trivial signal generation and samle correction. A further advantage of this choice is that since the trivial signal is generated by us, we are able to run the trivial counter even over max without forcing it to jum and aly the correction using that value. Indeed, substituting + 2 into (6) leads to (4). Therefore, there is no need to check whether the samle to be corrected is before or after the discontinuity and the two cases can be handled in the same way. When the transition region is detected, the corrected outut samle is comuted, and then the trivial signal jums, while the relative osition of the samle comared to the transition is irrelevant. This is shown in Fig. 5. The next source code shows how the algorithm can be rogrammed to generate a sawtooth signal ranging from - to. = + 2 T ; i f > T y = c o r r e c t ( ) ; = 2 ; e l s e y = ; The function correct() is resonsible for correcting the samle in the transition region. For the usual ± sawtooth waveform we simly use (4): correct() = T + T min. (7) For the general case, see Table. Note that the result would be the same if the trivial signal first jumed, then (6) was alied for correction. The resulting waveform is equivalent to the signal generated

5 region A, max, min (general case) A =, max =, min = linear region correct() + min max 2AT + min max 2 + (min max)2 T + T Table. Correction functions for the EPTR sawtooth algorithm. transition region AT AT max x x x 4 Figure 6. The DPW and EPTR sawtooth waveforms and their difference, and the sectrum of the signal generated with the DPW (c) and EPTR (d) algorithms. by the DPW and PTR algorithms as seen in Fig. 6. (The starting hase of the counter was offset by a half samle for the EPTR algorithm so that the two curves match erfectly). 4.3 Triangle 4.3. Derivation of the asymmetric triangle wave generation First let us consider comuting the maximum eak of the triangle signal. Due to symmetry, the minimum eak can be calculated similarly. Around the maximum eak the trivial signal ascends to the value max with gradient A, then it descents with gradient B. When the trivial signal has a value of [n] = before the eak, after its value is [n + ] = max + 2BT ( max + 2AT ) B/A. The waveshaers are (x 2 2 max)/a for the ascending and (x 2 2 max)/b for the descending regions, as discussed (c) (d) BT BT transition region Figure 7. EPTR algorithm of the triangle signal. in Section 2. Similarly to generating the sawtooth signal, the two adjacent values of the continuous signal are used which are at half samle distance from the outut samle. When the samle to be corrected is before the maximum eak ( > max AT ), the two adjacent values used are [n.5] = AT and [n +.5] = max + BT ( max ) B/A. The differentiation and scaling gives (([n +.5]) 2 2 max)/b (([n.5]) 2 2 max)/a = 4T = a a + a (8) which is included as the correctmax function in Table 2. Similarly, when the samle is after the eak ( > max + BT ), the used value to the right is [n +.5] = + BT and oint to the left is still before the eak, so its value is [.5] = max + AT ( max + 2BT ) A/B. However, if we aly the same trick as in Sec , that is, we are merging the trivial signal generation and samle correction, we are able to run the trivial counter above max, and one function (8) can handle both cases. The derivation is similar for the minimum eak. After determining the adjacent values, the differentiation and scaling gives (([n +.5]) 2 2 min )/A (([n.5])2 2 min )/B = 4T = b b + b (9) which is included as the correctmin function in Table 2. It is ossible that a high gradient section fits between two samles. The condition for this case is that A min

6 Linear region correctmax() a a + a correctmin() b b + b 5 Coefficient General Secial B A a 2 4A 2 T AT (A+B)+ a max(a B) 2A 2 T (B A)(AT a max) 2 4A 2 T A B b 2 4B 2 T BT (B+A)+ b min(b A) 2B 2 T (A B)(BT b min) 2 4B 2 T 4(A)T 2AT 4T +2 4(A)T (AT ) 2 4T (A) 4(B+)T 2BT +4T 2 4(B+)T (BT +) 2 4T (B+) Table 2. The olynomial correcting functions for the EPTR asymmetric traingle algorithm. In the secial case B = A/(A ), max =, min = Figure 8. The PTR (gray) and the EPTR (black) asymmetric triangle waveforms. /T = f s /f. This is not equivalent to the sawtooth signal in which the gradient is infinite, thus this case should be handled searately. When the trivial signal has a value of [n] = before the maximum eak, after it the value is [n + ] = + min ( A/B) max ( A/B). Since both of the adjacent samles are on linear sections with the same gradient, the calculation can be erformed similarly to the sawtooth waveform The EPTR asymmetric triangle wave algorithm Figure 7 exlains the algorithm for generating an asymmetric triangle signal. Similarly to the sawtooth signal, the trivial waveform generation and the corrections are merged, thus checking whether the trivial generator is in the transition region is sufficient. The next code segment shows the imlementation of the algorithm for a triangle waveform ranging from - to, with variable symmetry. i f d i r == / / c o u n t i n g u? = + 2 A T ; i f > A T / / t r a n s i t i o n r e g i o n? y = c o r r e c t M a x ( ) ; = + ( ) B /A; d i r = ; e l s e / / l i n e a r r e g i o n y = ; e l s e / / c o u n t i n g down = + 2 B T ; i f < B T / / t r a n s i t i o n r e g i o n? x x x 4 Figure 9. The sectrum of the trivial, the DPW and (c) the EPTR asymmetric triangle signal with 25% symmetry. y = c o r r e c t M i n ( ) ; = + ( + ) A/ B ; d i r = ; e l s e / / l i n e a r r e g i o n y = ; The correcting functions can be found in Table 2. First the counting direction must be determined, then the value of is checked. When is in the transition region, the corrected outut samle is comuted, the trivial counter is udated, and finally the counting direction is changed. If it is not in the transition region, the outut simly equals the trivial counter y =. The generated signal is equivalent to the DPW and PTR versions (see Fig. 8 and 9). The revious code assumed that the values of A and B are not higher than f s /f, so there is no region that fits between two samles. Although it is also ossible to imlement triangle waveforms with high gradient as discussed at the end of Sec. 4.3., it would result in a significantly more comlicated algorithm. The allowed highest gradient case is close enough to the sawtooth waveform, therefore imlementing the extra oerations is not rewarding. Figure shows the sectrum of a sawtooth with an infinitely shar transition and with a transition that lasts one samling instant. The only drawback of limiting the gradient is a slight attenuation at high frequencies, on the other hand, the aliasing is reduced, since now we are correcting two samles around the transition. So the asymmetric triangle with a one samle-time transition can be safely used instead of the sawtooth signal. However, if there is still a need for the secial case, the algorithm can be develoed according to Sec COMPARISON The advantage of the EPTR method over PTR is the reduced comutational load. For roviding a fair comarison we merged the trivial signal generation and the correction (c)

7 x x 4 Figure. Sectrum of the triangle signal with the highest allowed asymmetry and the sawtooth signal. The dashed line is the enveloe of the ideal sectrum. Sawtooth Add Mul Branch Total DPW PTR 2 T 3 + T EPTR + T T 2 + 2T Triangle Add Mul Branch Total DPW PTR 2 + 2T 6T T EPTR + 2T 6T T Table 3. Comutational load of DPW, PTR and EPTR for sawtooth and triangle signals (T = f /f s ). also for the PTR algorithm, although [] and the adherent source codes [3] were handling them searately. Table 3 comares the two algorithms in oerations er samle while generating sawtooth and asymmetric triangle waveforms. The PTR algorithm uses an addition oeration to increment the trivial counter. Then with a branch oeration it decides whether the current samle is in the linear or the transition region. Finally, in the linear region an addition for the offset is alied and in the transition region a multilication and an addition is necessary. When roducing a sawtooth waveform with the EPTR algorithm, using the shifted trivial signal eliminates the addition oeration in the linear region. Similarly, only the addition oeration of the trivial counter is needed in the linear region during the asymmetric triangle signal generation. The resulting waveforms generated with the two algorithms have the same sectrum, as we have seen in Sec CONCLUSIONS This aer has roosed a new version of the PTR algorithm. The Efficient Polynomial Transition Regions Algorithm requires around 3% lower number of oerations comared to the PTR algorithm, while results in exactly the same waveform as that of the DPW and PTR algorithms. Thus, it is the most efficient alias-reduced algorithm u to date, making it an ideal choice for systems with low comutational ower requirements. In addition, the aer has extended the DPW algorithm for generating triangle waves with variable symmetry, and its EPTR imlementation was also resented, allowing continuous transition between symmetric triangle and sawtooth signals. Future research includes the extension of the algorithm to higher orders, and to arbitrary waveforms comosed of line segments (e.g., traezoidal waves). Acknowledgments The work of Balázs Bank has been suorted by the Bolyai Scholarshi of the Hungarian Academy of Sciences. The authors are thankful to Prof. Vesa Välimäki for his valuable comments. 7. REFERENCES [] J. Pekonen and V. Välimäki, The brief history of virtual analog synthesis, in Proc. 6th Forum Acusticum. Aalborg, Denmark: Euroean Acoustics Association, June 2, [2] V. Välimäki and A. Huovilainen, Antialiasing oscillators in subtractive synthesis, IEEE Signal Processing Magazine, vol. 24, no. 2,. 6 25, March 27. [3] H.-M. Lehtonen, J. Pekonen, and V. Välimäki, Audibility of aliasing distortion in sawtooth signals and its imlications for oscillator algorithm design, Journal of the Acoustical Society of America, vol. 32, no. 4, , October 22. [4] T. Stilson and J. Smith, Alias-free digital synthesis of classic analog waveforms, in in Proc. International Comuter Music Conference, Hong Kong, August 996, [5] S. Tassart, Band-limited imulse train generation using samled infinite imulse resonses of analog filters, IEEE Transactions on Audio, Seech, and Language Processing, vol. 2, no. 3, , March 23. [6] E. Brandt, Hard sync without aliasing, in Proc. International Comuter Music Conference, Havana, Cuba, Setember 2, [7] V. Välimäki, J. Pekonen, and J. Nam, Percetually informed synthesis of bandlimited classical waveforms using integrated olynomial interolation, Journal of the Acoustical Society of America, vol. 3, no., , January 22. [8] J. Lane, D. Hoory, E. Martinez, and P. Wang, Modeling analog synthesis with DSPs, Comuter Music Journal, vol. 2, no. 4,. 23 4, Winter 997. [9] V. Välimäki, Discrete-time synthesis of the sawtooth waveform with reduced aliasing, IEEE Signal Processing Letters, vol. 2, no. 3, , March 25. [] V. Välimäki, J. Nam, J. O. Smith, and J. S. Abel, Alias-suressed oscillators based on differentiated olynomial waveforms, IEEE Transactions on Audio,

8 Seech, and Language Processing, vol. 8, no. 4, , May 2. [] J. Kleimola and V. Välimäki, Reducing aliasing from synthetic audio signals using olynomial transition regions, IEEE Signal Processing Letters, vol. 9, no. 2,. 67 7, February 22. [2] Moog Slim Phatty user s manual, Moog Music, Inc., 2, URL: htt:// default/files/slim hatty users manual.df. [3] J. Kleimola and V. Välimäki, Polynomial Transition Regions [Online], November 2, URL: htt://